A Century of Advancing Mathematics

Home , Albert Marden, Andreas Speiser, Andrew Granville, Bruce Kleiner, Douady rabbit, Dunham Jackson

A Century of Advancing Mathematics c 2015 by The Mathematical Association of America (Incorporated) Library of Congress Control Number: 2015936096 Print ISBN: 978-0-88385-588-1 Electronic ISBN: 978-1-61444-522-7 Printed in the United States of America Current Printing (last digit): 10987654321 A Century of Advancing Mathematics

Stephen F. Kennedy, Editor Associate Editors Donald J. Albers Gerald L. Alexanderson Della Dumbaugh Frank A. Farris Deanna B. Haunsperger Paul Zorn

Published and distributed by The Mathematical Association of America

Contents

Preface ...... ix Part I Mathematical Developments 1 The Hyperbolic Revolution: From Topology to Geometry, and Back ...... 3 Francis Bonahon A CenturyofComplexDynamics ...... 15 Daniel Alexander and Robert L. Devaney Map-ColoringProblems ...... 35 Robin Wilson SixMilestonesinGeometry ...... 51 Frank Morgan Defying God: the Stanley-Wilf Conjecture, Stanley-Wilf Limits, and a Two-GenerationExplosionof Combinatorics ...... 65 Eric S. Egge WhatIstheBestApproachtoCountingPrimes? ...... 83 Andrew Granville A CenturyofEllipticCurves ...... 117 Joseph H. Silverman Part II Historical Developments 133 The Mathematical Association of America: Its First 100 Years ...... 135 David E. Zitarelli The Stratiﬁcation of the American Mathematical Community: The Mathematical Association of America and the American Mathematical Society, 1915–1925 . . . 159 Karen Hunger Parshall Time and Place: Sustainingthe American Mathematical Community ...... 177 Della Dumbaugh Abstract (Modern) Algebra in America 1870–1950:A Brief Account ...... 191 Israel Kleiner Part III Pedagogical Developments 217 The History of the Undergraduate Program in Mathematics in theUnitedStates ..... 219 Alan Tucker

v vi A Century of Advancing Mathematics

Inquiry-BasedLearningThroughthe LifeoftheMAA ...... 239 Michael Starbird A PassporttoPleasure ...... 253 Bob Kaplan and Ellen Kaplan Strength in Numbers: Broadening the View of the Mathematics Major ...... 257 Rhonda Hughes A Historyof Undergraduate Research in Mathematics ...... 263 Joseph A. Gallian The CalculusReformMovement: A Personal Account ...... 275 Paul Zorn Introducing ex ...... 283 Gilbert Strang Part IV Computational Developments 295 Computational Experiences in the Pre-Electronic Days ...... 297 Philip J. Davis A CenturyofVisualization:One Geometer’s View ...... 301 Thomas F. Banchoff TheFutureofMathematics:1965to2065 ...... 313 Jonathan M. Borwein Part V Culture and Communities 331 Philosophy of Mathematics: What Has Happened Since G¨odel’s Results? ...... 333 Bonnie Gold Twelve Classics People who Love Mathematics Should Know; or, “Whatdoyoumean, youhaven’t read E.T.Bell? ...... 351 Gerald L. Alexanderson The Dramatic Life of Mathematics: A Centennial History of the Intersection of Mathematics and Theater in a Prologue, Three Acts, and an Epilogue ...... 365 Stephen D. Abbott 2007:TheYearofEuler ...... 379 William Dunham The PutnamCompetition:Origin,Lore, Structure ...... 387 Leonard F. Klosinski GettingInvolvedwiththeMAA:APath Less Traveled ...... 393 Ezra “Bud” Brown HenryL.Alder ...... 397 Donald J. Albers and Gerald L. Alexanderson LidaK.Barrett ...... 401 Kenneth A. Ross RalphP.Boas ...... 405 Daniel Zelinsky Contents vii

LeonardGillman—Reminiscences ...... 407 Martha J. Siegel PaulHalmos:NoApologies ...... 411 John Ewing IvanNiven ...... 415 Kenneth A. Ross GeorgeP´olyaandtheMAA ...... 421 Gerald L. Alexanderson

Preface

Searching, in the early years of the twentieth century, for a source of support for the fledg- ing and financially unstable American Mathematical Monthly Herbert Ellsworth Slaught conceived the idea of a new society. Our MAA was founded in 1915 to serve as a home for the Monthly. The full details of that story are contained in David Zitarelli’s contribution to this volume beginning on page 135. Within a decade we had published our first book, Calculus of Variations by G. A. Bliss. We now publishthree world-class print journals, one magazine, a newsmagazine, eight book series, and a host of online material. Our identity, from our inceptionrising from the Monthly’sneed for support up to the present-day centennial celebration, is firmly rooted in extraordinary exposition of mathematics. This makes it fitting to produce a volume of essays by some of the great mathematicians and expositors writing today as part of that celebration. But, Slaught’s vision was larger and the MAA’s mission has always been much more than just producing great expository mathematics. That mission—to advance the mathematical sciences, especially at the collegiate level—has been reflected in the Association’s deep engagement with pedagogical practices and public policy issues, with the history and philosophy of mathematics, and with the connections between mathematics and the arts and other sciences. It seems only natural for this volume to sample from the entire broad cultural sweep of mathematics throughout the century of the MAA’s existence. To organize this broad collection of material, the present volume is divided thematically into five sections. First, of course, comes the mathematics. The MAA century began, mathematically speaking, at the tail end of the move towards axiomatization and formalism that started in the nineteenth century. That movement, finished off by Kurt Gödel’s stunner in the early thirties, gave way to a movement towards deep abstraction and ever increasing distance between, so-called, pure and applied mathematics. The introduction of computing machines has, over the past decades, heightened interest in heuristic and algorithmicthinking and energized areas like chaotic dynamics and combinatorics where computing power can reveal new phenomena and prime new intuitions. Our section on mathematical developments includes some legendary, and some new, voices in mathematical exposition. Bob Devaney and Dan Alexander chronicle the early history of complex dynamics, touring the highlights up to the present day. Eric Egge takes on the challengingtask of givingus a lookat the explosionof interest in combinatoricsover the last few decades, focusing on the astounding developments surrounding the Stanley- Wilf Conjecture. Along the lines of famous conjectures that have fallen—Joe Silverman will show us some of the details that unraveled Fermat’s Last Theorem and Robin Wil- son takes us on a historico-mathematical magic ride through the story of the Four-Color

ix x A Century of Advancing Mathematics

Theorem. Frank Morgan chooses his favorite half-dozen geometric milestones of the last few decades and Andrew Granville gives us a new way to think about the Prime Number Theorem after a deep, and deeply historical, tour through our efforts to count the primes. Finally, Francis Bonahon pays tribute to the contributions of Bill Thurston in unearthing the deep connections between geometry, particularly hyperbolic geometry, and topology. From the very beginning the MAA has been about more than just world-class exposition; within three months of our founding the organization had constituted the National Committee on Mathematics Requirements charged with investigating the teaching of secondary and post-secondary mathematics, making curricular recommendations, and working towards improving the teaching of mathematics. It is fitting that we devote a section of this volume to the pedagogy of mathematics. A commitment to effective teaching and learning of mathematics is in the organization’s DNA; the perceived lack of such a commitment from our sister organization contributed to the movement that led to the MAA’s founding. Alan Tucker leads off with a fascinating history of the evolution of the undergraduate mathematics major. Joe Gallian, the founding father of the mathematics REU, recounts the early history of that pedagogical innovation. Michael Starbird writes historically and per- suasively about inquiry-based pedagogy, while Bob and Ellen Kaplan—the founders of the original Math Circle—write, passionately and lyrically, on the same topic. Rhonda Hughes offers advice on attracting underrepresented groups to our subject. Gil Strang tells us how we ought to teach the exponential function; and Paul Zorn, a key player in the Calculus Reform movement, tells us what that movement hoped to achieve. Zorn’s contributionhighlightshow calculus teaching was changed, even given new life, by computing technology. Egge and Alexander & Devaney made clear that massive computational power wrought seismic changes on mathematics itself—and on the kinds of mathematics we find interesting and exciting. In the Computing section of this volume we have Phil Davis reminiscing about his days in the pre-electronic era when he computed with analogue devices, many of which you will never have heard of. Tom Banchoff recounts some of the advances arising from our increased powers of visualization. Jon Borwein argues that the very nature of mathematics and mathematical practice have been revolutionarily ruptured by computing. He daringly speculates about where this disruption might lead. A centennial, of course, prompts reflection on our century of activity. In our History section David Zitarelli tells the story of the founding and growth of the MAA. Karen Par- shall relates the early history of MAA-AMS relations. Della Dumbaugh reflects on the establishment of an American mathematical infrastructure by considering the work of, and relationship between, Leonard Eugene Dickson and Oswald Veblen. Israel Kleiner practices history-by-biography by presenting a biography-driven tale of a century of progress in abstract algebra. In the final section on Cultures and Communities, Bill Dunham recalls the MAA’s Year of Euler, Jerry Alexanderson tells us about some of his favorite books, and Steve Abbott outlines the history of mathematically inspired theater. Bonnie Gold fills us in on what has happened in the philosophy of mathematics post-Gödel, and Leonard Klosinski muses on the history and structure of the Putnam. Finally, we offer a handful of reminiscences of and by leading figures in the MAA over the years. These short pieces not only succeed in introducingus to some of the giants of our Association, they also bring to lighta wonderful MAA tradition of mentoring and welcoming. Every single one of these pieces tells, inci- Preface xi dentally, one or more stories of a national MAA figure drawing a younger colleague into lifelong engagement with this organization. Taken together, these reflections tell a powerful story of the role of community in the lives of mathematicians, and in our discipline. It has been an extraordinary century for mathematics—more mathematics has been created and published since 1915 than in all of previous recorded history. We’ve solved age-old mysteries, created entire new fields of study, and changed our conception of what mathematics is. More mathematicians have lived and practiced in the MAA century than in all previous millennia combined. Those mathematicians have explored more than just mathematics; they have, as this volume tries to make evident, investigated mathematical connections to pedagogy, history, the arts, technology, literature, every field of intellectual endeavor. Mathematics is the most thrilling, the most human, area of intellectual inquiry. We hope you find in this volume compelling proof of that claim. The editors are grateful to the authors of the following essays. Their erudition, expository skill, and enthusiasm for this project have made the production of this volume a joyous labor. We also must express our profound gratitude to Ivars Peterson, Carol Baxter and, especially, Beverly Ruedi of MAA Press. Their collective encouragement, craftsman- ship, humoring of our foibles, and attention to detail are simply extraordinary and made this volume possible. —The Editors

Part I Mathematical Developments

The Hyperbolic Revolution From Topology to Geometry, and Back

Francis Bonahon1 University of Southern California

Dedicated to the memory of Bill Thurston (1946–2012)

The late nineteen seventies and early eighties saw a surprising convergence between topology and rigid geometry. This followed the groundbreaking work of Bill Thurston on the geometrization of three-dimensional manifolds, but this was also part of a larger trend that resulted in a period of intense cross-fertilization between topology, geometry, dynamical systems, combinatorial group theory, and complex analysis. First, we should begin with the traditional difference between topology and geometry. Both ﬁelds consider geometric objects, but topologists allow themselves to deform these objects and stretch distances, whereas geometers tend to focus on the ﬁne properties of these distances. As an illustration, it is well-known that topologists like to turn doughnuts into coffee mugs, whereas a typical result in geometry would be the Polyhedron Rigidity Theorem of Cauchy, which says that it is impossible to deform a convex polyhedron in euclidean space without changing the shape of any of its faces.

1 The hyperbolic space Among the geometries that can occur in dimension three, the more fundamental one is hyperbolic geometry. The n-dimensional hyperbolic space is the half-space Hn Rn 1 Œ0; / in Rn, D 1 endowed with the hyperbolicmetric defined as follows. First, for every differentiable curve n Œa; b H with .t/ x1.t/; x2.t/; : : : ; xn.t/ , we define its hyperbolic arc length W ! D ` . / b 1 n x .t/2 dt (differing from the usual arc length only by the 1 hyp a xn.t/ i 1 i0 xn.t/ D D n factor). The hyperbolicq distance dhyp.P; Q/ between two points P , Q H is then defined R P 2 as the infimum of the hyperbolic arc lengths `hyp. / over all curves joining P .a/ to D Q .b/. D What is not obviousfrom the above description isthat the hyperbolicspace Hn is highly symmetric. In fact Hn is homogeneous in the sense that, for every P , Q Hn, there is n 2 an isometry ' of the metric space .H ; dhyp/ that sends P to Q. It is even isotropic in the

1This work was partially supported by the grants DMS-1105402 and DMS-1406559 from the U.S. National Science Foundation,and by a Research Fellowship from the Simons Foundation (grant 301050).

3 4 A Century of Advancing Mathematics sense that we can require the isometry ' to send an arbitrary direction at P to an arbitrary direction at Q. In this regard, it is as symmetric as the usual euclidean space Rn. Hyperbolic geometry made its first appearance in the context of Euclid’s Fifth Postu- late, in the early nineteenth century. Henri Poincaré[34, 35] was the first to connect it to other major branches of mathematics, namely complex analysis and the theory of linear differential equations. About a century later, Thurston placed hyperbolic geometry at the center of three-dimensional topology.

2 Knots in space As an introduction to Thurston’s geometrization results for three-dimensional manifolds, let us focus on their applications to knot theory. The results are then easier to state, and give a good illustrationof the more general ideas. The author likes to say that knot theory is to the topologist what the fruit fly is to the biologist: a small laboratory example where big theories can be tested, which is easier to handle and to visualize than the long-term problems motivating these theories, and which nevertheless is sufficiently complex to offer a challenging testing ground. A knot is a closed curve K in the euclidean space R3 that is smooth, i.e., has a well- defined tangent (with no switchback) at each point, and that has no self-intersection. The main problem in knot theory is to decide when two knots K and K0 can be deformed to each other, that is whether there exists a continuous family of homeomorphisms .'t /t Œ0;1 3 2 of R such that '0 is the identity and '1.K/ K . D 0

Figure 1. A few knots.

Figure 1 offers a few examples. It is not immediately obvious that two of these knots can be deformed to each other, and deep mathematics is required to prove that no two of the remaining four knots can be deformed to each other. This situation is fairly typical. To tackle the challenge of rigorously proving that two knots that appear different cannot be deformed one to the other, mathematicians have traditionally used techniques of algebraic topology. One of the early successes of such an approach was due to James W. Alexander and Garland B. Briggs [2] who showed in 1927 that the knots of up to nine crossings listed in the nineteenth–century knot tables by Tait, Kirkman and Little [45, 16, 17, 20, 21] were actually different.2 They did so by comparing the homology groups of certain branched covers of these knot. The following decades saw the development of ever more sophisticated methods of algebraic topology to attack problems in knot theory. A less common approach to knot theory involved the cut-and-paste analysis of special surfaces in the complement of the knot, as in the innovative work of Horst Schubert [41, 42, 43].

2To be completely accurate, there were a very small number of exceptions that Alexander and Briggs could not settle. The Hyperbolic Revolution: From Topology to Geometry, and Back 5

Thurston’s Hyperbolization Theorem for knot complements provided a completely different type of knot invariants. To state this result, we need to mention a couple of classical constructions of knots. The ﬁrst one is that of torus knots. These are the knotsthat can be drawn on the surface of a standard torus in R3. More precisely, for coprime integers p and q, the p; q –torus f g knot is represented by the curve parametrized by

t .R r cos qt/ cos pt; .R r cos qt/ sin pt; r sin qt 7! C C for arbitrary 0 < r < R. For instance, Figure 2 represents the 5; 4 –torus knot, and f g the ﬁrst three knots of Figure 1 are the 1;0 , 2; 3 and 2; 3 –torus knots, respectively. f g f g f g Torus knots are very well understood. In particular, when p and q are different from 1, ˙ the p; q –torusknot can be deformed to the p ; q –torusknot if and only ifthe set p; q f g f 0 0g f g is equal to p ; q or to p ; q ; when p or q are equal to 1, the p; q –torus knot f 0 0g f 0 0g ˙ f g can be deformed to the unknot, the ﬁrst knot of Figure 1.

Figure 2. A torus knot.

The second construction that we need is that of satellite knots. Suppose that we are given a ﬁrst knot K R3 that cannot be deformed to the unknot, as well as another knot L contained in the standard solid torus

V .R cos v/ cos u; .R cos v/ sin u; sin v u; v; R; 0 r D C C I 2 Ä Ä n o consisting of those points which are at distance at most r from the horizontal circle C of radius R centered at the origin, for arbitrary r, R with 0 < r < R. We assume in addition that L is non-trivial in the solid torus V , in the sense that it cannot be deformed in V to a knot L0 which is disjoint from one of the cross-section disks where the coordinate u is constant, or to the central circle C of V .

3 3 The knot K R The knot L V The satellite K0 R Figure 3. A satellite knot.

We can then tie V as a tube around the knot K, and consider the image of L. More precisely, choose an injective continuous map V R3 which sends the central circle W ! C to the knot K. Assume in addition that ' is differentiable, and that its jacobian is everywhere different from 0, so that the image K '.L/ is now a new knot in R3. Any knot 0 D K0 obtained in this way is said to be a satellite of the knot K. 6 A Century of Advancing Mathematics

Theorem 1 (Hyperbolization Theorem for knot complements). Let K be a knot in R3, and let R3 R3 be obtained by adding to R3 a point at infinity. Then, exactly one D [ f1g 1 of the followingholds: 1. bK is a torus knot; 2. K is a satellite of a non-trivial knot; 3. the complement R3 K admits a complete metric d which induces the same topology as the euclidean metric of R3 and which is locally isometric to the hyperbolic metric of the hyperbolic spaceb H3. The first alternative is somewhat trivial, since torus knots are very well understood (and very rare). The second alternative can be essentially reduced to the other two, through a unique factorization process of satellite knots into non-satellite links [42, 14, 13, 5]. In practice, almost all knots satisfy the third alternative, and therefore admit a hyperbolic metric, that is, a metric d as in this third alternative. The Hyperbolization Theorem is greatly enhanced by the following earlier result of George Mostow [27, 28]. Theorem 2 (Mostow’s Rigidity Theorem). When the third case of Theorem 1 holds, the metric d is unique up to isometry. That is, for any two such metrics d and d 0, there exists a map R3 K R3 K such that d .x/; .y/ d.x; y/ for every x, y R3 K. W ! 0 D 2 The incredible power of the combination of Theorems 1 and 2 is that they turn the topologicalb problemb of deciding when two knots can be deformed to each otherb into the rigidgeometric problem of deciding when their associated hyperbolicmetrics are isometric. These metrics carry a lot of information. For instance they have a well-defined volume. Theorem 2 then shows that, if two knots satisfy the third conclusion of Theorem 1 and can be deformed to each other, then they must have the same volume. This simple test is remarkably efficient to show that two knots cannot be deformed to each other.

Figure 4. Two very similar knots.

A more powerful invariant of the hyperbolic metric of a knot complement is its Ford domain. This object was introduced in a two-dimensional setting [10] by Lester Ford,3 and generalized to knot complements by Bob Riley [38, 39, 40]. They provide a tessellation of the euclidean plane by polygons, which is invariant under two linearly independent trans- lations and which carries additional pairing information. See for instance [4 , 12.4]fora more precise description. It then follows from Theorem 2 that two knots can be deformed to each other if and only if there is a similitude (the composition of an isometry with a homothety) of the Euclidean plane that carries the tessellation associated to the ﬁrst knot

3Also famous for being the President of the Mathematical Association of America from 1947 to 1948. The Hyperbolic Revolution: From Topology to Geometry, and Back 7 to the tessellation associated to the second one, and that preserves the pairing information. The “if and only if” part of this statement is particularly impressive and useful. As an example taken from [4], consider the two knots of Figure 4. These are somewhat difﬁcult to distinguish with the tools of algebraic topology that were available before hyperbolic geometry techniques became available, and their complements even have the same hyperbolic volume. However a quick inspection, for instance counting the number of edges emanating from each vertex, shows that there is no similitude of the euclidean plane that exchanges their respective Ford tessellations, represented in Figure 5. It immediately follows that these two knots cannot be deformed to each other.

Figure 5. The Ford tessellations of the knots of Figure 4.

Another important property of these results is that they can be explicitly implemented on a computer. Following up on the early pioneering work of Bob Riley [38, 39], the software SnapPea developed by Jeff Weeks [53] has been particularly inﬂuential among researchers in knottheory; see [9] for a current incarnation, Python-based and called SnapPy, of the same tool.

3 Geometrization of general three-dimensional manifolds Theorem 1 is a special case of a more general geometrization result for three-dimensional manifolds. An n-dimensional (topological) manifold is a topological space that is locally homeomorphic to the usual n-dimensional euclidean space Rn. For instance, a surface is a two-dimensional manifold. The space R3 R3 of Theorem 1 is a three-dimensional D [f1g manifold, even near the point (and is homeomorphic to the three-dimensional sphere 1 S3 R4); a knot complement R3 Kbis also a three-dimensional manifold. More generally, an n-dimensional manifold-with-boundary is a topological space M locally homeomorphic to Rn 1b Œ0; /, and its boundary @M consists of the points that 1 go to Rn 1 0 under the corresponding local homeomorphisms. In particular, a manifold f g as defined in the previous paragraph is a manifold-with-boundary with empty boundary. The general Geometrization Theorem for three-dimensional manifold is a little difficult to state precisely while remaining within the scope of this article, and we just want to give the flavor of this result. 8 A Century of Advancing Mathematics

First of all, the Geometrization Theorem involves more geometries than hyperbolic geometry. A geometric structure on a manifold can be interpreted as a metric that is locally homogeneous, in the sense that any two points have isometric neighborhoods. These geometries are locally modeled on the homogeneous spaces associated to Lie groups and, in dimension three, the classification of Lie groups shows that there is a limited number of possible models. In fact, there are only eight geometries that are relevant for the Ge- ometrization Theorem: 1. the three isotropic geometries of the euclidean space R3, the three-dimensional sphere S3 R4, and the three-dimensional hyperbolic space H3; 2. the two product geometries S2 R and H2 R, product of the euclidean line R with the 2-dimensional sphere S2 R3 and the hyperbolic plane H2, respectively; 3. two suitably defined twisted product geometries H2 R and R2 R; 4. the Sol geometry, related to the unique three-dimensional solvable Lie group. e e See [36, 3] for a precise description of these geometries. Then, one needs a topological notion of triviality for a surface contained in a three- dimensional manifold. We cannot precisely describe this concept here, except by saying that the definition is consistent with the terminology: a trivial surface is obtained by a method that is too straightforward to be of much use. For instance, the boundary of a small ball in a three-dimensional manifold M is a trivial sphere in M ; similarly, the boundary of a thin tube around a simple closed curve in M gives a trivial torus. Theorem 3 (Geometrization Theorem for three-dimensional manifolds). Let M be a connected three-dimensional manifold that is topologically finite, in the sense that M D M @M is obtained by removing its (possibly empty) boundary from a compact manifold- with-boundary M . Then, at least one of the followingholds: 1. M admits a complete metric d which is locally isometric to one of the eight geometric models listed above; 2. M contains a non-trivial sphere, projective plane, torus or Klein bottle. In most cases, the geometry that occurs in the first case is that of the hyperbolic space H3, and the other geometries occur only for a limited array of three-dimensional manifolds. In that case, and under the additional hypothesis that each component of the boundary @M is a torus or a Klein bottle, the same Mostow’s Rigidity Theorem as in Theorem 2 guarantees that the hyperbolic metric on M is unique up to isometry. There is a small possible overlap between the two conclusions of Theorem 3, but no possible overlap in the case of hyperbolic geometry. In particular, non-trivial spheres, projective planes, tori or Klein bottles appear as topological obstructions to the existence of a hyperbolic geometric structure, and of several of the other geometric structures. We then benefit from two earlier pieces of work: one is the Kneser-Milnor [19, 23] unique factorization of a three-dimensional manifold into prime manifolds that contain no essential spheres or projective planes; the other one is the Waldhausen-Jaco-Shalen-Johannson [13, 14] canonical splitting (originally developed for completely different purposes) of a prime manifold into pieces that, either contain no essential tori or Klein bottles, or admit one of the seven non-hyperbolic geometries. In practice, this reduces the problem of the topological classification of three-dimensional manifolds to the isometric classification of The Hyperbolic Revolution: From Topology to Geometry, and Back 9 hyperbolic three-dimensional manifolds. Three-dimensional hyperbolic geometry is still very rich, but our discussion of Ford domains in 2 should give an idea of the powerful techniques that are available in this field. Thurston proved Theorem 3 in many cases, in particular when M is non-compact (which includes the case of knot complements considered in 2), in the late nineteen seventies. He also conjectured Theorem 3 in its full generality, which was then known as the Thurston Geometrization Program until Grigori Perelman proved it around 2000. Neither Thurston nor Perelman provided a complete exposition of their proofs, but they circulated partial preprints [48, 50, 51, 31, 32, 33] and gave enough lectures to enable others to fill in all the details; see for instance [15, 29, 30, 7, 8, 18, 25, 26].

4 A broader perspective: using geometry to prove results in topology and algebra The Geometrization Program took place in, and contributed to, a broader trend which in the last quarter of the twentieth century saw a closer integration between topology, differential geometry, dynamical systems and group theory. We already indicated how the combination of Thurston’s Hyperbolization Theorem and Mostow’s Rigidity Theorem translates topological problems to hyperbolic geometry questions, and can be used to prove theorems in knot theory. This interaction between topology and geometry occurs, not just in the consequence of these results, but also in the novel ideas introduced by Thurston for the proof of his Hyperbolization Theorem. Indeed, the flexibility of topology comes with the curse of a very large number of degrees of freedom. Geometry can be used to introduce some rigidity in a topological situation,in order to make it easier to handle. As an example, consider the two-dimensional analogue of knot theory which studies, in a surface S, all simple (smooth and without self- intersection) closed curves in S up to deformation. There is of course an overwhelming abundance of simple closed curves, and of deformations between them. However, we can take advantage of the following consequence of the Uniformization Theorem in complex analysis: if the topology of the surface S is complicated enough that it does not belong to a small finite number of exceptions such as the plane or the torus, the surface S can be endowed with a hyperbolicmetric d. Once such a hyperbolicmetric is chosen, every simple closed curve can be deformed to a unique simple closed curve that is geodesic, i.e., provides the shortest arc between any two of its points that are sufficiently close to each other (a hyperbolic geodesic is thus the hyperbolic equivalent of a straight line). This provides a one-to-one correspondence between, simple closed curves considered up to deformation on the one hand, and simple closed geodesics on the other hand. This greatly simplifies the original problem by eliminating the need to consider deformations, provided that we restrict attention to a very specific type of simple closed curves. Thurston took advantage of this construction to introduce a certain completion ML.S/ of the set S.S/ of simple closed curves in the surface S considered up to deformation. The elements of ML.S/ are measure-theoretic, or probabilistic, generalizations of simple closed geodesics and are called measured geodesic laminations. The space ML.S/ is endowed with a natural topology (for which, rather surprisingly, it is homeomorphic to a euclidean space Rn) and with a rescaling operation. In particular, given a sequence in 10 A Century of Advancing Mathematics

S.S/, it makes sense to talk of the limit of this sequence or, after suitably rescaling, of the asymptotic direction of this sequence in ML.S/. This method of taking limits of objects that are only defined up to deformation was a real conceptual breakthrough. It played a critical rôle in Thurston’s work on surface diffeomorphisms [49] and on three-dimensional hyperbolic geometry [46, 47, 52]. Together with similar rigidification techniques, it also provided the impetus and technical tools for much subsequent work by the low-dimensional geometry and topology community. At about the time when Thurston was pioneering the use of geometry to prove results in topology, Mikhail Gromov [12] was translating insights from geometry to abstract group theory. This comes from another situation where one considers closed curves up to deformation in a topologicalspace M , namely in the definitionof the fundamental group 1.M / of M . The Milnor-Svarcˇ Lemma [37, 24] asserts that, for a compact space M , the large– scale properties of the fundamental group 1.M / are essentially the same as those of the universal cover M . In particular, this has far–reaching consequences when M is a compact riemannian manifold of negative curvature. Building on the insights provided by the geometry, Gromov wasf able to identify the key algebraic features that yield these consequences, and to develop a purely algebraic theory of “groups that behave like fundamental groups of negatively curved manifolds” (now called Gromov hyperbolic groups or negatively curved groups); see [12], as well as for instance [6, 11]. This gave an important boost to the field of combinatorial group theory, further enhanced by the rich families of examples provided by the Geometrization Theorem. The Geometrization Program also greatly energized another area of mathematics. We already mentioned how Poincaréhad inserted two- and three-dimensional hyperbolic geometry intothe world of complex analysis. In the century that followed, the connection had become a little more tenuous (however, see for instance [1, 22]), but was greatly invigo- rated by Thurston’s Hyperbolization Theorem. Indeed, Thurston’s original proof combined both the complex analytic and the hyperbolic geometric aspects of kleinian groups. Con- versely, the three-dimensional point of view provided strong tools and insights for the corresponding complex analytic problems. These insights were pushed one step further, first by Thurstonand then by Dennis Sullivan,to the dynamics of rational maps on Riemann surfaces; see for instance Sullivan’s “dictionary” [44] between the theories of kleinian groups and of complex dynamics. Thurston’s Geometrization Program provided great results and tools that were used to solve many topological problems. However, its even more lasting impact may be the integration and cross-fertilization between numerous branches of mathematics that it triggered: topology, geometry, complex analysis, combinatorial group theory, dynamical systems, etc. From a sociological point of view, mathematics historians may trace the germs of these developments to the Berkeley mathematical school of the late nineteen sixties, where the same group of people were working on topology, dynamical systems and rigid geometry. How- ever, it is Bill Thurston’s extraordinary talent that initiated this technical and conceptual revolution, which led to one of the most productive periods in mathematics.

Bibliography [1] Lars V. Ahlfors, Fundamental polyhedrons and limit point sets of Kleinian groups, Proc. Nat. Acad. Sci. USA 55 (966), 251–254. The Hyperbolic Revolution: From Topology to Geometry, and Back 11

[2] James W. Alexander, Garland B. Briggs, On types of knotted curves, Ann. of Math. 28 (1926), 562–586. [3] Francis Bonahon, Geometric structures on 3-manifolds, in Handbook of geometric topology edited by R.J. Daverman and R.B. Sher, North-Holland, 2002, pp. 93–164. [4] ———, Low-dimensional geometry: from euclidean surfaces to hyperbolic knots, Student Math. Library 49, American Math. Soc., Providence, RI, 2009. [5] Francis Bonahon,LaurenceC. Siebenmann, New geometric splittings of classicalknots, and the classification and symmetries of arborescentknots, unpublished monograph, 1979–2009, available at www-bcf.usc.edu/˜fbonahon/Research/Preprints/BonSieb.pdf. [6] Martin R. Bridson, AndréHaefliger, Metric spaces of non-positive curvature, Grundlehren der Math. Wiss. 319, Springer-Verlag, 1999. [7] Huai-Dong Cao, Xi-Ping Zhu, A complete proof of the Poincaréand geometrization conjecture — application of the Hamilton-Perelman theory of the Ricci flow, Asian J. Math. 10 (2006), 165–492; Erratum, Asian J. Math. 10 (2006), 663. [8] B. Chow, S.-C. Chu, D. Glickenstein, C. Guenther,J. Isenberg, T. Ivey, D. Knopf, P. Lu, F. Luo, L. Ni, The Ricci flow: techniques and applications, Part I. Geometric aspects, Math. Surveys and Monographs 135, American Math. Soc., 2007; Part II. Analytic aspects, Math. Surveys and Monographs 144, American Math. Soc., 2008. [9] Marc Culler, NathanM. Dunfield,andJeffrey R.Weeks, SnapPy,a computerprogramfor studying the geometry and topology of 3–manifolds, snappy.computop.org. [10] Lester R. Ford, The fundamentalregion for a Fuchsiangroup, Bull. Amer. Math. Soc. 31 (1935), 531–539. [11] Etienne´ Ghys, Pierre de la Harpe (Eds.), Sur les groupes hyperboliques d’après Mikhael Gro- mov, Progress in Mathematics 83, Birkhäuser Boston, 1990. [12] Mikhail Gromov, Hyperbolic groups, in Essays in group theory, Math. Sci. Res. Inst. Publ. 8, Springer, 1987, pp. 75–263. [13] William H. Jaco, Peter B. Shalen, Seifert fibered spaces in 3–manifolds, Memoirs Amer. Math. Soc. 220, American Math. Soc., 1979. [14] Klaus Johannson, Homotopy equivalences of 3–manifolds with boundary, Lecture Notes in Math. 761, Springer-Verlag, 1979. [15] Michael Kapovich, Hyperbolic manifolds and discrete groups, Progress in Math. 183, Birkhäuser, 2001. [16] Thomas P. Kirkman, The enumeration, description, and construction of knots of fewer than 10 crossings, Trans. Roy. Soc. Edinburgh 32 (1883–84), 281–309. [17] ———, The 364 unifilar knots of ten crossings, enumerated and described, Trans. Roy. Soc. Edinburgh 32 (1884–85), 483–491. [18] Bruce Kleiner, John Lott, Notes on Perelman’s papers, Geom. Top. 12 (2008), 2587–2855. [19] Hellmuth Kneser, Geschlossene Flächen in dreidimensionalen Mannigfaltigkeiten, Jahr. Deutschen Math. Verein. 38 (1929), 248–260. [20] Charles N. Little, On knots, with a census for order 10, Trans. Connecticut Academy Sci. 18, Vol. 7 (1885), 27–43. [21] ———, Non-alternate ˙ knots of ordereight andnine, Trans.Royal. Soc. Edinburgh 35 (1890), 253–255. [22] Albert Marden, The geometry of finitely generated kleinian groups, Ann. of Math. 99 (1974), 383–462. 12 A Century of Advancing Mathematics

[23] John W. Milnor, A unique decomposition theorem for 3–manifolds, Amer. J. Math. 84 (1962), 1–7. [24] ———, A note on curvature and fundamental group, J. Diff. Geom. 2 (1968) 1–7. [25] John W. Morgan, Gang Tian, Ricci flow and the Poincaréconjecture, Clay Mathematics Mono- graphs 3, American Math. Soc., 2007. [26] ———, The Geometrization Conjecture, Clay Mathematics Monographs 5, American Math. Soc., 2014. [27] George D. Mostow, Quasi-conformal mappings in n-space and the rigidity of hyperbolic space forms, I.H.E.S.´ Pub. Math. 34 (1968), 53–104. [28] ———, Strong rigidity of locally symmetric spaces, Ann. of Math. Studies 78, Princeton Uni- versity Press, 1973. [29] Jean-Pierre Otal, Le théorème d’hyperbolisation pour les variétés fibrées de dimension 3, Astérisque, 235, SociétéMathématique de France, 1996. [30] ———, Thurston’s hyperbolization of Haken manifolds, in Surveys in differential geometry, Vol. III (Cambridge, MA, 1996), International Press, 1998, pp. 77–194. [31] Grigori Y. Perelman, The entropy formula for the Ricci flow and its geometric applications, unpublished preprint, arxiv.org/abs/math/0211159. [32] ———, Ricci flow with surgery on three-manifolds, unpublished preprint, arXiv:math/ 0303109. [33] ———, Finite extinction time for the solutions to the Ricci flow on certain three-manifolds, arXiv:math/0307245. [34] Henri Poincaré, Théorie des groupes fuchsiens, Acta Math. 1 (1882), 1–62. [35] ———, Mémoire sur les groupes kleinéens, Acta Math. 3 (1883), 49–92. [36] G. Peter Scott, The geometries of 3–manifolds, Bull. London Math. Soc. 15 (1983), 401–487. [37] Albert S. Schwartz(Svarc),ˇ A volume invariant of coverings (Russian), Dokl. Akad. Nauk SSSR 105 (1955), 32–34. [38] Robert F. Riley, Applications of a computer implementation of Poincaré’s theorem on fundamental polyhedra, Math. Comp. 40 (1983), 607–632. [39] ———, An elliptical path from parabolic representations to hyperbolic structures, in Topology of low-dimensional manifolds (Proceedings of the Second Sussex Conference, Chelwood Gate, 1977), Lecture Notes in Math. 722, Springer, 1979, pp. 99–133. [40] ———, Seven excellent knots, in Low-dimensional topology (Bangor, 1979), London Math. Soc. Lecture Note Series 48, Cambridge Univ. Press, 1982, pp. 81–151. [41] Horst Schubert, Die eindeutige Zerlegbarkeit eines Knotens in Primknoten, Sitz.-Ber. Heidel- berger Akad. Wiss. Math.-Nat. (1949), 57–104. [42] ———, Knoten und Vollringe, Acta Math. 90 (1953), 131–286. [43] ———, Knoten mit zwei Brücken, Math. Z. 65 (1956), 133–170. [44] Dennis P. Sullivan, Quasiconformal homeomorphisms and dynamics. I. Solution of the Fatou- Julia problem on wandering domains, Ann. of Math. 122 (1985), 401–418. [45] PeterG. Tait, On knotsI, Trans. Roy. Soc. Edinburgh 28 (1876-7), 145–190;II, Trans. Roy. Soc. Edinburgh 32 (1883-4), 327–342; III, Trans. Roy. Soc. Edinburgh 32 (1884–5),493–506. [46] William P. Thurston, The geometry and topology of three-manifolds, lecture notes, Princeton University, 1976-80. [47] ———, Three-dimensional manifolds, Kleinian groups and hyperbolic geometry, Bull. Amer. Math. Soc. 6 (1982), 357–381. The Hyperbolic Revolution: From Topology to Geometry, and Back 13

[48] ———, Hyperbolic structures on 3-manifolds, I: Deformation of acylindrical manifolds, Ann. of Math. 124 (1986), 203–246. [49] ———, On the geometry and dynamicsof diffeomorphisms of surfaces, Bull. Amer.Math. Soc. 19 (1988), 417–431. [50] ———, Hyperbolic structures on 3-manifolds, II: Surface groups and 3-manifolds which ﬁber over the circle, unpublished preprint, ca. 1980, arXiv:math/9801045. [51] ———, Hyperbolic structures on 3-manifolds, III: Deformations of 3-manifolds with incom- pressible boundary, unpublished preprint, ca. 1980, arXiv:math/9801058. [52] ———, Three-dimensional geometry and topology, Vol. 1 (Edited by Silvio Levy), Princeton Math. Series 35, Princeton University Press, 1997. [53] Jeffrey R. Weeks, SnapPea, computer program, www.geometrygames.org/SnapPea/ index.html.

Department of Mathematics, University of Southern California, Los Angeles, CA 90089-2532 [email protected]

A Century of Complex Dynamics

Daniel Alexander and Robert L. Devaney1 Drake University Boston University

1 Introduction

Like the MAA, the field of mathematics known as complex dynamics has been around for about one hundred years. Unlike the MAA, complex dynamics has had its ups and downs during this period. While the origins of complex dynamics stretch back into the late 1800s, the foundationsof the contemporary studywere established in the last years of World War I with the pioneering work of Gaston Julia and Pierre Fatou. Althoughone hundred years ago complex dynamics was a predominantly French field, there are some important American connections dating back to 1915, with some interesting historical connections to the MAA. Fatou and Julia continued to explore and expand complex dynamics in the 1920s, but as open questions were successfully addressed, developments slowed. After World War II, aside from a growing body of work by Irvine Noel Baker beginning in the early 1950s concerning the iteration of entire maps, and a few isolated papers, such as those by Hans Brolinand Thomas Cherry in the mid-1960s, interest in the subject dwindled,and toan outside observer the field appeared dormant. This changed dramatically around 1980 with the discovery of the Mandelbrot set when the availability of computers and computer graphics suddenly revealed the beautiful objects that Julia and Fatou could only see in their minds. Throngs of mathematicians (including Fields medalists John Milnor, William Thurston, Jean-Pierre Yoccoz, and Curt McMullen, as well as numerous other eminent individuals) entered the field and complex dynamics was reborn. In this paper we give a brief overview of the early and later history of the development of complex dynamics, including a discussion of the early American connections. For more historical details see [1], [2], and [3]. We also include a brief description of some of the major results that have come forward during the past century, and we describe briefly some of the dynamical behavior on what are now known as the Julia and Fatou sets, at least for the simplest types of complex functions, namely those with a single free critical orbit.

2 Preliminaries

In complex dynamics, the goal is to understand what happens when an analytic function in the complex plane C (or the Riemann sphere C) is iterated. Recently, this goal has been

1Robert L. Devaney was partially supported by Simons Fundation Grant #208870.

15 16 A Century of Advancing Mathematics expanded to include iteration in Cn as well, although we will not touch upon this subject in this paper. Different types of complex analytic functions—polynomials,rational maps, entire transcendental maps, and meromorphic functions—can lead to very different dynamical behaviors. For simplicity we will initially concentrate on polynomials, since many of the basic properties and definitions we describe for polynomials extend to other kinds of functions. We will also sketch some of the different behaviors that arise in other maps towards the end of this paper. Let P be a polynomial in the complex plane. The goal is to understand the behavior of this function when it is iterated. So let the second iterate of P be P 2 P P and, D ı inductively, let the nth iterate of P be P n P P n 1. Given z C, then the ques- D ı 2 tion is: what happens to the orbit of z, i.e., the sequence of points z; P.z/; P 2.z/; : : :? Many different types of orbits can occur. For example, the orbit of z could be periodic of period n; that is, for some n > 0 we have P n.z/ z. Or it could be eventually periodic, D meaning that P j n.z/ P j .z/ for some n; j > 0. The orbit could also tend to in the C D 1 plane. And, as we shall see later, there are various other possibilities for the behavior of these orbits. One of the most important objects in complex dynamics is the Julia set of P which we denote by J.P /. This set has several equivalent definitions. Since P is a polynomial, there is an open set surrounding in the Riemann sphere that consists of points whose 1 orbits simply tend to . This leads to a definition of the Julia set from a geometric pointof 1 view: J.P / is the boundary of the set of points whose orbits tend to . From a dynamical 1 systems point of view, the Julia set is also the closure of the set of repelling periodicpoints. Here a repelling (resp., attracting) periodic point is a point z for which P n.z/ z and D .P n/ .z/ > 1 (resp., .P n/ .z/ < 1). j 0 j j 0 j These two equivalent definitions imply that the Julia set is the chaotic set, for arbitrarily close to any pointin theJuliaset, there are pointswhoseorbitstend to as well as periodic 1 points whose orbitsreturn to themselves. This is sensitive dependence on initial conditions, the hallmark of chaotic behavior. The complement of the Julia set is called the Fatou set; this is the set where the dynamical behavior is usually quite tame. From a complex analysis point of view, J.P / is also the set of points in C at which the family of iterates of P fails to be a normal family in the sense of Montel. This means that, by Montel’s Theorem, any neighborhood of a point in J.P /, no matter how small, is eventually mapped over the entire complex plane (minusat most one point),which provides us with another way to see that the map P is extremely sensitive to initial conditions on its Julia set. There are other types of periodicpoints that will come up later in this paper. A periodic point z of period n is super-attracting if .P n/ .z/ 0. The periodic point is neutral if 0 D .P n/ .z/ e2i . When is rational, the periodic point is called parabolic (or rationally 0 D neutral) and the nearby dynamics are completely understood. But when is irrational the periodic point is irrationally neutral, and there are certain -values where we still have no idea what happens near z. Finally, a periodic point of period one is called a fixed point. Before going into more detail about the mathematics of complex dynamics, let’s first pause and turn back the clock to see how this field emerged one hundred years ago. A Century of Complex Dynamics 17

3 Complex dynamics through 1942 The early study of complex dynamics is dominated by French mathematicians. Nonethe- less significant early developments (and perhaps even its origins) occurred elsewhere in Germany, Poland, Italy, Japan and, in the year of the MAA’s birth, the United States. In order to set the stage for a discussion of the works of Fatou and Julia—as well as the work of the Americans—we will briefly discuss the origins of the field. Those curious to know more about the beginnings of complex dynamics should see [1]. To find find out more about the events discussed in this section, also see [2] and [3].

The origins Beginning in 1883 the French mathematician Gabriel Kœnigs wrote a series of papers out- lining the local theory of the iteration of a complex analytic function. He proved fundamental results involving the existence of repelling and attracting fixed points and developed a surprisingly robust local theory describing the dynamics of iteration on a neighborhood of an attracting (but not super-attracting) fixed point. Other French mathematicians, including mathematicians on whose dissertation committees he served, soon filled in details regarding the local behavior of super-attracting and rationally neutral fixed points. One of Kœnigs’s primary tools was the Schröder functional equation given by S ı f f .p/ S. Given a function f with an attracting (but not super-attracting) fixed D 0 point at p, Kœnigs rigorously demonstrated in [29] that an invertible function S exists on a neighborhood of p satisfying the Schröder equation. Since S f f .p/ S implies ı D 0 that f n.z/ S 1..f .p//n S.z//, solving the Schröder equation models iteration near D 0 p via the linear mapping z f .p/ z on a neighborhood of the origin. That is, in a 7! 0 neighborhood of p, f is analytically conjugate to this linear map. One of the major foci of post-Kœnigs study of iteration was the solution of related functional equations. Kœnigs, however, was not the first to consider iteration of complex functions in a dynamical context. In 1870–1871, the German mathematician and logician Ernst Schröder (of the Schröder-Bernstein Theorem) wrote two papers [40] and [41] on iterative algorithms for solving equations. His interest was piqued by the Newton’s method algorithm f .zn/ zn 1 zn 0 used to approximate solutions to f .z/ 0. When things go well, C D f .zn/ D Newton’s method generates a sequence zn converging to a root of f . f g Schröder’s curiosity about Newton’s method led him into a brief but insightful study of iteration on the complex plane where he discovered the phenomena of attracting fixed f .z/ points. Viewing Newton’s method as the complex function Nf .z/ z , he dis- D f 0.z/ covered that a possibly complex root p of f is also a super-attracting fixed point of Nf . This not only explained to Schröder why Newton’s method works, but led him to generalize Newton’s method and create a family of root-solvingalgorithms that continues to draw interest today. Schröder also became interested in the Schröder functional equation. Althoughhe could not solve it for arbitrary f , as Kœnigs did roughly fifteen years later, he used a Schröder 2z equation based on the trigonometric identity i tan.2 arctan.iz// to iterate 1 z2 D 2 C 1 z 2 Nq.z/ C , the Newton’smethod functionfor the quadratic q.z/ z 1. He showed D 2z D 18 A Century of Advancing Mathematics that on the left (resp., right) half-plane N n.z/ 1 (resp., 1). He observed sensitive q ! dependence on initial conditions on the imaginary axis and called attention to behavior we would now term chaotic. In the late 1870s Arthur Cayley independently used very different methods to obtain this same result in [8], but his examination did not involve general principles of iteration, as did Schröder’s. Buoyed by their successful examinations of Newton’s method for the quadratic, Cayley and Schröder each attempted without any success to the find the convergence regions for Newton’s method for higher degree polynomials. Both remarked that the obstacles to such a study were quite formidable.

The announcement of the 1918 Grand Prize While neither Kœnigs nor his immediate successors were able to describe iteration in the case where a periodic point p is irrationally neutral, Kœnigs’s greatest frustration appeared to be his inability to extend his study beyond the local behavior of iteration near a fixed point, a fact he explicitly lamented at the conclusion of his 1884 paper on iteration. Things had not improved by 1897, when Leau expressed a similar frustration that he could not find the full domains of solutions to functional equations such as the Schröder equation. Kœnigs wondered aloud whether it was possible to expand the study beyond a neighborhood of a fixed point. Leau thought such an attempt “impractical.” The reasons for Kœnigs and Leau’s failure to move their focus beyond the local are in large measure historical. In France, at least, set theory and point set topology were in their infancy. Important tools in analysis had yet to be invented: for example, Montel’s theory of normal families, which would prove instrumental to the successes of Fatou and Julia, would not be unveiled for almost another decade. It would then be another ten years before its applicability to complex dynamics would be understood, and even then Julia (and to a lesser extent Fatou) kept this insight under wraps for a bit longer. In Kœnigs’sinabilityto extend the study ofiteration beyond thefixed pointare the seeds of the works of Fatou and Julia. Not only did both mathematicians adopt the terminology and techniques of Kœnigs, including the study of functional equations,2 but his failure to extend knowledge of iteration beyond a neighborhood of a fixed point became a primary motivation for their studies. Fatou’s first published work regarding complex dynamics [13] appeared in 1906, and one of its accomplishments is a description of the global properties of iteration for the fam- zk ily of functions z . He continued to study the iteration of complex functions for 7! zk 2 several years before publishingagainC on the subject in 1917 (see [3] for more details about this).There were also other French mathematicians who studied iterationin theearly 1900s, most notably Samuel Lattès, who published several papers between 1903–1918 on iteration focusing on the iteration of functions with more than one (possibly complex) variable. The desire to extend the study of iteration beyond the neighborhood of a fixed point became central in late 1915 when the French Academy of Sciences announced that the 1918 Grand Prize in Mathematics would be devoted to the study of iteration of complex functions. The Academy cited Kœnigs’s work and suggested that entrants might want to

2Julia, however, intentionally postponed his use of functional equations until after the appearance of his 1918 monograph. A Century of Complex Dynamics 19 focus on the iteration of rational complex functions of a single variable. With the prize came 3000 francs, a tidy sum in its time. While Fatou had already begun a study of iteration almost ten years before the Grand Prize was announced, Julia had little if any previous experience in complex dynamics and was almost certainly inspired by the announcement.

The awarding of the prize When the contest was announced in 1915, Julia was in the midst of a long recovery from a terrible and disfiguring war injury that he later customarily covered with a nose patch. Julia had entered the war as an exceptionally promising 21 year-old mathematics student at the Ecole´ Normale Supérieure and suffered his wound in January 1915 in the battle of the Chemin des Dames. His recuperation was long and painful, and the severity of his injuries made it difficult for him to do mathematics for quite some time. However, as his recuperation progressed, he took up mathematics again to resounding success. He read mathematics deeply and in 1917 completed his doctoral thesis, which also earned him the Academy’s 1917 BordinPrize. Atsome pointin late 1916 or early 1917, he decided to enter the competition for the 1918 Grand Prix, and by spring 1917 his work was well underway. Meanwhile, Fatou had also been hard at work. The deadline for official entries was December 31, 1917, but results were often announced before formally submitting an entry, and in May 1917 Fatou published [14] which contains several preliminary findings that grew out of his 1906 publication. It is not knownif eithermathematician had suspected theother was planningtoenter the contest before this, but the results Fatou put forth in 1917 evidently startled Julia, who had already independently achieved many of them. At this point, Julia made the tactical decision to submit his own preliminary results to the Academy through a series of sealed letters that would remain unopened until Julia decided otherwise. There was nothing unusual in this, and the Academy even had a special registry dedicated to processing sealed submissions. By the end of May it seems that neither mathematician had thought to apply Montel’s theory of normal families. That changed on June 4 with a short publicationby Montel [33]. During its course, Montel applied his theory of normal families to a sequence of functions. Although neither the application nor the sequence had anything to do with iteration, it wouldbe difficult for eitherFatou and Julia tolookat the sequence that Montel expressed as “f1.z/; f2.z/; : : : ; fn.z/; : : : ” and not think of iteration. This publication evidently opened both men’s eyes to the potential of normal families. However, it seems that neither knew that the other had had the same insight, at least initially. Over the next few months both men found the theory of normal families powerful, and each, mostly likely operating in ignorance—but perhaps in suspicion—of what the other was doing, established a series of now fundamental results including the partitioning of the sphere into domains of normality (the Fatou set) and non-normality (the Julia set). While Julia submitted his preliminary findings to the Academy via the sealed letters, Fatou readied the short publication [15] announcing additional preliminary results that appeared on December 17. Since he had submitted his letters prior to the appearance of Fatou’s December 17 announcement, Julia no doubt felt he had established and deserved priority for the results 20 A Century of Advancing Mathematics they contained, and so on December 24 he asked the Academy in [23] for a formal priority judgment. Following established procedures, on December 31, the deadline for the contest, the Academy ruled in Julia’s favor, saying he had indeed communicated his results before the appearance of Fatou’s December 17 announcement. It is unclear, however, what advantage Julia gained by his tactics—unless his goal was to drive Fatou out of the contest—since Fatou decided not to submit an entry. It was a curious decision on Fatou’s part, but he evidently kept his own counsel, and the reasoning behind it remains a mystery. Michèle Audin argues in [3] that, had he entered, the Academy would have split the prize between Julia and Fatou, and at the meeting in late December 1918 when the results of the Grand Prize were announced, Fatou did receive a 2000 franc prize for his work in analysis throughout his career. Clearly, the Academy wanted to recognize Fatou, but perhaps they would have chosen the same route and still awarded the Grand Prize to Julia even if Fatou had submitted an entry. The events surrounding the prize proved controversial, and Audin presents a strong case that they were polarizing. For example, Montel and Lebesgue seemed to have had great sympathy for Fatou. On the other hand, other Parisian mathematicians, particularly Emile´ Picard, championed Julia. To complicate matters further, the personalities of Fatou and Julia were quite different. Although Fatou came from a prominent naval family, he suffered from ill health (and perhaps anxiety) much of his life, and consequently did not serve in the military. Despite his friendship with Montel and Lebesgue, he worked as an astronomer at the Paris Observatory rather than as academician in a department of mathematics. Fatou was by nature reticent, and Léon Bloch, a physicist and friend of Fatou, said that Fatou found it difficult to speak in front of an audience, which suggests, perhaps, a reason why he sought work at the Paris Observatory rather than a teaching position. However, this may not be entirely accurate since Fatou evidently applied to the Collège de France in the early 1920s. Perhaps Fatou’s career choice and lack of military service in the time of war made him a bit of an outsider. Julia on the other hand was a war hero, and during his recuperation from his battle wounds was often visited by Picard and Georges Humbert (the latter of whom issued the Academy’s priority judgement in favor of Julia). Moreover, Julia was a rising young star fifteen years Fatou’s junior. Many young French intellectuals had died in the war, Picard’s elder son included, and to many in the older generation, Julia represented the future. It is important to keep in mind, however, that there is no evidence that any of this came into play during the priority judgement or the awarding of the prize. Much of Humbert’s report cannot be debated. It stated that Julia’s results stemming from the theory of normal families were submitted first, which is true. It claims that the results from Fatou’s publication are by and large present in Julia’s sealed envelopes, also true. The only matter than can be debated is Humbert’s claim that Julia’s results are at times more general.

The work of Fatou and Julia Julia’s prize entry [24] is an almost 200-page monograph concerning the iteration of rational complex functions of a single variable that was published in 1918. Fatou’s monograph A Century of Complex Dynamics 21

[16], well over 200 pages, also focuses almost exclusively on rational functions and was published in three parts beginning in 1919. One assumes that at least part of Fatou’s monograph was originally intended to be submitted for the Grand Prize. Fatou and Julia’s monographs collectively form the bedrock of contemporary complex dynamics. Partitioning the plane into domains of normality and non-normality, they exploited the deep connections between Montel’s theory of normal families and complex dynamics. Fatou and Julia each understood the topological structure of the Julia and Fatou sets, as well as thedynamics of iterationon each, includingthefact that theforward orbitof a neighborhoodof a point in the Julia set encompasses the entire sphere, with the exception of at most two points. Likewise, they each showed that the domain of normality contains zero, one, two or infinitely many components.3 This last result helps explain the difficulty that Schröder and Cayley had in extending their analysis of Newton’s method to the cubic: since each of the three roots of a cubic corresponds to a separate component of the Fatou set, there must be infinitely many, and their methods were simply not up to the task of understanding this. Fatou and Julia each offered proofs that fractal Julia sets were the norm, not the exception, and explored many now famous examples. One fascinating aspect of their work was their ability to understand what now famous Julia sets looked like. Despite lacking the computational means to visualize such sets, they were able to explain what they perceived using existing examples from mathematics such as the Koch snowflake, which Helge von Koch introduced in 1906. Julia’s schematic of the Julia set for z . z3 3z/=2 (which 7! C bears some similarity to the left image in Figure 1), is based on the Koch curves, and Fatou invoked Koch as well. Fatou explored hypothetical regions that he called singular domains, that is, components of the Fatou set on which the family of iterates of f forms a normal family but is not contained in a domain of attraction for a periodic orbit.He was perfectly candid that he did know whether such regions even exist; nonetheless he established a limit upon them. The reader might recognize these regions as Siegel disks or Herman rings;4 to Fatou, however, their ultimate character was unknown, and he was careful not to speculate what they might look like. In contrast, Julia doubted the existence of such regions, and in a brief 1919 follow-up to his monograph outlined a proof that Siegel disks could not exist [25]. In the mid-1930s, Julia realized that his argument contained an error, yet this did not seem to shake his confidence that his claim was correct. While the studies of each man are remarkably similar there are differences. Most striking among them perhaps is Fatou’s openness about the possible existence of singular domains whose existence Julia denied. Interestingly, Fatou even remarked upon Julia’s denial and seemed not to take it as gospel. There were also differences in style: Julia wrote in an austere axiomatic style while Fatou’s account was looping and discursive, often revisiting ideas, much as a novelist might return to a character many times to better depict her maturation.

3At the time of Julia’s submission, Latt`es’ example of a function whose Julia set encompasses the entire Riemann sphere was unknown to Julia, although he speculated that such functions quite possibly exist. Once Latt`es’ result was known, both Fatou and Julia seemed rather nonplussed by it. 4A Siegel disk is a componentof the Fatou set on which the map is conjugate to rotation of a disk and will be discussed in more detail later in this section and in 4. A Herman ring is a componentof the Fatou set on which the map is conjugateto a rotation of an annulus. Herman rings will be discussed in 5. 22 A Century of Advancing Mathematics

Complex dynamics in the US: 1915–1917 Perhaps the most stubborn problem Julia and Fatou encountered involved iteration around an irrationally neutral fixed point p, that is, one whose derivative is f .p/ e2i with 0 D irrational. We know now that such a fixed point could be in the Fatou set, in which case a Siegel disk exists, or p could be in the Julia set, J.f /. The only result that either mathematician stated regarding this case was Julia’s mistaken proof that Siegel disks do not exist. Unbeknownst to them, however, a mathematician in the United States, George Pfeiffer, had already proved a substantial result. In April 1917 he published the paper [35] in the Transactions of the American Mathematical Society in which he found conditions on the derivative f 0.p/ of an irrationally neutral fixed point p that precluded the existence of a convergent solution S to the Schröder equation S f f .p/ S. In other words,he found ı D 0 conditions which imply that an irrationally neutral fixed point is in J.f /. Pfeiffer had already announced this result in presentations to the AMS in October 1915 and April 1916,as well as ina footnotein the 1915 paper [34] on conformal arcs published in the American Journal of Mathematics. His 1917 paper cited the work of Kœnigsas well as others who investigated the iteration of complex functions and the associated functional equations, and explicitly noted that his was the first to produce any definitive result in the case where the derivative of the fixed point f 0.p/ was an irrational root of unity. Pfeiffer constructed a function f with an irrationally neutral fixed point at p whose derivative f 0.p/ satisfies a convoluted recursion relation. Next, he deduced a function k S 1 sk.z p/ which algebraically satisfies S f f .p/ S by assuming D k 0 ı D 0 such a functionD exists, and then solving for its coefficients. He showed that the denomina- P tors of the sk become quite small as k forcing the coefficients to grow quite large ! 1 which causes S to diverge on any neighborhood of p. In other words, Pfeiffer constructed a function S with small divisors. He remarked in his paper that he had received a helpful (but unspecified) suggestion from George David Birkhoff. Birkhoff was no doubt familiar with small divisors problems in celestial mechanics, and perhaps he gave Pfeiffer advice on treating them. Pfeiffer observed that he became interested in Schröder equation via the lectures of another American mathematician, Edward Kasner,5 a founding member of the MAA who taught Pfeiffer at Columbia. Kasner’s lectures involved conformal invariants, which link to the Schröder equation, though not in the context of complex dynamics. In 1918 Pfeiffer published a follow-up [36] to his 1917 paper concerning a related functional equation, g2 f , where f is given, but was known more as a teacher than D a researcher, although he did serve as an editor for the Annals in the 1920s. Pfeiffer later taught at Princeton before settling in at Columbia where he taught until his death in 1943. It is probably not surprising that Julia and Fatou worked in ignorance of Pfeiffer’s results. The war no doubt made the transportation of American journals and mathematical ideas problematic, and it is not clear that they even looked to America for help.

5Kasner is perhaps most famous for his association with the words “googol” and “googolplex”, which he coined, he says, after asking his nephews,who were young children at the time, what they might call a very large number. Others might know him in conjunction with his co-authoring of Mathematics and the Imagination with James R. Newman. A Century of Complex Dynamics 23

Nonetheless, there was a burgeoning interest in iteration in America in 1915 which, in addition to Pfeiffer’s announcements, saw the publication of papers involving iteration by two other American mathematicians. As was the case with Pfeiffer’s announcements, these works also predated the French Academy of Sciences’ December 1915 Grand Prize announcement. The first paper [4] was written by another founding member of the MAA, Albert A. Bennett, and the other [37] by the mathematician Joseph Fels Ritt, who later published periodically in the MAA circle of magazines and became a life-long member of the MAA in the early 1920s. Bennett’s paper, appearing in the Annals in September 1915, came before Ritt’s and represents the first American research paper to look at the iteration of complex functions. While Bennett’s paper does not contain any important new results, its greatest benefit was an introductionto US readers of the results of Kœnigs and others. Bennett followed up this paper with [5] the next year on the iteration of functions of several variables. While Pfeiffer traced his interest in functional equations to problems arising out of lectures by Kasner, it is not clear what sparked Bennett’s interest in iteration. A 1914 letter written by Oswald Veblen to Birkhoff discussed a conversation he had with Bennett while they were both in Paris in which he had urged Bennett to seek new mathematical directions. Perhaps Bennett’s paper is the fruit of that discussion. Bennett went on to a distinguished mathematical career at Brown after teaching at Princeton. He also served the MAA in several capacities including as a member of the Council (equivalent to the current-day Board of Governors), Vice-President, Trustee, and Editor-in-Chief of the Monthly. In 1967 he wrote a history of the pre-World War II MAA [6] that appeared in the Fiftieth Anniversary Issue of the Monthly.He died in 1971. Ritt’s paper appeared in the Annals in December 1915 and concerned the so-called Babbage functional equation, f n f for a real function f , an equation that the British D logician and mathematician George Babbage examined in early 1800s. This paper was the first of several by Ritt to concern iteration, some of which made lasting contributionsto the field, especially his 1923 paper [39] in the Transactions on complex permutable functions, that is, functions f and g which satisfy f g g f which Fatou and Julia had also ı D ı studied. Setting g f n, it follows that f n f f f n, so permutable functions are D ı D ı linked to the process of iteration. Ritt, an important American mathematician who enjoyed a long career at Columbia untilhisdeath in 1951,was a student of Kasner. He publishedhis first results onthe iteration of complex functions in France in early 1918 in the same journal that Fatou published his preliminary results [38]. Since Ritt’s interest in iteration stems back to 1915 prior to the announcement of the 1918 Grand Prize, one wonders if he considered submitting an entry to it. While the American interest in iteration waned, it did not disappear. As we will see, a paper fundamental to the study of complex dynamics was published in the Annals in 1942, although the author was not an American mathematician.

1920–1942 Following their great monographs on the iteration of rational complex functions, Julia and Fatou each studied dynamics well into the 1920s, writinghundreds of pages and over forty 24 A Century of Advancing Mathematics publications between them. While none of these works had the majesty of their monographs, there were important works among them. Beginning in 1919 Julia applied techniques involving normal families honed in his study of iteration to the so-called “curves of Julia,” which result from examining the values a function f takes along an angle whose vertex is an isolated essential singularity [26]. In 1922 Fatou examined the dynamics of a particular kind of algebraic function in [17], and in 1926 published a foundational work on the iteration of transcendental functions [20], each of which opened new lines of inquiry. In the early 1920s, both wrote important papers on permutable functions [18], [27] (another topic introduced by Kœnigs!) and explored the iteration of functions of more than one variable [19], [28]. There were others abroad who were inspired by their studies. The renowned Japanese complex analyst, Kiyoshi Oka, became intrigued by complex dynamics in the late 1920s, and even travelled to Paris where he began a long, still unpublished paper on permutable functions that drew upon the studies of Ritt, Fatou and Julia.6 In Germany, Hubert Cremer steeped himself in Fatou and Julia’s monographs and in 1924 gave a presentation at the Mathematics Colloquiumat the University of Berlin that introduced their ideas to a German audience. Cremer’s interest in the subject grew. Beginning in 1927, he took up the study of irrationally neutral ﬁxed points, which he continued through a series of papers over the next decade. Unlike Fatou and Julia, he read and acknowledged Pfeiffer’s work, and Cremer’s best known result is actually a reﬁnement of Pfeiffer’s discovery of (to use the contemporary point of a view) irrationallyneutral points p that belong to the Julia set [9].

The conditionsthat Pfeiffer placed on f 0.p/ defy concise explanation. It was Cremer’s genius to ﬁnd conditions that can be easily expressed: Let f be a rational function of a single complex variable of degree s with an irrationally neutral ﬁxed point at p. If

sn n lim inf .f 0.p// 1 0; n 1;2;::: j j D D p then a convergent solution to the associated Schr¨oder equation S f f .p/ S does not ı D 0 exist. A few years later, in connection with his interest in maps of annuli, Cremer obtained another important result in [10], namely, that if a singular domain (a component of the Fatou set that is not part of a domain of attraction) exists for a rational function, its degree of connection is at most two. Moreover, he also showed that doubly connected singular domains could not exist if f was entire. While he did not prove such domains exist, his result serves as an anticipation of Herman rings. Like Fatou, Cremer remained agnostic towards the existence of Siegel disks or Herman rings throughout his study but seemed skeptical of the validity of Julia’s proof that Siegel disks could not exist. It seems reasonable to assume that Cremer tried to show the existence of irrationally neutral ﬁxed points that were not in the Julia set, that is, that Siegel disks exist. If so, he was unsuccessful, but his work suggests an explicit connection between number theory and the center problem, one that was also implicit in Pfeiffer’s paper: Cremer showed that the

6This paper is available on the web at www.lib.nara-wu.ac.jp/oka/ikou/s19/p000-1.html. It is speculated, but not documented,that Oka met and perhapsstudied with Julia. A Century of Complex Dynamics 25 lim-inf conditions that he imposed upon f .p/ e2i (stated above) forces to be a 0 D Liouville number, which are said to be well-approximated by rational numbers.

It turns out that in order for a Siegel disk to exist, the conditions on f 0.p/ need to be flipped: if is “highly irrational,” that is, the continued fraction expansion of consists of a collection of integers that are bounded above, then a Siegel disk surrounding p exists and p is in the Fatou set. Indeed, in 1942 Karl Ludwig Siegel, a German mathematician who came to Princeton to escape Nazi Germany, published a remarkably important paper whose slenderness—six pages—belies its impact [24]. Siegel showed that if is highly irrational, then a convergent solution to the Schröder equation S f f .p/ S exists. In other words, iteration around ı D 0 such an irrationally neutral fixed point is conjugate to an irrational rotation by 2. Siegel’s construction of S relies on delicate bounds on the coefficients of S whose denominators are quite small. Siegel’s solution thus represents a successful resolution of a small divisors problem—itself an important achievement. Indeed, Jürgen Moser, who after the war was a student of Siegel’s back in Germany, found inspiration in Siegel’s work for his own studies of what was to become known as KAM theory.7 Siegel’s result closed a door on a phase in the development of complex dynamics that began with Fatou’s 1906 paper. The center problem in complex dynamics was arguably the most obvious of the problems Fatou and Julia left unresolved in their monographs. While research in the iteration of complex dynamics never completely stopped—soon after the war Paul Charles Rosenbloom published a short paper on fixed points of entire functions and Irvine Noel Baker began his own exploration of entire functions in 1955—it is safe to say that the subject no longer received the attention it had prior to the war, nor would it for quite some time.

4 The renaissance of complex dynamics

While there certainly was some work going on in the field of complex dynamics in the period 1942–79, nothing compares to what happened in 1979. At that time, Benoit Man- delbrot was working at the IBM Thomas J. Watson Research Center, home to some of the most powerful computers of the day. Interestingly,BenoitMandelbrot had an uncle, Szolem Mandelbrojt, who was also a mathematician. Szolem was a student of Jacques Hadamard and later succeeded him as a Professor at the Collège de France. Mandelbrojt worked in the field of complex analysis and was familiar with the work of Julia and Fatou. He eventually informed Benoit Mandelbrot about the interesting objects that Julia and Fatou had thought about so many years earlier, and so Mandelbrot decided to have a look at these objects using computer graphics. What he saw astounded him (as well as the rest of the mathematical community). Mandelbrot decided to concentrate for simplicity on quadratic polynomials. It is well known that any such quadratic map is dynamically equivalent to one of the form Pc .z/ D z2 c where c is a complex parameter. Now, when c 0, the Julia set of z2 is the unit C D circle; all pointsoutside the unit circle have orbitsthat tend to , while all pointsinsidethe 1 unit circle have orbits that tend to 0, which is therefore an attracting fixed point. Similarly,

7For more about Siegel’s solution and its connection to KAM theory, see [2]. 26 A Century of Advancing Mathematics the Julia set of z2 2 is the interval Œ 2; 2 on the real axis in C, though this is a little harder to prove. It turns out that these are the only two “computable” Julia sets for z2 c; C all other Julia sets for z2 c are fractals. C Without going into details, a fractal object is a set that is everywhere self-similar (if you zoom in on the set, you see the same structure over and over again) and that also has the property that its “fractal” dimension (usually the Hausdorff dimension) exceeds its topological dimension. For Julia sets of z2 c, the topological dimension is just 1 if this set C is connected and it is 0 otherwise, but when c 0; 2, the fractal dimension is often not ¤ an integer. For example, the Julia set for c 1 is the “basilica” and for c :12 :75i D D C it is the “Douady rabbit.” See Figure 1. Zooming in to the rabbit shows that the rabbit’s ears have ears, and those sub-ears have ears, etc., etc. That is self-similarity. Mandelbrot, the father of fractal geometry, was intrigued.

Figure 1. The Julia sets for z2 1 (the basilica) and z2 0:12 C :75i (the Douady rabbit). Colored points have orbits that escape to 1 while black points have orbits that tend to a periodic orbit of period 2 in the basilica case and of period 3 in the rabbit case. So the Julia sets here are the boundaries of the black and colored regions.

Mandelbrot plunged more deeply into the quadratic case. Julia and Fatou knew that the Julia set of z2 c was either a connected set or else a Cantor set, i.e., a totally disconnected C set. There are no quadratic Julia sets that consist of 2 or 20 or 200 components; either the Julia set is one piece or it consists of uncountably many pieces, each of which is a point. And Julia and Fatou also knew that, amazingly, it was the orbit of 0 that determines this: if n P .0/ , then J.Pc / is a Cantor set, but if the orbit of 0 behaves otherwise, J.Pc / is c ! 1 a connected set. The reason that the orbit of 0 determines this is that 0 is the only critical point for z2 c and the fate of the “critical orbits” essentially determines everything in C complex dynamics, something Fatou and Julia both understood well. (For higher degree polynomials, there are usually more critical orbits and so the structure for these maps is more “complex.”) So Mandelbrot decided to draw the picture of all those c-values in the complex plane for which the orbit of 0 does not tend to . What astonishingly comes out is one of the 1 most famous and most beautiful objects in all of mathematics, the set that now bears his name, the Mandelbrot set. See Figure 2. A Century of Complex Dynamics 27

Figure 2. The Mandelbrot set and a magniﬁcation. Colored points are c-values for which the orbits of 0 escape to 1; black points are c-values for which this does not happen. So the Mandelbrot set is the black region in these images.

The black bulbs visible in the Mandelbrot set each contain parameters for which there is an attracting periodic orbit of some given period. For c-values in the large main cardioid, each Pc has an attracting ﬁxed point, and the corresponding Julia set is a simple closed curve. The large bulb to the left of the main cardioid is actually an open disk of radius 1=4 centered at c 1 and c-values here give rise to an attracting cycle of period 2 (the D basilica is the Julia set that arises when c is at the center of this disk). And the two large disks above and below the main cardioid correspond to parameters for which there is an attracting cycle of period 3; the Douady rabbit sits at the center of the northern period-3 bulb. After the appearance of the Mandelbrot set, many mathematicians jumped in and continued the work of Fatou and Julia. Luckily, the areas of mathematics known as dynamical systems and complex analysis had made important strides forward during the prior ﬁfty years, and many new tools were therefore available to extend the earlier results. One of the most important new results was Sullivan’s No-Wandering Domains Theorem [45] from 1985. In this paper Sullivan showed that any component of the Fatou set must be eventually periodic in the case of polynomials or rational maps.8 In particular, it then follows that there are only three types of Fatou components in the polynomial case: 1. Attracting basins, in which all points tend to a particular attracting periodic orbit which therefore lies in the Fatou set; 2. Parabolic basins, in which all pointstend to a periodic orbit of period n that now lies in the Julia set and for which the derivative of P n is of the form exp.2i.p=q//; 3. Siegel disks. Along the boundaries of the bulbs in the Mandelbrot set are the c-values for which n Pc has a cycle that is neutral, i.e., the derivative of P is of the form exp.2i/. If is rational, then we are in the case of a parabolic basin. If is “highly irrational” we are in

8Although it is a natural to ask if wandering domains exist, neither Fatou nor Julia seem to have raised this question in their published works. 28 A Century of Advancing Mathematics the Siegel disk case. See [32] for the precise technical definitions of highly irrational. But there are certain irrational values of , for example the ones Pfeiffer and Cremer found, for which we do not have a Siegel disk. What happens here dynamically is stillnot understood. Think about this: if the quadratic function z2 c has a fixed point whose multiplier is a C not-so-irrational number, we still do not know what happens near this fixed point. This is one of the major open problems in complex dynamics. Another important contribution in the 1980s was made by Douady and Hubbard [12]. It is well known that the basin of attraction Bc of in the Riemann sphere is an open disk 1 when c lies in the Mandelbrot set. Hence, by the Riemann Mapping Theorem, there is an analytic homeomorphism c Bc D that takes to 0 and for which .0/ a > 0. W ! 1 c0 D Douady and Hubbard showed that this map actually conjugates the map z2 on the disk D 2 to Pc in Bc. That is, c.Pc .z// .c .z// . This implies that Pc behaves dynamically 2 D 2 on Bc just like z does in D. Since z interchanges the straight rays lying in the unit disk and given by exp.2it/, the curves that are mapped to these straight rays by c are also interchanged by Pc. These curves are called external rays of angle and we denote them by .t/. If the limit as t 1 of .t/ is a unique point in J.Pc /, then we say that ! the external ray lands at this limit point. And, if all such external rays land, then we essentially know the dynamics on the Julia set since the straight rays are permuted by z2. This may not happen, however. For example, if the Julia set is a locally connected set, then all of the external rays do indeed land. But if J.Pc / is not locally connected, then some rays may only accumulate on a portion of J.Pc /. This is what may happen when we have those not-so-irrational parameters in the boundary of the Mandelbrot set. More importantly, the same external ray construction can be carried over to the parameter plane. Let ˆ be the map defined on the complement of the Mandelbrot set in the c-plane given by ˆ.c/ c .c/. Douady and Hubbard [12] also show that ˆ is an analytic D homeomorphism onto D. The main open problem involving the Mandelbrot set is then: Conjecture. The boundary of the Mandelbrot set is a locally connected set. If this conjecture is true, then all of the corresponding external rays in the parameter plane land at unique points in the boundary of the Mandelbrot set. In this case, we would then get a complete map of the Mandelbrot set that tells us everything about its structure: how the bulbs containing periodic cycles are arranged; how the parameters along the “antennas” on the bulbs are situated, etc. Despite the fact that we are dealing here with the relatively simple map z2 c, this conjecture seems to be a long way from being re- C solved. In particular, a recent result of Buff and Chéritat [7] shows that certain Julia sets for quadratic polynomials that contain those not-so-irrationalfixed points can have positive Lebesgue measure. This means that things are even more complicated than most complex dynamicists had thought back in the 1980s. Furthermore, a result of Shishikura [42] states that the Hausdorff dimension of the boundary of the Mandelbrot set is 2, so this boundary is also an extremely complicated object.

5 Rational maps Complex rational maps are naturally more complicated than polynomials, primarily because there often is no basin of attraction at and there are usually many more critical 1 A Century of Complex Dynamics 29 orbits. So, to simplify matters, we concentrate here on the family of degree 2n rational maps given by

n F.z/ z D C zn n where we assume n 2. It turns outthat is an attracting ﬁxed point in C since F z 1 when z is large, so we do have an immediate basin B of . Also, one checks easily j j 1 that the critical points are given by 1=2n. However, there are only two critical values given by 2p. And, just as in the case of z2 c, there really is only one critical orbit (up to ˙ C symmetry). This follows from the fact that, if n is even, both critical values then map to the same point, whereas, if n is odd, the map is symmetric under z z, so the orbitsof both 7! 2p behave symmetrically. ˙ The Fatou set for a rational map can now contain a kind of set that does not occur with a polynomial, namely a Herman ring. First discovered by Herman in 1979 [22] right

1=4 1=16 D D

0:01 0:001 D D Figure 3. Various Sierpinski curve Julia sets drawn from the family z2 C=z2; all of thesesetsare homeomorphic, but it is known that the dynamical behavior on each of these sets is very different. The red regions are the preimages of B. 30 A Century of Advancing Mathematics around the time the Mandelbrot set was ﬁrst observed (although, as noted above, Cremer anticipated Herman rings in the early 1930s), these regions are (eventually) periodic an- nular regions of period n in which all points rotate around distinct simple closed curves under a given irrational rotation. The reasons these types of Fatou domains do not occur for polynomials is that there has to be a pole inside one of these Herman rings; otherwise, all points inside these annuli would be mapped to corresponding points inside the image annuli. So the iterates would form a normal family in these disks. Unlike quadratic polynomials where there were only two types of Julia sets depending on the escape behavior of the critical orbit, there is now an “escape trichotomy” for this family. Since we have a basin of attraction B at and a pole at 0, there is a neighborhood 1 of 0 that is mapped to B. If this neighborhood is disjoint from B we call it the trap door T since any orbit that eventually enters B must do so by passing through T. Then there are three possible types of Julia sets depending upon the behavior of the critical orbit:

1. If the critical values lie in B (if one does, the other must also due to the z z 7! symmetry), then the Julia set of F is a Cantor set;

2. If the critical values lie in T, then J.F/ is a Cantor set of simple closed curves surrounding the origin [31];

3. In all other cases J.F/ is a connected set, and if the critical values do not lie in B or T but the critical orbit eventually enters B, then J.F/ is a Sierpinski curve. A Sierpinski curve is a planar set that is homeomorphic to the well-known Sierpinski carpet fractal. See Figure 3 for several Sierpinski curve Julia sets that arise in the family z2 =z2. These sets are very important from a topologicalpoint of view in that they form C a dictionary of all possible plane curves. More precisely, given any one-dimensional plane continuum, this curve can be homeomorphically manipulated so that it can be embedded in the Sierpinski carpet [44].

6 Entire functions Now we turn to the very different case of entire transcendental functions where the possibilities for Fatou sets (as well as Julia sets, as we will soon see) become even richer. Wandering domains are now possible. These are Fatou domains that are never eventually periodic. For example, the map z z 2 sin z has a wandering domain. The vertical 7! C lines given by Re z 2k for k Z are easily seen to be invariant under this map and D 2 each lies in the Julia set. However, neighborhoodsof the critical points given by =2 2k C lie in the Fatou set and all wander off to . 1 Another new possibilitythat arises for the Fatou set is a Baker domain. These are open sets extending to in which all orbits tend to . But, unlike the quadratic polynomial 1 1 case, there is no longer an open disk that completely surrounds the point at . Since is 1 1 an essential singularity there are points in the Julia set that are arbitrarily close to . An 1 example of this arises in the map z z e z 1, where points in the right half-plane 7! C C tend to . This was shown by Fatou in 1926 in [20]. 1 The analogue of quadratic polynomials in the entire transcendental case is the exponential function E.z/ exp.z/. There are no critical points for this function, but there D A Century of Complex Dynamics 31

Figure 4. A small piece of the Cantor bouquet for E with < 1=e and the ensuing explosion when > 1=e. Colored points again have orbits that escape to 1 and so, in this case,lie in the Julia set. is what is known as an asymptotic value, namely 0. This is the omitted value for the exponential maps and, moreover, any curve tending to in the far left half-plane is mapped 1 to a curve that limits on the asymptotic value. As a consequence, 0 plays the same role as the critical points do in the case of polynomials or rational maps. But now, a very different phenomenon occurs. A theorem of Goldberg, Keen, and Sullivan [21], [45] says that, if the orbit of the asymptotic value 0 tends to , then J.E/ is now the entire complex plane. 1 Now consider the family E where R . The graphs of E along the real axis show 2 C that, if > 1=e, then the orbitof 0 (in fact, all orbitson the real axis) tend to . So, in this 1 case, J.E/ C. But, if < 1=e, there exist one attracting and one repelling fixed point D in RC. Moreover, it is easy to check that all points to the left of the repelling fixed point in RC are contracted into a disk lying to the left of this half-plane. So, by the Contraction Mapping Principle, these points all have orbits that tend to the attracting fixed point, and so this half-plane lies in the Fatou set. In fact, when 1=e, it is known that J.E/ is a Ä Cantor bouquet, i.e., a collection of infinitely many disjoint smooth curves with endpoints that extend to in the right half-plane. 1 So there is an amazing explosion in the Julia sets for these maps when passes through 1=e. See Figure 4. When 1=e, the Julia set lies in the right half-plane, but when Ä > 1=e, suddenly J.E/ C. No new periodic points are born as passes through 1=e; D rather, all of the repelling periodic points move continuously and suddenly become dense in the complex plane. What a change! See Figure 4.

7 The future of complex dynamics The natural question is: where is the ﬁeld ofcomplex dynamics heading in the next century? Already there have been many excursions into areas outside of polynomial dynamics, like the study of rational and entire maps alluded to above. But much more is beginning to happen and likely to expand in the future. This includes the study of other complex maps (like meromorphic functions) as well as higher dimensional complex analytic maps. Another recent topic of interest is algebraic dynamics where questions involving algebraic aspects 32 A Century of Advancing Mathematics

(rather than the dynamical behavior) of iterated functions arise. And much more is on the horizon. The beauty of this expansion includes the fact that many distinct areas of mathematics now enter the picture, including dynamical systems, complex analysis, topology, number theory, and algebraic geometry. And one final note. Complex dynamics is a field that is quite accessible to undergraduate students. After all, the primary topic of interest is the simplest nonlinear function, z2 c. Many undergrads can begin by studying the quadratic family and then move on C to investigate other families of functions (their own choice: cubics, quartics, trigonometric functions, etc.) The beauty of this is, while the complete understanding of these maps will certainly be elusive, nonetheless, in many cases, the students become the first mathematicians to see the interesting behavior in their chosen family of interest. This definitely sparks their interest in research-level mathematics.

Bibliography [1] D. Alexander, A History of Complex Dynamics: From Schröder to Fatou and Juila, Aspects of Mathematics, Vol. 24, Vieweg, Heidelberg, 1994. [2] D. Alexander, F. Iavernaro, and A. Rosa, Early Days in Complex Dynamics: A History of Com- plex Dynamics in One Variable During 1906–1942, History of Mathematics Vol. 38, American Mathematical Society, Providence, and London Mathematical Society, London, 2011. [3] M. Audin, Fatou, Julia, Montel, The Great Prize of the Mathematical Sciences of 1918, and Beyond, Lecture Notes In Mathematics Vol. 2014, History of Mathematics subseries vol. 1, Berlin, Springer-Verlag, 2011. [4] A. A. Bennett, The iteration of functions of one variable. Ann. Math. 17 no. 1 (1915), 23–60. [5] ———, A case of iteration in several variables. Ann. Math. 17 no. 4 (1916), 188–196. [6] A. A. Bennett, Brief history of the Mathematical Association of America before World War II. MAA Monthly 74 no. 1, pt 2: Fiftieth Anniversary Issue (Jan, 1967), 1–11. [7] X. Buff and A. Chéritat, Julia sets with positive measure. Ann. Math 176 2012, 673–746. [8] A. Cayley, Applications of the Newton–Fourier method to an imaginary root of an equation. J. Math. Pur. Appl. 16 (1879), 179–185. [9] H. Cremer, Zum zentrumproblem, Math. Ann. 98 (1927), 151–163. [10] ———, Uber¨ die Schrödersche funktionalgleichung und das Schwarzsche eckenabbil- dungsproblem. Leipziger Berichte 84 (1932), 291–324. [11] R. L. Devaney, D. M. Look, and D. Uminsky, The escape trichotomy for singularly perturbed rational maps. Indiana Univ. Math.J. 54 (2005), 1621–1634. [12] A. Douady, and J. Hubbard, Itération des polynomes quadratiques. C. R. Acad. Sci. Paris 294 (1985), 123–126. [13] P. Fatou, Sur les solutions uniformes de certaines équations fonctionnelles. C. R. Acad. Sci. Paris 143 (1906), 546–548. [14] ———, Sur les substitutions rationnelles. C. R. Acad. Sci. Paris 164 (1917), 806–808. [15] ———, Sur les substitutions rationnelles. C. R. Acad. Sci. Paris 165 (1917), 992–995. [16] ———, Sur les équations fonctionnelles. Bull. SMF 47 (1919), 161–271; Bull. SMF 48 (1920), 33–94 and 208–314. [17] ———, Sur l’itération de certaines fonctions algébriques. Darboux Bull. 2 no. 46 (1922), 188– 198. A Century of Complex Dynamics 33

[18] ———, Sur l’itération analytique et les substitutions permutables. J. Math. Pur. Appl. 2 (1923), 343–384; 3 (1924), 1–49. [19] ———, Substitutions analytiques et équations fonctionnelles àdeux variables. Ann. Sc. Ec.´ Norm. Sup. 3 no. 41 (1924), 67–142. [20] ———, Sur l’itération des fonctions transcendantes entières. Acta Math. 47 (1926), 337–370. [21] Goldberg, L. and Keen, L. A finiteness theorem for a dynamical class of entire functions. Er- godic Theory and Dynamical Systems 6 (1986), 183–192. [22] M. Herman, Sur la conjugasion différentiable des difféomorphisms du cercle àdes rotations. Publ. Math. de IHES 49, (1979), 5–233. [23] G. Julia, Sur les substitutions rationnelles. C. R. Acad. Sci. Paris 165 (1917), 1098–1100. [24] ———, Mémoire sur l’itération des fonctions rationnelles. J. Math. Pur. Appl. 8 no. 1 (1918), 47–245. [25] ———, Sur quelques problèmes relatifs àl’itération des fractions rationnelles. C. R. Acad. Sci. Paris 168 (1919), 147–149. [26] ———, Sur quelques propriétés nouvelles des fonctions entières ou méromorphes. Ann. Sc. Ec.´ Norm. Sup. 3 no. 36 (1919), 93–125;37, 3 no. 37 (1920), 165–218; 3 no. 38 (1921), 165–181 [27] ———, Mémoire sur la permutabilitédes fractions rationnelles. Ann. Sc. Ec.´ Norm. Sup. 3 no. 39 (1922), 131–215. [28] ———, Sur les substitutions rationnelles àdeux variables. C. R. Acad. Sci. Paris 175 (1922), 1182–85; 176 (1923), 58–60 [29] G. Kœnigs, Recherches sur les intégrales de certaines équations fonctionnelles. Ann. Sc. Ec.´ Norm. Sup. 3 no. 1 (1884), 1–41. [30] L. Leau, Etude´ sur les équations fonctionelles àune ou plusieurs variables. Ann. Fac. Sci. Toulouse 1 no. 11 (1897), 1–110. [31] C. McMullen, Automorphisms of rational maps. Holomorphic Functions and Moduli. Vol. 1. Math. Sci. Res. Inst. Publ. 10. Springer, New York, 1988. [32] J. Milnor, Dynamics in One Complex Variable, Third Ed. Princeton University Press, 2006. [33] P. Montel, Sur la représentation conforme. C. R. Acad. Sci. Paris 164 (1917), 879–881. [34] G. A. Pfeiffer, On the conformal geometry of analytic arcs. Amer. J.. Math. 37 (1915), 395–430. [35] ———, On the conformal mapping of curvilinear angles. The functional equation ˚Œf .x/ D a1˚.x/. Trans. AMS 18 (1917), 185–198. [36] ———, The functional equation f Œf .x/ D g.x/. Ann. Math. 2 no. 20 (1918), 13–22. [37] J. F. Ritt, On certain real solutions of Babbage’sfunctional equation. Ann. Math. 2 no. 17 (1915), 113–123. [38] ———, Sur l’itération des fonctions rationnelles. C. R. Acad. Sci. Paris 166 (1918), 380–381. [39] ———, Permutable rational functions. Trans. AMS 25 (1923), 399–448. [40] E. Schroder, Uber¨ unendlich viele algorithmen zur auflösung der gleichungen. Math. Ann. 2 (1870), 317–365. [41] ———, Uber¨ iterirte funktionen. Math. Ann. 3 (1871), 296–322. [42] M. Shishikura, The Hausdorff dimension of the boundary of the Mandelbrot set. Ann. Math. 147 (1987), 225-267. [43] C. L. Siegel, Iterations of analytic functions. Ann. Math. 43 (1942), 607–612. [44] W. Sierpinski, Sur une courbe Cantorienne qui contient une image biunivoque et continue de toute courbe donnée. C. R. Acad. Sci. 162 1916, 629–632. 34 A Century of Advancing Mathematics

[45] D. Sullivan, Quasiconformal homeomorphisms and dynamics I: Solution of the Fatou-Julia problem on wandering domains. Ann. Math. 122 (1985), 401–418.

Daniel Alexander Mathematics Department, Drake University, 2507 University Avenue, Des Moines, IA 50311 [email protected] Robert L. Devaney Mathematics Department, Boston University, 111 Cummington Mall, Boston, MA 02215 [email protected] Map-Coloring Problems

Robin Wilson Open University, UK

Dedicated to the memory of Kenneth Appel

In 1852 Augustus De Morgan was asked whether all plane maps can be colored with just four colors in such a way that neighboringcountries are always colored differently. In 1976 Kenneth Appel and Wolfgang Haken answered this question in the afﬁrmative. But why did this easily stated question take 124 years to be answered, and what was involved in its solution? And since maps drawn on a plane are equivalent to maps drawn on a sphere, can the problem be extended to the coloring of maps drawn on other surfaces? And once these problems were solved, what was there left for map-colorers to do?

1 Early days The earliest reference to map-coloring problems occurs in a letter dated October 23, 1852 from Augustus De Morgan, Professor of Mathematics at University College, London, to Sir William Rowan Hamilton, the distinguished Irish mathematician and astronomer. In this letter, De Morgan described how one of his students had asked whether every map can be colored with only four colors so that neighboring countries are colored differently. A student of mine asked me today to give him a reason for a fact which I did not know was a fact — and do not yet. He says that if a figure be anyhow divided and the compartments differently coloured so that figures with any portion of common boundary line are differently coloured — four colours may be wanted, but not more ...Query cannot a necessity for five or more be invented...My pupil says he guessedit in colouring a map of England. The more I think of it, the more evident it seems. The student was later identified as Frederick Guthrie, who claimed that the problem was due to his brother Francis. Francis Guthrie went on to a mathematical career, but never published anything on the problem. De Morgan became intrigued with the problem and communicated it to other mathematicians, so that it soon became part of mathematical folklore. He also stated his belief that the sufficiency of four colors could not actually be proved, but had to be taken as a postulate—a “necessary truth.” The first known printed reference to the four-color problem was a note by F. G. in The Athenaeum of June 10, 1854.It is not known who “F. G.” was: it may have been one of the Guthrie brothers, or possibly Francis Galton, the geographer and gentleman of science who was seeking admission to the Athenaeum Club around that time.

35 36 A Century of Advancing Mathematics

But it was not until after De Morgan’s death that any progress was made in solving it. On July 13, 1878, at a meeting of the London Mathematical Society, the Cambridge mathematician Arthur Cayley enquired whether the problem had yet been solved. Soon afterwards, at Galton’srequest, he wrotea short note[6] for the Royal Geographical Society in which he attempted to explain where the difﬁculties lay. Cayley also showed that when trying to prove that four colors are always sufﬁcient, one may assume that exactly three countries meet at each point of intersection. To see this, suppose that at each point where more than three countries meet, we stick a circular “patch” over the point, as shown in Figure 1. The resulting map is cubic—it has three countries meeting at each point. If we now color the new cubic map with four colors, we can then obtain a four-coloringof the originalmap by shrinkingeach patch down to a point.In view of this, when desirable, we can assume that all maps under consideration are cubic maps.

1 2 1 2

4 3 1 3 1 2 2

original map add patch color map shrink patch Figure 1. It sufﬁces to consider only cubic maps.

2 Kempe’s ‘‘proof ’’ We now come to one of the most famous mistakes in mathematics. It was made by Alfred Kempe, a London barrister who had studied with Cayley in Cambridge. Kempe published his “proof” that all maps can be four-colored in the newly founded American Journal of Mathematics, edited by Cayley’s friend J. J. Sylvester [11]. Although Kempe’s argument was incorrect, it nevertheless contains some important ideas that resurfaced in the eventual solution to the problem. Indeed, his error is somewhat subtle and his proof was regarded as correct by the mathematicians of the day. Before summarizing the main ideas of Kempe’s fallacious argument, we need a simple lemma. Lemma 1. Every map contains at least one country with at most ﬁve neighbors. Proof We use an important result of Leonhard Euler, who observed in 1750 that, for any polyhedron with F faces, V vertices, and E edges, F V E 2: By projecting C D C the polyhedron onto a plane we deduce that, for any map with C countries, P points of intersection, and L boundary lines, C P L 2: C D C Using Euler’s formula, we deduce that L 3C 6: For, at least three lines meet at Ä each of the P intersection points, so 2L 3P (the factor 2 arising since each line has two ends and is counted twice), and so P 2 L; substituting this into Euler’s formula and Ä 3 rearranging gives the desired inequality. To prove the lemma we now assume that every country has at least six neighbors. Counting up the lines bounding all the countries, we obtain 6C 2L (the factor 2 arising Ä Map-Coloring Problems 37 since each line borders two countries and is counted twice); so 3C L 3C 6, which Ä Ä is a contradiction. This proves the result. We can now present Kempe’s argument. Theorem 2. Every map can be colored with four colors in such a way that neighboring countries are assigned different colors. Proof (Kempe) We assume that the result is false and that a map M is a minimal counterexample—thus, M cannot be four-colored but any map with fewer countries can be. It follows from the above lemma that M contains a digon, triangle, square, or pentagon (see Figure 2). We now show that each possibilityleads to a contradiction.

digon triangle square pentagon Figure 2. Every map contains one of these.

Digon or triangle If the map M contains a triangle (a digon is similarly treated), we remove one of its sides, merging the triangle with one of its neighbors. The resulting map has fewer countries and so, by our assumption, this map can be four-colored. When we reinstate the triangle, we can color it with a color not used for its (at most three) neighbors (see Figure 3). This gives a coloring of M —a contradiction.

r b r b y

g g

original map obtain new map color new map color original map Figure 3. Triangle case.

Square If the map M contains a square, we remove one of its sides, merging the square with one of its neighbors. The resultingmap has fewer countries and so, by our assumption, this map can be four-colored. A difﬁculty may then arise when we try to reinstate the square, since the countries surrounding the square may all be colored differently, leaving no “spare” color to color the square (see Figure 4).

r r

y b y ? b

g g

original map obtain new map color new map try to color original map Figure 4. The problem with the square case. 38 A Century of Advancing Mathematics

To get out of this difﬁculty we use a Kempe-chain argument, as follows. Consider the red and green countries above and below the square S and investigate whether the map M contains a chain of red and green countries joining the red country and the green country next to S (see Figure 5).

r r g g r g g g r g r r r b b g y S b y S b y y r b g b g g r r g r g r

case 1 case 2 Figure 5. Two situations for the square case.

If there is no such red-green chain of countries,as in case 1,then we can interchangethe colors of the red and green countries above S without affecting the colors of those below S. The square is then surrounded by only the three colors green, yellow, and blue, and so can be colored red. This gives a coloring of M in this case—a contradiction. If, however, there is a connecting chain of red and green countries, as in case 2, then interchanging the colors red and green does not help. But, in this case, we can consider the blue and yellow countries on the left and right of the square. There can be no chain of blue and yellow countries connecting these countries since the red-green chain gets in the way (see Figure 6). So instead we interchange the colors of the blue and yellow countries to the right of S without affecting the colors of those to the left of S. The square is then surrounded by only the three colors yellow, green, and red, and so can be colored blue, giving rise to a coloring of M in this case—a contradiction.

r r g g r g g g r g r r r r b b g g y S b y S y y r b r b g b g g g r g r g r g r r g r

Figure 6. Completing the square case.

Pentagon If the map M contains a pentagon, we remove one of its boundaries, giving a map with fewer countries which, by our assumption, can be four-colored. As for the square, a difﬁculty arises when we try to reinstate the pentagon, since the countries surroundingthe pentagon may use all four colors, as shown in Figure7. To get around this difﬁcultyKempe used two Kempe-chain arguments. First consider the yellow and red countries above and below the pentagon P . If there is no yellow-red chain connecting these countries, then we can interchange the colors of the yellow and red countries above P without affecting those below P . We can then color the Map-Coloring Problems 39

y g y g

? b b b b

r r

original map obtain new map color new map try to color original map Figure 7. The problem with the pentagon case. pentagon yellow, giving a contradiction. So we may assume that there is a yellow-red chain connecting these countries. Similarly, we consider the green and red countries above and below the pentagon P . If there is no green-red chain connecting these countries, then we can interchange the colors of the green and red countries above P without affecting those below P . We can then color the pentagon green, giving a contradiction. So we may assume that there is a green-red chain connecting these countries. The situation now looks like Figure 8.

y r r g y g r r y P g y b b g r g r y r y y g g r r r y g r

Figure 8. Double Kempe-chain picture for the pentagon case.

We next observe that blue-yellow part to the right of P is separated from the blue- yellow part to the left of P , since the red-green chain gets in the way. Thus, we may interchange the colors of the blue-yellow countries on the right of P without affecting those on the left of P , Similarly, we may interchange the colors of the blue-green countries to the left of P without affecting those on the right of P . The pentagon is then surrounded by only the three colors red, green, and yellow (see Figure 9), so the pentagon can be colored blue and we get a four-coloring of the original map M —a contradiction.

y r r g y g r r y P g y g y g r b r b r y y g g r r r y g r

Figure 9. Double Kempe-chain picture for the pentagon case with the colors switched.

Since we have considered all possibilities and obtained a contradiction in each case, no such counterexample M can exist. This proves the theorem. 40 A Century of Advancing Mathematics

3 Heawood’s bombshell In 1890 Percy Heawood, a lecturer in mathematics at the Durham Colleges (later Durham University), published an article [9] in the QuarterlyJournalofPureandAppliedMathe- matics. In it he apologetically pointed out a fundamental ﬂaw in Kempe’s accepted proof of the four-color theorem, and salvaged enough from it to deduce that every map drawn on a plane or sphere can be colored with ﬁve colors. He also generalized the map-coloring problem to surfaces other than the sphere, as we see later. Kempe admitted his error, but could not correct it. He spent the rest of his life as a very effective treasurer of the Royal Society of London, being rewarded with a knighthood. As Heawood observed, where Kempe went wrong was in his treatment of the pentagon. Using a Kempe-chain argument twice, Kempe had made two interchanges of color. Either on its own is valid, but doing both at once may cause two neighboring countries, formerly colored differently, to receive the same color—which is disallowed. An example in which this can occur was presented by Heawood (see Figure 10). In this map every country has been colored red, blue, yellow, or green, except for the central pentagon.

Figure 10. Heawood’s example.

Heawood’s argument was as follows. In Figure 11(a) the blue and yellow neighbors of the pentagon P are connected by a blue- yellow chain, so the red and green countries above P are separated from the red and green countries below P . So we can interchange the reds and greens above P without affecting those below P , as shown in Figure 11(b). Alternatively, in Figure 11(c) the blue and green neighbors of P are connected by a blue- green chain, so the red and yellow countries above P are separated from the red and yellow countries below P . So we can interchange the reds and yellows below P without affecting those above P , as shown in Figure 11(d). Either of these color interchanges is permissible on itsown. Kempe’s error was in trying to do them simultaneously. For, if we interchange the reds and greens and the reds and Map-Coloring Problems 41

g g r

y y y b r b r b g g y g g y g r y r r b r b r b r b g b g b P P P y r y r y r g g g b b b r r r b y g b y g b y g g y g y g y r b r b r b y y y

(a) (b)

g g g

y y y b r b r b r g y g g y g g y g r b r b r b r b r b r b P P P y r y r y y g g g b b b r r y b y g b y g b r g g y g y g r r b r b y b y y r

(c) (d) Figure 11. Heawood’s argument. yellows as described above, then the green country A and the yellow country B in Figure 12 both become red, which is disallowed (see p. 42). So Kempe’s proof, which allowed the possibility of such simultaneous color interchanges, is invalid.

4 From 1890 to 1960

Progress on the four-color problem came slowly, with a number of mathematicians misled by its apparent simplicity. Several investigators chose to study the properties of maps that cannot be colored with four colors, hoping either to prove that such maps cannot possibly exist or to obtain so many restrictions on one that it could actually be constructed. In particular, the American mathematician Philip Franklin [7] proved that such a map must have at least 26 countries, and over the ensuing decades a succession of investigators gradually increased this number to 95. Thus, any counterexample to the four-color theorem would necessarily be complicated. 42 A Century of Advancing Mathematics

A r y originally green b g r y r g b g b P y y g b y b r g g r y b originally yellow

r B

Figure 12. Heawood’s argument.

The eventual proof of the four-color theorem in 1976 depended on two complementary ideas, both of which have their origins in Kempe’s 1879 paper. These fundamental ideas, of an unavoidable set and a reducible conﬁguration,were developed by a number of mathematicians during the years 1890 to 1976. We now look at these two approaches and see how they led to an eventual solution to the problem. In what follows we assume that all maps under consideration are cubic maps; the justiﬁcation for doing so was given above. Under this assumption we can deduce from Euler’s formula the following useful counting formula which generalizes Lemma 1. Theorem 3. Given a cubic map with C countries, L boundary lines, and P points of intersection, let Ck be the number of countries with k boundary lines. Then

12 4C2 3C3 2C4 C5 C7 2C8 .6 k/Ck : D C C C D k 2 X Proof On counting up the countries, we get

C C2 C3 C4 C5 C6 C7 : D C C C C C C Counting up the boundary lines around each country, and noting that each boundary line bounds two countries and is therefore counted twice, we have

2L 2C2 3C3 4C4 5C5 6C6 7C7 : D C C C C C C Counting up the points of intersection around each country, and noting that each point lies on the boundary of three countries and is therefore counted three times, we have

3P 2C2 3C3 4C4 5C5 6C6 7C7 : D C C C C C C Substituting these values for C , L, and P into Euler’s formula gives the result. A consequence of the counting formula is that every cubic map must contain a digon, triangle, square, or pentagon, since otherwise (with C2 C3 C4 C5 0) 12 would D D D D equal a negative quantity. We express this by saying that this set of countries forms an Map-Coloring Problems 43 unavoidable set. More generally, a set of conﬁgurations of countries is an unavoidable set if every map must contain at least one of them. In 1904 Paul Wernicke [19] proved that the set of conﬁgurations in Figure 13 is also an unavoidable set—so if a cubic map contains no digon, triangle, or square, then it must contain either two adjacent pentagons or a pentagon adjacent to a hexagon.

digon triangle square two pentagons pentagon/hexagon Figure 13. An unavoidable set.

To see why, suppose that there exists a cubic map containing none of these configurations. Then every pentagon adjoins only countries bounded by at least seven edges. Let us assign to each country in the map an “electrical charge”: if the country has k boundary edges, we assign the charge 6 k; thus pentagons receive a charge of 1, hexagons receive zero charge, heptagons receive a charge of 1, and so on. By the above counting formula, the total charge on the countries of the map is 12. Now transfer one-fifth of a unit of charge from each pentagon to its five negatively charged neighbors. Then the total charge on the map remains 12, but each pentagon now has zero charge, each hexagon still has zero charge, each heptagon still has a negative charge (since a heptagon can acquire a positive charge only if it has at least six neighboring pentagons, but two of these pentagons would then have to be adjacent, which is not permissible), and similarly octagons, nonagons, etc., retain a negative charge. Thus, each country now has a negative or zero charge, and so the total charge on the map cannot be 12. This contradiction proves that the above set of configurations is an unavoidable set. A reducible configuration of countries in a map is an arrangement of countries that cannot appear in a counterexample to the four-color theorem—in particular, as Kempe showed, a digon, triangle, and square are all examples of reducible configurations. Thus, a configuration of countries is reducible if a four-coloring of the rest of the map can be extended to the configuration. A celebrated example of a reducible configuration is the Birkhoff diamond (see Fig- ure 14), introduced by the American mathematician G. D. Birkhoff [5], who initiated the subject of reducibility in 1913.

6 1

5 2

4 3

Birkho diamond Figure 14. A reducible conﬁguration. 44 A Century of Advancing Mathematics

To see why theBirkhoffdiamond is reducible,let us remove itfrom a map, givinga new map which we can assume has been colored with the four colors red, green, blue, and yellow. We now investigate whether every such four-coloring can be extended to the diamond. To do so, we list all the thirty-one essentially different colorings of the six surrounding countries 1–6. These are: rgrgrg rgrbrg* rgrbgy* rgbrgy rgbryb rgbgbg* rgbyrg rgbygy* rgrgrb* rgrbrb rgrbyg* rgbrbg* rgbgrg* rgbgby rgbyby rgbybg* rgrgbg rgrbry rgrbyb* rgbrby rgbgrb* rgbgyg rgbyry* rgbyby rgrgby* rgrbgb* rgbrgb rgbryg rgbgry* rgbgyb rgbygb

Sixteen of these colorings (indicated by *) can immediately be extended to the diamond. It can also be shown that the remaining colorings can be extended to the diamond by using Kempe-chain arguments to interchange specific pairs of colors. Thus, all thirty-one colorings of the surrounding countries can be extended to the diamond, either directly or after a change of colors, and so the Birkhoff diamond is reducible. It was noted by the German mathematician Heinrich Heesch that, in order to prove the four-color theorem, it is sufficient to find an unavoidable set consisting entirely of reducible configurations. For, since the set is unavoidable, all maps must contain at least one of the configurations, but since all of the configurations are reducible, none of them can appear in a minimal counterexample. Thus, no such counterexample can exist and the theorem is proved.

5 Enter the computer By the early 1970s large numbers of reducible configurations had been discovered, as were a few methods for constructing unavoidable sets, but no one had been able to find an unavoidable set of reducible configurations. (One writer on unavoidable sets was Henri Lebesgue [12], mainly known for his work on integration.) Probabilistic arguments had indicated that such sets should exist, but that they might contain many thousands of configurations, far too many for the largest computers then available. The usual approach was to generate large collections of reducible configurations and then to try to package them up into an unavoidable set, but no one had succeeded in doing so. Two mathematicians from the University of Illinois, Kenneth Appel and Wolfgang Haken, took a different approach. Over several years they developed the knack of being able to recognize at sight (with over eighty per cent accuracy) whether a given configuration is reducible. They then focused on unavoidable sets, constructing sets of “likely- to-be-reducible” configurations and then testing them for reducibility, modifying the unavoidable set as necessary. This procedure saved them a great deal of time and, together with a graduate student named John Koch, they eventually obtained an unavoidable set of 1936 configurations (later reduced to 1482), thereby settling the 124-year-old problem (see [2, 3, 4]). The methods employed by Appel and Haken were basically those described above, although the technical details were far more complicated and involved massive testing on a computer, totaling some 1200 hours of computer time. In particular, while developing methods for moving units of charge around a map, they used much trial and error, together Map-Coloring Problems 45 with a great deal of insight and experience, and eventually obtained a systematic procedure that could be applied by hand. As mentioned above, the configurations were then tested for reducibility, and those that were not reducible were replaced by other configurations until an unavoidable set of reducible configurations was obtained. In fact, Appel and Haken’s method produced many thousands of unavoidable sets of reducible configurations, thereby yielding thousands of proofs of the four-color theorem. Minor errors were located and corrected, and the computer programs were made more efficient. The achievement of Appel and Haken in settling the four-color problem was a substantial one. At first many mathematicians were unhappy with a computer-assisted proof, as we shall see, but as the underlying methods of proof became clearer and computers came more into general use the solution gradually came to be accepted by the mathematical community. At the University of Illinoisthere was general rejoicing, and the proofof the four-color theorem was commemorated by a specially designed postmark (see Figure 15).

Figure 15. The University of Illinois’s celebratory postmark.

6 Maps on other surfaces By stereographic projection, coloring maps on a plane corresponds to coloringmaps on the surface of a sphere, and this suggests investigating the coloring of maps on other surfaces. The number of colors needed for maps on orientablesurfaces (that is, spheres with handles) was raised by Percy Heawood in the 1890 paper [9] in which he pointed out the ﬂaw in Kempe’s proof of the four-color theorem. Heawood began with the torus. For a map with C countries, L boundary lines, and P points of intersection embedded on a torus, Euler’s formula takes the form C L C P 0. Using arguments similar to those for the plane, one can easily prove that every D toroidal map contains a digon, triangle, quadrilateral, pentagon, or hexagon, and thus (as Heawood observed), every map embedded on a torus can be colored with seven colors. He also presented a toroidal map that needs seven colors (see Figure 16).

Figure 16. A toroidal map requiring seven colors. 46 A Century of Advancing Mathematics

Heawood then went on to consider orientable surfaces of genus g (that is, spheres with g handles), for which the corresponding version of Euler’s formula is C L P 2 2g. C D He deduced that, for each g 1, every map embedded on such a surface can be colored with H.g/ colors, where H.g/ .7 p1 48g/=2 ; this formula yields seven colors D b C C c for a torus and eight for a double-torus. Unfortunately, except for the torus, Heawood failed to prove that this formula is best possible—that is, on every orientable surface of genus g there are maps that need H.g/ colors. In 1891 Lothar Heffter [10] obtained proofs for several small values of g, but proving this in general turned out to be incredibly difﬁcult, and the unproved statement became known as the Heawood conjecture. Proving the Heawood conjecture took 78 years. The eventual proof split up into twelve separate arguments, and was ﬁnally completed in 1968 after a long struggle by Gerhard Ringel and J. W. T. Youngs and others (see [13]). For non-orientable surfaces, H. Tietze [17] proved in 1910 that every map drawn on a projective plane can be colored with six colors, and that there are projective plane maps that need six colors. Later, in 1934, Philip Franklin [8] proved that every map drawn on a Klein bottlecan be colored with six colors, and that there are Klein bottlemaps that require six colors. The general result was obtained in 1952 by Gerhard Ringel, who considered non- orientable surfaces of genus k (that is, spheres with k cross-caps), for which the corresponding version of Euler’s formula is C L P 2 k. He deduced that, for k 1 C D D and k 3, every map embedded on such a surface can be colored with H.k/ colors, where H.k/ .7 p1 24k/=2 , and there are maps that need this number of colors. When D b C C c k 2 (the Klein bottle), the formula gives 7, whereas the correct answer is 6, as noted D above. It is surprising that the non-orientablecase proved to be simpler than the orientable case and was solved earlier, and also that the number of colors needed for orientable surfaces of genus h, for all h 1, took less time to solve than the case h 0 (the sphere)—but that is D all part of the fascination of map coloring!

7 Is it a proof? In the 1970s the use of a computer to prove mathematical results was highly controversial. Don Albers [1] described the scene when Wolfgang Haken presented the four-color theorem to a joint meeting of the American Mathematical Society and the Mathematical Association of America at the University of Toronto in August 1976: The elegant and old lecture hall was jammed with mathematicians anxious to hear Professor Haken give the proof. It seemed like the perfect setting to announce a great mathematical result. He proceededto outline clearly the computer-assistedproof that he and his colleagues had devised. At the conclusion of his remarks I had expected the audience to erupt with a great ovation. Instead, they responded with polite applause. Mathematician after mathematician expressed uneasiness with a proof in which a computer played a major role. They were bothered by the fact that more than 1000 hours of computer time had been expended in checking some 100,000 cases and often suggested (hoped?) that there might be an error buried in the hundreds of pages of computer printouts. Beyond that concern was a hope that a much shorter proof could be found. Map-Coloring Problems 47

The computer-assisted proof of Appel and Haken did indeed raise a number of issues. Several small errors were found in it, but these were quickly corrected. More serious was the philosophical question as to whether a mathematical proof can be considered correct if it cannot readily be checked by hand. The fact that everything that the computer had done was essentially routine work that could in principle be done by hand (althoughnot quickly) was largely discounted. In contrast it is interesting to compare the different standards tolerated by the mathematical community towards purported proofs of the four-color theorem and two major results in group theory. In the 1960s the 250-page proof of the Feit–Thompson theorem on solvable groups was enthusiastically accepted, even though its length might indicate a substantial likelihood of human error (indeed, its proof initially contained a number of minor errors, later corrected), so that hardly anyone could check the details. Later, the classification of finite simple groups involved many thousands of pages of work by hundreds of contributors, and parts of it made substantial use of computers. But in the 1980s most mathematicians were happy to accept the classification as correct, even though there were gaps in the argument that were still being filled more than twenty years later. Whatever the philosophical issues raised, it was undoubtedly the case that Appel and Haken’s methods had been somewhat ad hoc, and in the mid-1990s a more systematic approach was developed by Neil Robertson, Daniel Sanders, Paul Seymour, and Robin Thomas. Using essentially the same general method as Appel and Haken, they produced a shorter and more robust proof [14, 15] involving only 633 reducible configurations— indeed, the steps in their proof can be externally verified on one’s home computer in just a few hours. Moreover, their methods led to a quadratic algorithm for map coloring: the running time required to color a map with n countries is proportional to n2, whereas for Appel and Haken it had been n4. RobinThomas later wrote an article [16] outliningthe main ideas of the improved proof and emphasizing the four-color problem’s importance by listing some results from other fields that are equivalent to it—these include results on the algebra of three-dimensional vectors and the divisibilityof integers and one relating to matrices and tensors. The correctness of a large computer-assisted proof is difficult to verify. In 2004 the French computer scientist Georges Gonthier provided a fully machine-checked proof of the four-color theorem: this was a formal language implementation and machine verification of the approach of Robertson and his co-workers. Gonthier’s first step was to provide an axiomatic formulation of the terms “map” and “four-colorable.” A respected and widely used machine proof-checker called Coq then verified some sixty thousand lines of formal language proof before pronouncing the proof valid. The doubters could no longer doubt!

8 Conclusion Once the four-color problem had been solved, what else was there for map-colorers to do? The English combinatorialist Bill Tutte [18] asked this question back in 1978: I imagine one of them outgribing in despair crying ’What shall I do now?’ To which the proper answer is ‘Be of good cheer. You can continue in the same general line of research.’ For the four-colortheorem is by no means the end of the line—indeed, there are several current mathematical problems that extend the four-color theorem and develop its ideas in 48 A Century of Advancing Mathematics new and exciting directions. Solving the four-color problem is just one special instance of these much harder problems, such as Hadwiger’s conjecture and the five-flow conjecture, on which good progress has already been made. With these thoughtsof the future, we leave our last poetic musings to Bill Tutte: The Four Colour Theorem is the tip of the iceberg, the thin end of the wedge and the first cuckooof Spring.

Acknowledgement Most of the material in this chapter is described in much greater detail in Robin Wilson, Four Colors Sufﬁce (revised color edition), Princeton Science Li- brary, Princeton University Press, 2014. The pictures in this chapter are all taken from this book and are reproduced with the permission of the publisher.

Bibliography [1] D. Albers, Polite applausefor a proof of one of the great conjectures of mathematics: what is a proof today?, College Mathematics Journal 12 (2) (March 1981), 82. [2] K. Appel and W. Haken, Every planar map is four colorable, Bulletin of the American Mathe- matical Society 82 (1976), 711–712. [3] K. Appeland W.Haken,Every planarmap is four colorable,Part I: Discharging, Illinois Journal of Mathematics 21 (1977), 429–490. [4] K. Appel, W. Haken, and J. Koch, Every planar map is four colorable, Part II: Reducibility, Illinois Journal of Mathematics 21 (1977), 491–567. [5] G. D. Birkhoff, The reducibility of maps, American Journal of Mathematics 35 (1913), 115– 128. [6] A. Cayley, On the colouring of maps, Proceedings of the Royal Geographical Society 1 (1979), 259–261. [7] P. Franklin, The four color problem, American Journal of Mathematics 44 (1922), 225–236. [8] P. Franklin, A six color problem, Journal of Mathematical Physics 13 (1934), 363–369. [9] P. J. Heawood, Map colour theorem, Quarterly Journal of Pure and Applied Mathematics 24 (1890), 332–338. [10] L. Heffter, Ueber das Problem der Nachbargebiete, Mathematische Annalen 38 (1891), 477– 508. [11] A. B. Kempe, On the geographical problem of the four colours, American Journal of Mathe- matics 2 (1879), 193–200. [12] H. Lebesgue,Quelques consequencessimple de la formule d’Euler, Journal de Mathématiques Pures et Appliquées 9 (1940), 27–43. [13] G. Ringel, Map Color Theorem, Springer, 1974. [14] N. Robertson, D. Sanders, P. Seymour, and R. Thomas, A new proof of the four-colour theorem, Electronic Research Announcements of the American Mathematical Society 2 (1) (August 1996), 17–25. [15] P. Seymour, Progress on the four-color theorem, Proceedings of the International Congress of Mathematicians, Zürich, Birkhäuser, 1995. Map-Coloring Problems 49

[16] R. Thomas,An updateon the four-color theorem, Notices of the AmericanMathematical Society 45 (7) (August 1998), 848–859. [17] H. Tietze, Einer Bemerkungen über das problem des Kartenfärbens auf einseitingen Flächen, Jahresber.Deut. Math.-Verein. 19 (1910), 155–159. [18] W. T. Tutte, Colouring problems, The Mathematical Intelligencer 1 (1978), 72–75. [19] P. Wernicke, Uber¨ den kartographischen Vierfarbensatz, Mathematische Annalen 58 (1904), 413–426.

Emeritus Professor of Pure Mathematics, The Open University, UK, and Emeritus Professor of Geometry, Gresham College, London [email protected]

Six Milestones in Geometry

Frank Morgan Williams College

Dear MAA, What would you say are the six largest advances in geometry during your 100-year lifetime? You say you remember only the more recent ones? Me too. Here are my choices.

1 Regularity of area-minimizing surfaces An oldtheme in mathematics is that nice problems shouldhave nice solutions,although this can be hard to prove and is sometimes false. The nicest problem about curves has the nicest possible solution: the shortest distance between two points is a straight line. Similarly the nicest problem about surfaces has a nice solution: given a smooth curve in R3, a least-area surface bounded by the curve as in Figure 1 is a nice, smooth surface (Fleming [7], 1962).

Figure 1. A least-area surface is smooth. Wikimedia Commons.

In 1966, Fred Almgren [2] extended the result to R4, proving that a least-area hypersurface is smooth by showing singularities unstable. In 1968 Jim Simons [32] extended the result through R7. Then in 1969 E. Bombieri, E. De Giorgi, and E. Giusti [4] gave a counterexample in R8 (the cone over S3 S3, with a singular point at the origin). In Rn, an area-minimizing hypersurface can have a singular set of dimension up to n 8. In higher codimension (much more difﬁcult), an m-dimensional area-minimizing surface in Rn can have a singular set of dimension up to m 2. For example, two orthogonal discs in R4 are area minimizing with an isolated singularityat the origin. Regularity in general codimension was the subject of Fred Almgren’s 995-page Big Regularity Paper [1]. We are still far from any classiﬁcation of singularities. An early fundamental question asked which unions of oriented m-planes are area minimizing. The answer depends on the

51 52 A Century of Advancing Mathematics geometric relationship between the planes, described by m angles

0 ˛1 ˛2 ˛m: Ä Ä Ä Ä The Angle Conjecture said that the planes are area minimizing if and only if the largest angleisless thanorequal tothesumof theothers.For m 1 thisoccurs onlywhen ˛1 0, D D i.e., when the two lines coincide. For m 2 this occurs only when the two angles are D equal, i.e., when the two planes are simultaneously complex for some orthogonal complex structure on their span. For m 3 this occurs for open sets of planes. This is one conjecture D for which both directionswere difficult. Sufficiency was proved in 1987 by “calibration” by Dana Nance (Mackenzie) [28], now a popular mathematics writer. Necessity was proved in 1989 by my PhD student Gary Lawlor [23], by providing beautiful families of comparison surfaces. A still open question asks whether the singular set in an area-minimizing surface is a stratified manifold. At the other extreme, for all we know, the singular set could be fractional dimensional. The subject of regularity has many technical subtleties, suppressed here. For some details and references, see Morgan’s Geometric Measure Theory book [26].

2 Kepler’s sphere-packing conjecture (Thomas Hales) In 1611 Johannes Kepler [21] conjectured that the standard way of packing unit spheres (Figure 2) is actually the densest. In 1993 the International Journal of Mathematics published a purported proof by Wu-Yi Hsiang [18]. The incredible features were that the proof considered only close neighbors (centers within 2.18) and used mainly trigonometry. As far as I know, there are no counterexamples to the method and the ﬁrst observed mistakes have been repaired, but Hsiang’s proof has not been generally accepted by the mathematics community. A few years later Thomas Hales submitted his proof [13] to AnnalsofMathematics. In his expository article “Cannonballs and Honeycombs” in the April 2000 Notices [12], he reported that, A jury of twelve referees has been deliberating on the proof since September 1998.

Figure 2. The standard “cannonball” way of packing spheres is the densest. Wikimedia Commons. Six Milestones in Geometry 53

They did have a toughjob. It was a momentous result. After the controversies over Hsiang’s published proof, they had to be careful. And it was a difficult body of work to referee, consisting of six papers, including the thesis of Hales’s PhD student Samuel Ferguson. The proof involved extensive computer analysis. Many cases initially would have required more computation time than the age of the universe. The proof had to be continually modified and aided by intricate analysis and geometry. In their paper on “A formulationof the Kepler conjecture,” Hales and Ferguson [14] wrote: As our investigations progressed, we found that it was necessary to make some adjustments. However, we had no desire to start over, abandoning the results of “Sphere Packings I” and “Sphere Packings II.” “A Formulation” gives a new decomposition of space [but] shows that all of the main theorems from “Sphere Pack- ings I” and “Sphere Packings II” can be easily recovered in this new context with a few simple lemmas. After years of effort, the referees gave up. Meanwhile, Hales and the world were waiting for the referees’ conclusions. Annals finally decided on an unprecedented course of action: to publish the work with a disclaimer that the referees had been unable to verify the proof. At the Joint Mathematics Meetings in Baltimore in January 2003, Hales received the Chauvenet Prize of the Mathematical Association of America for his Notices article. In his acceptance speech, he read from a letter he received from the editors of the Annals, leaving the impression that they would be unable to publish his result. According to Hales, Annals had written [32]: The news from the referees is bad.... They have not been able to certify the correctness of the proof, and will not be able to certify it in the future, because they have run out of energy ... One can speculate whether their process would have converged to a definitive answer had they had a more clear manuscript from the beginning, but this does not matter now. In the lively discussions after the prize ceremony, it was apparent that Hales was not at all satisfied with the Annals delays and with its eventual decision to publish the work with a disclaimer. Annals then decided to call on another referee, who got Hales to reorganize his papers into more readable, checkable mathematics. Annals finally published without disclaimer a single paper with the overall mathematical strategy of the proof. The entire mathematical proof, in six papers, appeared in a special issue of Discrete and Computational Geometry, edited by some of the referees. (See [14].) Some of the computer programs and data are on an Annals website. Meanwhile, Hales launched a world-wide cooperative project called “Flyspeck” (based on the letters FPK for “Formal Proof of Kepler”) to produce a verification of the proof by computer. This sounded nearly impossible to me, but Hales [11] now reports that the work has been completed. Shortly after Hales finished his proof of Kepler’s Conjecture, Denis Weaire recommended to him an even older problem, the Hexagonal Honeycomb Conjecture. It says that regular hexagons as in Figure 3 provide the least-perimeter way to partition the plane into unit areas. Widely believed and often asserted as fact, even by such notables as Hermann Weyl [41], it was the longest standing open problem in mathematics, going back thousands of 54 A Century of Advancing Mathematics

Figure 3. The Hexagonal Honeycomb Conjecture said that regular hexagons provide the least- perimeter way to partition the plane into unit areas. Wikipedia. years. Around 36 BC, before his death, Marcus Terentius Varro [39] wrote an epistle to his young wife on how to take care of their farming estate, including honey bees. He gave two reasons for the hexagonal shape in their honeycombs: ﬁrst, that a bee has six feet; second, The geometricians prove that this hexagon inscribed in a circular ﬁgure encloses the greatest amount of space.

Actually the Greek mathematician Zenodorus ( 200 BC) probably had considered only hexagons, triangles, and parallelograms [17, pp. 206–212]. Varro’s knowledge of bees was not perfect. He also observed that, They follow their king wherever he goes. The fact that the leader is not a king but a queen was not discovered until the seventeenth century. So Weaire recommended the Hexagonal Honeycomb problem to Hales: “Given its celebrated history, it seems worth a try.” Hales promptly dispatched it in under a year. “In contrast with the years of forced labor that gave the proof of the Kepler Conjecture, I felt as if I had won a lottery.” [35, Chapt. 14] One major difﬁculty in proving regular hexagons optimal is that the result is not true locally. A hexagon is not the least-perimeter way to enclose unit area: a circle is. Of course you cannot partition the plane into unit circular regions. For a single region, a circle is best, but its favorable outward convexity would cause adjacent regions to have unfavorable inward concavity, so such outward convexity should carry a penalty, and inward concavity a corresponding credit. Similarly polygons with more than six edges can do better than hexagons, but by Euler the average number of edges should be at most six, so that extra edges should also carry a penalty and fewer edges a credit. Using such penalties and credits, Hales created a new problem in which the regular hexagons are best locally as well as globally. Since globally the penalties and credits must all cancel out, hexagons also solve the originalHexagonal Honeycomb Conjecture. The proof involves careful considerationof maybe a dozen cases and subcases, depending for example on the size of the penalty terms. Just as for the Hexagonal Honeycomb Conjecture, a major difﬁcultyin provingKepler’s Conjecture is that the result is not true locally. The densest way to pack spheres around one central sphere is modeled on the regular dodecahedron, but such an arrangement cannot be continued, because dodecahedra do not tile space. Hales added local penalties and credits Six Milestones in Geometry 55

(which cancel out globally) to produce a new problem for which the standard packing would even locally beat the dodecahedral and all other packings. The appropriate penalties are very hard to find. Simple convexity and extra faces do not work; for starters, there is no formula for the average number of faces of a polyhedral partition of space. Earlier workers had tried to fix on features of the associated polyhedral partition into so-called Voronoi cells. (Each Voronoi cell consists of the set of points in space closer to the center of one particular sphere than to the center of any other.) Hales originally had the idea of using instead features of the so-called Delaunay triangulation, with vertices at the centers of the spheres, dual to the Voronoi decomposition. His main conceptual breakthrough may have been when he decided to use both. Unlike for the planar Hexagonal Honeycomb Conjecture, there were thousands of cases to check, some too difficult for the computer. As he advanced to more and more difficult cases, Hales had to make intricate revisionsof the penalties to get the proof to work. The set of penalties finally used was arrived at in collaboration with Ferguson, whose PhD thesis handled the most difficult case. So despite similarities,Kepler’s Conjecture on sphere packing in R3 was orders of mag- nitudemore difficult than the Hexagonal Honeycomb Conjecturein R2, mainly because R3 provides so many more geometric possibilities than R2. But it was easier in one aspect: it is a packing problem, whereas the Honeycomb Conjecture is a partitioning problem. The optimal two-dimensional packing, with six circles fitting perfectly around every circle, is relatively easy and was proved in 1890 by Thue. Packing problems are in general much easier because you “just” have to determine where to put the centers of the circles or spheres. For partitioningproblems, you have to find the shape or shapes of the regions: hexagons in R2, stillopen in R3, despite a milestone result, the subject of the next section. This section is closely based on my review [27] of Kepler’s Conjecture [35] by G. Szpiro. For more details and references, see my review, Szpiro’s book, and my Geometric Measure Theory book [26].

3 Weaire-Phelan counterexample to Kelvin conjecture In 1994 came striking news of the disproof of Lord Kelvin’s 100-year-old conjecture by Denis Weaire and Robert Phelan [40] of Trinity College, Dublin. Kelvin sought the least- area way to partition all of space into regions of unit volumes. His basic building block was a truncated octahedron, with its six square faces of truncation and eight remaining hexagonal faces, which packs perfectly to ﬁll space as suggested by Figure 4. (The regular

Figure 4. Lord Kelvin conjectured that the least-area way to partition space into unit volumes uses relaxed truncated octahedra. Graphics by Ken Brakke in his Surface Evolver [5]. 56 A Century of Advancing Mathematics

Figure 5. Kelvin loved his truncated octahedron, constructed models, and exhibited the pictured stereoscopic images. (Crossing your eyes to superimpose the two images produces a three- dimensional view.) [37, p. 15] dodecahedron, with its twelve pentagonal faces, has less area, but it does not pack.) The whole structure relaxes slightly into a curvy equilibrium, which is Kelvin’s candidate. All regions are congruent. Kelvin loved this shape, constructed models, and exhibited stereoscopic images as in Figure 5. Weaire and Phelan recruited a crystal structure from certain chemical “clathrate” com- pounds, which uses two different building blocks: an irregular dodecahedron related to Fool’s Gold and a tetrakaidecahedron with twelve pentagonal faces and two hexagonal faces. The tetrakaidecahedra are arranged in three orthogonal stacks, stacked along the hexagonal faces, as in Figure 6. The remaining holes are ﬁlled by dodecahedra. Again, the structure is allowed to relax into a stable equilibrium. Computation in the Brakke Evolver [5] shows an improvement over Kelvin’s conjecture of approximately 0.3%. The rigorous proof, by Kusner and Sullivan [22], proves only approximately 0.01%. Weaire and Phelan [40] thus provided a new conjectured minimizer. Proving the Weaire-Phelan structure optimal looks perhaps a century beyond current mathematics to me, but I understand that Hales is already thinkingabout it.

Figure 6. The relaxed stacked tetrakaidecahedra and occasionaldodecahedraof Weaire and Phelan beat Kelvin’s conjecture by approximately 0.3%. Graphics by Ken Brakke in his Surface Evolver [5]. Six Milestones in Geometry 57

The counterexample was almost discovered many times earlier. The clathrate com- pounds that inspired Weaire and Phelan had just three years earlier inspired counterexamples by Tibor Tarnai [36] to related conjectures on the optimal way to cover a sphere withdiscs. As early as 1890, J. Dana [6] described similar structures in volcanic lava, more recently observed in popcorn. Brakke spent hours at his grandfather’s old desk seeking counterexamples. Had he reached for his father’s copy of Linus Pauling’s classic, The Na- ture of the Chemical Bond, it would doubtless have fallen open to the illustration, in the clathrate compound section, of the chlorine hydrate crystal, essentially the Weaire-Phelan counterexample. R. Williams [42], after spending years seeking a Kelvin counterexample, finally gave up and later published a well-illustrated The Geometrical Foundation of Natu- ral Structure: A Source Book of Design. In one of his figures, he picturedthe Weaire-Phelan counterexample without realizing it. Of course, it would have been difficult to check without Brakke’s Surface Evolver. In 1988 at the Geometry Center, John M. Sullivan, inspired by Fred Almgren, computed Voronoi cells of equal volumes, but Weaire-Phelan requires weighted Voronoi cells (with the distance to each point weighted differently). Weights also play an essential role in Gary Lawlor’s new proof of the Double Bubble Theorem, featured in the next section, and in Perelman’s proof of the PoincaréConjecture in the following section. This section is closely based on Chapter 15 of my Geometric Measure Theory book [26], which can be consulted for further details and references.

4 Double bubble conjecture A round spherical soap bubble provides the least-area way to enclose a given volume of air, as was proved mathematically by H. Schwarz in 1884. Similarly, the familiar double soap bubble of Figure 7a, consisting of three spherical caps meeting at 120 degrees, provides the least-area way toenclose and separate twogiven volumes of air. But the proofof that double bubble conjecture had to wait until 2002. Indeed, it was not stated as an open mathematical

(a)

(b)

Figure 7. (a) The double bubble provides the least-area way to enclose and separate two given volumes of air. It consists of three spherical caps meeting at 120 degrees. If the volumes are equal, the central cap is planar. Computer graphics copyright c John M. Sullivan; color version at www.math.uiuc.edu/˜jms/Images/. (b)Foisyetal.[9] proved that the planar double bubble provides the least-perimeter way to enclose and separate two given areas. 58 A Century of Advancing Mathematics conjecture until a 1991 undergraduate thesis by Joel Foisy [8] at Williams College. The previous summer, Foisy, Manuel Alfaro, Jeff Brock, Nickelous Hodges, and Jason Zimba [9], my NSF undergraduate research Geometry Group, had proved the planar version: that three circular arcs as in Figure 7b meeting at 120 degrees provide the least-perimeter way to enclose and separate two given areas. Fred Almgren and Jean Taylor (see [3]) had proved that a least-area doublebubbleexists and consists of smooth, constant-mean-curvature surfaces meeting in threes along curves. Furthermore, by symmetry, it had to be a surface of revolution, although the argument was a bit tricky. First you use a ham sandwich theorem to get a plane that splits both volumes in half, so that by regularity you can assume that it has reflectional symmetry. Then you similarly get reflectional symmetry across an orthogonal plane. Composing the two symmetries yields symmetry under 180-degree rotation about say the z-axis. Therefore all vertical planes through the z-axis split both volumes in half. By regularity, it must meet all these planes orthogonally. Therefore it must be a surface of revolution. Thus symmetry reduced the problem to one about curves in the plane. No one realized how hard it was. The main difficulty was that in principle each of the two regions could have many components. Even if you tried to make the problem easier by not allowing several components, in the limit the minimizer might have several components. What made the problem accessible was a theory developed by one of my former undergraduate research students, Michael Hutchings [19], now Professor of Mathematics at UC Berkeley. Hutchings realized he could generalize the symmetry argument to get bounds on the number of components. In the case of equal volumes, each region had just one component. For this case, Joel Hass and Roger Schafly [16] were able to finish off the proof computationally.At the time it was a cutting-edge use of computers, requiring careful error estimates. In the end they reduced the problem to 200,260 integrals, which a PC accomplished in about twenty minutes. The result appeared in Annals of Mathematics in 2000. The proof [20] of the general case of unequal volumes appeared two years later. Hutch- ings was one of my coauthors. The other two, Manuel Ritoréand Antonio Ros hailed from the University of Granada, which has become the world center for minimal surface theory. The computer analysis was replaced by an instability argument. My undergraduate Geometry Group [31] promptly generalized the result to R4, where the difficult cases multiplied from six to about two hundred. The group leader, Ben Re- ichardt [30], eventually generalized the result to Rn, so perfecting the instability argument that the Hutchings bound on the number of components became unnecessary. Recently Gary Lawlor [24] has found a much simpler proof based on a deep, new idea, which he calls unification. He replaces the infinite family of problems of minimizing surface area for given volumes to a single problem: minimize the ratio of surface area to the surface area of the standard double bubble over all volumes. Now not only would a counterexample have less total surface area than the standard double bubble, but every one of its interfaces would have less area and smaller mean curvature than the corresponding interfaces of the standard double bubble. This makes it relatively easy to derive a contradiction by comparing areas via the Gauss map. Lawlor’s proof works in a larger context where the various interfaces carry weights, as occurs physically when the two regions and the exterior are immiscible fluids with the energy of an interface depending on which pair of fluids it separates. Six Milestones in Geometry 59

Along the way Lawlor discovered a beautiful, simple, new symmetry argument using stretching to prove that the minimizer S is a surface of revolution, a rather astonishing accomplishment given the thousands of years of close attention to symmetry arguments. Roughly as in the standard symmetry argument he gets a “quarter” of S in say x; y 0 f g with a quarter of each volume and a quarter of the surface area. But now he maps the polar coordinate to 2 to get a surface with half of each volume in y 0 and less than half f g the surface area, unless it is a surface of revolution. Reﬂection across y 0 now yields f D g a surface with the original volumes and less than the original surface area (contradiction), unless the portion came from a surface of revolution. A big advantage of Lawlor’s new symmetry stretching argument is that it does not use regularity, which is not known in the larger context of weighted interfaces. For more details and references on this section, see my Geometric Measure Theory book [M1] and Lawlor’s new paper [24].

5 Poincaré conjecture (Grigori Perelman) It has long been known that among connected compact two-dimensional manifolds, such as the sphere, the torus, and the two-holed torus, the sphere is characterized by the fact that any loop can be contracted to a point, whereas a loop around a torus, for example, cannot be contracted to a point. The PoincaréConjecture, suggested by Henri Poincaréin 1904, proposed the analogous result for three-dimensional manifolds: a simply connected compact three-dimensional manifold must be a sphere. At the 2006 International Congress of Mathematicians, Grigori Perelman [29] was awarded the Fields Medal for its proof, althoughhe declined to accept it. In 2010 the Clay Mathematics Institutionoffered him their million-dollar Millennium prize, but he turned that down too. An article in The Moscow Times (29 April 2011) reported that Perelman said that his research is too interestingdue to its vast implications—both practical and philosophical—to spend time on other matters. “I know how to control the universe. Tell me, why would I need to chase a million [dollars]?” For a Russian these days, such money could well be more trouble than it’s worth. High-dimensional versions of the PoincaréConjecture, with more space to do geometric constructions, are easier. Stephen Smale [33] proved the analogous conjecture for dimensions at least five and won the Fields Medal in 1966. Michael Freedman [10] proved the four-dimensional case and won the Fields medal in 1986. The basic idea of Perelman’s proof, due to Richard Hamilton [15], is to start with any simply connected compact three-manifold and let it shrink at each point in each direction at a rate proportionalto its Ricci curvature. If you can show that you eventually end up with a round sphere, with perhaps other spheres pinched off along the way, you can conclude that you must have started with a (deformed) sphere. Unfortunately singularities sometimes form, as in Figure 8. So the fundamental difficulty is to obtain some control over the formation of singularities. One has global estimates (Perelman’s “monotonicity of energy”), but one needs local control. Here I’d like to explain a point not emphasized by other expositors, but the starting point in Perelman’s paper. As in Lawlor’s [24] new proof of the Double Bubble Theorem, the key technique is using a weighting function. To focus attention about a point p of concern, Perelman gives the manifold large weight or density about that point and lets 60 A Century of Advancing Mathematics

1995 M. Fried Institut f. Angewandte Mathematik Uni Freiburg Figure 8. The Ricci flow applied to a dumbbell shape leads to a singularity. mathematik.uni-freiburg.de the metric flow by an associated generalized Ricci curvature. If one fixes the measure, then the density evolves as a modified backwards heat equation and approaches a delta function at p. His general monotonicity of energy now provides the requisite local information instead of the usual global information.(Actually, in the proof, to facilitate surgery, Perelman moves to a localized version of the density called the length function.) Remarkably, modulo diffeomorphisms, this generalized Ricci flow is equivalent to the standard Ricci flow. As Perelman puts it, “The remarkable fact here is that different choices of [density] lead to the same flow, up toa diffeomorphism; that is, the choice of [density]is analogous tothe choice of gauge.” This means that duringthe proof one can choose any density for convenience. In summary, weightings provide a convenient technical context for applying diffeomorphisms to focus attention on regions of concern. In his paper Perelman concludes: “...we apply our monotonicity formula to prove that for a smooth solution on a finite time interval, the injectivity radius at each point is controlled by the curvatures at nearby points. This result removes the major stumbling block in Hamilton’s approach to geometrization.” Perelman’s proof used geometry to solve a topologyproblem,much as the original1896 proofs of the prime number theorem by Hadamard and de la Vallé-Poussin used complex analysis. It was another half century until Selberg and Erd˝os provided elementary proofs. I think it will be another half century until someone gives a purely topological proof of Poincaré, fulfilling the promise of earlier work of Thurston. In fair exchange, the next section includes a crucial role for topology in the proof of a geometric result, the Willmore Conjecture.

6 Willmore conjecture (Fernando Marques and Andre Neves) In 2012 mathematicians Fernando C. Marques of IMPA in R´ıo de Janeiro, Brazil, and Andr´eNeves [25] of Imperial College, London, proved the 1965 Willmore Conjecture for the best shape for a torus or doughnut,pictured in Figure 9, with the narrow holeonly about 17 percent (3 2p2) of the width of the torus. “Best” means minimizing the integral of the mean curvature squared, a physically natural elastic bending energy, used for example to model certain cell membranes. Six Milestones in Geometry 61

Figure 9. Marques and Neves prove the torus or doughnut which minimizes the integral of the mean curvature squared. Left image courtesy of geom.uiuc.edu/˜banchoff, thanks to Tom Banchoff, all rights reserved; right image from Wikipedia.

Thomas J. Willmore [43], an English geometer (see Figure 10), guessed the best shape in a 1965 paper. He died February 20, 2005, seven years, almost to the day, before its solution.

Actually since the curvature energy is invariant under the conformal group of symmetries, there are inﬁnitely many equally good shapes, as in Figure 11. Such “Dupin cyclides” are being used in modeling and architecture, where they piece together to design beautiful structures.

A key to the proof of the Willmore Conjecture is moving the problem from R3 to the compact sphere S3, conformally related by stereographic projection. On the sphere the energy becomes area plus the integral of mean curvature squared, and a minimizer is the Clifford torus T S1 S1, a minimal submanifold with zero mean curvature. A major D step along the way is to show that among minimal tori T has least area. Of course, among all surfaces, T does not minimize area since it can shrink or degenerate into a sphere in a five-parameter family of ways, but it does minimize area in directions orthogonal to that family—just as a mountain pass does not minimize height, although it does in the direction orthogonal to the highway. F. Urbano [38] had earlier proved that T was the only minimal torus with that much stability. The theory of such hybrid “mini-maxima” was developed by my own late thesis advisor Fred Almgren of Princeton University and his PhD student Jon Pitts now of Texas A&M. The key to the proof of the Willmore Conjecture is a precise understanding of the topology of a partially singular five-parameter space associated with any competitor. Thus in fine counterpoint to Perelman’s geometric proof of Poincaré’s topological conjecture, topology is the key to this new proof of the (geometric) Willmore Conjecture. Still an open question is the optimal shape for a two-holed torus, perhaps something like the two-holed torus of Figure 12. Robert Kusner of the University of Massachusetts at Amherst has a conjecture for the optimal torus of any given number of holes, a family of minimal surfaces in S3 discovered in 1970 by Blaine Lawson of the State University of New York at Stony Brook.

Bibliography [1] F. J. Almgren, Jr., Q-valued functions minimizing Dirichlet’s integral and the regularity of area- minimizing rectifiablecurrentsup to codimension2. World Scientific Publishing Co., Inc., River Edge, NJ, 2000. [2] ——— Some interior regularity theorems for minimal surfaces and an extension of Bernstein’s theorem, Ann. of Math. 84 (1966) 277–292. [3] F. J. Almgren, Jr. and J. Taylor, Geometry of soap films, Sci. Am. 235 (1976) 82–93. [4] E. Bombieri, E. De Giorgi, and E. Giusti, Minimal cones and the Bernstein problem, Invent. Math. 7 (1969) 243–268. [5] K. Brakke,The SurfaceEvolver, www.susqu.edu/brakke/evolver/evolver.html. [6] K. Cashman and H. Wright, Pattern and structure of basaltic reticulite: foam formation in lava fountains, Geophys. Res. Abstr. 8 (2006) 05398. [7] W. H. Fleming, On the oriented Plateau problem, Rend. Circ. Mat. Palermo (2) 11 (1962) 1–22. [8] J. Foisy, Soap Bubble Clusters in R2 and R3. Undergraduate thesis. Williams College, Williamstown, MA, 1991. [9] J. Foisy, M. Alfaro, J. Brock, N. Hodges, and J. Zimba, The standard double soap bubble in R2 uniquely minimizes perimeter, Pac. J. Math. 159 (1993) 47–59. [10] M. Freedman, The topology of four-dimensional manifolds, J. Diff. Geom. 17 (1982) 357–453. [11] T. Hales, Announcement of completion of Flyspeck, code.google.com/p/flyspeck/ wiki/AnnouncingCompletion. [12] ———, Cannonballs and honeycombs, Notices Amer. Math. Soc. 47 (2000) 440–449. [13] ———, A proof of the Kepler conjecture, Ann. of Math. (2) 162 (2005) 1065–1185. [14] T. Hales and S. Ferguson, A formulation of the Kepler conjecture, Discrete Comput. Geom. 36 (2006) 21–69. [15] R. Hamilton, Three-manifolds with positive Ricci curvature, J. Diff. Geom. 17 (1982) 255–306. [16] J. Hass and R. Schlafly, Double bubbles minimize, Ann. of Math. 151 (2000) 459–515. [17] Sir Thomas Heath, A History of Greek Mathematics Vol. II. Oxford University Press, London, 1921. [18] W.-Y. Hsiang, On the sphere packing problem and the proof of Kepler’s conjecture, Internat. J. Math. 4 (1993) 739–831. [19] M. Hutchings, The structure of area-minimizing double bubbles, J. Geom. Anal. 7 (1997) 285– 304. [20] M. Hutchings, F. Morgan, M. Ritoré, and A. Ros, Proof of the Double Bubble Conjecture, Ann. of Math. 155 (2002) 459–489. [21] J. Kepler, The Six-cornered Snowflake. Oxford Clarendon Press, Oxford, 1966. [22] R. Kusner and J. M. Sullivan, Comparing the Weaire-Phelan equal-volume foam to Kelvin’s foam, Forma 11 (1996) 233–242; reprinted in The Kelvin Problem. Foam Structures of Minimal Surface Area (D. Weaire, ed.), Taylor & Francis, London, 1996. [23] G. Lawlor, The angle criterion, Invent. Math. 95 (1989) 437–446. [24] ———, Double bubbles for immiscible fluids in Rn, J. Geom. Anal., 24 (2014) 190–204. [25] F. Marques and AndréNeves, Min-max theory and the Willmore conjecture, Ann. of Math. (2) 179 (2014) 683–782. [26] F. Morgan, Geometric Measure Theory. Academic Press, San Diego, 2009. 64 A Century of Advancing Mathematics

[27] ———, Review of Kepler’s Conjecture by G. Szpiro, Notices Amer. Math. Soc. 52 (2005). [28] D. Nance, Sufficient conditions for a pair of n-planes to be area-minimizing, Math. Ann. 279 (1987) 161–164. [29] G. Perelman, The entropy formula for the Ricci flow and its geometric applications, arXiv.org (2002). [30] ———, Proof of the double bubble conjecture in Rn, J. Geom. Anal. 18 (2008) 172–191. [31] B. Reichardt, C Heilmann, Y. Lai, and A. Spielman, Proof of the double bubble conjecture in R4 and certain higher dimensional cases, Pac. J. Math. 208 (2003) 347–366. [32] J. Simons, Minimal varieties in Riemannian manifolds, Ann. of Math. 88 (1968) 62–105. [33] S. Smale, Generalized Poincaré’s conjecture in dimensions greater than four, Ann. of Math. (2) 74 (1961) 391–406. [34] G. Szpiro, Does the proof stack up? Nature 424 (July 3, 2003) 12–13. [35] ———, Kepler’s Conjecture:How Some of the GreatestMinds in History Helped Solve One of the Oldest Math Problems in the World. John Wiley & Sons, Inc., 2003. [36] T. Tarnai, The observed form of coated vesicles and a mathematical covering problem, J. Mol. Biol. 218 (1991) 485–488. [37] W. Thomson (Lord Kelvin), On the homogeneous division of space, Proc. R. Soc. London 55 (1894) 1–16. [38] F. Urbano, Minimal surfaces with low index in the three-dimensional sphere, Proc. Amer. Math. Soc. 108 (1990) 989–992. [39] M. Varro, On Agriculture. The Loeb Classical Library. Harvard University Press, Cambridge, MA, 1934. [40] D. Weaire and R. Phelan, A counter-example to Kelvin’s conjecture on minimal surfaces, Phil. Mag. Lett. 69 (1994) 107–110. [41] H. Weyl, Symmetry. Princeton University Press, Princeton, NJ, 1952; Princeton Sci. Lib. ed., 1989. [42] R. Williams, The Geometrical Foundation of Natural Structure: A Source Book of Design. Dover, New York, 1979. [43] T. Willmore, Note on embedded surfaces, An. Sti. Univ.“Al. I. Cuza” Iasi Sect. I a Mat. (N.S.) 11B (1965) 493–496.

Department of Mathematics and Statistics, Williams College, Williamstown, MA 01267 [email protected] Defying God The Stanley-Wilf Conjecture, Stanley-Wilf Limits, and a Two-Generation Explosion of Combinatorics

Eric S. Egge Carleton College

In 2005, at the Third International Conference on Permutation Patterns in Gainesville, Florida, Doron Zeilberger declared that “Not even God knows a1000.1324/.” Zeilberger’s claim raises thorny theological questions, which I am happy to ignore in this article, but it also raises mathematical questions. The quantity a1000.1324/ is the one-thousandth term in a certain sequence an.1324/. God may or may not be able to compute the thousandth term in this sequence, but how far can mortals get? If we can’t get beyond the fortieth or fiftieth term, can we at least approximate the one-thousandth term? How fast does an.1324/ grow, anyway? And what does an.1324/ even mean? The answers to these questions involve fast computers, fascinating mathematics, and remarkable human ingenuity. But their stories, which are ongoing, also reflect important undercurrents and developments that have influenced all of mathematics, but especially combinatorics, over the past two generations and more.

1 Knuth’s railroad problems

The story of a1000.1324/ begins with a gap in the railroad literature, which Donald Knuth began tofillin 1968inthefirst editionof thefirst volumeof hismasterpiece The Art ofCom- puter Programming. In the second section of Chapter 2, Knuth included several exercises exploring a problem involvingsequences of railcars one can obtain using a turnaround.One of Knuth’s exercises is equivalent to the following problem.

At dawn we have n railroad cars positionedon the rightside of the track in Figure 1, numbered 1 through n from rightto left. Duringthe day we gradually move the cars to the left side of the track, by moving each car into and back out of the turnaround area. There can be any number of cars in the turnaround around area at a time, and at the end of the day the cars on the left side of the track can be in many different orders. Each possible order determines a permutation of the numbers 1; 2; : : : ; n. Show that a permutation 1; : : : ; n (this time reading from left to right along the tracks) is attainable in this way if and only if there are no indices i < j < k such that i < k < j .

65 66 A Century of Advancing Mathematics

p p ... p p ... 1 2 n1– n n n–1 2 1

Figure 1. Knuth’s railroad tracks.

The solution to this problem is a fun exercise in careful bookkeeping. If such a subsequence exists, then consider the situation when car i enters the turnaround. Since i is the smallest of our three car numbers, cars j and k have already entered the turnaround, in that order. Furthermore, in order for car i to appear to the left of cars j and k, cars j and k must both still be in the turnaround when car i enters. But now cars j and k will leave the turnaround in the wrong order.

Conversely, suppose we have a target permutation 1; : : : ; n with no subsequence of the forbidden type. We can always move 1 into position, and when car 1 leaves the turnaround, the cars in the turnaround are, from bottom to top, n; n 1; : : : ; 1 1. Now C notice that 2 cannot be larger than 1 1, since thiswouldmean 1, 2, and 1 1 form a C C forbidden subsequence. So if 2 is in theturnaround,then it is the top car there. Either way, we can move car 2 intoposition.In general, ifwe have just moved car j into position,and b is the smallest entry greater than j which has not yet left the turnaround,then j 1 b, C Ä since otherwise would have a forbidden subsequence j , j 1, b. Therefore, if j 1 has C C entered the turnaroundthen it is the top car there, and we can move it into place. Knuth was interested in this railcars problem because it models the data structure commonly called a stack, which arises in numerous programming problems, so he introduced no particular notation for the permutations he obtained. Indeed, no general notation for these permutations appeared in print until 1985, when Simion and Schmidt [31] published the first systematic study of permutations with forbidden subsequences of the type Knuth uses. Today, if and are permutations of lengths n and k respectively, then we say a subsequence of of length k has type whenever its entries are in the same relative order as the entries of . For example, the subsequence 829 of the permutation 718324695 has type 213, since its smallest entry is in the middle, its largest entry is last, and its middle entry is first. In this context, we say avoids , or is -avoiding, whenever has no subsequence of type , and we write Sn./ to denote the set of all permutations of length n which avoid . We might also say that is a forbidden subsequence or a forbidden pattern. With this terminology, the permutations Knuth obtains with his railcars are exactly the 132-avoiding permutations, and the term a1000.1324/ that Zeilberger’s God finds so perplexing is none other than the size of S1000.1324/. In Table 1 we have the first ten values of Sn.1324/ . j j Defying God: the Stanley-Wilf Conjecture, Stanley-Wilf Limits, and . . . 67

n 0 1 2 3 4 5 6 7 8 9

Sn.1324/ 1 1 2 6 23 103 513 2762 15793 94776 j j Table 1. The ﬁrst ten values of jSn.1324/j. 2 A scattered history

Of course, combinatorial problems predate computers, the MAA, and even railroads. In fact, we can find evidence of people studying combinatorial problems nearly as far back in time as we can find evidence of people doing mathematics. For example, in the following quotation from Plutarch’s Table-Talk [25, VIII.9, 732], we find Chrysippus (circa 200 BCE) and Hipparchus (circa 300 BCE) discussing how to form logical expressions. Chrysippus says that the number of compound propositions that can be made from only ten simple propositions exceeds a million. (Hipparchus, to be sure, refuted this by showing that on the affirmative side there are 103; 049 compound statements, and on the negative side 310,952.) Plutarch doesn’t say what a compound proposition is, but it seems reasonable to assume it’s a combinatorial object of some kind. Indeed, in 1994 David Hough, who was then a graduate student at George Washington University, observed that 103,049 is the tenth small Schröder number [34, 17, 1]. This means 103,049 is, among other things, the number of ways to parenthesize a sequence of eleven letters. To give a sense of what this means, there are three ways to parenthesize a sequence of three xs, namely xxx, .xx/x, and x.xx/, and eleven ways to parenthesize a sequence of four xs, namely xxxx, .xxx/x, ..xx/x/x, .x.xx//x, x.xxx/, x..xx/x/, x.x.xx//, .xx/.xx/, .xx/xx, x.xx/x, and xx.xx/. Plutarch’s account of Chrysippus and Hipparchus’s debate over the number of compound propositions is not the most ancient combinatorial reference, nor does it involve the most common combinatorial quantities. In Sushruta Samhita, a sixth century BCE Sanskrit text on surgery attributed to the Indian physician Sushruta, the author observes that we can make sixty-three combinations out of six different tastes, when we take them one at a time, two at a time, etc. This gives us all but one of the entries of the sixth row of Pascal’s triangle, more than two millennia before Pascal. And Sushruta’s discussion of what would come to be called the binomial coefficients is not an isolated occurrence in ancient Indian literature. The Bhagabati Sutra, a religious text of the Jains which appeared around 300 BCE, contains a more general rule for computing binomial coefficients, and the Jain mathematician Mahavira gives a completely general rule in his Ganita Sara Sangraha, which was written around 850 AD [20, p. 27]. Combinatorics is ancient, but for much of its history it has also been scattered, arising in diverse contexts but having few or no adherents of its own. Hipparchus solves a combinatorial problem to refute Chrysippus, but he does it to make a larger point. Sushruta solves a combinatorial problem in the midst of a landmark text on surgery, which barely contains any other mathematics at all. And this is how it goes for centuries: Euler invents graph theory to solve the Königsberg bridge problem, Pascal (re)discovers the binomial coefficients in his quest to resolve interrupted games of chance, Kempe “proves” the four color theorem to answer a question posed almost thirty years earlier by a student trying to color a map of England, Cayley studies partitions as a tool in invariant theory, Young 68 A Century of Advancing Mathematics introduces the tableaux that now bear his name to build on Cayley’s work in invariant theory, and Kirchhoff proves his matrix-tree theorem for counting spanning trees in a graph to solve a problem in electrical engineering. To be sure, some combinatorial problems were studied for their own sake a century or two ago, and there were even people we would recognize today as combinatorialists. For instance, in the early 1880s J. J. Sylvester and his students devoted themselves to the study of partitions, pioneering the use of Ferrers diagrams to develop a general theory [24]. But in the nineteenth century and the first half of the twentieth, Sylvester’s work and life are the exceptions, not the rule, as most combinatorial problems were treated as isolated curiosities, not fundamental examples around which one might build a theory.

3 How fast does the number of Knuth railcar permutations grow?

Fortunately for us, the problem of estimating the rate at which Sn.1324/ grows is not j j an isolated curiosity, but is instead part of a general theory, which means we can gain some insight by looking at a related problem. For instance, since 132 is part of 1324, every permutation which avoids 132 also avoids 1324, so let’s try to first estimate Sn.132/ . j j To get a crude upper bound on this quantity, first recall that a left-to-right minimum in a permutation is an entry which is smaller than every entry to its left.For example, the left- to-right minima in 694853127 are 6, 4, 3, and 1. Somewhat surprisingly, if avoids 132, then it is determined by the values and positions of its left-to-right minima. For example, suppose S9.132/ has left-to-right minima 6, 3, and 1, which are in the first, third, and 2 fifth positions. Since the left-to-right minima must be in decreasing order, we can start to construct as in Figure 2. Now the second entry of must be greater than 6, since it’s

6 3 1

Figure 2. A permutation 2 S9.132/ with prescribed left-to-right minima. not a left-to-rightminimum. But if we put 8 (resp. 9) there, then the 6, the 8 (resp. 9), and the 7, which must appear somewhere to the right, will form a subsequence of type 132. Therefore, the second entry must be 7. Similarly, the fourth entry must be the smallest available number greater than 3, and each successive entry is the smallest available number which is greater than the nearest left-to-right minimum on its left. Following this recipe, we find that in our example 673412589. D Returning to our bound on Sn.132/ , the leftmost entry of a permutation is always a j j n 1 left-to-right minimum, as is 1, so there are at most 2 sets of values for the left-to-right n 1 n 1 minima, and 2 sets of positions for these values. Therefore, Sn.132/ 4 when j j Ä n 1. Since 4n 1 is much less than nŠ when n is even reasonably large, the chances we can put our railcars into any particular order is almost zero for any train more than a few cars long. n 1 We expect our 4 bound on Sn.132/ to be terrible, because there are many ways a j j given choice of numbers and positions can fail to be the values and positions of the left- to-right minima for any permutation. For example, we made no effort to ensure that we have the same number of left-to-right minima as we have positions for them. And even Defying God: the Stanley-Wilf Conjecture, Stanley-Wilf Limits, and . . . 69 when we have the same number of values as positions, the values and positions may not be compatible. For instance, if we insist on having left-to-right minima 1, 5, and 7 in the first, third, and eighth positionsof a permutationof length 9, then we’re goingto have a bad time, since at least one of 2, 3, and 4 must appear between the 5 and the 1, and this entry will be another left-to-right minimum. Nevertheless, a miracle of sorts occurs: while it is 4 n 1 possible to improve our bound on Sn.132/ to 4 , the base 4 of the exponential j j n3=2p factor cannot be replaced with a smaller number. To see why no base less than four will work, we return to Knuth, who doesn’t bother to estimate Sn.132/ at all in his solution to the railroad problem. Instead, he notes that we j j can encode each permutation in the set uniquely by writing down the sequence of moves that generate it. In particular, if we write N each time we move a railcar intothe turnaround and E each time we move a railcar out of the turnaround on the other side, then we get a bijection between Sn.132/ and the set of sequences of n N s and n Es in which every initial segment has at least as many N s as it has Es. Such sequences are called ballot sequences, and it was already well known in 1969 that they are counted by the Catalan 1 2n Cn 4n 2 Cn number Cn n 1 n . In particular, we have C n 1 , so limn C 4 and n1 !1 n1 n D C D C D limn Sn.132/ 4. !1 j j D Introducingp the notion of the type of a subsequence, and the associated idea of pattern avoidance, opens a panorama of fruitfulgeneralizations and questions.For example, for any permutation of length n, let r denote the reverse of , which has r .j / .n 1 j / 1 D C for 1 j n. Similarly, let denote the group-theoretic inverse of . Then it’s not Ä Ä r r 1 hard to show that avoids if and only if avoids , which occurs if and only if 1 avoids . Combining these, we find Sn.132/ Sn.213/ Sn.231/ Sn.312/ j j D j j D j j D j j and Sn.123/ Sn.321/ for all n. Furthermore, it’s already implicit in work of Percy j j D j j MacMahon [21] in 1915 and Craige Schensted [30] in 1961 that Sn.123/ Cn. As a n j j D result, we have limn Sn./ 4 for all S3. All of which leads to the natural !1 j j D 2 question: what happens if is bigger? p

4 The rise of the machines in combinatorics

Computer programming questions motivated Knuth’s railcars problem, and they continued to drive the study of patterns in permutations for the next decade and a half. Rotem [29] was the next person to look at permutations which can be produced with one pass through a stack. He called these permutations stack-sortable, and in his main results he gave a bijection from these permutations to binary trees, which he used to analyze the average length of a monotonic subsequence in these permutations. Even after Simion and Schmidt kicked off the study of restricted permutations for their own sake, others continued to investigate restricted permutation problems arising from data structures. The next major step in this direction came in Julian West’s thesis [39], where he studied permutations one can obtain using two passes through a stack, which he called 2-stack-sortable. Relying on the language of pattern-avoidance, West showed that these permutations are the ones which avoid 2341 and whose only subsequences of type 3241 are those which are contained in a subsequence of type 35241. West also conjectured that the number of 2-stack-sortable 70 A Century of Advancing Mathematics

2 3n permutations of length n is , which Zeilberger proved [41] shortly .n 1/.2n 1/ n C C ! after hearing West describe his conjecture at a talk in May of 1991. Since then a thriving industry has developed, in which people use the language of permutation patterns to study permutations generated by parallel queues, dequeues, networks of stacks, ordered stacks, token passing in graphs, and even forklifts, to name just a few. There are more relevant references than I can comfortably list here, but the interested reader would do well to start with the summary in Section 2.1 of Kitaev’s encyclopedic monograph [19]. Questions from computer science lead to interesting problems throughout combinatorics, but for the past thirty years computers have played a more explosive role in the field, letting mathematicians generate reams of data that would be impossible to produce by hand. These data, in turn, suggest new conjectures, and eventually lead to new theorems. Simion and Schmidt’s paper, for example, enumerates more than a dozen families of restricted permutations, with sequences like powers of two, the Fibonacci numbers, the n central binomial coefficients n , and the triangular numbers plus one. With a computer, 2 one can essentially peek in theb c back of the book, generating the first ten or so terms of each sequence, and then seeing what needs to be proved. West’s conjecture on the number of 2-stack sortable permutations of length n would also be much harder to discover without computational assistance: one could certainly write down the 91 2-stack-sortable permutations of length five in half an hour or so, and perhaps get the 408 2-stack-sortable permutations of length six in another hour, but the 1938 2-stack-sortable permutations of length seven would present a challenge, and examining all 40320 permutations of length eight to find the 9614 which are 2-stack-sortable would take days, or even weeks, to do by hand. With a computer, obtaining all of these values and more is the work of a pleasant afternoon. When evening comes, and our programming work has told us that our sequence begins with the terms 1;2;6;22;91;408;1938, and 9614, we still need to formulate a conjecture about the general term before we can make more progress. Here, too, computers have become indispensable. Forty years ago we might have asked a dozen of our closest friends whether they had ever seen this sequence. Sometimes, this actually worked! For example, David Robbins first discovered that alternating-sign matrices are connected with the descending plane partitions first introduced by George Andrews by asking Richard Stanley whether he had ever seen the sequence 1;2;7;42;429;7436;::: [28]. But most of the time, our friends’ memories are no better than our own. In the area of remembering integer sequences, computers have completely replaced human beings. Neil Sloane began this process in 1964, when he started collecting sequences he encountered on index cards, and in 1973 he published his collection in his book A Hand- book of Integer Sequences, which included 2372 sequences. In 1995 Sloane and Simon Plouffe published a second edition of Sloane’s book, with the new title The Encyclopedia of Integer Sequences, which included 5487 sequences. Books can’t keep up with computers, though, and in 1996 Sloane established the On-Line Encyclopedia of Integer Sequences [23], which he singlehandedly maintained until 2002. The OEIS, as it is affectionately known, is now run as a wiki by a devoted group of users. It grows by more than 10000 entries each year, and would require more than 750 volumes of more than 500 pages each if it were published in book form today. Defying God: the Stanley-Wilf Conjecture, Stanley-Wilf Limits, and . . . 71

When we enter our terms 1;2;6;22;91;408;1938;9614 into the OEIS search box, we are rewarded with a description of the sequence that includes several more terms, a general formula, code for generating even more terms, a list of related references, comments from other users about the sequence, families of objects the sequence is known or conjectured to count, and more. In hindsight it’s amusing that Sloane called his book “A Handbook,” as though there might be competitors. There are none, and the OEIS is a required stop for anyone who encounters an integer sequence they don’t recognize. It’s no exaggeration to observe that in certain parts of combinatorics, the OEIS alone has increased the rate of new discoveries by an order of magnitude.

5 The Stanley-Wilf conjecture In keeping with the prevailing sense that combinatorics was just an offshoot of other subjects, few papers even mentioned patterns in permutations before 1985, and those that did generally followed Knuth in approaching the subject from a computer science point of view. Nevertheless, throughout the 1970s and into the 1980s, combinatorics was becoming a proper, and popular, subject of study in its own right, and the study of patterns in permutations was a current in this wave of acceptance. In particular, by 1980 a handful of people behind the scenes had begun to ask what sorts of sequences can appear as Sn./ , and in j j particular, how fast such a sequence can grow. Just as we bounded Sn.132/ with an exponential function, we can also bound j j Sn.12 : : : k/ with an exponential function in an elementary way. Inspired by the clas- j j sic pigeonholeprinciple proof of the Erd˝os-Szekeres theorem on increasing and decreasing subsequences in permutations, we ﬁrst label each entry m of a given Sn.12 : : : k/ with 2 the length of the longest increasing subsequence of whose last entry is m. In Figure 3 each entry of the permutation 381426975 has its label above it in bold. Since has no

1 2 1 2 2 3 4 4 3 3 8 1 4 2 6 9 7 5 Figure 3. Labeling the entries of the permutation 381426975. increasing subsequence of length k, all of these labels will be among 1; 2; : : :; k 1. In addition, for each j the entries with label j will be in decreasing order: if an entry labelled j has a larger entry to its right,then that larger entry must be labelled at least j 1. Therefore C is a disjoint union of k 1 or fewer decreasing subsequences. There are .k 1/n ways to choose one of these subsequences for each of 1; 2; : : : ; n to be in, and there are no more than .k 1/n ways to choose which positions belong to which subsequences. Therefore, 2n Sn.12 : : : k/ .k 1/ . In fact, in 1981 Regev used [27] analytical machinery to prove j j Ä results which imply n 2 lim Sn.12 : : : k/ .k 1/ : (1) n j j D !1 However, the relationship betweenp Regev’s work and subsequences of permutations is not visible to the naked eye, in part because his actual results have to do with certain sums arising in the study of the representations of the symmetric group. Unaware at ﬁrst of Regev’s work and the bound on Sn.12 : : : k/ , around 1980 Herb n j j Wilf asked whether Sn./ .k 1/ for all Sk. At nearly the same time, Richard j j Ä C 2 72 A Century of Advancing Mathematics

n 2 Stanley independently asked whether limn Sn./ .k 1/ for all Sk. Per- !1 j j D 2 haps responding to Stanley’s conjecture or Regev’s results, Wilf soon asked whether there p n exists, for each permutation , a finite constant c./ such that limn Sn./ c./. !1 j j D Eventually, two conjectures emerged: the upper bound conjecture and the limit conjecture, p which together came to be known as the Stanley-Wilf conjecture. Conjecture (The Stanley-Wilf Upper Bound Conjecture). For every permutation , there n is a real number c./ such that Sn./ c./ . j j Ä Conjecture (The Stanley-Wilf Limit Conjecture). For every permutation , there is a real n number c./ such that limn Sn./ c./. !1 j j D It’s routine to show that the limitp conjecture implies the upper bound conjecture, and in 1999 Arratia showed [4] that these two conjectures are equivalent. More specifically, Arra- tia gave a simple combinatorial proof that for any permutation and all m; n 1, we have Sn m./ Sn./ Sm./ . This implies that f .n/ ln. Sn./ / has the property j C j j jj j D j j that f .n m/ f .n/ f .m/ for all m; n 1; we call such functions subadditive [40]. C Ä C Now we can use a result, Fekete’s subadditivity lemma, which says that if f is subadditive f .n/ ln. Sn./ / then limn n exists, though it might be . This means limn j n j exists, !1 1 !1 n 1 n 1 so limn S ./ exists. The upper bound conjecture implies that S ./ is bounded !1 n n 1 j j j j below by c./q, and the limit conjecture follows. q For nearly a quarter of a century the Stanley-Wilf conjecture stood against all who tried to prove it. Indeed, by the dawn of the new millenniumno proof was in sight,though many of the problem’s sharp corners had been chipped away. Miklós Bóna sheared off one of the sharpest of these corners in 1999, when he showed that the Stanley-Wilf conjecture holds for layered permutations [7]. Bóna’s result is an extension of the fact that the conjecture holds for monotone permutations, since a layered permutation is one obtained by first listing the smallest `1 numbers in decreasing order, then listing the next smallest `2 numbers in decreasing order, etc. For example, 321765498 is the layered permutation with `1 3, D `2 4, and `3 2. Noga Alon and Ehud Friedgut [3] chiseled away another sharp cor- D D ner in 1999, when they showed that the Stanley-Wilf conjecture holds for all permutations consisting of an increasing sequence followed by a decreasing sequence, as well as for all permutations which avoid 123. Fissures one might use to break the problem open were also noted throughoutthe1990s. For instance, Alon and Friedgut showed [3] that the Stanley-Wilf conjecture almost holds, .n/n by showing that Sn./ c./ for a certain slow-growing function , which they j j Ä constructed from the inverse of the Ackermann function. In the same paper, Alon and Friedgut also proved that if there is a linear upper bound on the lengths of certain words over an ordered alphabet, then the Stanley-Wilf conjecture will follow. Another fissure in the problem appeared in the early 1990s, but it was only visiblefrom a certain angle. To describe this new point of view, we first recall that each permutation Sn has an associated permutation matrix, namely the n n matrix of 0s and 1s 2 whose ij th entry is 1 if .i/ j and 0 if .i/ j . It is not difficult to translate the D ¤ notions of pattern avoidance and containment we’ve been discussing into the language of permutation matrices, but our new point of view actually depends on a different, more permissive definition of containment. Specifically, we say an mA nA matrix A of 0s and 1s contains an mB nB matrix B of 0s and 1s whenever there is an mB nB submatrix Defying God: the Stanley-Wilf Conjecture, Stanley-Wilf Limits, and . . . 73

B1 of A such that if the ij th entry of B is 1 then the ij th entry of B1 is 1. Note that 0s in B impose no restriction on the corresponding entry of B1. For example, the matrix 1 1 1 1 1 1 0 1 1 0 1 contains the matrix exactly twice: once in bold here 1 0 1 and 0 1 1 1 0 1 1 1 0 Â Ã 1 1 0 @ A 1 1 1 @ A once in bold here 1 0 1 . 01 1 01 One can ask the@ same questionA about matrices that we have asked about permutations, namely, how many m n matrices do not contain a given matrix? However, the fact that our matrices need not contain any particular number of 1s inspired Zoltán Füredi and Péter Hajnal [16] to ask a different question: if we have a certain matrix C , all of whose entries are 0s and 1s, how many 1s can we put into an n n matrix before it must contain C ? Füredi and Hajnal answered this question for a variety of specific C , some of which were permutation matrices, and others of which were not. At the end of their paper, tucked in among several other open questions, they asked whether there exists, for any permutation matrix C , a constant c.C / such that the number of 1s an n n matrix can contain before it must contain C is bounded above by a linear function of n, namely c.C / n. Füredi and Hajnal asked the question, but it was Martin Klazar who, in 2000, promoted Füredi and Hajnal’s question to the status of a conjecture. Conjecture (The Füredi-Hajnal Conjecture). For every permutation matrix C , there is a real number c.C / such that if an n n matrix of 0s and 1s containsat least c.C / n entries equal to 1, then it contains C . Klazar’s promotion of this idea from a question to a conjecture may have been justified by his main result [20], which is that the Füredi-Hajnal conjecture implies the Stanley-Wilf conjecture. The problem had another hairline crack, but it wasn’t clear whether anyone could get some explosives, or a crowbar, or even a chisel, into it.

6 More machines in combinatorics As we mentioned earlier, over the past two generations computers have become essential tools that many mathematicians, and especially combinatorialists, use to generate mathematical data. These data lead to conjectures, and, if all goes well, to theorems. But computers can be more than just data-generating devices; more and more frequently, computers play a central role in generating conjectures, and even in proving theorems. The first, and certainly most famous, proof in which computers played a substantial role is Appel and Haken’s 1976 proof of the four-color theorem. Although several people have found ways to streamline this proof over the past forty years, its overall structure remains the same: this is a proof by several hundred cases. More specifically, to prove the four-color theorem we first show that if G is a planar graph which cannot be properly colored with four colors, and every planar graph with fewer vertices than G can be properly colored with four colors, then G must contain one of several hundred configurations of vertices. We then show that for each of these configurations, it’s possible to reduce the number of vertices in G to obtain a smaller counterexample. The computer’s role in this proof is twofold: it uses human-generated heuristics to help find a family of unavoidable configurations of vertices, and it aids in constructing proofs that each configuration can be reduced. (For more details 74 A Century of Advancing Mathematics on the four-color theorem, its history, and its proof, see Robin Wilson’s contributionto this volume.) Since Appel and Haken’s work, combinatorialists enumerating pattern-avoiding permutations have found ways to use computers that go beyond generating data (though this remains an important computer task) or checking numerous cases. For example, in the first few steps of his proof of West’s formula for the number of 2-stack-sortable permutations of length n, Zeilberger used Maple to find a functional equation for the generating function for these permutations with respect to length and another statistic. But Zeilberger took the ability of computers to do mathematics to a new level in 1998, when he taught his computer Shalosh B. Ekhad to enumerate pattern-avoiding permutations. More specifically, Zeilberger wrote code that enables his computer to generate, for each set R of forbidden patterns an object called an enumeration scheme, which can be readily converted into a recursive formula for Sn.R/ [42]. Following Zeilberger’s construction, Shalosh B. Ekhad j j was able to recover most of the enumerations then known, and even to discover (and in the process prove) some new ones. Since then Vince Vatter [37], Lara Pudwell [26] and others have refined Zeilberger’s work on pattern-avoiding permutations, and extended it to more general objects and definitions of pattern containment.

7 The proof of the Füredi-Hajnal conjecture: too nice to be true

At the turn of the millennium the Stanley-Wilf conjecture was widely seen as one of the most important and difficultopen problems in the study of permutation patterns, and several of the top researchers in the field were trying everything they could to crack the problem. In spite of their efforts, by late 2003 they had made no further progress. Meanwhile, unbeknownst to anyone studying permutation patterns, Adam Marcus and Gábor Tardos had become interested in a nice collection of questions involving 0-1 matrices that Füredi and Hajnal had posed a decade before. Although Marcus and Tardos didn’t know it yet, these questions included the one Klazar had elevated to conjecture status. It didn’t seem to Mar- cus and Tardos that these questions had attracted much attention, even though they were intriguing and approachable. Marcus and Tardos didn’t care about permutations in particular, and they had never heard of the Stanley-Wilf conjecture, but they did find they could make progress on one of Füredi and Hajnal’s questions involving permutation matrices. Soon they had settled one of Füredi and Hajnal’s questions: they had shown that if C is a permutation matrix then there is a real number c.C / such that if an n n matrix of 0s and 1s contains at least c.C /n entries equal to 1, then it contains C [22]. But they didn’t hear the crowd cheering their result until weeks later, when Marcus found some of Klazar’s other work related to Füredi and Hajnal’s paper, and Klazar told him about the Stanley-Wilf conjecture. Marcus and Tardos’s proof of the Füredi-Hajnal conjecture was a major advance in the study of patterns in permutations, but it’s not a long proof, and it doesn’t involve any complicated technical machinery. It is a product of human ingenuity, not fast computers. Nevertheless, it took the permutation patterns community by storm. Bóna had just submitted the first edition of his book Combinatorics of Permutations [8] to the publisher when Defying God: the Stanley-Wilf Conjecture, Stanley-Wilf Limits, and . . . 75 he learned of Marcus and Tardos’s proof. Rather than publish a book that would be out of date before it went to press, Bóna insisted that the publisher wait for him to add a new section on Marcus and Tardos’s work before moving ahead with publication. The proof is accessible enough that Zeilberger has given a one-page account in the personal journal of Ekhad and Zeilberger [43], and Tardos gave the entire proof in detail, on the blackboard, in a 45-minute talk at the Second International Permutation Patterns conference in Nanaimo, British Columbia, in early July of 2004. Marcus and Tardos’s proof is so elegant that when they first discovered it, Marcus and Tardos themselves thought it was too simple to be correct. They gave themselves the weekend to find the error they were certain was hidden within. We need not give all of the details of Marcus and Tardos’s proof here, but we can outline it. Suppose C is a k k permutation matrix, and for each positive integer n, let f .n/ denote the largest number of 1s an n n matrix of 0s and 1s can contain without containing a copy of C . For simplicity, let n be a multiple of k2. If M is an n n matrix of 0s and 1s which does not contain C , then divide it into contiguous k2 k2 blocks. Call a block wide whenever it has at least k columns which contain a 1, and call a block tall whenever it has at least k rows which contain a 1. Essentially, Marcus and Tardos show that there are few wide blocks and few tall blocks, and that blocks which are neither wide nor tall contain few 1s. Let’s look at the blocks which are neither wide nor tall first. Any block with more than .k 1/2 1s must be tall or wide, so the number of 1s in a block which is not tall or wide is at most .k 1/2. But we can also bound the number of these blocks which contain any f n 1s at all. In particular, if there are more than k2 blocks which contain a 1, then there is a pattern of type C of these blocks. By choosing appropriateÁ rows and columns, we can select each of the 1s we need to form a copy of C in M , since C has exactly one 1 in f n every row and column. Therefore, there are at most k2 blocks which are neither tall nor wide, but which still contain a 1. Combiningthese observations, Á we find that the blocks which are neither wide nor tall contain no more than .k 1/2f n 1s in total. k2 Now let’s consider the wide blocks and the tall blocks. Each blockÁ is k2 k2, so each wide block and each tall block contains at most k4 1s. Marcus and Tardos use the pigeon- n k2 hole principle to show that there are no more than k k wide blocks and no more than n k2 2 k2 k k tall blocks, so these blocks contain at most 2k k n 1s in total. Combining these estimates with our earlier work, we find n k2 f .n/ .k 1/2f 2k2 n: 2 Ä k C k ! Á Now it’s not hard to show by induction that

k2 f .n/ 2k4 n: (2) Ä k !

The coefﬁcient of n on the right side of (2) is large as a function of k, but we have what F¨uredi and Hajnal (through Klazar) promised: a linear bound on f .n/. 76 A Century of Advancing Mathematics

8 Undergraduates in combinatorics

Marcus and Tardos’s proof of the Füredi-Hajnal conjecture is remarkable in many ways, one of which is that Marcus had barely finished his undergraduate work when they found it. In particular, Marcus spent his junior year in 2001–02 as an undergraduate in the Budapest Semesters in Mathematics program, and after graduating from Washington University, he returned to Budapest on a Fulbright fellowship in 2003. It was during Marcus’s Fulbright year that he and Tardos proved the Füredi-Hajnal conjecture, which places their work in the middle of another crucial development in the explosion of combinatorics that has taken place over the past two generations: student research, and especially undergraduate research. The tradition of involving students in combinatorics research actually goes back more than a century: Sylvester did much of his foundationalwork on the theory of partitions[36] in collaboration with the nine graduate students who were taking a class with him on the subject at Johns Hopkins University in the spring of 1882. But the practice was rare in the United States when the National Science Foundation began funding the URP (Undergrad- uate Research Program) in 1958, and undergraduate research in mathematics was still not widespread when the NSF began funding mathematics REUs in 1987. Since then, however, undergraduate research in mathematics has blossomed: there were eight NSF-funded REU sites in mathematics in 1987, and by 2003 there were more than fifty. Combinatorics has played a central role in this flowering. To see how extensively combinatorics is intertwined with undergraduate research, one need look no further than the NSF’s own website: of the forty-seven REU sites listed there as being funded for the summer of 2014, twenty-three include projects on combinatorics and/or graph theory. (For more details on the history of undergraduate research in mathematics in the United States, see Joseph Gallian’s contribution to this volume.) Combinatorics and graph theory are also well-represented among undergraduate research award winners: in three of the last four years a Morgan Prize winner or runner-up has been recognized for her or his substantial work in combinatorics. Not only is undergraduate research ubiquitous in combinatorics and vice versa, but the quality of the contributions undergraduates have made to the subject has been remarkably high. In 1989 Bill Doran contributed [13] a key piece of theenumeration of the totallysym- metric self-complementary plane partitions which fit in a 2n 2n 2n box, by explaining how to reformulate these objects as collections of nonintersectinglattice paths, which could then be counted using determinants or permanents of matrices of binomial coefficients. In 2002 Joshua Greene won the Morgan Prize for his simplified proof of the Kneser conjecture, which involves the chromatic numbers of certain Kneser graphs. Undergraduates have also contributed substantially to the study of patterns in permutations, even beyond Mar- cus and Tardos’s work. For instance, in 2004 Reid Barton won the Morgan Prize for his work on the number of copies of a given pattern which can be packed into a permutation of a given length, and in the early 1990s Zvezdelina Stankova gave combinatorial proofs [32, 33 ] that Sn.4132/ Sn.3142/ and Sn.1234/ Sn.4123/ for all n 0. These j j D j j j j D j j last two results will save us a substantial amount of work when we start to look beyond the n results of Marcus and Tardos, by investigating the actual values of limn Sn./ for !1 j j various permutations . p Defying God: the Stanley-Wilf Conjecture, Stanley-Wilf Limits, and . . . 77

9 Stanley-Wilf limits Marcus and Tardos’s proof of the Füredi-Hajnal conjecture made it official: for every per- n mutation there is a constant L./ such that limn Sn./ L./. Today we refer !1 j j D to the number L./ as a Stanley-Wilf limit, and we’d liketo know the value of L./ for as p many permutations as possible. We have seen that L.123/ L.132/ 4, and we’ve D D also seen how to use some easy symmetries to show that L./ 4 for every S3. In D 2 addition, we have seen evidence that L.1234:::k/ .k 1/2, a fact which follows from D the work of Regev [27]. The values of L./ given above are the only values that were known by the mid 1990s, and they all support the idea that L./ depends only on the length of . In other words, it appears that all permutations of a given length are equally difficult to avoid, at least asymp- totically. In fact, these data are also consistent with Stanley’s much older conjecture that L./ .k 1/2 for any permutation of length k. It turns out, though, that making true D conjectures about Stanley-Wilf limits is much harder than making false ones. Indeed, in his thesis [6] Bóna used a connection between 1342-avoiding permutations and plane trees to find an exact formula for Sn.1342/ as a sum of terms involving binomial coefficients, and j j from this formula he extracted the fact that L.1342/ 8. This is shocking! Not only does D this result disprove two longstanding conjectures at once, it also says that at a fundamental level, 1342 is harder to avoid than 1234. With L.1234/ and L.1342/ in hand, we might thinkthat we have 22 more permutations of length four to consider. However, thanks to the symmetries we mentioned earlier and work of West and Stankova, we know that if is a permutationof length four, then Sn./ j j is equal to one of Sn.1234/ , Sn.1423/ , or Sn.1324/ for all n. As a result, we have j j j j j j just one more Stanley-Wilf limit to compute for a forbidden pattern of length four, namely L.1324/. All of which means that we have returned to our original question: how fast does an.1324/ grow, anyway? We can start to get a feel forthe growthrate of an.1324/ by lookingat the data in Table n 2, where we have the values of Sn.1324/ when n is a multiple of four. Our table ends j j at n 36, because this is the largest value for which Sn.1324/ is currently known. All D p j j of these terms are available in the On-Line Encyclopedia of Integer Sequences; the terms with n 31 are due to recent work of Fredrik Johansson and Brian Nakamura [18], and Ä the rest are due to even more recent work of Andrew Conway and Anthony Guttmann [12]. n 4 n 5=2 It is known that Sn.1234/ 9 n and Sn.1342/ 8 n , so we might ask for a j j j j n constants and for which Sn.1324/ is asymptotic to a function of the form n . In j j fact, Johansson and Nakamura, with a computational assist from Shalosh B. Ekhad, have found that if Sn.1324/ is asymptotic to a function of this form, then 10:45 and j j 8:64 are the most consistent with the values of Sn.1324/ for n 31. If Johansson j j Ä and Nakamura’s values of and are in the right neighborhoods, then the fact that the value of for Sn.1324/ is so much greater than the corresponding values for the other j j j j n two sequences explains the slow convergence we see in the data for Sn.1324/ . j j p n 4 8 12 16 20 24 28 32 36

n Sn.1324/ 2:19 3:348 4:141 4:728 5:186 5:559 5:869 6:134 6:363 j j n p Table 2. Values of jSn.1324/j for small n. p 78 A Century of Advancing Mathematics

Unfortunately, Johansson and Nakamura’s values might not be in the right neighborhoods. Even more troubling, as Johansson and Nakamura themselves point out, it’s un- n likely that there is an asymptotic formula for Sn.1324/ of the form n at all. Indeed, j j in May of 2014 Conway and Guttmann gave compelling evidence that Sn.1324/ actually n pn g j j behaves like a function of the form B 1 n for constants B, , 1, and g. In addition, they estimated that 11:60 0:01, 1 0:0398 0:0010, g 1:1 0:2, and D ˙ D ˙ D ˙ B 9:5 1:0. Conway and Guttmann’s methods strongly suggest they have the right D ˙ asymptotic form for Sn.1324/ , but these methods do not seem powerful enough to prove j j a claim like this. Which means that if we want to understand how fast an.1324/ grows, then we should probably try to bound it.

There is a natural lower bound on L.1324/: since Sn.132/ Sn.1324/, we must have L.1324/ 4. Beyond this, little progress was made on the problem of bounding L.1324/ below until 2006, when Michael Albert, Murray Elder, Andrew Rechnitzer, Paul Westcott, and Mike Zabrocki [2] used a finite automaton which accepts only sequences constructing certain elements of Sn.1324/ to show that L.1324/ 9:47. This result was the first nontrivial lower bound on L.1324/, but it also showed conclusively that 1234 is neither the most restrictive nor the least restrictive forbidden pattern of length four, a fact which is at odds with our intuitionabout which permutations are at the extremes of the set of all permutations. We continue to see incremental progress on this problem: in early June of 2014, David Bevan submitted a paper to arXiv [5] in which he uses interleaved trees and Łukasiewicz paths to construct a large class of permutationsin Sn.1324/, thus showing that L.1324/ 9:81. One of the nice features of Marcus and Tardos’s proof of the Stanley-Wilf conjecture, in addition to its simplicity, is that it gives an explicit upper bound on L./ in terms of the length of : if has length k, then the proof tells us that L./ 15a, where a 4 k2 Ä D 2k k . Even better, in 2009 Josef Cibulka improved [10] this general bound significantly, 2 k2 to L./ 2:88 4k8 . Since Ä k k2 k2.k2 1/.k2 2/ .k2 k 1/ C k D k.k 1/.k 2/ 2 1 !

2 k2 is at least a polynomial of degree k in k, the factor k in Cibulka’s bound is at least 2k ln k. Ever since Stanley and Wilf’s ﬁrst conjectures about L./, people had hoped this bound could be improved to some polynomial in k. But Jacob Fox dashed these hopes in the fall of 2013, by showing [15 ] that L./ is exponential in k for almost all permutations of length k. Nevertheless, Cibulka’s work gives us a starting point in our quest to bound L.1324/ in particular: it says L.1324/ 2:50078 1012. However, because this bound Ä is general, it’s not at all sharp, so we turn our attention to ﬁnding tighter upper bounds on L.1324/ in particular. To start to improve our upper bound on L.1324/, recall that we saw earlier how to reconstruct a permutation which avoids 132 from the positions and values of its left-to- right minima. This observation effectively divides the set of all permutations of a given length into classes, each of which contains no more than one element of Sn.132/. Since n 1 n 1 there are at most 4 classes, there are at most 4 elements of Sn.132/. In his thesis Defying God: the Stanley-Wilf Conjecture, Stanley-Wilf Limits, and . . . 79

Bóna expanded on this idea, by proving that each of these classes contains at most 8n elements of Sn.1324/. Thus, L.1324/ 32. n Ä The numerical data regarding Sn.1324/ in Table 2 suggest that our upper bound j j on L.1324/ is worse than our lower bound, but Bóna’s thesis work represented the state p of the art for nearly a decade and a half. A new idea was required, and in 2012 Anders Claesson, V´ıt Jel´ınek, and Einar Steingr´ımsson provided one [11]. To each Sn.1324/, 2 we associate a sequence of colors, red or blue, one for each entry of . To do this, we first color the leftmost entry 1 red. Proceeding from left to right, if we have colored the entries 1; : : : ; j 1, then we color the entry j according to the following rules. 1. If coloring j red would create a red subsequence of type 132, then we color it blue.

2. If one of 1; : : : ; j 1 is blue, and is less than j , then we color j blue. 3. If neither of the first two rules applies, then we color j red. For example, if 749538612 then we would color the entries as in 749538612 (here the D red entries are in bold), obtaining the color sequence RRRBRBBRR. Note that we color the 5 blue because otherwise the 4, the 9, and the 5 would create a red subsequence of type 132. On the other hand, we color both the 8 and the 6 blue for two reasons: otherwise they would be part of a red subsequence of type 132, and they each have a smaller blue entry (namely, the 5) to their left. We can show (go ahead! it’s not hard) that if avoids 1324, then its sequence of red entries avoids 132 and its sequence of blue entries avoids 213. Claesson, Jel´ınek, and Ste- ingr´ımsson use some standard computations with binomial coefficients to show that this implies L.1324/ 16, but we can also follow Bóna [9] to really see where this bound Ä comes from. Having colored each entry of Sn.1324/ red or blue, we now separate 2 the entries of each color into two classes. To do this, first replace each red entry which is a left-to-right minimum in with the letter A, and then replace each remaining red entry with the letter B. Similarly, inspired by the fact that 1324 is its own reverse-complement and 132 and 213 are reverse-complements of each other, replace each blue entry which is a right-to-left maximum with the letter C , and replace each of the other blue entries with the letter D. For example, we’ve seen that if 749538612 then our color sequence is D RRRBRBBRR. Following our new prescription, we find that the associated string of As, Bs, C s, and Ds is AABDACCAB. We have now seen that for each Sn.1324/, we obtain a string w./ of As, Bs, 2 C s, and Ds by listing which of these letters is assigned to the first entry, which to the second, etc. Similarly, we obtain a second such string z./ by listing which of these letters is assigned to the entry 1, which to the entry 2, etc. For instance, if 749538612 then D we’ve seen that w./ AABDACCAB, and we also have z.p/ ABAADCACB. D D Note that our constructions of w./ and z./ amount to an elaboration on our earlier use of the positions and values of the left-to-right minima in a permutation Sn.132/. 2 Bóna’s main result is that while some pairs of strings correspond to no permutation, no pair of strings corresponds to two or more permutations. Since there are 16n pairs of strings of length n, we have L.1324/ 16. In fact, Bóna also shows that none of these strings Ä contains a B which is immediately followed by a C , so some standard computations with linear recurrence relations gives us L.1324/ 7 4p3 13:928, a result which Bóna Ä C has recently improved to L.1324/ 13:73718. Ä 80 A Century of Advancing Mathematics

10 New frontiers: getting closer to God For more information on the state of the permutation pattern art, I would refer the reader to B´ona’s book [8], Kitaev’s monograph [19], and Steingr´ımsson’s survey of open problems [35]. In the meantime, Table 3 summarizes the Stanley-Wilf limits we’ve discussed in this paper, and the reader can ﬁnd more information about these limits in Section 6.1.4

L./ 132 4 1342 8 1324 Œ9:81; 13:73718 123 : : : k .k 1/2 Table 3. The Stanley-Wilf limits in this paper. of Kitaev’s monograph. We still have much to discover about these quantities. In addition, there are several closely related topics that fall outside the scope of this article. For example, while it follows from the Marcus-Tardos theorem (as the Stanley-Wilf conjecture is now known) that for every set R of forbidden patterns there is a constant c.R/ such that n n Sn.R/ c.R/ , it is still not known whether limn Sn.R/ exists for every set R j j Ä !1 j j of forbidden patterns. Nevertheless, this limit does exist for many sets R, and Vatter has p found some amazing structure in the set of real numbers which can appear as one of these limits [38]. Finally, Steingr´ımsson has recently stepped up to challenge Zeilberger’s original claim about the difficulty of finding S1000.1324/ , saying, “I’m not sure how good Zeilberger’s j j God is at math, but I believe that some humans will find this number in the not so distant future.” In fact, in 2013 Steingr´ımsson and Zeilberger made a bet about whether someone will find S1000.1324/ by 2030. If someone finds this value in year n for n 2030, j j Ä then Zeilberger will pay Steingr´ımsson e 10.2030 n/. Otherwise, Steingr´ımsson will pay Zeilberger e 170. Until Zeilberger and Steingr´ımsson settle their bet, we’ll have to let Conway and Guttmann have the last word on the subject. They write, “While making no Messianic claims, our asymptotics permit the approximate answer 4:6 101017.” Acknowledgements Several people gave me helpful and extensive comments, suggestions and corrections on various drafts of this paper, and brought me up to date on recent developments related to Stanley-Wilf limits. In particular, I would like to thank Miklós Bóna, Stephen Kennedy, Darla Kremer, Adam Marcus, Lara Pudwell, Kailee Rubin, Einar Steingr´ımsson, Gábor Tardos, and Doron Zeilberger for their thorough and thoughtful feedback on various drafts of this article. Their comments led to a much-improved final version. Of course, blame for any remaining errors or shortcomings is mine alone.

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Department of Mathematics and Statistics, Carleton College, Northﬁeld, MN 55057 [email protected] What Is the Best Approach to Counting Primes?

Andrew Granville Universit´ede Montr´eal

As long as people have studied mathematics, they have wanted to know how many primes there are. Getting precise answers is a notoriously difficult problem and the first suitable technique, due to Riemann, inspired an enormous amount of great mathematics, the techniques and insights permeating many different fields. In this article we will review some of the best techniques for counting primes, centering our discussion around Riemann’s seminal paper. We will go on to discuss its limitations, and then recent efforts to replace Riemann’s theory with one that is significantly simpler.

1 How many primes are there? Predictions

You have probably seen a proof that there are inﬁnitely many prime numbers, and were perhaps curious as to roughly how many primes there are up to a given point. With the advent of substantial factorization tables,1 it was possible to make predictions supported by lots of data. On December 24th, 1849, Gauss wrote to his “most honored friend,” Encke, describing his own attempt to guess at an approximation as to the number of primes up to x (which we will denote throughout by .x/). Gauss describes his work:

First beginning ...in 1792 or 1793 [when Gauss was 15 or 16] ...I ...directed my attention to the decreasing frequency of prime numbers, to which end I counted them up in several chiliads [blocks of 1000 consecutive integers] and recorded the results ...I soon recognized ...it is nearly inversely proportional to the logarithm, so that the number of all prime numbers under a given boundary x were nearly expressed through the integral x dt log t Z2 where the logarithm is understood to be the natural logarithm.

x dt Gauss went on to compare his guess 2 log t , which we denote by Li.x/, with .x/, the actual count of the number of primes up to x: R 1See Appendix 1.

83 84 A Century of Advancing Mathematics

Under .x/ # primes x Li.x/ Error D f Ä g ˙ 500000 41556 41606:4 50:4 1000000 78501 79627:5 126:5 1500000 114112 114263:1 151:1 2000000 148883 149054:8 171:8 2500000 183016 183245:0 229:0 3000000 216745 216970:6 225:6 Table 1. Primes up to various points and a comparison with Gauss’s prediction.

In his Th´eorie des Nombres, Legendre proposed x log x A with A 1:08366 as a good approximationfor .x/, in which case the comparative errors D are 23:3; 42:2; 68:1; 92:8; 159:1; and 167:6; C C C C C respectively. These are smaller than the errors from Gauss’s Li.x/, though both seem to be excellent approximations. Nevertheless Gauss retained faith in his prediction: These differences are smaller than those with the integral, though they do appear to grow more quickly than [the differences given by the integral] with increasing x, so that it is possible that they could easily surpass the latter, if carried out much farther. Today we have data that goes “much farther”:

x .x/ # primes x Gauss’s error term Legendre’s error term D f Ä g 1020 2220819602560918840 222744644 2981921009910364 1021 21127269486018731928 597394254 27516571651291205 1022 201467286689315906290 1932355208 254562416350667927 1023 1925320391606803968923 7250186216 2360829990934659157 Table 2. Comparing the errors in Gauss’s and Legendre’s predictions.

It is now obvious that Gauss’s prediction is indeed better, that Legendre’s error terms quickly surpass those of Gauss and keep on growing bigger. Table 3 has some of the most recent data and a comparison to Gauss’s guesstimate, Li.x/. When looking at this data, compare the widths of the righttwo columns. The rightmost column is about half the width of the middle column ... How do we interpret that? Well, the width of a column is given by the number of digits of the integer there, which corresponds to the number’s logarithm in base 10. If the log of one number is half that of a second number, then the first number is about the square-root of the first. Thus this data suggests that when we approximate .x/, the number of primes upto x, by Gauss’s guesstimate, Li.x/, the error isaround px, which is really tiny in comparison to the actual number of primes. In other words, Gauss’s prediction is terrific. What Is the Best Approach to Counting Primes? 85

x .x/ # primes x Overcount: ŒLi.x/ .x/ D f Ä g 103 168 10 104 1229 17 105 9592 38 106 78498 130 107 664579 339 108 5761455 754 109 50847534 1701 1010 455052511 3104 1011 4118054813 11588 1012 37607912018 38263 1013 346065536839 108971 1014 3204941750802 314890 1015 29844570422669 1052619 1016 279238341033925 3214632 1017 2623557157654233 7956589 1018 24739954287740860 21949555 1019 234057667276344607 99877775 1020 2220819602560918840 222744644 1021 21127269486018731928 597394254 1022 201467286689315906290 1932355208 1023 1925320391606803968923 7250186216 Table 3. Primes up to various x, and the overcount in Gauss’s prediction.

We still believe that Gauss’s Li.x/ is always that close to .x/. Indeed in section 6, we will sketch how the, as yet unproved, Riemann Hypothesis implies that

.x/ Li.x/ x1=2 log x (RH1) j j Ä for all x 3. This would be an extraordinary thing to prove as there would be many beautiful consequences. For now we will just focus on the much simpler statement that the ratio of .x/ Li.x/ tends to 1 as x . Since Li.x/ is well-approximated by x= logx,2 W ! 1 this quest can be more simply stated as

x lim .x/ exists and equals 1: x log x !1 ı This is known as the Prime Number Theorem, and it took more than a hundred years, and some earth-shaking ideas, to prove it (as we’ll outline in sections 4 to 7 of this article).

2To prove this, try integrating Li.x/ by parts. 86 A Century of Advancing Mathematics

2 Elementary techniques to count the primes

It is not easy to ﬁnd a way to count primes at all accurately. Even proving good upper and lower bounds is challenging. One effective technique to get an upper bound is to try to use the principle of the sieve of Eratosthenes. This is where we “construct” the primes up to x, by removing the multiples of all of the primes px. One starts by removing the multiplies of 2, from a list of all Ä of the integers up to x, then the remaining multiples of 3, then the remaining multiples of 5, etc. Hence once we have removed the multiples of the primes y we have an upper Ä bound:

# p prime y < p x # n x p − n for all primes p y : f W Ä g Ä f Ä W Ä g At the start this works quite well. If y 2 the quantity on the right is 1 x 1, and so D 2 ˙ bounded above by 1 x 1. If y 3 then we remove roughly a third of the remaining 2 C D integers (leaving two-thirds of them) and so the bound improves to 2 1 x 2. For y 5 3 2 C D we have four-ﬁfths of the remaining integers to get the upper bound 4 2 1 x 4. And, in 5 3 2 C general, we obtain an upper bound of no more than

1 1 x 2.y/ 1: p C p y Â Ã YÄ The problem withthisbound is thesecond term ...as onesieves by each consecutive prime, the second term, which comes from a bound on the rounding error, doubles each time and so quickly becomes larger than x (and thus this is a useless upper bound). This formula does allow us, by letting y slowly with x, to prove that ! 1 .x/ lim 0 x x D I !1 that is the primes are a vanishing proportionof the integers up to x, as x gets larger.3 There has been a lot of deep and difficult work on improving our understanding of the sieve of Eratosthenes, but we are stillunable to get a very good upper bound for the number of primes in this way. Moreover we are unable to use the sieve of Eratosthenes (or any other sieve method) to get good lower bounds on the number of primes up to x The first big leap in our ability to give good upper and lower bounds on .x/ came from an extraordinary observation of Chebyshev in 1851. The observation (as reformulated 2n by Erd˝os in 1933) is that the binomial coefficient n is an integer, by definition, and is divisible once by each prime p in .n; 2n, since p is a term in the expansion of the numerator .2n/Š, but not of the denominator, nŠ2. Therefore

2n p : Ä n p prime ! n

3 1 To deduce this we need to know that limy!1 pÄy 1 0, a fact established by Euler. p D Q Á What Is the Best Approach to Counting Primes? 87

Now, by the binomial theorem, 2n 2n 2n .1 1/2n 22n. Moreover for each n Ä j 0 j D C D p .n; 2n we have p > n and so D 2 P 2n n.2n/ .n/ n p 22n: D Ä Ä n Ä p prime n

22n 2n pep 2n .2n/.2n/: 2n 1 Ä n D Ä D C ! p prime p prime pY2n pY2n Ä Ä 2n log 2 log.2n 1/ Taking logarithms gives us the lower bound .2n/ C , and therefore log.2n/ x .x/ .log 2 / log x

4 if x is sufﬁciently large. Hence we have shown that there exist constants c2 > 1 > c1 > 0, such that if x is sufﬁciently large then x x c1 .x/ c2 : (2) log x Ä Ä log x The Prime Number Theorem, that is the conjecture of Gauss and Legendre estimating the number of primes up to x, can be re-phrased as the claim that these inequalities hold for

4 In factour proof shows that we can take any c1 < log 2 and c2 > log 4. 88 A Century of Advancing Mathematics

any constants c1 and c2 satisfying c2 > 1 > c1 > 0; and in particular we can take both c1 and c2 arbitrarily close to 1. Can the method of Chebyshev and Erd˝os be suitably modified to prove the result? In other words, perhaps we can find some other product of factorials that yields an integer, and in which we can track the divisibilityof the large prime factors, so that we obtain constants c1 and c2 in (2) that are closer to 1. We might expect that the closer the constants get to 1, the more complicated the product of factorials, and that has been the case in the efforts that researchers have made to date.5 There is one remarkable identity that gives us hope. First note that in our argument above we might replace 2n by ŒxŠ=Œx=2Š2, where x 2n.6 n D Hence the correct factorials to consider take the shape Œx=nŠ for various integers n. Our remarkable identity is: p Œx=nŠ .n/: (3) D p prime n x eY1 YÄ pe x Ä This needs some explanation. The left-hand side is the product over the primes p x, each kp Ä p repeated kp times, where p is the largest power of p that is x. On theright-handside Ä we have the promised factorials, each to the power 1; 0 or 1. Indeed the Möbius function .n/ is defined as

0 if there exists a prime p for which p2 divides n .n/ I D ( . 1/k if n is squarefree, and has exactly k prime factors The difﬁculty with using the identity (3) to prove the prime number theorem is that the length of the product on the right side grows with x, so there are too many terms to keep track of. One idea is to simply take a ﬁnite truncation of the right-hand side; that is

Œx=nŠ .n/ n N YÄ for some fixed N . The advantage of this is that, once x > N 2, then this product is divisible by every prime p Œx=.N 1/; x to the power 1. The disadvantage is that the product 2 C is often not an integer, though we can correct that by multiplyingthrough by a few smaller factorials. We can handle this, and other difficulties that arise, to obtain (2) with other values of c1 and c2, which each appear to be getting closer and closer to 1. However when we analyze what it will take to prove that the constants (which we now denote by c1.N / and c2.N / since they depend on N ) tend to 1 as N , we find that the issue lies in the ! 1 average of the exponents .n/. In fact one can prove that the constants c1.N / and c2.N / do tend to 1 if 1 lim .n/ exists and equals 0: (5) N N !1 n N XÄ That is, (5) is equivalent to the prime number theorem. The problem in (5) certainly looks more approachable than the prime number theorem itself, even though the problems are equivalent. It can be restated as: There are roughly

5Which is not to say that someone new, not overly influenced by previous, failed attempts, might not come up with a cleverer way to modify the previous approaches. 6Here Œt denotes the largest integer less than or equal to t. What Is the Best Approach to Counting Primes? 89 the same number of squarefree integers with an even number of prime factors, as there are squarefree integers with an odd number of prime factors. This seems very plausible, and leads to many elementary approaches, as we will discuss in section 9, and beyond. One of the most famous old problems about primes was Bertrand’s postulate, to prove that for every n > 1 there is always a prime p for which n < p < 2n. This follows easily 2n from suitable modifications of the above discussion with the binomial coefficient n . This beautiful proof, due to Erd˝os at age 20, announced his arrival onto the mathematical stage and inspired the lines: Chebyshev said it, and I say it again: There is always a prime between n and 2n. Up to now we have proved that .x/ lies between two multiples of x= logx, and we have looked to see whether the ratio of .x/ to x= logx tends to 1, as predicted by Gauss and Legendre. Is any other behavior possible, given what we know already? There are two possibilities: As x grows larger, (i) The ratio of .x/ to x= log x oscillates, never tending to a limit. (ii) The ratio of .x/ to x= log x tends to a limit, but that limit is not 1. Our goal in the rest of this section is to show that option(ii) is not possible. Indeed we will show that if there is a limit then that limit would have to be 1. Yet again the trick is to study factorials both algebraically (by determining their prime factors), and analytically (by analyzing their size). By definition, N log NŠ log n: D n 1 XD The right-hand side is very close to the integral of log t over the same range. To see this note that the logarithm function is monotone increasing, which implies that

n n 1 log t dt < log n < C log t dt n 1 n Z Z for every n 1. Summing these inequalities over all integers n in the range 2 n N Ä Ä (since log1 0), we obtain that for N 2, the value of log NŠ equals D N log t dt Œt.log t 1/n N.log N 1/ 1; D 1 D C Z1 plus an error that is no larger, in absolute value, than log N . On the other hand NŠ is the product of the integers up to N , and we want to know how often each prime divides thisproduct. The integers less than or equal to N that are multiples of a given integer m (which could be a prime or prime power) are m;2m;:::;ŒN=mm, since ŒN=mm is the largest multiple of m that is less than or equal to N , and therefore there are ŒN=m such multiples. Now the power of prime p dividing NŠ is given by the number of integers less than or equal to N that are multiples of p, plus the number of integers less than or equal to N that are multiples of p2, etc., which yields a total of N N N ::: p C p2 C p3 C Ä Ä Ä 90 A Century of Advancing Mathematics

Hence, by studying the prime power divisors of NŠ we deduce that

N N log NŠ log p ::: : D p C p2 C p prime ÂÄ Ä Ã pXN Ä

The total error created by discarding those ŒN=pk terms with k 2, and by replacing each ŒN=p by N=p, adds up to no more than a constant times N . Comparing our two estimates for log NŠ, and dividing through by N we deduce that there exists a constant C for which

log p log N C (7) ˇ p ˇ Ä ˇp N ˇ ˇ XÄ ˇ ˇ ˇ ˇ ˇ for all N 1. ˇ ˇ Now let’s suppose that there exists a constant , such that for any > 0

x x . / .x/ . / ; log x Ä Ä C log x for all sufﬁciently large x (say > x). Our goal is to show that 1. D We will work with the following (easy to verify) identity:

log p log N N log x 1 dx p D N C x2 p N p N ( Zp ) XÄ XÄ log N N log x 1 .N / .x/ dx; D N C x2 Z2 log p inserting our assumed bounds on .x/ to obtain bounds on p N p . The part of the Ä integral with x x is bounded by some constant that only depends on , call it C1./. Ä P Therefore we obtain an upper bound

log p log N N log x 1 C1./ .N / .x/ dx p Ä C N C x2 p N Zx XÄ N log N N x log x 1 C1./ . / . / dx Ä C C log N N C C log x x2 Zx N dx C2./ . / . / log N C2./; Ä C C x Ä C C Zx for some constant C2./, that only depends on . This implies that 1 else we let .1 /=2 and this inequality contradicts (7) for N sufﬁciently large. An analogous D proof with the lower bound implies that 1. We deduce that if .x/ x tends to a Ä log x limit as x then that limit must be 1. ! 1 ı What Is the Best Approach to Counting Primes? 91

3 A first reformulation: Introducing appropriate weights

So far, we have counted primes by estimating the size of the product of the primes in some interval. Taking logs, this means that we bounded

log p: p prime pXx Ä We denote this by .x/; and we also deﬁne its close cousin,7

.x/ log p: D W p prime mX1 pm x Ä We will see that when we do calculations, these functions seem to be more natural than .x/ itself. This ﬁts rather well with Gauss’s original musings in his letter to Encke. The key phrase is:

I soon recognized, that under all variations of this frequency [of prime numbers], on average, it is nearly inversely proportional to the logarithm.

We re-word this as “The density of primes at around x is 1= log x.” Then we would expect that the number of primes, each weighted with log p (that is, the sum .x/) should be well-approximated by

x dt x log t dt x 2: log t D D Z2 Z2 Occam’s razor tells us that, given two choices, one should opt for the more elegant one. There can be little question that x is a more pleasant function to work with than the complicated integral Li.x/, and so we will develop the theory with logarithmic weights, and therefore use the function .x/ rather than .x/.8 We believe that

log p x x1=2.log x/2; (RH2) ˇ ˇ Ä ˇp x ˇ ˇXÄ ˇ ˇ ˇ since thisis equivalent to our conjectureˇ (RH1)ˇ on .x/ Li.x/. The prime number theorem is equivalent to the much weaker assertion that

1 lim log p exists and equals 1: x x !1 p x XÄ

7The reader might verify that .x/ and .x/ do not differ by more than a boundedmultiple of px. 8Using partial summation, it is not difﬁcult to show that a good estimate for one is equivalent to an analogous estimate for the other, so there is no harm done in focusing on .x/. 92 A Century of Advancing Mathematics

4 Riemann’s memoir In a nine-page memoir written in 1859, Riemann outlined an extraordinary plan to attack the elementary question of counting prime numbers using deep ideas from the theory of complex functions. His approach begins with what we now call the Riemann zeta-function: 1 .s/ : WD ns n 1 X To make sense of an inﬁnite sum it needs to converge, and preferably be absolutely convergent, meaning that we can rearrange the order of the terms without changing the value.9 The sum deﬁning .s/ is absolutely convergent only when Re.s/ > 1. This is especially interesting when we apply the Fundamental Theorem of Arithmetic to each term in the sum: Every integer n 1 can be factored in a unique way and, vice-versa, every product of primes yields a unique positive integer. Then we can write

n 2n2 3n3 :::; D where each nj is a nonnegative integer, and only ﬁnitely many of them are nonzero. Hence

1 .s/ D .2n2 3n3 5n5 :::/s n2;n3 ;n5 ;::: 0 X 1 1 1 1 : D 0 .pnp /s 1 D ps p prime np 0 p prime Â Ã Y X Y @ A This product over primes is an Euler product — indeed, it was Euler who ﬁrst seriously explored the connection between .s/ and the distributionof prime numbers, though he did not penetrate the subject as deeply as Riemann. The Euler product provides a connection between .s/ and prime numbers, and this was exploited by Riemann in an interesting way. Since the sum deﬁning .s/ is absolutely convergent when Re.s/ > 1, it is safe toperform calculus operations on .s/ in thisdomain. By taking the logarithmic derivative we have

.s/ d d 1 0 log .s/ log 1 .s/ D ds D ds ps p prime X Â Ã using the Euler product. Using the chain rule, we then obtain

0.s/ log p log p .s/ D ps 1 D pms I p prime p prime X mX1 and notice that the sum of the coefﬁcients of 1=ns on the right side, for n up to x, equals

m log p .x/. As we remarked above, this is a close cousin of .x/, and we have p x D now observed,Ä like Riemann, that it arises naturally in this context. P 9Riemann proved that if one has a convergentbut not absolutely convergentsum then one might get different limits if one rearrangesthe order of the terms in the sum. What Is the Best Approach to Counting Primes? 93

5 Contour integration One of the great discoveries of nineteenth-century mathematics is that itis possible to con- vert problems of a discrete ﬂavor, in number theory and combinatorics, into questions of complex analysis. The key lies in ﬁnding suitable analytic identities to describe combinatorial issues. For example, if we ask whether two integers, a and b, are equal, that is “does a b?”, then this is equivalent to asking “is a b 0?” and therefore we need some D D analytic device that will distinguish 0 from all other integers. This is given by the integral of the exponential function around the circle: 1 1 if n 0; e2i nt dt D 0 D 0 otherwise: Z ( So, for example, if we want to determine thenumber of pairs p; q of primes n, which add Ä to give the even integer n, we create an integral that gives 1 if p q n, and 0 otherwise, C D and then sum over all such p and q. Therefore we have 1 # p; q n p; q primes, p q n e2i.p q n/t dt f Ä W C D g D C p;q n Z0 p;qXprimesÄ and this can be rearranged as 2 1 2i nt 2ipt e 0 e 1 dt: Z0 p n Bp XprimeÄ C B C This is of course an approach to Goldbach’s@ conjecture (thatA every even integer greater than or equal to 4 is the sum of two primes), and it is (arguably) surprising that understanding this integral is equivalent to the original combinatorial number theory question. So we have seen how to analytically identify when two integers are equal, and why that is useful. Next we will show how to analytically verify a proposed inequality between two real numbers. As you might have guessed, we start by noting that asking whether u < v is the same as asking whether v u > 0, and so we restrict our attention to determining whether a given real number is less than 0, equal to 0, or greater than 0 (though we are less interested in the middle case). Here the trick is that for any > 0 we have

0 y < 0 y. it/ if 1 1 e I C dt 81=2 if y 0 2 it D ˆ D I Z1 C <ˆ1 if y > 0; which is Perron’s formula. It is convenient to writeˆs for it and, instead of the limits of : C the integral, we write “Re.s/ ,” understanding that we take s along the line Re.s/ , D D that is, s it as t runs from to . Hence we integrate eys=s. Moreover if we let D C 1 1 z ey, the formula can be rephrased as D 0 if 0 < z < 1 1 zs I ds 81=2 if z 1 2i Re.s/ s D ˆ D I Z D <ˆ1 if z > 1; ˆ :ˆ 94 A Century of Advancing Mathematics for any > 0. In number theory, the most common use of Perron’s formula is to identify when an integer n is < x, that is when x=n > 1.

We are interested in estimating .x/ m log p. We extend the sum to all prime D p x powers, multiplying by 1 if pm x, and by 0Äotherwise, which we achieve by using mÄ P Perron’s formula with z x=p . The outcome is .x/ which has the same value as D 1 .x/ except that we subtract 2 log x if x is a prime power. Therefore we have 1 .x=pm/s .x/ log p ds; D 2i Re.s/ s p prime Z D mX1 for any > 0. Now we would like to swap the order of the summation and the integral, but there are convergence issues. Fortunately these are easily dealt with when the sum is absolutely convergent, as happens when > 1. Then we have, after a little re-arrangement,

1 log p xs .x/ ms ds D 2i Re.s/ p s Z D p prime mX1 1 .s/ xs 0 ds: (11) D 2i Re.s/ .s/ s Z D This seems, at first sight, to be a rather strange thing to do. We have gone from a perfectly understandable question like estimating .x/, involving a sum that is easily interpreted, to a rather complicated integral, over an infinitelylong line in the complex plane, of a function that is delicateto workwithin thatit is onlywell-defined when Re.s/ > 1.Itisbynomeans obvious how to proceed from here, as we will discuss in more detail in the next section.

The proofof (11)did notusemany properties of 0.s/=.s/. In fact if an is any sequence s of real numbers with each an 1 then deﬁne the Dirichlet series A.s/ an=n , j j Ä D n 1 to obtain 1 xs P a A.s/ ds: n 2i s n x D Re.s/ XÄ Z D 6 Riemann’s genius How do we evaluate the integral in (11)? In complex integrationtheidea is to shiftthe path of the integral to one on which the integrand is “very small.” Then the value of the integral is given by the sum of the residues of the integrand at its poles. There is a lot to explain here — indeed the main points of a ﬁrst course in complex analysis. Rather than get into all of these details, let me just say that the poles are the points where the function goes to , like the point s 1 for the function 1=.s 1/. And if the function f .s/ has a pole at, 1 D say, s 1, whereas .s 1/f .s/ equals r C at s 1, then we say that f .s/ has a simple D 2 D pole at s 1 with residue r. D So what new path should we take from i to i to be able to apply this 1 C 1 strategy to the integral in (11)? If we are going to choose the same path for each value of x, then we want to ensure that the integrand (on the path) does not grow large with x. Now xs xRe(s) x , so the smaller is, the better. In fact if we make negative then the j j D D xs in the integrand will ensure that the integral over this line gets smaller as x . Or, ! 1 What Is the Best Approach to Counting Primes? 95

rather, that would be true if .s/ and .0=/.s/ are defined in this region, which they are not, for now. Under the right conditions,functions that are naturally defined only in part of the complex plane (like .s/), can be re-defined so that the function can be appropriately extended to the rest of the complex plane. This is called an analytic continuation, which involves what can be a deep and subtle theory. In such circumstances, one can express the function in terms of a Taylor series,10 or a Laurent series if there is a pole. Analytic continuations are a little bit mysterious — for example the theory allows for more than one apparently different way that one can define the function on the rest of the complex plane, but it will turn out that any two such definitionswill have equal values everywhere they are bothdefined.11 Anyway, we can analytically continue .s/ to all of the complex plane, except for its pole at s 1, and to do this, Riemann discovered some remarkable properties of .s/ (which D we do not pursue here). There are several subtleties involved in bounding the contribution of the integrand on the new contours, and Riemann succeeded in doing that. Finally one needs to find the poles of .s/ xs 0 ; .s/ s and to compute their residues: Evidently xs has no poles in the complex plane, and 1=s has a simple pole at s 0, which contributes a residue of D .s/ xs .s/ .0/ .0/ lim s 0 lim 0 xs 0 x0 0 s 0 .s/ s D s 0 .s/ D .0/ D .0/ ! ! to the value of the integral. The poles of 0.s/=.s/ are the poles and zeros of .s/. The only pole of .s/ is at s 1, and so this contributes the residue D .s/ xs 1 x1 lim .s 1/ 0 lim .s 1/ x; s 1 .s/ s D s 1 .s 1/ 1 D ! ! Â Ã the expected main term, to the value of the integral (since .0/ 0). The Euler product ¤ representation of .s/ converges in Re.s/ > 1, so there can be no zeros of .s/ in this half-plane. Otherwise the zeros of .s/ are rather mysterious. All we can really say is that if .s/ looks like c.s /m, near to the zero , for some integer m 1 (and non-zero constant c) then .s/=.s/ looks like m=.s /, and therefore the residue at s is 0 D .s/ xs m x x lim .s / 0 lim .s / m : s .s/ s D s .s / D ! ! Â Ã If we count such a zero of multiplicity m, m times in the sum, then we have evaluated the integral so as to yield Riemann’s remarkable explicit formula:

x .0/ .x/ x 0 : D .0/ ./ 0 W XD

10 That is, for f .s/ at s0 C, there exist constants a0; a1;::: and some constant r, such that if s s0 < r 2 2 j j then a0 a1.s s0/ a2.s s0/ is absolutely convergent, and converges to f .s/. 11ThisC allows us to giveC the analytic continuationC the same name as the original function, since we know that it can only be analytically continued in one way, if at all. 96 A Century of Advancing Mathematics

The left-hand side is a step function, with a jump at each prime power, whereas the right- hand side isthe sum of infinitelymany smooth functions.Somehow these smooth functions, which correspond to the zeros of .s/, conspire to stay constant as x varies, other than to jump at exactly the prime powers. Surprising, it may be, but is it useful? We have gone from a simple question like counting the number of primes up to x, to a sum over all of the zeros of the analytic continuationof .s/. Ever since Riemann’s memoir, mathematical researchers have struggled to find a way to fullyunderstand the zeros of .s/, so as to make this “explicit formula” useful. We have had some, rather limited, success. Riemann showed that there are infinitely many zeros of .s/ so we have a problem in that the sum over , in Riemann’s explicit formula, is an infinite sum and one can easily show that it is not absolutely convergent. So to evaluate it directly, we would need to detect cancellation amongst the summands, something that we are not very skilled at. Instead, one can modify Riemann’s argument to show that one can truncate the sum, taking only those in the box up to height T ,

B.T / C 0 Re./ 1; T Im./ T ; WD f 2 W Ä Ä Ä Ä g in our sum. This turns out to be a ﬁnite set, and we get the explicit formula,

x .x/ x a small error; D C ./ 0 W XB.TD / 2 where the “small error” is small if T is appropriately chosen (as a function of x; typically T px). Then we can bound .x/ x , by taking absolute values in the sum over zeros D j j , above. Again the key issue is that x xRe./ so that j j D

x x xRe./ 1 ˇ ˇ xm.T / ˇ ˇ Ä D Ä ˇ ./ 0 ˇ ./ 0 ˇ ˇ ./ 0 j j ./ 0 j j ˇ W XD ˇ W XD ˇ ˇ W XD W XD ˇ B.T / ˇ B.T / ˇ ˇ B.T / B.T / ˇ 2 ˇ 2 ˇ ˇ 2 2 ˇ ˇ where ˇ ˇ m.T / max Re./: WD B.T / 2 The sum over zeros can be shown to be bounded by a multiple of .log T/2, so if we can get a good bound on m.T / then we will be able to deduce the prime number theorem. By “good bound” here we mean that m.T / must be somewhat less than 1, in fact

3 log log T m.T / 1 Ä log T will do. Riemann made a few calculations of the zeros of .s/ and all the real parts seemed to be 1=2.This led tohim to:12

12 1 Riemann actually wrote: “It is very probablethat all rootsare [on the 2 -line]. Certainly one would wish for a stricter proof here; I have meanwhile temporarily put aside the search for this after some ﬂeeting futile attempts, as it appears unnecessaryfor the next objective of my investigation.” What Is the Best Approach to Counting Primes? 97

The Riemann Hypothesis If ./ 0 with 0 Re./ 1, then Re./ 1 . D Ä Ä D 2 Even though the Riemann Hypothesis remains unproven today, more than 150 years after Riemann’s article, most mathematicians believe that it is true. There have been exten- 1 sive calculations, provingthat the first ten billionzeros above the real axes lie on the 2 -line. More persuasive is that it fits so well with so many other ideas that the world would be a 1 much uglier place if it is not true. The reason that we are drawn to the 2 -line is Riemann’s remarkable functional equation which shows that .s/ can easily be determined in terms of .1 s/; in particular once we understand .s/ for Re.s/ 1 then we understand it on the 2 whole complex plane. If the Riemann Hypothesis is true then each x has absolute value x1=2 or is less than 1, and one can deduce, via the argument that we have just sketched, the estimates (RH1) and (RH2) for .x/ and .x/. Actually there is a very intimate link between upper bounds for the real parts of the zeros of .s/ and boundson the error term in the prime number theorem, and one can show that if either (RH1) or (RH2) is true then the Riemann Hypothesis follows. This connection goes much further. For example, fix 1 > ˇ > 1=2. Then all zeros of .s/ satisfy

ˇ Re.s/ < ˇ if and only if log p x Cˇ x ; ˇ ˇ Ä ˇp x ˇ ˇXÄ ˇ ˇ ˇ for some constant Cˇ > 0. How strange! Hereˇ we are in twoˇ different worlds, counting primes, and zeros of the analytic continuation of a function, and yet a key part of understanding each is equivalent. This is the bedrock on which mathematics is formed. Surpris- ing connections between fields that have no obvious right to be related, and yet they are, at some fundamental level. Riemann’s work gave one of the first results of this type, and now every field of research in pure mathematics is full of such links. Riemann’s connection is not restricted to this one question. Indeed, using the explicit formula, one can reformulate many different problems about primes, as problems about zeros of zeta-functions, upon which we can use the tools of analysis. Mathematicians love bringingfields together that seem so distant,hopefullyallowing a more rounded perspective of both. These observations are so seductive that they have been the thrust of almost all research into the distribution of prime numbers ever since. Moreover there are many other good questions about prime numbers, number fields, finite fields, curves and varieties that can be re-cast in terms of appropriate zeta-functions, so there is no end to what can be investigated by such methods.

7 The coup de grâce in the proof of the prime number theorem What we have sketched above is not quite the end of the story of the proof of the prime number theorem. Although Riemann came up with the whole idea, and made many spectacular advances in his short memoir, he could not give an unconditional proof. He left several steps to be completed. These turned out to be very difﬁcult indeed, and it was only 37 years later that Hadamard and de le Vall´ee Poussin did so, independently, in 1896. The 98 A Century of Advancing Mathematics

final step that was left to them, was to show that .s/ has no zeros on the line Re.s/ 1, D which we call the 1-line from now on. Both of their proofs, and most that followed, show that .s/ cannot have a zero at 1 it by showing: C If .1 it/ 0 then .1 2it/ (13) C D C D 1I that is, .s/ has a pole at 1 2it. However we have already noted that .s/ only has a C pole at s 1, and therefore t 0. This yields a contradiction to the assumption that D D .1 it/ 0. The proofs of Hadamard and de le Vallée Poussin are complicated, and C D the proof of Mertens that can be found in every textbook is relatively easy without being enlightening. Nonetheless it is not difficult to get an intuitive feel for why (13) should be true: Since .s/ is an analytic function, if it equals 0 at 1 it, then it must be well- C approximated by the leading term in its Taylor series, c.s .1 it//r , when s is sufficiently C close to 1 it, for some integer r 1 and some non-zero constant c. For example, C 1 if s 1 it then .s/ c=.log x/r ; D C C log x which is pretty small.13 Since Re.s/>1 we can determine .s/ in terms of its Euler product, and one can approximate .s/ well at s 1 it 1 by truncating the Euler product at x. In other D C C log x words 1 1 .s/ 1 : p1 it p prime Â C Ã pYx Ä Now the pth term in this Euler product has absolute value

1 1 1 1 1 1 ; 1 it (17) p C C p ˇ ˇ and one can deduce from (7) thatˇ ˇ

1 1 c 1 0 ; C p log x p prime Â Ã pYx Ä for some constant c . This implies the lower bound .s/ c = logx. 0 j j 00 Comparing these two estimates for .s/ allows us to deduce that j j c=.log x/r c = log x 000 for all sufﬁciently large x. Hence r 1, but we know that r is an integer greater than or Ä equal to 1, and so r 1. Therefore (17) must be close to equality, most of the time; that is D 1=p1 it 1=p, and therefore C pit 1 for “most primes p,” a concept we will make precise in section 9. In fact we will deduce this directly from (5) by more elementary methods.

13Throughoutwe will use the notation “ ” to mean “is approximately equal to.” Typically I’ll avoid being too precise as this can introduce uninteresting yet substantial technicalities. What Is the Best Approach to Counting Primes? 99

Squaring pit 1, one sees that p2it . 1/2 1 for most primes p, which tells us D that if s 1 2it 1 then D C C log x 1 1 1 1 .s/ 1 1 c log x; p1 2it p 00 p primeÂ C Ã p primeÂ Ã pYx pYx Ä Ä and hence, letting x , we deduce that .s/ has a pole at s 1 2it. ! 1 D C This is not the only way to show that we cannot have pit 1 for most primes p. My favorite technique is to take logarithms and to show that if pit 1 for most primes p then these primes are clustered in intervals of the form

Œ.1 /e.2k 1/= t ; .1 /e.2k 1/= t C j j C C j j where k is an integer, and is very small. One can then use sieve techniques (the direct descendants of the sieve of Eratosthenes, speciﬁcally the Brun-Titchmarsh Theorem) to show that primes cannot be clustered into intervals at more than double the expected density, and thus we obtain a contradiction. Later we will study (5) to show that “pit 1 .p/ for most primes p” is impos- D sible.

8 Selberg’s elementary approach The explicit formula which directly relates the primes to the zeros of .s/ suggests a tautology between primes and zeros. This persuaded no lesser authorities than Hardy, Ingham and Bohr to assert that it would be impossible to ﬁnd an elementary proof of the prime number theorem. After all, how could it be possible? The prime number theorem implies restrictions on the zeros of the analytic continuationof .s/ — how could one have a proof of that which does not use analysis? As Hardy said in Copenhagen in 1921: No elementary proof of the prime number theorem is known, and one may ask whether it is reasonable to expect one. Now we know that the theorem is roughly equivalent to ... the theorem that Riemann’s zeta function has no roots on a certain line. A proofof such a theorem, not fundamentally dependent on the theory of functions, seems to me extraordinarily unlikely. It is rash to assert that a mathematical theorem cannot be proved in a particular way; but one thing seems quite clear. We have certain views about the logic of the theory; we think that some theorems ... ‘lie deep’ and others nearer to the surface. If anyone produces an elementary proof of the prime number theorem, [s]he will show that these views are wrong, that the subject does not hang together in the way we have supposed, and that it is time for the books to be cast aside and for the theory to be rewritten. And, as Ingham wrote in the introductionof his 1932 book [20]: Every known proof of the prime number theorem is based on a certain property of the complex zeros of .s/, and this conversely is a simple consequence of the prime number theorem itself. It seems therefore clear that this property must be used (explicitly or implicitly) in any proof based on .s/, and it is not easy to see how this is to be done if we take account only of real values of s. 100 A Century of Advancing Mathematics

The key to Selberg’s elementary approach14 is Selberg’s formula:

log x log p log p log q 2x log x Error: (19) p x C pq x D C pXprimeÄ p; q XbothÄ prime

Here the “Error” term is bounded by a multiple of x. So instead of getting an accurate estimate for the weighted number of primes up to x, Selberg gets an accurate estimate for the weighted number of primes and P2’s up to x (where a “P2” is an integer that is the product of two primes). Moreover Selberg [30] gave an elementary proof that (19) is true using combinatorial methods; and it is temptingto believe that it shouldnot then be difﬁcult to remove the P2’s from the equation. But ﬁrst we ask, how can a formula like (19) hold without any hint of the zeros of .s/? Selberg does not indicate how he came up with such a formula, and why he would have guessed that it would be true, so we can only speculate. Selberg was a master analyst, so it is plausible that he reasoned as follows: The main problem in using Riemann’s formula is that if a zero, , of .s/ has real part equal to 1 (which is not easy to disprove), then the corresponding error term, x=, has size cx for some non-zero constant c, a positive fraction of the main term. So, can we come up with a formula, for a quantity similar to the primes, where one such “bad zero” cannot have such a damning effect? One way to approach this is to try to produce an integrand that is similar to the one that Riemann worked with, but for which there is a double pole at s 1, and no new D higher order poles elsewhere. The simple way to get a double pole is from the function

00.s/=.s/. This also has the feature that if is a simple zero of .s/ then it is a simple pole of 00.s/=.s/. This is not the case with double or higher order zeros, but we expect them to be rare. Hence if we consider the integral

1 .s/ xs 00 ds 2i Re.s/ .s/ s Z D then we have the double pole at s 1. One can compute the residue, using the Taylor D expansion, at s 1, to be 2x.logx 1 /. If is a simple zero of .s/ then its residue D is cx , for some constant c, and witha bit of luck, all of these on the 1-line will sum up to no more than a constant times x. In other words, one might guess that the above integral should equal 2x log x plus an error which is bounded by at most some multiple of x, even if there are zeros on the 1-line. Evaluating the integral is tricky in its current form, but once we note that 2 .s/ .s/ 0 .s/ 00 0 0 ; .s/ D .s/ C .s/ Â Ã Â Ã we can rewrite the integral as

s 2 s 1 .s/ 0 x 1 .s/ x 0 ds 0 ds 2i Re.s/ .s/ s C 2i Re.s/ .s/ s Z D Â Ã Z D Â Ã

14I asked Selberg, in around 1989, how he would deﬁne “elementary.” He responded that there is no good deﬁnition, but that it is perhaps best expressedas “what a good high school student could follow.” What Is the Best Approach to Counting Primes? 101 which, by Perron’s formula, equals

m.log p/2 logp log q: C pm x pq x pXprimeÄ p;qXprimesÄ m 1 It is easy to show that the prime powers do not contribute much to the ﬁrst sum, and that log p is close to logx for most of the primes p counted in the sum. Hence we get the left-hand side of (19). This is perhaps why Selberg believed that something like (19) holds, and why it should be accessible to an elementary proof. However to make this argument elementary required substantial ingenuity (see [30]). How can we deduce the prime number theorem from (19)? The ﬁrst thing to do is to recast this in terms of the function .x/ or, even better, the error term E.x/ .x/ x. WD Using (7) and (19) we obtain

E.x/ log x E.x=p/ log p Error (23) C p x D XÄ where the Error is bounded by a multiple of x. Dividing through by x log x we obtain

E.x/ 1 E.x=p/ log p Error; x x x=p p D log p x C XÄ where the Error 0 as x . It is a little difﬁcult to appreciate what this tells us. The ! ! 1 right-hand side can be viewed as 1 times the suitably weighted average of E.t/=t for t x=2 (use (7) to see that this really is a weighted average). But this says that E.x/=x Ä is minus the average of E.t/=t, which is only consistent if that average is 0, and therefore, we would hope to deduce that E.x/=x 0 as x , as desired. One can make this ! ! 1 deduction if one can prove that E.x/=x does not change value quickly as x varies, which is not straightforward. On the other hand, this argument is easily adapted to deduce that

E.x/ E.x/ lim inf lim sup : x x D x x !1 !1 This is as far as Selberg had gone using (19) when Erd˝os heard about Selberg’s formula and started to workfrom it. Indeed both Erd˝os and Selberg went on to deduce the prime number theorem using entirely elementary methods.15 We will not describe their proofs here since we will now take these ideas in a different direction.

15An unfortunate and unpleasant controversy arose as to who deserved credit for this ﬁrst elementary proof of the prime number theorem. The establishment (as represented by the opinions of Weyl) judged that Erd˝os had “muscled in” on Selberg’s breakthrough, that Selberg would have found the route to the elementary proof in time by himself. However Goldfeld [10] provides an account of the controversyin which one cannot help but be sympathetic to Erd˝os. To my mind, the controversy reﬂects two different perspectives on what is appropriate when one hears about the latest research of others, and what is not. Moreover what is appropriate changes over time and I do not think anyonewould have questioned Erd˝os’s behaviour today, nor would have been so unkind as Weyl. (See also [1] and [12].) 102 A Century of Advancing Mathematics

9 Mean values of multiplicative functions We explained in section 2 that the prime number theorem is equivalent to the statement that the mean value of .n/, the Möbius function, for n up to N , tends to 0 as N (which ! 1 is formulated in (5)). The beauty of reformulating the prime number theorem like this is that .n/ is a multiplicativefunction, and this opens up many possibilities.A multiplicative function f is one for which f .mn/ f .m/f .n/ whenever m and n are coprime integers. D Other important examples include nit , for fixed t R; 2 .n/, where is a Dirichlet character, which comes up when one studies arithmetic progressions; .n/, which counts the number of divisors of n; .n/, the sum-of-divisors function, which arises when studying perfect numbers; etc. In all these cases we might ask for the function’s mean value as we take the average up to infinity; that is 1 lim f .n/: N N !1 n N XÄ One should ask first whether this limitexists and, if so, whether we can determine its value. And, more importantly for the prime number theorem, can we come up with a simple classification of those multiplicative functions which have mean value 0? In fact the mean value, up to N , of .n/ tends to 0 as N , of .n/ tends to ! 1 log N , and of .n/ tends to cN for some non-zero constant c. The most interesting of our examples is nit with t 0, since its mean value is ¤ 1 N 1 N N it nit uit dt : N N D 1 it n 1 Z0 XD C That is, the mean value does not tend to a limit as N , but rather rotates steadily ! 1 around a circle of radius 1=p1 t 2. We see here that the period works on a logarithmic C 2= t scale, that is, we get roughly the same mean value for N and Ne j j. In 1971 Halász resolved the key issue of determining which multiplicative functions do not have mean value tending to 0. Restricting attention to multiplicative functions f for which f .n/ 1 for all n, there is the obvious example 1, or any example much like j j Ä 1 (e.g., in which we perturb the value at each prime by just a small amount). There is the generalization nit , for any real number t (as we have just seen), and any small perturbations of that. Halász proved that these are essentially all the examples: The only multiplicative functions whose mean value do not tend to 0 are ones that look a lot like nit for some real number t, that is, pretend to be nit . His proof involves Dirichlet series to the right of 1 and Parseval’s identity, but never uses analytic continuation. We now apply this to (5). If .n/ does not have mean value 0, then Halász’s theorem tells us that .n/ must pretend to be nit for some real number t. Hence .n/2 must pretend to be n2it , which implies that t 0, and hence .n/ pretends to be 1, a contradiction. D Formulating what “pretends” means takes a little bit of doing: If f and g are multiplicative functions with absolute value 1, then f pretends to be g (meaning that they are not too What Is the Best Approach to Counting Primes? 103 different, at least in an appropriate average sense) if and only if f g pretends to be 1. Since the values of a multiplicativefunction only depend on its values at primes and prime powers, we can restrict our attention to these. Now if h pretends to be 1 then we might measure that, by seeing how small 1 h.p/ is, averaged in some way over the primes, or j j even 1 Re h.p/, which turns out to be more natural (because of (31), below). Thus we define the distance, D.f; g x/, between f and g for n x, by: I Ä 1 Re .f .p/g.p// D.f; g x/2 : p I D p x XÄ We say that f is g-pretentious if D.f; g x/ is less than some small constant. This allows I us to formalize what we meant earlier (e.g., after (17)) when we wrote “pit 1 for most primes p” — now we simply writethat D.; nit x/ is bounded. I D.f; g x/ is not trulya distance; for example, it is only 0 if f g and, also, f .n/ I D j j D 1 for all n. But for us it is more important that our notion of distance satisfies the triangle inequality D.f; h x/ D.f; g x/ D.g; h x/: (29) I Ä I C I In particular we deduce that D.f 2; 1 x/ 2 D.f; x/, and from this we easily deduce I Ä I the prime number theorem. For an elegant proof of this triangle inequality, see Appendix 2. There is a direct connection between D.f; nit x/ and the Dirichlet series F.s/ s 1 I WD n 1 f .n/=n : If 1 log x then D C P F. it/ log x exp D.f; nit x/2 : (31) j C j I where the symbol “ ” means that the ratio of the two sides is bounded, above and below, by positive constants. Halász’s Theorem gives an upper bound for the mean value of f in terms of the minimum of D.f; nit x/ as we range over t in some box, t T , where T I j j Ä is a power of log x (that is, the minimum occurs at that t, with t T , for which nit is j j Ä “closest” to f .n/). From (31) this t can also be thoughtof as the value at which F. it/ j C j is largest (up to a constant) out of those t for which t T . j j Ä Besides the distance function, another key tool in working with mean values of multiplicative functions is a generalization of the argument we used in obtaining (7). Now we are interested in S.x/ n x f .n/. The trick is to evaluate WD Ä P f .n/ log n (37) n x XÄ in two ways, one analytic, the other algebraic. First analytically, note that (37) equals x S.t/ S.x/ log x dt: t Z1 Since S.t/ f .n/ t, the integral here is x 1dt x, so the value is j j Ä n t j j Ä Ä 1 D S.x/ log x plus an errorÄ bounded by x, in absolute value. The second way to evaluate (37) P R involves again writing log n as the sum of the logarithms of its prime and prime power divisors. As before we can bound the contribution of the prime power divisors, and so we are left with the “identity” S.x/ log x f .p/ log p S.x=p/ Error (41) D p x C XÄ 104 A Century of Advancing Mathematics where the Error is bounded by a multiple of x. If we look at the special case f and D write M.x/ n x .n/ then we obtain D Ä P M.x/ log x M.x=p/ log p Error C p x D XÄ where, analogously, the Error is bounded by a multiple of x. Notice that this is exactly the same functional equation as (23), with M.:/ replaced by E.:/ (which was the error term in the prime number theorem). This was a lot easier to derive than (23), and from here we can also prove that M.x/=x 0 as x , much like Erd˝os and Selberg deduced that ! ! 1 E.x/=x 0 as x from (23). This yields another “elementary” proof of the prime ! ! 1 number theorem.

10 What else do we count about primes? If one writes down the primes, it soon appears as if there are roughly equal numbers that end in a 1; 3; 7 or 9; in other words, in each residue class a .mod 10/ where .a; 10/ 1. D And it appears that, in general, for each fixed q, there should be roughly equal numbers of primes in each arithmetic progression a .mod q/ for any positive integer a with .a; q/ 1. D However, even proving that there are infinitely many primes in any such arithmetic progression is a rather tough challenge. It was only in 1837 that Dirichlet did so, showing that the primes are equidistributed16 in the arithmetic progressions if one weights them with a 1=p. In doing this, Dirichlet invented a generalization of the Riemann-zeta function, called a Dirichlet L-function:17 Dirichlet characters, .mod q/, are multiplicative functions Z C that are periodic with minimal period q, and .n/ 0 if .n; q/ > 1. The W ! D most interesting are the real characters (i.e. those characters that can only take the values 1; 1 and 0, and do, in fact, take each of those values) like the Legendre symbol : . For q each Dirichlet character we create the Dirichlet L-function Á .n/ L.s; / ; WD ns n 1 X which is absolutely convergent when Re.s/ > 1. This can be analytically continued to the whole complex plane with no poles. Dirichlet’s proof is an elegant piece of combinatorics which easily leads to his theorem that there are infinitelymany primes a .mod q/ Á whenever .a; q/ 1 provided one can prove that D L.1; / 0 for all real characters .mod q/: ¤ This is a lot harder to prove than one might guess, even though there are many different proofs. The most interesting proof, due to Dirichlet himself, shows that L.1; / can be determined as a simple multiple of the size of a certain group that comes up in algebra.

16By “equidistributed” we mean that there are roughly the same number of primes, up to x, in each arithmetic progression a .mod q/ with .a; q/ 1. 17The astute reader might ask howD Dirichlet could “generalize” the Riemann-zeta function, 22 years before Riemann’spaper! The fact is that .s/ was consideredat lengthby Euler aboutone hundredyearsbefore Dirichlet; it was later named after Riemann, in honor of his trailblazing work. What Is the Best Approach to Counting Primes? 105

This Dirichlet’s class number formula was the first deep connection found between algebra and analysis and is the pre-cursor of so many of the great theorems and conjectures of the last thirty years in number theory.18 The proof of the prime number theorem was soon modified, using Dirichlet series, to show that, whenever .a; q/ 1 and .b; q/ 1, D D # p x p prime, and p a .mod q/ lim f Ä W Á g exists and equals 1 x # p x p prime, and p b .mod q/ I !1 f Ä W Á g that is, the primes are equidistributed amongst the plausible arithmetic progressions mod q. This is called the prime number theorem for arithmetic progressions mod q. Approaching this with the Möbius function, one can show that the prime number theorem for arithmetic progressions holds mod q if and only if

1 lim .n/ .n/ 0 N N D !1 n N XÄ for every Dirichlet character .mod q/. As in the classical proof, this is easily proved using Hal´asz’s Theorem when takes complex values (since then .p/.p/ is often quite far from the real line). If is a real character it also follows immediately from Hal´asz’s Theorem provided L.1; / 0. So the “pretentiousproof” hinges on the same issue as the ¤ classical proof.

11 Is a pretentious proof an elementary proof? The prime number theorem can be phrased as: For all > 0, we have .x/ x < x j j once x is sufﬁciently large. However we believe that much more is true, namely

.x/ x < x1=2.log x/2; j j so the question becomes how close we can get to our belief. We have seen that improving the error term in the prime number theorem is equivalent to exhibitingwider regions to the left of the “1-line” that contain no zeros of .s/, so-called zero-free regions. Proving such results requires complicated analysis of various explicit formulas involving the zeros of .s/ (details can be found, for instance, in [4] and [2]). The key idea is that we understand the values of .s/ well to the rightofthe 1-line, and we can use that understanding,via such explicit formulas, to get some control over .s/ justtothe left ofthe 1-line (we write “just,” since the zero-free regions that have been obtained are so very narrow). These are beautiful and subtle proofs but they give relatively weak results. Moreover, the main application is to discuss issues about prime numbers that are, essentially, questions that arise in the elementary world to the right of the 1-line. It seems strange to work so hard to extrapolate our knowledge of .s/ to the right of the 1-line, in order to get a meager understanding just to the left of the 1-line, so as to answer questions to the right of the 1-line. Why on earth should we cross the 1-line at all? The goal of these methods is to render that arduous journey unnecessary.

18Like Wiles’ Theorem, the Birch-Swinnerton Dyer conjecture, etc. 106 A Century of Advancing Mathematics

Pretentious methods use non-trivial techniques of complex analysis (in particular Per- ron’s formula, as we will see), but not analytic continuation, nor Cauchy’s Theorem and residue computations, nor subtle calculations of zeros of analytic continuations. On the other hand, the calculations used in pretentious techniques can be challenging, though usually they can be reduced to completely elementary techniques, at the cost of further complications. So, technically, one could invoke Selberg’s deﬁnition to say that these are elementary techniques, though that misses the point. The main issue is that these methods avoid needing the Dirichlet series F.s/ to be analytically continuable,19 and so are much more widely applicable.

12 Primes in arithmetic progressions (questions involving uniformity) If we begin computing primes in arithmetic progressions mod, say, 101, we notice that, quite soon, the primes are roughly equidistributedin all of the 100 possible progressions.20 So there is an important, new question for primes in arithmetic progressions:

When can we expect roughly equal numbers of primes in each arithmetic progression mod q? After a lot of computing, researchers guess that there should be roughly equal numbers of primes up to x, in each arithmetic progression a .mod q/, for each a with .a; q/ 1, once 1 D x > cq C , for some constant c, for any ﬁxed > 0. This is far out of reach of what we can prove. Indeed, the best result we have a plan of how to prove is that the primes are 2 roughly equidistributed mod q once x > cq C . This plan, though, involves proving the Generalized Riemann Hypothesis,21 which seems very far out of reach. So what can we prove unconditionally?In both the classical theory, and the pretentious approach, the issue is how close L.1; / is to 0 for real characters . A lower bound of the shape L.1; / > c=pq follows immediately from Dirichlet’s class number formula, and leads to the result that the primes are roughly equidistributed mod q once x > cepq for some constant c > 0 that one can determine. This lower boundfor x is far bigger than what we expect to be true. In 1936 Siegel improved this lower bound to: For all > 0 there exists a constant c > 0 such that L.1; / > c=q , and so the primes are roughly equidistributed mod q q once x > e . But there is a catch. The method of proof does not allow one to determine either c or : Note that we are not saying that these constants have not been computed, but rather that these constants cannot be computed. The proof is very surprising in that Siegel splits his considerations into two complementary cases:

Either, the Generalized Riemann Hypothesis is true, so it is easy to compute c.

19Which is a rare property for a Dirichlet series, a technical property that sometimes seems unreasonably convenient,though it is conjectured to be true for most L-functions of direct arithmetic interest. 20For example,by the time there are 100 primes, on average, in each arithmetic progressionmod 101, the least is 87 primes in some arithmetic progression mod 101, andthe mostis 109. By the time there are 1000 primes, on average, in each arithmetic progression, the least is 968 and the most is 1030. By the time there are 10; 000 primes, on average,in each arithmetic progression, the least is 9912 and the most is 10070. 21That is, if is a zero of any Dirichlet L-function with 0 Re./ 1, then Re./ 1 . Ä Ä D 2 What Is the Best Approach to Counting Primes? 107

Or, the Generalized Riemann Hypothesis is false, in which case it is easy to compute c in terms of any given counterexample. The problem with this dichotomy is the second case. The Generalized Riemann Hypothesis is unresolved, and were the Generalized Riemann Hypothesis to be false, then Siegel’s proof can only provide a constant once some counterexample to the Generalized Riemann Hypothesis is known. All this talk of Riemann Hypotheses in the proof of Siegel’s theorem means that we are involving zeta-functions to the left of the 1-line, and so I had believed that this result could only be obtained by classical means. That was my prejudice, until my postdoc, Dimitris Koukoulopoulos,22 came up with a very subtle elementary argument that allowed him to completely replace Siegel’s argument by a purely pretentious one, with no analytic continuations in sight. One can ﬁnd links between his proof and that of Siegel’s (as developed by Pintz [28]) and so, rather amazingly, we now have an “elementary proof” of Siegel’s theorem. Koukoulopoulos [23] also showed that the primes are not only equidistributed mod q q once x > e , but that theratio is very close to 1 (as in the Siegel-Walﬁsz Theorem). This in turn allows one to use the large sieve to prove the Bombieri-Vinogradov Theorem. The Bombieri-Vinogradov Theorem can be interpreted as stating that the primes are more-or- 1=2 less equidistributed mod q for almost all q < x . This is the consequence we expect from the Generalized Riemann Hypothesis, but we obtain it only for most q, not necessarily all q.

13 Pretentiousness is repulsive

Seemingly, one of the deepest results about L-functions is that their zeros “repel” each other. That is, they do not like to be too close together. In particular one cannot have zeros of two Dirichlet L-functions both close to 1, and this can be rephrased as saying that there is at most one real character .mod q/, amongst all the real characters with q in the range Q < q 2Q, for which L.1; / < c= logQ. Hence L.1; / c= logq for all of the other Ä real characters with q in this range, and therefore one can state a strong prime number theorem for the arithmetic progressions for all these other moduli.23 In fact with such a strong lower bound on L.1; / one can show that the primes are roughly equidistributed mod q once x > qA for some sufﬁciently large A (how large depends on the constant c, and how nearly you want the primes to be equidistributed). Rather surprisingly, these repulsion results are much easier to prove in the pretentious world. Basically L.1; / being very small means that is .n/nit -pretentious for some real number t, and so if L.1; / is also small then is .n/niu-pretentious for some real number u. Now if is very close to .n/nit as well as to .n/niu, then they are close to each other (which formally follows from our triangle inequality), and therefore the i.u t/ Dirichlet character is n -pretentious, which is easily shown to be impossible. This all goes to show that pretentiousness is repulsive.

22And now my faculty colleague. 23I have simpliﬁed here a little bit, rather than get in to the technicalities of primitive and induced characters. For more on this, see Davenport’s book [4]. 108 A Century of Advancing Mathematics

14 The pretentious large sieve Perhaps the deepest proofs in the classical analytic number theory approach to the distribution of prime numbers are the proofs of Linnik’s Theorem; that is that there exist constants c > 0 and L > 0 such that for any positive integers a and q, there is a prime less than or equal to cqL which is congruent to a .mod q/. Linnik’s 1944 proof [26] has been improved many times (e.g., in Bombieri’s [2]) but remains delicate and subtle. Inspired by a new, technically elementary proof in November 2009 given by Friedlander and Iwaniec [9], Soundararajan and I went on to develop an idea we had for a pretentious large sieve [14], and we ended up giving what is surely the shortest and technically easiest proof of Linnik’s Theorem, though bearing much in common with an earlier proof of Elliott [6]. This new technique has enormous potential, because it can replace some very subtle classical techniques, and yet does not require the function involved to have an L-function that can be analytically continued. We made one application, with de la Br´eteche [15], to better understand the solutions to Pythagoras’ equation a2 b2 c2 mod p (as well as to C D several other additive number theory problems). With Adam Harper we can now prove a weak form of Hoheisel’s deep theorem on primes in short intervals: That is, there exists a 1 ı constant ı > 0 and a constant cı > 0 such that if x < y x then Ä y # p prime x < p x y cı : f W Ä C g log x

15 From a collection of ad hoc results, to a new approach to prime numbers Our easy proof of Linnik’s Theorem suggested to Soundararajan and me that we should be able to prove all of the basic results of analytic number theory without ever using analytic continuation. Since early 2010 we have been working on developing this new approach. Our goal is to reprove all of the key results in the standard classical books [4] and [2] using only “pretentious methods.” Within a year we found that we could prove some version of all of the results, perhaps not as strong, but much the same in principle. In doing this we have stood ﬁrmly on the shoulders of giants. Many analytic number theorists have developed ideas about multiplicative functions, over the last 50 years, that have allowed them to prove results on different aspects of prime numbers. Those who have been most central to our description of the subject are Erd˝os, Selberg, Wirsing, Bombieri, Delange, Daboussi, Hildebrand, Maier, Hall, Pomerance,Tenenbaum, Pintz, Elliott, Montgomery and Vaughan, Friedlander, Iwaniec and Kowalski,. . . . Despite being able to prove some version of all of the principal results known on the distribution of primes, we increasingly found ourselves frustrated for three reasons: (1) Although we could show the prime number theorem, we could not show that convergence is anywhere like as fast as had been shown by classical means. In fact we could not see how one might use pretentious methods to even prove something as (relatively) weak as .x/ x x= log x for x sufﬁciently large. j j Ä (2) There are many strong results in the subject that are proved assuming the Riemann Hypothesis. We could not conceive of proving analogous results since the Riemann What Is the Best Approach to Counting Primes? 109

Hypothesis is a conjecture about the zeros of the analytic continuation of .s/, something we are trying to avoid discussing at all. (3) All proofs of Hal´asz’s Theorem, which lies at the center of the whole theory, were not only complicated but also hard to motivate. We had modiﬁed this proof in several ways, for example in the proof of the pretentiouslarge sieve, and this led to a lot of the theory seeming somewhat obscure, even if technically straightforward. Fortunately several of the best young people in analytic number theory got interested in these issues, and they have satisfactorily resolved all of them, as I will now describe:

16 The strongest known form of the prime number theorem The prime number theorem can be phrased as .x/=x 1 as x . The proofs of 1896 ! ! 1 immediately yielded that for any fixed A > 0 we have x .x/ x j j Ä .log x/A for all sufficiently large x. In fact de la Vallée Poussin proved the much stronger result that there exists a constant c > 0 for which

.x/ x x= exp c log x j j Ä p Á for all sufﬁciently large x. The strongest version proved unconditionally is from 1959 (by Korobov and Vinogradov), and gives

.log x/3=5 .x/ x x= c : exp 1=5 (43) j j Ä .log log x/ ! There has been no improvement on this in over 50 years, yet it is so far from what we believe to be true, and can prove assuming the Riemann Hypothesis. In directly using Halász’s Theorem, applied to the multiplicative function .n/ as described above, one can prove results like .x/ x x=.log x/ for sufficiently large x, j j Ä for some < 1, and one can show that it is impossible to obtain larger as an immediate corollary. This is a lot weaker result than the simplest results that one obtains from classical methods. Selberg showed that if f .p/ ˛ for all primes p (with ˛ 1), then the bounds D j j Ä given by Halász’s Theorem are very near to the truth, unless ˛ 0 or 1 in which case D the mean value is much smaller. This includes the example of the mean value of .n/. In fact, Delange went on to show that if the mean value of f .p/ is ˛ then much the same result holds; and hence the mean value of .n/ .n/ converges rapidly to 0 for any Dirichlet character . The Koukoulopoulos converse theorem [24] goes one big step forward, stating that if the mean value of f .n/ is small then f .p/ must average 0 or 1 over the primes. This opens the door to getting much stronger upper bounds for the mean value of f , via appropriate modifications of Halász’s Theorem. Indeed Koukoulopoulos was then able to prove the strongest known versions of both the prime number theorem, as in (43), and the 110 A Century of Advancing Mathematics prime number theorem for arithmetic progressions, using only pretentious methods, never venturing to the left of the 1-line. At first sight it is surprising that he could not do better than the classical proofs. After all, if Koukoulopoulos’s proofs are so different from the classical proofs, then why would he also come up with such an unlikely bound as in (43)? The reason is that, despite ap- pearances, the proofs are fundamentally the same. The classical proof uses deep tools of analysis, which are stripped away in Koukoulopoulos’ proof, suggesting that the use of zeros of .s/ is artificial.

17 The pretentious Riemann Hypothesis The Riemann Hypothesis tells us that the zeros of (the analytic continuation of) .s/ are far 1 into the domain of analytic continuation,that is, they are all on the “ 2 -line.” Does .s/ feel the effect of thisto the right of the “1-line”? Can we recognize the Riemann Hypothesis to the right of the 1-line?

To count primes we looked at 0.s/=.s/. The Riemann Hypothesis is equivalent to this not having any poles, other than at s 1, to the right of the 1 -line. One can remove the D 2 pole at s 1 by working instead with D .s/ 1 0 ; .s/ C s 1 or even .s/ 0 .s/ : (47) .s/ C

If this function’s Taylor series converges around any given point s0 to therightofthe 1-line, within the ball 1 1 B s0; s s s0 < ; 2 WD W j j 2 Â Ã then there can be no poles in that region. The union of all those balls equals the domain to 1 the right of the 2 -line; that is 1 1 B s0; s Re.s/ > : 2 D W 2 s0 Re.s0 />1 Â Ã W [ Therefore, we have proved that the Riemann Hypothesis holds if the Taylor series for (47) 1 at s s0 converges within B s0; 2 , for any s0 C to the right of the 1-line. D 2 .k/ The kth coefﬁcient of the Taylor series for f .s/ at s s0 is given by f .s0/=kŠ. D 1 We can therefore guarantee that the Taylor series converges absolutely within B s0; 2 , .k/ k if f .s0/ c.s0/kŠ2 for every integer k 0, for some constant c.s0/ which may j j Ä depend on f and s0. We conjecture such a hypothesis for the function in (47):

The Pretentious Riemann Hypothesis For all > 0 there exists a constant c > 0 such that for every integer k 1 we have .k/ 0 1 k .s/ ckŠ2 .1 t / C s 1 Ä C ˇ Á ˇ ˇ ˇ ˇ ˇ What Is the Best Approach to Counting Primes? 111 uniformly for s it with 1 < 2 and 0 t ek. D C Ä Ä Ä Using Koukoulopoulos’ methods one can show that if the pretentious Riemann Hy- 1=2 pothesis is true then .x/ x < x for any ﬁxed > 0, which in turn implies the j j C Riemann Hypothesis. On the other hand, the Riemann Hypothesis implies

.k/ 0 .s/ .s/ c kŠ2k log t C Ä ˇ Á ˇ ˇ ˇ for s it with 1ˇand t 1, whichˇ is somewhat more than we asked for in the D C pretentious Riemann Hypothesis. Together these remarks imply:

The Riemann Hypothesis holds if and only if the Pretentious Riemann Hypothesis holds.

18 A re-appraisal of the use of Perron’s formula Soundararajan and I had written up as palatable a proof as we could of Halász’s Theorem for thefirst draftsof ourbook [14], but even we had to admit that it was difficult to motivate. So I was delighted when, in January 2013, my new postdoc, Adam Harper, suggested a new, simpler path to a proof of Halász’s Theorem. Subsequently we have developed his idea with him, and find ourselves re-appraising the use of Perron’s formula when summing coefficients of Dirichlet series. In Riemann’s approach one takes the formula (11), and shifts the line of integration far to the left side of the complex plane. In the pretentious approach one stays with the same line of integration. But then how can one get an accurate estimate, or even a decent upper bound, since the integral of the absolute value of the integrand is usually much larger than the value of the integral? There are several important observations involved. First though, let’s look at this in more generality, with the identity 1 xs f .n/ F.s/ ds; (53) 2i s n x D Re.s/ XÄ Z D where F.s/ f .n/=ns and 1 1 . One can give a version of this (like WD n 1 D C log x in the proof of the prime number theorem), with the values of s running over only those t P where t T , for some suitably chosen T (taking T as a power of log x will do). Then j j Ä we can take absolute values in the integrand, noting that xs x ex to get an upper j j D D bound 1 xs 1 1 F.s/ j jdt 3x max F. it/ dt s it 2 j j s Ä t T j C j 2 t T it Dt CT j jÄ Z j jÄ j j Zj jÄ j C j x max F. it/ .log T 1/: Ä t T j C j C j jÄ This would more-or-less be Halász’s Theorem (via (31)) if the upper bound was divided through by log x. This is encouraging since our approach in getting this upper bound was very crude, and we can surely refine it a bit. Studying the integrand F.s/xs =s, we might expect that F.s/x =s does not change much while xit rotates once around the unit circle (which requires an interval, for t, of 112 A Century of Advancing Mathematics length 2= log x). The easiest way to pick up this cancellation is to integrate by parts, so that (53) becomes:

1 xs 1 xs F.s/ 2 F 0.s/ : D 2i Re.s/ s log x 2i Re.s/ s log x Z D Z D The “log x” in the denominator is the cancellation. The ﬁrst term, because of the s2 in the denominator, is sufﬁciently small to be ignored, and so we are left with

s s 1 x 1 F 0.s/ x F 0.s/ F.s/ : (59) 2i log x Re.s/ s D 2i log x Re.s/ F.s/ s Z D Z D If we take absolute values here, much as we did in (53), then we get the desired bound so long as F .s/=F.s/ is “small” in a certain average sense. It is indeed this small for many j 0 j multiplicative functions f of interest to us, but not all, so another idea is needed. This is where the key new idea of Harper comes in. Going from (53) to (59), we gained 0 a factor of 1 F .s/ , which does improve things and gaining another such factor would log x F .s/ be enough to get us to our goal. We cannot quite do this, but a variant gets us there, using the ﬂexibility of slightly varying the (vertical) line of integration:

1 1 iT F F xs ˇ C 0 .s ˇ/ 0 .s ˇ/ F.s ˇ/ ds dˇ 0 i 1 iT F F C C s ˇ ! Z Z C where is a suitably chosen multiple of 1= log x. There are similarities between this and the formula used in the usual proof of Hal´asz’s Theorem, but it is now much clearer how we got here (which means that this new technique is much more ﬂexible). In particular it allows us to obtain asymptotic formulae if the mean value is “reasonably well-behaved.”

19 New results Our goal in this project is to reprove all of the results of classical analytic number theory from our new perspective. This is a worthwhile project but typically mathematical researchers look forward to what comes next, not to what has been. So to truly justify developing these methods, one might ask whether we can prove results that classical methods could not. In fact, these pretentious techniques were not born from trying to reprove old results, but rather from proving new results on old problems and seeing a pattern emerge in our proofs. The inspiration was the first “big” improvements in bounds for sums of characters in ninety years [16]. By understanding that a character sum could be large only if the character pretended to be a different character with much smaller modulus, we were able to find our improvement. Moreover we found new inter-relations between large character sums that had not previously been known to exist, or even guessed at. There have been other results: other questions on character sums [17, 11], large L- function values [18], least non-residues [25], convexity problems for L-functions [32], and most spectacularly, Soundararajan’s work, with Holowinsky, completing the proof of Arith- metic Quantum Unique Ergodicity [32, 33, 19], which had been a famous conjecture. Very What Is the Best Approach to Counting Primes? 113 recently, Matomäki and Radiziwill [27] have used such techniques to study sign changes in the coefficients of holomorphic Hecke cusp forms. Halász’s theorem is bound to be a better tool to study more general analytic problems than classical analytic methods since the Dirichlet series arising from the given multiplicative function do not need to be analytic, which was the whole point of using zeta-functions. On the other hand zeta-functions have a rich history, and are central to many key themes in mathematics. Dirichlet L-functions are the zeta-functions of weight one, the simplest class. Next come the weight two zeta-functions, which include the L-functions associated to elliptic curves. There is much to do to establish a pretentious theory here. The classical theory can prove much less with these L-functions, so we can hope that pretentious techniques might have significant impact on arithmetic questions associated to these L- functions.

Acknowledgements Thanks are due to Eric Naslund and Dimitris Koukoulopoulos, as well as the anonymous referee and Frank Farris the editor, who read and commented helpfully on a preliminary version of this article.

Bibliography [1] N.A. Baas and C.F. Skau, The lord of the numbers, Atle Selberg. On his life and mathematics. Bull. Amer. Math. Soc. 45 (2008) 617–649. [2] E. Bombieri, Le grandcrible dans la théorie analytique des nombres. Astérisque,18 (1987/1974) 103. [3] Chernac, Cribrum Arithmeticum, 1811. [4] H. Davenport, Multiplicative number theory. Springer Verlag, New York, 1980. [5] P.D.T.A. Elliott, Multiplicative functions on arithmetic progressions. VII. Large moduli. J. Lon- don Math. Soc. 66 (2002) 14–28. [6] ———, The least prime primitive root and Linnik’s theorem, in Number theory for the millennium, I (Urbana, IL, 2000), A K Peters, Natick, MA,2002, 393–418. [7] P. Erd˝os, On a new method in elementary number theory which leads to an elementary proof of the Prime Number Theorem. Proc. Nat. Acad. Sci 35 (1949) 374–384. [8] ———, On a Tauberian theorem connectedwith the new proof of the Prime Number Theorem. J. Ind. Math. Soc. 13 (1949) 133–147. [9] J.B. Friedlanderand H. Iwaniec, Opera de Cribro. American Mathematical Society Colloquium Publications, #57, AMS, Providence, RI, 2010. [10] D. Goldfeld, The elementary proof of the prime number theorem: An historical perspective, in Number Theory (New York 2003, Springer, New York, 2004, 179–192. [11] L. Goldmakher, Multiplicative mimicry and improvements of the Pólya-Vinogradov inequality. Algebra and Number Theory 6 (2012) 123–163. [12] R. Graham and J. Spencer,The Elementary Proof of the Prime NumberTheorem. Mathematical Intelligencer 31 (2009) 18–23. [13] A. Granville, Harald Cramér and the distribution of prime numbers. Scandanavian Actuarial J. 1 (1995) 12–28. 114 A Century of Advancing Mathematics

[14] A. Granville and K. Soundararajan, Multiplicative number theory: The pretentious approach, to appear. [15] R. de la Bretèche, A. Granville and K. Soundararajan, Exponential sums with multiplicative coefficients, to appear. [16] A. Granville and K. Soundararajan,Large character sums: pretentious characters and the Pólya- Vinogradov theorem. J. Amer. Math. Soc. 20 (2007) 357–384. [17] ———, Large Character Sums: Pretentious characters, Burgess’s theorem and the location of zeros, to appear. [18] ———, Extreme values of j.1 C it/j, in The Riemann zeta function and related themes: papers in honour of Professor K. Ramachandra, Edited by R. Balasubramanian and K. Srinivas, Ramanujan Mathematical Society, Bangalore, 2006. 65–80. [19] R. Holowinsky and K. Soundararajan, Mass equidistribution for Hecke eigenforms. Ann. of Math. 172 (2010) 1517–1528. [20] A.E. Ingham, The distribution of prime numbers. Cambridge Math Library, Cambridge, 1932. [21] ———, MR0029410/29411. Mathematical Reviews 10 (1949) 595–596. [22] H. Iwaniec and E. Kowalski, Analytic number theory. Amer. Math. Soc., Providence, RI, 2004. [23] D. Koukoulopoulos, Pretentious multiplicative functions and the prime number theorem for arithmetic progressions. Compos. Math. (2013), to appear. [24] ———, On multiplicative functions which are small on average, (2013), to appear. [25] Xiannan Li, The smallest prime that does not split completely in a number field. Algebra Num- ber Theory 6 (2012) 1061–1096. [26] U.V. Linnik, On the least prime in an arithmetic progression. II. The Deuring-Heilbronn phenomenon. Rec. Math. [Mat. Sb.] N.S. 15 (1944) 347–368. [27] K. Matomäki and M. Radiziwill, Sign changes of Hecke eigenvalues,to appear. [28] J. Pintz, Elementary methods in the theory of L-functions. II. On the greatest real zero of a real L-function. Acta Arith. 31 (1976), no. 3, 273–289. [29] ———, Cramér vs. Cramér. On Cramér’s probabilistic model for primes. Funct. Approx. Com- ment. Math. 37 (2007) 361–376. [30] A. Selberg, An elementary proof of the Prime Number Theorem. Ann. of Math. 50 (1949) 305–313. [31] ———, On elementary methods in prime number-theory and their limitations. Cong. Math. Scand. Trondheim 11 (1949) 13–22. [32] K. Soundararajan, Weak subconvexity for central values of L-functions. Ann. of Math. 172 (2010) 1469–1498.

[33] ———, Quantum unique ergodicity for SL2.Z/ n H . Ann. of Math. 172 (2010) 1529–1538. [34] J. Spencer and R. Graham, The elementary proof of the prime number theorem. Math. Intelli- gencer 31 (2009) 18–31.

Appendix One Factorization tables In Gauss’s December 24th, 1849 letter to Encke, he wrote: The 1811 appearance of Chernac’s Cribrum Arithmeticum gave me great joy, and (since I did not have the patience for a continuous count of the series) I have very often employed a spare unoccupied quarter of an hour in order to count up a chiliad What Is the Best Approach to Counting Primes? 115

Figure 1. A page from Chernac’s 1811 Cribrum Arithmeticum.

here and there; however, I eventually dropped it completely, without having quite completed the ﬁrst million.

Figure 1 is a photograph of Chernac’s Table of Factorizations of all integers up to one million, published in 1811, which was used by Gauss. My colleague, Anatole Joffe, kindly presented his copy of these tables to me when he retired. Nowadays I also happily distract myself from boring office-work, by flipping through to discover obscure factorizations! There are exactly one thousand pages of factorization tables in the book, each giving the factorizations of one thousand numbers. For example, page 677, seen here, enables us to factor all numbers between 676000 and 676999. The page is split into five columns, each in two parts, the ten half columns on the page each representing those integers that are not divisibleby 2, 3 or 5, in an interval of length 100.24 On the left sideof a columnis a number like 567, which represents, on this page, the number 676567 to be factored. On the right side of the column we see 619 1093 which gives the complete factorization of 676567. On the other hand for 589, which represents the prime number 676589,the right column simply contains “——” and hence that number is prime. And so it goes for all of the numbers in this range. It only takes a minute to get used to these protocols, and then the table becomes very useful if you do not have appropriate factoring software at your disposal.

24Chernac trusted that the reader could easily extract those factors for him- or herself. 116 A Century of Advancing Mathematics

Appendix Two A proof and a challenge The triangleinequality,(29), liesat the heart of thisnew theory. There were several proofs at the time I placed the preliminaryversion of thisarticle on the arxiv (arXiv:1406.3754),none of which were particularly elegant. The best was due to Eric Naslund, which he created during an undergraduate research project in 2011. In that preprint I gave Naslund’s proof and announced a competition to ﬁnd the most elegant proof. I am happy here to give the winner, due to Oleksiy Klurman, a PhD student at the University of Montreal.

Proof. (Oleksiy Klurman) Deﬁne .u/ 1 u 2, so that 2.u; v/2 .u/2 D j j D C .v/2 u v 2. The result follows from applying the usual triangle inequality to the C j j p vector addition

.w z; .w/; .z/; 0/ .z y; 0; .z/; .y// .w y; .w/; 0; .y//; C D in R4.

Universitéde Montréal, Pavillon André-Aisenstadt, Département de mathématiques et de statistiques, CP 6128 Succursale Centre-Ville, Montréal QC H3C 3J7, Canada. [email protected] A Century of Elliptic Curves

Joseph H. Silverman1 Brown University

The ubiquity of elliptic curves Elliptic curves appear in many places and in many guises throughout mathematics, including algebraic geometry, algebra, number theory, topology, cryptography, real and complex analysis, and mathematical physics, to name just a few [25].My aim inthisbrief essay is to provide an overview of the study of the number-theoretic side of elliptic curves during the past century (plus epsilon). Considerations of space and taste have determined the particular topics covered. Those seeking further information will ﬁnd many paths to follow. For a few suggestions, see the list of references preceding the bibliography.

1 What is an elliptic curve? There are many ways to define an elliptic curve, ranging from the mundane (a cubic equa- tionwith a marked point)to the sublime (a propercommutative group scheme of dimension one). We will take the concrete approach and initially define an elliptic curve E to be the set of solutions to a cubic equation2 f .x; y/ ax3 bx2y cxy2 dy3 ex2 f xy gy2 hx iy j 0; (1) D C C C C C C C C C D together with a marked solution f .x0; y0/ 0. If the coefficients of the polynomial f D and the coordinates of the point .x0; y0/ are in a field k, then we say that the ellipticcurve is defined over k. For example, the field k might be C or R or Q or a finite field Fp. We write E.k/ for the set of solutions .x; y/ to (1 ) with x;y k. 2 The equation defining E is a polynomial of degree 3, so if we intersect E with a line L, then generally we expect E L to consist of 3 points (at least if we work over C). This \ is not quite true for several reasons. The first, and most important, is that we are missing some points “at infinity.” The solution is to work instead in the projective plane P 2, which is the usual affine plane A2 together with an extra line at infinity. In P 2, any two distinct lines intersect in exactly one point; there is no longer a problem with “parallel lines” not intersecting. A second difficulty is that tangent lines don’t intersect E in enough points. We can fix this problem by counting such points with an appropriate multiplicity. For example, there

1The author’s research is supported by DMS-0854755. 2This definition is a lie, although it contains much that is truthful. As we progress, we will refine the definition until it is accurate.

117 118 A Century of Advancing Mathematics are nine so-called flex points on E for which the tangent line intersects E at only one point, so we count these flex intersection points with multiplicity three. A third difficulty is that the equation f .x; y/ 0 may define a singular curve, that is, D it may have points where @f=@x @f=@y 0. Such points would cause problems for D D later constructions, which finally leads us to an accurate definition. An elliptic curve E is the set of solutions to a cubic equation (1) having no singular points, together with points at infinity (also nonsingular)so that every line intersects E in exactly three points (counted with appropriate multiplicity), together with a marked point O E. 2 One can show that there is a change of coordinates so that the equation for E takes the form E y2 x3 Ax B with 4A3 2B2 0. W D C C C ¤ The marked point O has been moved to infinity, and it is the only point of E at infinity. The intuitionis that O lies on every vertical line, and on no other lines. The quantity E 3 2 D 4A 2B is called the discriminant of E, and the assumption that E 0 means that C ¤ the curve E is nonsingular. Equivalently, it says that the cubic polynomial x3 Ax B C C has no repeated roots. Three examples of E.R/ are illustrated3 in Figure 1, which shows that E.R/ may have either one or two connected components.

y2 x3 3x 3 y2 x3 x y2 x3 x D C D C D

Figure 1. Three elliptic curves.

A fundamental axiomof plane geometry isthat throughany two distinctpoints P and Q thereis auniqueline LPQ going through P and Q. Then the fact that every line intersects E in exactly three points(counted with multiplicity)means that for any two (distinct)points P and Q on E, we can uniquely determine a third point R by the rule

E LPQ P;Q;R : (2) \ D f g

If P Q, then we take LPQ to be the tangent line to E at P , and then the rule given D by (2) still works. (See Figure 2.) It is tempting to deﬁne R to be some sort of “sum” of P and Q, but this summation rule does not have nice properties. Instead we deﬁne

P Q reﬂection of R about the x-axis. ˚ D 3The author thanks Springer Science+Business Media for their permission to include Figures 1 and 2, which are Figures 3.1 and 3.3 from his book The Arithmetic of Elliptic Curves, GTM 106,2009. A Century of Elliptic Curves 119

(If R O, then we set P Q O.) We also deﬁne D C D P reﬂection of P about the x-axis, i.e,. .x; y/ .x; y/. « D « D With this “addition” law, it is an amazing fact that the points of E have the structure of an abelian group. Theorem 1. The additionlaw on E has the following properties: (a) P O O P P for all P E. C D C D 2 (b) P . P/ O for all P E. C « D 2 (c) P Q Q P for all P;Q E. C D C 2 (d) P .Q R/ .P Q/ R for all P;Q;R E. C C D C C 2 All of Theorem 1 is easy to prove except for associativity, which can be tediously checked using explicit formulas, or with less work using fancier methods. Example 2. The elliptic curve E y2 x3 17 W D C has several obvious points whose coordinates are integers, for example

P1 . 2; 3/; P2 . 1; 4/; and P3 .2; 5/: D D D

The line L12 through P1 and P2 is y x 5, and the thirdpointof intersectionin E L12 D C \ is the point P4 .4; 9/, so P1 P2 .4; 9/. (Don’t forget that we have to reflect the D ˚ D third intersection point across the x-axis.) Similarly, the line L23 through P2 and P3 is y 1 13 8 109D 3 x 3 . Then elementary algebra yields E L23 P2;P3;P5 with P5 9 ; 27 C 8 109 \ D f g D so P2 P3 9 ; 27 . To add P1 to itself, we first compute the tangent line L11 to E ˚ D 2 at P1. Using dy=dx 3x =2y, we find that L11 is the line y 2x 7, and then P1 P1 D D C 19 ˚522 turns out to be .8; 23/. Repeating the process, we find that P1 P1 P1 ; . ˚ ˚ D 25 125 We make two observations about this example: The sum of two points with rational coordinates is again a point with rational coordinates. The sum of two points with integer coordinates need not have integer coordinates. It is convenient to indicate repeated addition of E by

nP P P P : D ˚ ˚ ˚ n summands

This is for n > 0. We also deﬁne 0P „O, andƒ‚ for n <… 0, we set nP n P . D D «j j 2 Why are elliptic curves special? What is it that makes cubic equations so special? The answer lies in the geometry of the solution set, but to really understand this geometry we must look at solutions in complex numbers. In general, if F.x; y/ CŒx; y is a polynomial,then the set of complex solutions 2 .x; y/ C2 F.x; y/ 0 D 2 W D ˚ « 120 A Century of Advancing Mathematics

R s R Q s P s P s s T s

P Q s ˚ 2T D O 2P s P ˚ Q ˚ R D O

Addition of distinct points Adding a point to itself Figure 2. Examples of the addition law on an elliptic curve. is a surface, since if we identify C2 R4, then the real and imaginary parts of the equation D F.x; y/ 0 give two real equations satisﬁed by the coordinates in R4. If we add a few D extra points “at inﬁnity”to make our surface complete, and if we smooth out any singularities,4 then becomes a Riemann surface, i.e., it is a g-holed torus. The number g is called the genus of . The surface is an elliptic curve if its genus is one, and it is only in this case that has a natural structure as a group. Elliptic curves are also special when studying number theory. So we now suppose that F QŒx; y has rational coordinates, and we let .Q/ denote the set of solutions 2 to F.x; y/ 0 in rational numbers x; y Q. Assuming that .Q/ is not empty, we can D 2 give a qualitative description of .Q/ solely in terms of the genus of :

g .Q/ 0 Q plus a few extra points 1 a (finitely generated) abelian group 2 a finite set Thus the rational points on .Q/ look very different in the three cases. The case of g 0 D is now well understood.For g 1, much is known, but there are stillmany open questions, D which justifies making these curves the focus of this article. There are also many interesting open questions when g 2, and indeed, the fact that .Q/ is finite in this case was a famous conjecture of Mordell that was proven by Faltings [7].

3 The group of rational points

We noted in Example 2 that the sum of points having rational coordinates seemed again to have rational coordinates. This general fact seems to have first been observed by Poincaré. Theorem 3. (Poincaré, 1901 [21]) Let E be an elliptic curve defined over Q, i.e., given by an equation y2 x3 Ax B D C C with A and B in Q, and let P1 and P2 be points on E whose coordinates are in Q. Then the coordinates of P1 P2 are again in Q. ˚ 4“Smoothing out” singularities means breaking apart nodes and flattening cusps. A Century of Elliptic Curves 121

Theorem 3 is proven using explicit formulas for the sum P1 P2 that can be derived ˚ using elementary algebra and calculus. For example, writing P1 .x1; y1/ and P2 D D .x2; y2/, one ﬁnds that the x-coordinate of P1 P2 (excluding some special cases) is ˚ given by the formula

2 y2 y1 x1 x2 if P1 P2, x2 x1 ¤ x.P1 P2/ 8Â Ã (3) ˚ D ˆx4 2Ax2 8Bx A2 <ˆ 1 1 1 3 C if P1 P2. 4x1 4Ax1 4B D ˆ C C ˆ These formulas involve only addition,: subtraction, multiplication, and division, so they transform rational numbers into rational numbers. Definition. Let E be an elliptic curve defined over a field k. We write E.k/ for the set of points on E having coordinates in k,

E.k/ .x; y/ k2 y2 x3 Ax B O : D 2 W D C C [ f g Poincaré’s theorem (Theorem˚ 3) says that if E is defined over« Q, then E.Q/ is closed under addition, and hence E.Q/ is a subgroup of E.C/. More generally, if E is defined over k, then the addition formula (3) shows that E.k/ is a group. A famous theorem of Mordell, later vastly generalized by Weil to number fields and abelian varieties of arbitrary dimension, describes the structure of the group E.Q/. Mordell’s theorem marked the dawn of a new age in the study of Diophantine equations, an epoch that continues to this day. Theorem 4. (Mordell, 1922 [19]) Let E be an elliptic curve defined over Q. Then E.Q/ is a finitely generated abelian group. Remark 5. In more prosaic terms, Mordell’s theorem says that there is a finite set of points P1;:::;Pk E.Q/ such that every point P E.Q/ can be writtenin the form 2 2

P n1P1 n2P2 nkPk for some n1; : : : ; nk Z. D ˚ ˚ ˚ 2 A big open problem is to make Mordell’s theorem effective, i.e., to find an effective algorithm that takes as input the elliptic curve E and gives as output a set of generators P1;:::;Pk for E.Q/. f g Definition. Mordell’s theorem and the structure theorem for finitely generated abelian groups imply that r E.Q/ E.Q/tors Z ; Š where the torsion subgroup E.Q/tors is a finite group and the integer r is the rank of E.Q/. Example 6. We give some examples of different sorts of groups of rational points, where we write CN for a cyclic group of order N : 2 3 (i) The elliptic curve E y x x has E.Q/ C2 C2. The points .0; 0/ W D Š and .1; 0/ generate E.Q/. 2 3 (ii) The elliptic curve E y x 1 has E.Q/ C6. The point .2; 3/ gener- W D C Š ates E.Q/. 122 A Century of Advancing Mathematics

2 3 (iii) The elliptic curve E y x 5x 18 has E.Q/ C2 Z. The point . 2; 0/ W D C C Š has order 2, and together with the point .2; 36/ generates E.Q/. (iv) The elliptic curve E y2 x3 17 has E.Q/ Z2. The points . 1; 4/ W D C Š and . 2; 3/ are independent and generate E.Q/. Of course, proving that these statements are true is not at all trivial. A beautiful theorem of Nagell and Lutz (independently) from the 1930s gives a way to compute E.Q/tors. Theorem 7. (Nagell 1935 [20], Lutz 1937 [16 ]) Let E y2 x3 Ax B be an elliptic W D C C curve with A; B Z. (If E is deﬁned over Q, one can always change coordinates to 2 make A; B Z.) Let P .x; y/ E.Q/tors. Then either y 0, in which case 2P O, 2 D 2 D D or else y2 divides 16.4A2 27B2/. C 2 2 Recall that the discriminant E 4A 27B is nonzero, so Theorem 7 gives a ﬁnite D C list of y values to test. However, as A and B grow, the number of possibilities could also grow. Mazur proved a strong uniformity result on the allowable size of E.Q/tors. Theorem 8. (Mazur 1977 [17 ]) Let E.Q/ be an elliptic curve. Then the torsion subgroup E.Q/tors has one of the followingforms:

Z=N Z with 1 N 10 or N 12, E.Q/tors Ä Ä D Š ( Z=2Z Z=2N Z with 1 N 4. Ä Ä

In particular, E.Q/tors has order at most 16. The proof of Mazur’s theorem, and its generalization to number fields by Kamienny and Merel, uses deep methods from modern algebraic geometry. Theorems 7 and 8 give us an excellent understanding of the elements of E.Q/ having finite order. The rank, by way of contrast, is very mysterious. For example, we still do not have an effective algorithm that takes as input the equation of an ellipticcurve and outputs the rank of E.Q/, although there are algorithms that seem to work well in practice as long as the coefficients are not too large. Notwithstanding the difficulty of computing the rank, there is the following longstanding conjecture. Conjecture. For every r there exists an elliptic curve E defined over Q such that E.Q/ has rank at least r. Conjecture 3 has led to an ongoingfriendly competition to find elliptic curves of higher and higher rank, similar in some ways to the search for large Mersenne primes (primes of the form 2p 1). The elliptic curve search has involved a combination of theoretical advances due to Mestre, Nagao, Elkies, and others, as well as on the availability of ever increasing amounts of computational power. The current record as of 2012 is the following curve of rank at least 28 found by Elkies (2006 [6]):

y2 C xy C y D x3 x2 20067762415575526585033208209338542750930230312178956502x C 3448161179503055646703298569039072037485594435931918 0361266008296291939448732243429.

Needless to say, such examples are not constructed by randomly choosing coefﬁcients and hoping that the rank is high! A Century of Elliptic Curves 123

4 Integer points on elliptic curves For this section we assume that our elliptic curve E is given by an equation5

y2 x3 Ax B with integer coefﬁcients A; B Z. D C C 2 It then makes sense to talk about the set of integer points, i.e., points with integer coordinates, which we denote by

E.Z/ .x; y/ Z2 y2 x3 Ax B : D 2 W D C C What should we expect the˚ set E.Z/ to look like? We’ve seen« that if P and Q are in E.Q/, then P Q is again in E.Q/, but theformula (3) for P Q suggests that P Q ˚ ˚ ˚ is unlikely to be in E.Z/, even if P and Q are in E.Z/. Thus it is not surprising that the following result is true, but the proof turns out to be surprisingly difficult. Theorem 9. (Siegel 1928 [24 ]) Let E be an ellipticcurve given by an equationwithinteger coefficients. Then E.Z/ is a finiteset. Example 10. The set of integer points E.Z/ on the elliptic curve E y2 x3 17 contains quite a few integer points, including one that’s quite large: W D C

.2; 3/; .1;4/; .2;5/; .4;9/; .8;23/; .43;282/; .52;375/; .5234;378661/:

We mentioned earlier that we don’t yet have an effective algorithm for computing generators of the group E.Q/. The original proof of Theorem 9 suffered from the same defect; it showed that E.Z/ is finite, but it did not give an effective method for finding all of the points in E.Z/. This flaw was remedied by Baker in 1966 as an application of his effective bounds for linear forms in logarithms. The following version of Baker’s theorem is not the best known, but it illustrates both the effectivity and the size of the upper bounds coming from Baker’s method. Theorem 11. (Baker [2, page 45]) Let E y2 x3 Ax B be an elliptic curve with W D C C integer coefficients. Then every point .x; y/ E.Z/ satisfies 2 106 max x ; y < exp 106 max A ; B : j j j j fj j j jg ˚ « Á The bound in Theorem 11 is certainly not optimal, but all known bounds are at least exponential in the coefficients of the elliptic curve. This contrasts with what is believed to be true. Conjecture. (Hall–Lang) There are absolute constants c and so that for every elliptic curve E y2 x3 Ax B with integer coefficients and every point .x; y/ E.Z/, we W D C C 2 have max x ; y < c max A ; B : j j j j j j j j Since the set E.Z/ is finite,˚ it is natural« to ask˚ how large« it can be. The answer is that we can make it as large as we want. Take any curve E y2 x3 Ax B, and W D C C 5The changeof variables .x; y/ .u2x; u3y/ transforms the equation for E into y2 x3 u4Ax u6B, so choosing an appropriate u !lets us assume that the equation defining E has integer coefficients.D C C 124 A Century of Advancing Mathematics

let .x1; y1/; : : : ; .xn; yn/ be points in E.Q/. Now choose a large integer so that all of the numbers Dx1; Dy1; : : : ; Dxn; Dyn are integers. Then the points

2 3 2 3 .D x1;D y1/; : : : ; .D xn;D yn/ are integer points on the elliptic curve

E y2 x3 D4Ax D6B; 0 W D C C so E0.Z/ has at least n points. However, if we rule out this “cheating” method of creating integer points, which can be done by putting an appropriate condition on A and B, then Lang has conjectured a surprising relationship between the size of E.Z/ and the rank of E.Q/. Conjecture. (Lang 1978 [15 ]) Thereisaconstant C with the following property. Let E W y2 x3 Ax B be an elliptic curve with A and B integers such that gcd.A3;B2/ is D C C not divisible by any nontrivial twelfth-powers. Then

#E.Z/ C 1 rank E.Q/: Ä C The deep uniformity in Conjecture 4 lies in the fact that the constant C does not depend on the elliptic curve E. A weaker result in which C depends on the number of primes dividing the denominator of the rational number j.E/ 4A3=.4A3 27B2/ was proven by D C Silverman [26], and Hindry and Silverman [10] proved that Conjecture 4 is a consequence of the abc-conjecture of Masser and Oesterl´e.

5 Elliptic curves over finite fields A fundamental tool in studying integral or rational solutions to polynomial equations is to first study solutions modulo m. The Chinese remainder theorem reduces the problem to studying solutions modulo prime powers, and at least for elliptic curves, it suffices to consider solutions modulo primes. e Alternatively, we can start with a finite field Fq having q p elements and consider D an elliptic curve defined over Fq, i.e.,

2 3 E y x Ax B with A; B Fq. W D C C 2 3 2 Of course, we require as usual that E 4A 27B 0. (For simplicity, we will also D C ¤ assume that p, the characteristic of Fq, is not 2.) Then E.Fq/ denotes the set of points

2 2 3 E.Fq/ .x; y/ F y x Ax B O ; D 2 q W D C C [ f g and the explicit addition formulas˚ (3) can be used to give E.Fq«/ the structure of a (ﬁnite abelian) group. How large would we expect E.Fq/ to be? For each x Fq, we get twopointsin E.Fq / 3 2 3 if x Ax B is a (nonzero) square in Fq, and we get no pointsin E.Fq/ if x Ax C C C C B is a not square in Fq. Since half the nonzero elements of Fq are squares, we would expect #E.Fq/ to be approximately q 1. A theorem of Hasse quantiﬁes this heuristic C argument. A Century of Elliptic Curves 125

Theorem 12. (Hasse) Let E be an elliptic curve defined over a finite field Fq. Then

#E.Fq/ q 1 2pq: Ä ˇ ˇ If E is deﬁned over Fq, thenˇ we can lookat itsˇ pointswhosecoordinates lie in extension ﬁelds Fqn . It is standard practice to encode the resultingsequence of numbers into a formal power series.

Deﬁnition. The zeta function of E=Fq is the power series

1 1 Z.E=Fq;T/ exp #E.Fqn / : D n n 1 ! XD

The sequence of orders #E.Fqn / for n 1; 2; : : : is surprisingly regular, which leads D to the ﬁrst part of the following result.

Theorem 13. Let E be an ellipticcurve defined over a finite field Fq.

(a) (Rationality) The zeta function Z.E=Fq;T/ is a rational function with integer co- efﬁcients. More precisely, it has the form

1 aT qT 2 Z.E=Fq;T/ C ; D .1 T /.1 qT / where a q 1 #E.Fq/. D C (b) (Duality) The zeta function Z.E=Fq;T/ satisﬁes the functional equation

Z.E=Fq;T/ Z.E=Fq; 1=qT /: D (c) (Riemann Hypothesis) Factor 1 aT qT 2 .1 ˛T /.1 ˇT / in CŒT . Then ˛ C D and ˇ are complex conjugates satisfying ˛ ˇ pq. j j D j j D It is an instructive exercise to show that Theorem 13 implies that

n n n #E.Fqn / q 1 ˛ ˇ for all n 1. D C Furthermore, Hasse’s theorem (Theorem 12) is equivalent to the Riemann hypothesis for Z.E=Fq;T/ (Theorem 13(c)). Hasse’s theorem gives an estimate for #E.Fq/, but if q is large, it is not clear how we might efficiently compute this number. A major advance in the computational theory of elliptic curves was the discovery by Schoof [23] in 1985 of a polynomial time6 algorithm to compute #E.Fq/. Subsequent improvements were made by Elkies and Atkins, so it is often call the SEA Algorithm. The past 25 years have seen a surprising real-world application of elliptic curves over finite fields. This application is in the area of cryptography,and ellipticcurves are being used to protect your bank transactions and your personal privacy. The key to these applications is the computational difficulty of the following problem.

6In this context, an algorithm is polynomial time if its running time is a polynomial in log q, where log q is essentially the number of bits in the number q. 126 A Century of Advancing Mathematics

Definition. Let E be an elliptic curve defined over a finite field Fq, and let P and Q be points in E.Fq/. The elliptic curve discrete logarithm problem (ECDLP) is the problem of determining an integer m such that Q mP (assuming that such an integer exists). D In principle it is easy to solve the ECDLP, we simply compute P;2P;3P;::: until we get to Q. A more efficient method is to use a collision algorithm. We choose random integers n1; n2;::: and make two lists,

List 1: n1P; n2P; n3P;:::;

List 2: n1P Q; n2P Q; n3P Q;:::: C C C

When we find a match, say ni P nj P Q, then Q .ni nj /P . An elementary analysis D C D shows that collision algorithms of this sort usually take around O.pq/ steps. For general elliptic curves, variants of this collision algorithm are the most efficient ways known to solve the ECDLP. Thus in practice the ECDLP takes exponential (in log q) time to solve, making it much harder than the analogous problem for the multiplicative group7, and also much harder than the integer factorization problem underlying RSA, for which subexpo- nential algorithms are known. How is the ECDLP used in cryptography? Alice and Bob first agree on a curve E and point P E.Fq/. (In practice, a standards body such as NIST would publish curves and 2 points to use.) Alice chooses a secret integer m. She computes mP E.Fq / and publishes 2 the value of mP . Bob uses mP to encrypt his message, but the system is set up so that in order to decrypt,one needs toknow thevalue of m, which onlyAlice knows. And finding m from the data .Fq ; E; P; mP / is exactly the problem of solving the ECDLP. Lack of space prevents us from giving further details, but the interested reader will find elliptic curve cryptography discussed in many books, including for example [11, Chapter 5] and [33].

6 L-series, modularity, and Fermat’s last theorem

As we saw in Section 5, if E is an elliptic curve defined over a finite field Fq, then #E.Fq / looks like q 1, with an error no larger than 2pq. The behavior of the error C

aq.E/ q 1 #E.Fq/ D C is of fundamental importance in the study of elliptic curves. The quantity aq.E/ is called the trace of Frobenius, because one can compute it as the trace of the linear map induced by the Frobenius map .x; y/ .xq; yq/ on a certain (co)homology space. ! We now take an elliptic curve E given by an equation y2 x3 Ax B having D C C integer coefficients, so for each prime p 3 not dividing 4A3 27B2 we can reduce the C coefficients of E to obtain an elliptic curve E defined over the finite field F . We then Qp p look at the trace of Frobenius at p, which by definition is8

E/ap. ap.Ep/ p 1 #Ep.Fp/: D Q D C Q

It is natural to inquire how the values of ap.E/ vary as p varies. The answer is given by a famous conjecture of Sato and Tate.

7 m The discrete logarithm problem for Fq is, for given a; b Fq , to ﬁnd an exponent m such that b a . 8 3 2 2 D For p 2 or p dividing 4A 27B , there is a more complicated way to deﬁne ap .E/. D C A Century of Elliptic Curves 127

Conjecture. (Sato–Tate Conjecture) For each prime p, Hasse’s theorem says that the quantity ap.E/=2pp is between 1 and 1, so there is a unique angle 0 p.E/ Ä Ä such that ap.E/ 2pp cos p.E/. Then for all 0 ˛ ˇ , D Ä Ä Ä ˇ # p X ˛ p.E/ ˇ 2 lim f Ä W Ä Ä g sin2 t dt: X X= log X D ˛ !1 Z

In other words, the collection of angles p.E/ is equidistributed with respect to a sine- squared measure. ˚ « Recent progress by Clozel, Harris, Shepherd-Barron, and Taylor [5, 9, 31] has shown that this conjecture is true for all elliptic curves for which the rational number j.E/ D 4A3=.4A3 27B2/ is not an integer. The proof is very difﬁcult. C The Sato–Tate conjecture describes the average distribution of the ap.E/ values. An- other way to study these values is to put them into a generating function. Deﬁnition. The zeta function of E=Q is the product

s 1 2s s 1 ap.E/p p .E=Q; s/ Z.Ep=Fp; p / C : D Q D .1 p s/.1 p1 s/ p prime p prime Y Y (The second equality comes from Theorem 13 .) The L-functionof E=Q is the product

1 L.E; s/ s 1 2s : D 1 ap.E/p p p Yprime C s Using the standard product expansion of the Riemann zeta function .s/ 1 n D n 1 D .1 p s/ 1, one easily sees that D p P Q .s/.1 s/ .E=Q; s/ ; D L.E; s/ so all of the information about E encoded in the function .E=Q; s/ is already contained in L.E; s/.

It follows easily from Hasse’s estimate ap.E/ 2pp that the product defining Ä L.E; s/ is absolutely convergent for all s C in the half-plane Re.s/ > 3 . A long- 2ˇ ˇ 2 standing conjecture that was proven by Wilesˇ and othersˇ during the 1990s is that L.E; s/ admits an analytic continuation to the entire complex plane. Theorem 14. (Wiles et al. [3, 32, 34 ]) The L-series L.E; s/ has an analytic continuation to the entire complex plane and satisfies a functional equation. More precisely, there is an integer N 1 and sign w 1 such that the function .E=Q; s/ s=2 s 2 f˙ g D N .2/ .s/L.E; s/ satisfies

.E=Q; s/ w.E=Q; 2 s/ for all s C. D 2 (Here .s/ is the classical -function.) 128 A Century of Advancing Mathematics

Theorem 14 is an immediate consequence of the modularity conjecture of Shimura, Taniyama, and Weil.9 To state the conjecture, which is what Wiles et al. actually proved, we ﬁrst expand the product deﬁning L.E; s/ to obtain a Dirichlet series

1 a L.E; s/ n ; D ns n 1 XD and then we use the an to deﬁne a function that is holomorphicon the uppper half-plane,

1 2in fE ./ ane for H z C Im.z/ > 0 . D 2 D f 2 W g n 1 XD Theorem 15. (Wiles et al. [3, 32, 34 ]) There isaninteger N 1 (which can be easily computed from the equation for E) so that the function fE .z/ is a modularform of weight 2 a b for 0.N /. What thismeans isthat for every matrix SL2.Z/ with c divisible by N , c d 2 the function f .z/ satisfies the transformation formula10 E a b 2 fE C .c d/ fE ./ for all H. (4) c d D C 2 Â C Ã It is not hard to express L.E; s/ in terms of the modular form fE ./, and then the modularity transformation formula (4) gives both the analytic continuation and functional equation of the L-series. The proof of Wiles’ theorem is very difficult, and even a brief sketch would be far beyond the scope of this survey. We now have many important quantities associated to an elliptic curve, including the rank and torsion subgroups of E.Q/ and the orders #E.Fp/ for primes p. These orders, in the guise of the ap.E/ p 1 #E.Fp/ values, are encoded into the function L.E; s/, D C which we know has an analyticcontinuationto all of C. The function L.E; s/ encapsulates information about the points of E modulo primes p, so the following conjecture is quite remarkable because it says that the ap.E/ values determine the rank of the groupof rational points E.Q/. Conjecture. (Birch and Swinnerton-Dyer) Let E be an elliptic curve defined over Q. Then

ords 1 L.E; s/ rank E.Q/; (5) D D 11 where ords 1 L.E; s/ dentoes the order of vanishing of L.E; s/ at s 1. D D Special cases of the Birch and Swinnerton-Dyer conjecture, together with a description of the leading coefﬁcient of the Taylor series expansion of L.E; s/ around s 1, have D been proven by Coates–Wiles, Gross–Zagier, Rubin, Kolyvagin, and others, but the full conjecture remains open. Indeed, the rank formula given by (5) is one of theClay Institute’s MillenniumProblems, which comes attached with a million-dollar prize for the ﬁrst person to supply a proof. We close this brief survey with a spectacular application of the modularity theorem, namely the proof of Fermat’s “Last Theorem.”

9There is considerable controversyas to which names to attach to this conjecture. 10There is also a growth condition at the cusps that we do not attempt to describe. 11In general, the order of vanishing of a complex analytic function function '.s/ at s a is the integer 0 D such that the Taylor series expansion of ' around a looks like '.s/ c.s a/ (higher order terms) with D C leading coefﬁent c 0. ¤ A Century of Elliptic Curves 129

Theorem 16. (Fermat’s Last Theorem) Let n 3. Then the equation an bn cn has C D no solutions in integers a; b; c with abc 0. ¤ One easily reduces to the case that n p is prime (the case n 4 having been done D D already by Fermat), but it is not at all clear how the equation ap bp cp is related to C D elliptic curves. Frey [8] suggested using a putative solution .a; b; c/ to Fermat’s equation to build an elliptic curve

2 p p Ea;b;c y x.x a /.x b /: W D C

A simple computation shows that the discriminant of Ea;b;c is a pth-power (up to some powers of 2), p p p p p E a b .a b / .abc/ : a;b;c D C D This seemed sufficiently unusual that Frey wondered whether Ea;b;c could be modular, i.e., could its associated function fEa;b;c ./ be a modular form? Serre gave a more precise form of Frey’s conjecture, and Ribet [22] proved that indeed Ea;b;c could not be modular. What Ribet showed is that if Ea;b;c is modular, then the associated value of N in Theorem 15 would divide 4, but it was well-known that there are no such modular forms with N 10. Ä This did not, at the time, prove Fermat’s theorem, because it was not known that every elliptic curve defined over Q is modular. That last step was supplied by Wiles [32, 34] (with assistance from Taylor) when he proved Theorem 15 for semistable elliptic curves.12 Since all of the Ea;b;c curves are semistable, this sufficed to prove Fermat’s last theorem.

Further Reading The following is a short list of books and articles that provide further reading about elliptic curves and their arithmetic: [1, 4, 11, 12, 13, 14, 15, 18, 27, 28, 29, 30, 33].

Bibliography [1] A. AshandR. Gross, Elliptic Tales: Curves, Counting, and Number Theory. Princeton Univer- sity Press, Princeton, 2012. [2] A. Baker, Transcendental number theory. Cambridge Mathematical Library. Cambridge Uni- versity Press, Cambridge, second edition, 1990. [3] C. Breuil, B. Conrad, F. Diamond, and R. Taylor, On the modularity of elliptic curves over Q: wild 3-adic exercises. J. Amer. Math. Soc., 14(4): 843–939 (electronic), 2001. [4] J. W. S. Cassels, Lectures on Elliptic Curves, volume 24 of London Mathematical Society Student Texts. Cambridge University Press, Cambridge, 1991. [5] L. Clozel, M. Harris, and R. Taylor, Automorphy for some l-adic lifts of automorphic mod l Galois representations. Publ. Math. Inst. Hautes Etudes´ Sci., (108): 1–181, 2008. With Appendix A, summarizing unpublished work of Russ Mann, and Appendix B by Marie-France Vignéras. [6] N. Elkies, Z28 in E.Q/. Number Theory Listserver, May 2006. [7] G. Faltings, Endlichkeitssätze für abelsche Varietäten über Zahlkörpern. Invent. Math., 73(3): 349–366, 1983.

12Roughly speaking, an elliptic curve y2 x3 Ax B is semistable if A and B are relatively prime. D C C 130 A Century of Advancing Mathematics

[8] G. Frey, Links between stable elliptic curves and certain Diophantine equations. Ann. Univ. Sarav. Ser. Math., 1(1): iv+40, 1986. [9] M. Harris, N. Shepherd-Barron, and R. Taylor, A family of Calabi-Yau varieties and potential automorphy. Ann. of Math. (2), 171(2): 779–813, 2010. [10] M. Hindry and J. H. Silverman, The canonical height and integral points on elliptic curves. Invent. Math., 93(2): 419–450, 1988. [11] J. Hoffstein, J. Pipher, and J. H. Silverman, An Introduction to Mathematical Cryptography. Undergraduate Texts in Mathematics. Springer-Verlag, New York, 2008. [12] D. Husemöller, Elliptic Curves, volume 111 of Graduate Texts in Mathematics. Springer- Verlag, New York, second edition, 2004. [13] A. W. Knapp, Elliptic Curves,volume 40 of Mathematical Notes. Princeton University Press, Princeton, NJ, 1992. [14] N. Koblitz, Introduction to Elliptic Curves and Modular Forms, volume 97 of Graduate Texts in Mathematics. Springer-Verlag, New York, second edition, 1993. [15] S. Lang, Elliptic Curves: Diophantine Analysis, volume 231 of Grundlehren der Mathematis- chen Wissenschaften. Springer-Verlag, Berlin, 1978. [16] E. Lutz, Sur l’equation y2 D x3 ax b dans les corps p-adic. J. Reine Angew. Math., 177: 237–247, 1937. [17] B. Mazur, Modular curves and the Eisenstein ideal. Inst. Hautes Etudes´ Sci. Publ. Math., (47): 33–186 (1978), 1977. [18] J. S. Milne, Elliptic Curves. BookSurge Publishers, Charleston, SC, 2006. [19] L. Mordell, On the rational solutions of the indeterminate equations of the third and fourth degrees. Math. Proc. Cambridge Philos. Soc., 21: 179–192,1922. [20] T. Nagell, Solution de quelque problèmes dans la théorie arithmétique des cubiques planes du premier genre. Wid. Akad. Skrifter Oslo I, 1935. Nr. 1. [21] H. Poincaré, Sur les surfaces de translation et les fonctions abéliennes. Bull. Soc. Math. France, 29: 61–86, 1901. [22] K. A. Ribet, On modular representations of Gal.Q=Q/ arising from modular forms. Invent. Math., 100(2): 431–476, 1990. [23] R. Schoof, Elliptic curves over finite fields and the computation of square roots mod p. Math. Comp., 44(170): 483–494, 1985. [24] C. L. Siegel, Uber¨ einige Anwendungen diophantischer Approximationen (1929). In Collected Works, pp. 209–266. Springer-Verlag, 1966. [25] J. H. Silverman, The ubiquity of elliptic curves. MAA Invited Lecture, JMM, Baltimore, 2003. www.math.brown.edu/˜jhs/UbiquityOfEllipticCurves.ppt. [26] ———, A quantitative version of Siegel’s theorem: integral points on elliptic curves and Cata- lan curves. J. Reine Angew. Math., 378: 60–100, 1987. [27] ———, Advanced Topics in the Arithmetic of Elliptic Curves,volume 151 of Graduate Texts in Mathematics. Springer-Verlag, New York, 1994. [28] J. H. Silverman, The Arithmetic of Elliptic Curves, volume 106 of Graduate Texts in Mathe- matics. Springer-Verlag, New York, second edition, 2009. [29] J. H. Silverman and J. Tate, Rational points on elliptic curves. Undergraduate Texts in Mathe- matics. Springer-Verlag, New York, 1992. [30] J. T. Tate, The arithmetic of elliptic curves. Invent. Math., 23: 179–206, 1974. [31] R. Taylor, Automorphy for some l-adic lifts of automorphic mod l Galois representations. II. Publ. Math. Inst. Hautes Etudes´ Sci., (108): 183–239, 2008. A Century of Elliptic Curves 131

[32] R. Taylor and A. Wiles, Ring-theoretic properties of certain Hecke algebras. Ann. of Math. (2), 141(3): 553–572, 1995. [33] L. C. Washington, Elliptic Curves. Discrete Mathematics and Its Applications (Boca Raton). Chapman & Hall/CRC, Boca Raton, FL, second edition, 2008. [34] A. Wiles, Modular elliptic curves and Fermat’s last theorem. Ann. of Math. (2), 141(3): 443– 551, 1995.

Mathematics Department, Box 1917, Brown University, Providence, RI 02912 [email protected]

Part II Historical Developments

The Mathematical Association of America Its First 100 Years

David E. Zitarelli Temple University

This article presents an overview of the history ofthe Mathematical Association of America as part of the celebration of its centennial in 2015. It describes events this author regards as the most important over the century but the account is certainly not exhaustive; for example, it makes little mention of competitions conducted under the aegis of the Association or of the expanded book publication program. Our account begins with the founding of the MAA and then describes its sections, governance, and meetings. Overarching activities are outlined in two distinct periods, 1916–1955 and 1955–2014, with an explanation for the separation into disjoint stages. The article discusses prizes and awards before ending with a brief mention of MAA headquarters.

Founding One of the most historicmoments for mathematics in America occurred with the establishment of a national organization on the last two days of 1915. It is rather miraculous that the Mathematical Association of America (MAA) was founded amidst World War I, a year after Canada entered the fray as a Dominion of the British Empire and sixteen months before the US Congress declared war. It is important to note that the use of “America” in the title of the Association includes both Canada and the US. As Albert Bennett wrote upon the fiftieth anniversary of the MAA in 1965, “The phrase ‘of America’ was interpreted from the start to include Canada and indeed the North American continent” [3, p. 1]. Since that time, members living in the Caribbean areas belong to the Florida Section of the MAA. The founding of the MAA took the reverse of the usual route whereby an organization is established first and creates its official journal later. For instance, the American Statis- tical Association was formed in 1839 but did not establish a publication for another 49 years. The American Mathematical Society (AMS) was quicker, being founded in 1888 but creating the Bulletin as its first periodical three years later. Yet the MAA’s official journal, The American Mathematical Monthly, was initiated more than twenty years before the As- sociation was founded. So in this case a journal (the Monthly) spawned an association (the MAA). After the MAA assumed the reins of the Monthly, its masthead read, “Founded in 1894 by Benjamin F. Finkel, published by him until 1913. From 1913 to 1916 it was owned and published by representatives of fourteen Universities and Colleges in the Middle West.”

135 136 A Century of Advancing Mathematics

Benjamin F. Finkel (1865–1947) Founder of The American Mathematical Monthly

Benjamin Franklin Finkel was a graduate of Ohio Northern University who had taught in secondary schools before being appointed professor of mathematics and physics at Drury College in Missouri. Finkel pursued graduate studies during summers at the University of Chicago before earning his PhD at the University of Pennsylvania. Finkel’s earlier classroom experience had made him keenly aware of poor instruction in elementary mathematics, and that inspired him to establish a journal “devoted solely to mathematics and suitable to the needs of teachers of mathematics in these schools”[8, p. 309]. He consulted with numerous high-school teachers and college professors but few teachers responded favorably, whereas he received enthusiastic support from several notable professors. The first issue of the Monthly had appeared in January 1894 with the title The American Mathematical Monthly: A Journal for Teachers of Mathematics in the Col- legiate and Advanced Secondary Fields. The task of physically producing a journal was not easy, especially mathematical typesetting, and it became a family affair, with Finkel carving most of the woodcuts himself while his wife Hannah Cokeley Finkel served as proofreader. Finkel met Leonard Dickson while studying at the University of Chicago in the summer of 1895. Seven years later Finkel invited Dickson to become an editor of the Monthly, and called his acceptance “a red-letter day in the history of the Monthly”[8, p. 314]. The cover of the January 1903 issue reads, “Edited by B. F. Finkel and Leonard E. Dickson.” By then it was apparent that college and university instructorsevinced a much greater interest in the Monthly than its intended audience, especially the problems department, which was central at the outset and has been a mainstay to this day. A “second red-letter day in the history of the Monthly”[8, p. 314] occurred in 1907 when Herbert Ellsworth Slaught replaced Leonard Dickson as editor along with Finkel. Slaught was a graduate of Colgate College who taught at the Peddie School near Princeton The Mathematical Association of America: Its First 100 Years 137 before matriculating at the University of Chicago, where he obtained his PhD in 1898 under E. H. Moore, four years after joining the mathematics faculty. Slaught was in charge of submissions during 1907 and 1908, an experience that heightened his awareness of the need for financial support for the Monthly beyond subsidies provided by Drury College and the University of Chicago. He soon obtained a matching subsidy from the University of Illinoiswiththe helpof a thirdeditor, G. A. Miller.But in 1912Benjamin Finkel confided to Slaught that the printer could no longer afford the low-cost services he had been providing. Undaunted, by the end of the year Slaught,along with Earle Hedrick (MAA president 1916) and Florian Cajori (MAA president 1917), arranged for eleven Midwestern universities and colleges to help defray the costs and to pass the Monthly legally from the private possession of Benjamin Finkel to the Board of Editors. Moreover, during 1913–1915 Slaught gained the conviction that more had to be done for the average mathematics teacher within the field of collegiate mathematics. Accord- ingly, he conducted an informal discussion at the dinner of the April 1914 meeting of the Chicago Section of the AMS about the role of collegiate mathematics in America. It was felt that, on one end of the spectrum, secondary mathematics was being handled well by existing secondary associations throughout the country, while at the other end, research interests were being fortifiedby the AMS. Yet, as Slaught wrote, “the great intermediate field of collegiate mathematics ... so far has had no organized attention” [20, p. 251]. Karen Parshall describes this division of labor as “the stratification of the American mathematical community” [17]. In late 1914 the resolute Slaught appealed to the Council of the AMS to appoint a committee to “consider the general relation of the Society to the promotion of teaching, especially in the collegiate field” [12, p. 20]. Over the next year two critical elements nudged closer together: the Monthly and a movement to emphasize collegiate mathematics. The

Herbert Ellsworth Slaught (1861–1937) MAA president 1919 138 A Century of Advancing Mathematics impetus was the search for a source of dedicated support for the Monthly, but Slaught had conceived a grander idea of founding a national body as well. Thereupon the AMS Coun- cil appointed the requested committee, which voted three-to-two that the AMS should take neither control nor responsibility for publishing the Monthly but resolved [2, p. 79; 12, p. 20]: It is deemed unwise for the American Mathematical Society to enter into the activities of the special field now covered by the American Mathematical Monthly; but the Council desires to express its realization of the importance of the work in this field and its value to mathematical science, and to say that should an organization be formed to deal specifically with this work, the Society would entertain toward such an organization only feelings of hearty good will and encouragement. [Emphasis added.] This resolution was endorsed by an overwhelming majority of the Council, which then adopted it. Events moved quickly after that decision. Herbert Slaught and a loyal band of supporters gave wide publicity to the idea of forming the kind of body the AMS resolved to encourage with good will. In June 1915 he mailed letters to mathematicians throughoutthe US and Canada soliciting feedback on this idea, enclosing a reply postcard. It met with unbridled support, but also sprinkled opposition. Finally, Slaught circulated a form letter seeking to identify those who favored such an organization, resulting in a call to an organizational meeting signed by 450 mathematicians representing every state in the US and province in Canada. Where and when should the organizational meeting be held? The obvious choice, it would seem, would be at the annual AMS meeting that December 27–28. However, the AMS met at Columbia University in New York City, whereas the majority of support for the new body came from the Midwest. Instead, the organizational meeting was held at Ohio State University on December 30–31, 1915, in conjunction with the annual meetings of Section A of the American Association for the Advancement of Science (AAAS) and the Chicago Section of the AMS. Ironically, the AAAS meeting had been scheduled for Toronto by organizer J. C. Fields but the outbreak of WWI cancelled those plans and the AAAS moved to Columbusinstead. Otherwise the MAA might have held its organizational meeting in Canada. Two of David Hilbert’s American students played decisive roles in the meeting—Earle Hedrick (who presided) and Will Cairns (who served as secretary pro tem). Overall 104 delegates attended, including ten women. Cairns reported, “After three hours of patient and painstaking deliberation, all mooted questions were settled except the name of the new organization” [4, p. 3]. A committee of three was appointed to select a name from amongst the eighteen that were submitted. The next day they voted independently, only to discover a “remarkable unanimity of purpose” [1, p. 29], as all three favored the Mathematical Associ- ation of America. Therefore the centennial of the MAA will occur on December 31, 2015. The response to the formation of the new organization was overwhelming. Charter membership was closed on April 1, 1916, at 1045. (While that number might seem small today, it represented 1/100,000 of the US population at the time.) By contrast, AMS membership that year was 732. Because the first names of women were recorded, I counted 132 females (12.6%), a total that is in line with a study that asserted, “About twelve percent of The Mathematical Association of America: Its First 100 Years 139

Earle Raymond Hedrick (1876–1943) MAA president 1916 the over 1000 charter members were women” [19, p. 29]. Twenty-nineof the 132 held PhD degrees at the time, accounting for 22.1% of 131 charter members who had obtained that degree. It is harder to distinguish African-Americans; the only charter member I know was Dudley Weldon Woodard. With its Midwestern origins one might assume that the majority of charter members wouldhail from there, yet the four leading states were New York (120), Ohio (72), Massachusetts (70), and Pennsylvania (67). A mere fifteen came from Canada. It was telling that only 145 of the 1045 were high-school teachers, confirming once again the wisdom of Benjamin Finkel’s emphasis on collegiate-level mathematics. Founding the Association obviated the need for university subsidies that had kept the Monthly afloat the previous three years. Thus Slaught’s idea to form an organization, with members’ dues paying for the journal, was brilliant. Herbert Slaught was truly the father of the MAA. To reinforce this designation, the title page for the January 1938 issue of the Monthly asserted: In appreciation of his exceptional services to the Mathematical Association of Amer- ica and to this Monthly, this volume is dedicated to the late Herbert Ellsworth Slaught, who at the time of his death was Honorary President of the Association and had served continuously as an editor of the Monthly for thirty years. The second day of the organizational meeting continued with passage of a constitution that had been tentatively prepared beforehand. In addition, a nominating committee presented a slate of officers: Earle Hedrick, president; E. V. Huntington and G. A. Miller, vice-presidents; and W. D. Cairns, secretary. Huntington, Miller, and Cairns were elected president for 1918, 1921, and 1943–1944, respectively. The MAA’s indebtedness to Ger- many was immediate—Hedrick and Cairns had obtained doctorates at Göttingen, Hunting- ton at Strassburg, and Miller had studied with Sophus Lie at Leipzig after receiving a PhD from Cumberland University (TN). 140 A Century of Advancing Mathematics

An appointed twelve-person Executive Council selected Alexander Ziwet and Karl Swartzel to negotiate with the owners of the Monthly to secure it as the ofﬁcial journal of the Association. As a result the Monthly was formally transferred to the MAA in time for the appearance of the January 1916 issue. This action underlines the fact that it was the journal that had spawned the Association! The organizational meeting ended with an illustrated address titled “The story of algebra” by L. C. Karpinski. In 1965 Brown University emeritus professor A. A. Bennett lyrically described the work of those mathematicians involved in establishing the MAA [3, p. 1]: Our Association was founded under especially auspicious circumstances. The many favorable factors were not accidents, nor miracles, nor achieved through serendipity. Some were the end results of a chain of events, not always desired, not always with the eventual outcome in view. But in large part they were secured through wise planning, tactful compromise, cajoling of the apathetic, courageous facing of pessimists in high places, and unremitting work. The MAA developed in several directions at once, perhaps the most inﬂuential being the establishment of MAA sections.

MAA sections In his November 1915 Monthly article “The teaching of mathematics,” Herbert Slaught stated that the object of the journal was “to stimulate activity on the part of college teachers . . . that may lead to production”[21, p. 291]. But he quickly added that “the formation of smaller groups ... will provide a far-reaching stimulus to individual activity” and he hoped “to see the college teachers of mathematics organized in every state, or even in some smaller groups.” Those smaller groups had a model to follow—theAMS. Its first section (Chicago) was established in 1896, a dozen years after its founding. Two sections followed: San Fran- cisco (1902) and Southwestern (1906). How long did it take the MAA to form sections? Minutes! No sooner had the constitution been approved than three states submitted formal applications to become sections—Kansas, Missouri, and Ohio. Which of the three state organizations became the first MAA section? Organizationally savvy, the Ohio group created a special committee at the conclusion of the first day of the two-day meeting to prepare itsown constitution.ConsequentlyOhio beat Missouri by a few minutes in the heated race; Kansas placed third. Therefore the Council of the MAA (the Board of Governors since 1938) acted expediently and granted Ohio’s application as the first section on March 1, 1916. But the honor of holding the first official meeting reverted to Kansas, which met in March. The Ohio Section held its initial meeting a month later for two days, a tradition that has continued every spring since then (except the war year 1945). The Missouri Section waited until November to hold its initial gathering. Four more MAA sections were formed during 1916, three from Midwestern states (Iowa, Indiana, and Minnesota) and the first one with a non-state geographical boundary (Maryland-Virginia-District of Columbia). Ten years later, Slaught singled out the seventeenth section—Philadelphia—as the first to bear the name of a city instead of a state or union of regions, because he had feared “the seeming apathy or lethargy” in the Atlantic The Mathematical Association of America: Its First 100 Years 141 states [27, p. 7]. Unlike Midwestern mathematicians, those in New England felt adequately represented by the Association of Teachers of Mathematics in New England until 1955, when the Northeastern Section was formed to also include the Canadian provinces of New Brunswick, Newfoundland, Nova Scotia, and Prince Edward Island. There were 28 sections when the MAA celebrated its semicentennial in 1965. Only one has been added since that time, the Intermountain Section, founded in 1975 when the Rocky Mountain Section was partitioned.

Officers and governance Since its foundingthe MAA has had a core of four officers: president, vice president, secretary, and treasurer. Its onlinesite lists all presidents with photographsand short biographies: www.maa.org/about-maa/governance/maa-presidents. During the first ten years the presidency was an honorary office held for one year. However, the term limit was increased to the present two years starting in 1927. Dorothy Bernstein was the first woman president, elected for 1979–1980. Due to increasing membership, today’s officers include, in addition to the core: president-elect, first vice president, second vice president, associate secretary, and associate treasurer.

Dorothy Bernstein (1914–1988) MAA president 1979–1980

By 1938 many MAA leaders had become concerned about the tenuous relationship between the Association and its sections, so the Trustees appointed the Committee to Review the Activities of the Association. This committee reported its ﬁndings in the Monthly two years later, whereupon the Trustees accepted the report and discharged the committee with appreciation. Its ﬁve recommendations provided the most extensive analysis of the structure and operations of the MAA since its founding in 1915. One of those recommendations replaced the Trustees with the now familiar Board of Governors to administer and control all 142 A Century of Advancing Mathematics

MAA scholarly and scientific activities. To exercise its fiduciary responsibility, the Board of Governors votes on dues and on reports from the treasurer, Budget Committee, and Au- dit Committee. The Board also approves the editor and editorial board of the Monthly, and some other journals. The Board was initially composed of six national officers, six governors elected at large, and fourteen governors elected by region. The term for governors at large was three years, so two were elected annually. However, the designation of regional governors soon proved to be unnecessarily cumbersome and so in 1945 the Board began phasing them out in favor of sectional governors serving three-year terms. Since that time the Executive Committee was formed to act on behalf of the governors on matters that arise between Board meetings. This Committee has also made recommendations on the management, policies, and activities of the MAA.

National and sectional meetings There are three kinds of mathematicians—those who can count and those who can’t. The assignment of numbers to initial MAA annual meetings seems to bear out this joke. The December 1915 organizational meeting was regarded as the first annual meeting because the next one was called second in the Monthly report.The 1917 annual meeting was called the third. Yet the 1918 annual meeting was also referred to as the third. Subsequent meetings were numbered after this one. The 1942 meeting was cancelled at the request of the Office of Defense Transport, the only time in MAA history that this annual affair did not take place, so the fortieth occurred in December 1956. However, after that meeting the Board voted “to hold the Annual Meeting normally scheduled for December 1958 during the latter part of January 1959”[9, p. 213].Up to that point,most annual meetings had been scheduled between Christmas and New Year’s Day (except three held in November during WWII) but they have all been held in January since the 41st in 1958. This explains why the 98th annual meeting took place in January 2015. Back in 1916, the second annual meeting was held December 28–30 at Columbia Uni- versity with 184 attending, including 141 MAA members. It was run in affiliation with the AAAS but with neither the AMS nor the American Astronomical Society, both of which met December 27–28 at the same place. The program consisted of a joint dinner on Thurs- day evening, four lectures on Friday, and one on Saturday morning. The first “third annual meeting” met in Chicago in conjunction with the Chicago Section of the AMS but it was not until the second such meeting in 1918 that the two professional organizations of mathematics met jointly. The philosophyof including Canada in the titleof the MAA was reinforced at the sixth annual meeting held in Toronto in 1921 in conjunction with the AMS, AAAS, and the APS (American Physical Society). This suggests that the titles of all four organizations implicitly included Canada. The MAA met jointly with the AMS and AAAS in 1922. Although Harvard was the official host, lectures were given at MIT as well, especially those held in conjunction with Section A of the AAAS. At a joint dinner Herbert Slaught spoke about the founding of the MAA and its prospects for future activities. Only three papers were delivered at the joint session but the MAA sponsored an additional six and the AMS 53. A symposium on mathematical statistics was also conducted at the meeting. The Mathematical Association of America: Its First 100 Years 143

The MAA held its first summer meeting September 1–2, 1916, at MIT, sandwiched between the organizational and second annual meetings. (“Annual meeting” refers to the December/January gathering, as opposed to “summer meeting,” though summer meetings are held annually as well.) Attendees wore badges with identifying names and institutions for the first time, a custom followed to this day. The nine invited lectures were delivered in MIT’s spanking new buildings while Harvard Yard dormitories provided accommodations. At the welcoming dinner on Friday night (9/1), a cablegram with fraternal greetings was received from Mittag-Leffler in Sweden. Except for three years during WWII, summer meetings were held after the U.S. Labor Day until 1952, when they were switched to the week before the national holiday. Twenty-five years later these gatherings were scheduled for earlier in August because many universities and colleges began their fall semesters the week before the three-day holiday. Moreover, a mandatory group photograph was shot through 1948. The largest meeting of the MAA (through 2014) was held in January 2012 in Boston with 7199 registrants. The attendance of 750 persons in 1958 encouraged both the MAA and AMS to meet later in January from that time forward. Moreover the MAA began to enlarge its program dramatically. Longtime MAA secretary H. M. Gehman reported, “The reasons for the expanded programs of meetings is due to the increase in the membership of the Association, the broadening of interests of the membership, and the increase in the activities in which the M.A.A. is engaged” [10, p. 579]. A historicchange for MAA national meetings took place in 1996 when the AMS voted to disband its summer gatherings. The MAA decided to continue alone, adopting the name “MathFest” starting in 1997, and has sponsored this meeting every summer since then. The MAA has sponsored two meetings a year up to the present time with only a few cancellations. We mentioned the only annual meeting cancelled (in 1942) but seven summer meetings were not held. Four of these were cancelled whenever an International Congress of Mathematicians was held in North America (1924 in Toronto, 1950 in Cambridge, 1974 in Vancouver, and 1986 in Berkeley) while the other three were in 1938 for the AMS semicentennial celebration, 1945 for WWII, and 1992 for the Canadian Mathematical Society. Today most sessions and lodgings of annual meetings are held in hotels, often in conjunction with a convention center, and not on a college campus. The first time the MAA did not hold a winter meeting on a college campus was 1929, when the site was the Hotel Fort Des Moines, but that did not happen again until Chicago in 1960. Only a few annual winter meetings have met at universities since then. There have been at least two controversial locations of annual meetings. Negotiations for sites take place years beforehand, and those for the January 1970 meeting began five years earlier. Yet in late October 1969, just ten weeks before the affair was scheduled to begin, officials in Miami (FL) informed meeting organizers that they could not supply the number of rooms agreed upon earlier. Fortunately Henry L. Alder (MAA president 1977– 1978) informed organizers about the sparkling new Convention Center in San Antonio, whereupon the location was hastily changed to that Texas city (and its renowned River Walk), which has now hosted more annual meetings than any other site. Another controversial annual meeting was held in Las Vegas in 1972. The combination of smoke-filled rooms and an “ambiance of gambling [that some found] intrusive and oppressive (and a handful found too tempting)” caused MAA and AMS leaders to resolve 144 A Century of Advancing Mathematics that “the Society not meet again in Las Vegas in this century” [18, p. 36]. Nonetheless, the AMS Western Section has held meetings there, and one is scheduled for 2015. The locations of some section meetings were controversial because of discrimination against African-Americans, particularly in the Southeastern Section of the MAA. As far as is known, no African-American attended a meeting in this five-state section up to 1951, when Fisk University chair Lee Lorch and three African-American colleagues attended the gathering in Nashville. However, they were prohibited from registering for the banquet to hear a talk by Saunders Mac Lane (MAA president 1951–1952). Shortly thereafter the Board of Governors responded to a letter from Fisk University faculty, not by changing its bylaws, as the group had requested, but by affirming its intention to conduct the affairs of the MAA without discrimination. Nonetheless, no African-American mathematicians attended Southeastern meetings again until 1960. However, when a group from Atlanta University was refused rooms at the meeting site, the Wade Hampton Hotel in Columbia (SC), they left in protest. It was not until the late 1960s and early 1970s that African- Americans felt comfortable enough to attend Southeastern meetings on a regular basis. Since that time minority mathematicians have participated in sectional governance and have presented invited addresses. In 1995 the report, “A history of minority participation in the Southeastern Section,” concluded [7, p. 10]: The Southeastern Section has been the birthplace of protest against discrimination on the basis of race. Confrontationshere have helped to move the MAA forward in the elimination of barriers in the mathematics community, not only for African- Americans, but for women, gays and lesbians, and other minorities. Today the Southeastern Section stands as a leader and a model in its determination to encourage and promote the full participation of all of its members.

MAA activities I: 1916–1955 Once the MAA had been established in 1915 itscharter members launched several projects, forming six committees in 1916 alone. The Committee on Relations with the Annals of Mathematics, chaired by E. H. Moore, entered into an agreement with Princeton, the journal’s publisher, whereby the MAA provided an annual subvention in exchange for the publication of expository articles and subscription rates at half price for MAA members. The MAA entered into a similar agreement with the Duke Journal when it was established in 1935, but both subventions were discontinued at the end of 1942 due to lack of resources. The most significant initial project was carried out by the National Committee on Math- ematics Requirements (NCMR), formed in March 1916 “to investigate the whole field of mathematical education from the secondary school through the college and to make recommendations looking toward a desirable reorganization of courses and the improvement of teaching” [25, p. 5]. NCMR issued a preliminary report at the MAA’s first summer meeting that September and a final report seven years later—a detailed, 650-page document titled The Reorganization of Mathematics in Secondary Education. Part I consisted of separate sets of recommendations for mathematics to be taught in grades 7–9 and 10–12. Part II was made up chiefly of reports by specialists, like Earle Hedrick’s chapter stressing functionsas a unifying theme across the entire curriculum. The last chapter contained a bibliographyof 569 items on the teaching of mathematics that had been published from 1911 through1921. The Mathematical Association of America: Its First 100 Years 145

The NCMR report was influential in establishing the basic contour of the high-school curriculum for several decades but jousting over the aims in teaching mathematics continues to this day. However, no major change in practice occurred through the 1930s due mainly to the Great Depression and the inertia of maintaining traditional educational practices. One important byproduct of this work was the founding of another professional organization of mathematicians in 1920, the National Council of Teachers of Mathematics (NCTM). Like the MAA, the NCTM adopted an existing publication as its official journal, The Mathematics Teacher. The impact of the NCMR was generally positive but not the work of the Committee on Standard Departments, charged with formulating standards for undergraduate departments. As we know from NCTM documents formulated in the 1990s, the term “standards” can be controversial, and it was no different back then, as the committee made little progress and was allowed to lapse within two years. Another ill-fated action from the early years of the MAA was the approval of a resolution by L. C. Karpinski in favor of the metric system of measure. Two other MAA committees from the early years met some success but ultimate disappointment. The Com- mittee on Libraries languished in 1939 because few members availed themselves of its loan services and its holdings were shelved in the much larger AMS library at Columbia University. However, even the AMS library was discontinued in 1950. Another initiative whose perceived needs never came to fruition was the Dictionary Committee, which proposed in 1917 that the MAA publish a two-volume dictionary covering all terms that would be used chiefly by college students. This project initially met a favorable response but funds were never allocated and although the committee met as late as 1937, it had been mostly inactive during those twenty years. The MAA never published such a dictionary but in 1942 Glenn James published Mathematics Dictionary commer- cially and it ran through five editions. These few disappointments paled in comparison with MAA successes, like the establishment of mathematics clubs at the undergraduate level. Reflecting this development, the Monthly initiated a column “Undergraduate Mathematics Clubs” in 1918 that continued under different names until 1954. The printing costs problem raised its ugly head again in 1920, when the MAA was forced to increase dues by a dollar from the initial $3 per year. However, the resulting $4 dues remained in place until 1958, causing Lester R. Ford (MAA president 1947–1948) to joke that the three most important constants in mathematics were , e, and 4 [11, p. 105]. To address rising costs, the MAA was incorporated in 1920 under the statutes of the state of Illinois, an important move that allowed the Association to receive donations and bequests; the AMS followed suit three years later. The MAA took advantage of this status right away when Herbert Slaught announced a sizable gift from Mary Carus, the editor of Open Court Publishing Co. The purpose of her gift was to fund a series of books to publish mathematical exposition at nominal cost. The first Carus Mathematical Monograph was Calculus of Variations by G.A. Bliss (1925) and the second Functions of a Complex Variable by D. R. Curtiss (1926); Curtiss was MAA president 1935–1936. By 2014, 31 books had appeared in this popular series. The MAA engaged in one unusual enterprise when it published the two-volume Rhind Mathematical Papyrus in 1929. The translation of the oldest mathematics book in existence 146 A Century of Advancing Mathematics was carried out by 84-year-old Arnold B. Chase, who added elaborate mathematical commentary. He was aided by his wife (an artistwho copied the hieratic scriptand produced the corresponding hieroglyphic transcription), L. S. Bull (an Egyptologist with the Metropoli- tan Museum of Art), and Brown mathematician H. P. Manning (who had become an expert in hieratic script in his sixties and had taught Chase mathematics back in the 1890s). In his retiring presidential address published in 1932, John Wesley Young (MAA president 1929–1930) called for the MAA to (1) provide funds to send national leaders to speak at sectional meetings in remote locales, (2) launch a second periodical with particular appeal to undergraduate students, (3) initiate a Committee on Publicity, and (4) form a competitive national examination like the one given in Hungary [26]. All four proposed initiatives have now become part and parcel of MAA activities, but none was instituted at once. The MAA’s second journal grew out of the effort of S.T. Sanders of Louisiana State University (LSU) to encourage high school teachers to join the Association. In 1926 he began publishinga monthly pamphlet and the next year expanded it into a magazine. How- ever, financial support from LSU was terminated 1942 due to university budget constraints. Deficits mounted alarmingly! The MAA provided subsidies but even that dried up in 1945, whereupon the journal abruptly ceased publication. Fortunately, in 1947 Glenn James assumed sponsorship, resumed publication, and shortened the title to Mathematics Magazine from the name National MathematicsMagazine that had been used since1934.Butby 1959 deteriorating eyesight caused him to negotiate and the December 1960 issue revealed the complete transfer to the MAA. In 1974 the MAA initiated its third periodical, now called The College Mathematics Journal, which had been published by Prindle, Weber & Schmidt as The Two-Year College Mathematics Journal the previous four years. (The name change came in 1984 to better reflect the evolving content.) During the interwar years the MAA sponsored several studies of undergraduate mathematics courses, especially at the freshman and sophomore levels. At the 1922 summer meeting, J.W. Young proposed a general one-year course for freshmen that would meet the needs of those pursuing upper-level mathematics as well as those not continuing. Like today, there was little agreement on the contents of such an integrated course. Five years later an MAA committee headed by A. A. Bennett supported traditional instruction during the first two years but suggested that a certain amount of historical and philosophical background could provide additional mathematical concepts. The committee also drew up a suggested list of readings, a precursor to today’s Basic Library List. The Great Depression exacted a distressing toll on the profession but the MAA addressed the employment problem directly. In 1932 the Board discussed the training of PhD students for teaching at junior colleges or high schools due to the scarcity of available professorships. The next year those Trustees asked Arnold Dresden (MAA president, 1933–1934) to appoint a Commission on the Training and Utilization of Advanced Stu- dents in Mathematics. Under the leadership of Northwestern University chair E. J. Moulton, the Commission cautioned graduate students considering teaching in secondary schools to satisfy legal requirements of any state where they might teach. In 1934 the Commission issued a report describing the unemployment situation facing the 180 mathematics PhDs seeking positions for 1934–1935. Only fourteen were unemployed but many others held “makeshift” positions [16, p. 143]. Yet the Commission forecasted correctly that demand The Mathematical Association of America: Its First 100 Years 147 would soon exceed the supply of PhDs so no further action was taken and the situation improved markedly in later years. The other big activity taken in response to external demands of the profession tookplace when the MAA acted swiftly after war broke out in Europe in September 1939. Within a month the Board of Trustees authorized W. B. Carver (MAA president 1939–1940; chair at Cornell) to consult with AMS president G. C. Evans (the chair at UC Berkeley) about appropriate measures regarding national defense. The result was the joint War Prepared- ness Committee that presented its report at the summer 1940 meeting held at Dartmouth College [15]. The report listed three recommendations: 1) competent secondary students should take the maximum amount of mathematics courses available, and colleges should offer courses in 2) mechanics, probability, surveying, navigation, and other essentials of military science, and 3) applied mathematics at the graduate level. This Committee also compiled and maintained a register of vacancies and availability of mathematicians for service throughout the war.

MAA activities II: 1955–2014

What is the distinctive role of the MAA? The founders viewed the pedagogy of collegiate mathematics as its distinguishing characteristic yet over its ﬁrst forty years the Association mainly held meetings, elected ofﬁcers, selected hour speakers, and published a journal, all of which duplicated AMS actions. The critical event that separated these two national organizations occurred during the presidencies of Edward J. McShane and William L. Duren Jr. 1953–1956. These two leaders had been friends from their undergraduate years at Tu- lane and graduate studies at Chicago, where both earned PhDs in 1930. Duren returned to Tulane upon graduation and taught there until 1955. He then joined McShane at Virginia, where the latter had been since 1935.

Edward J. McShane (1904–1989) MAA president 1953–1954 148 A Century of Advancing Mathematics

William L. Duren, Jr. (1905–2008) MAA president 1955–1956

Both McShane and Duren had established solid reputations in research by 1955, which was typical of most MAA leaders. But their experiences in teaching propelled them into roles that altered the character of the MAA. Before assuming the MAA presidency in Jan- uary 1953 McShane had settled upon the idea that the cornerstone of his term would be to dramatically improve undergraduate instruction in mathematics. His first action was to establish the Committee on the Undergraduate Mathematical Program (CUP), where by “program” he meant not only the curriculum but faculty and students as well. He appointed Duren to chair CUP. The idea of organizing a national committee to deal with the undergraduate program seems commonplace today but in retrospect this was an unusual move for the MAA and would be the first in a series of broad strokes that would distinguishit from the AMS. This was not the first MAA committee to deal with curricular issues, as noted above. However, even the massive NCMR report from 1923 dealt only with secondary education. Duren was the perfect choice to head CUP. As chair at Tulane he had established a graduate program in 1947 with grants from the Office of Naval Research administered by Mina Rees. But the program had a low success rate, which Duren attributed to poor undergraduate training. His idea was to replace the traditional freshman college algebra course by calculus or by a general analysis course including calculus. CUP acted with dispatch and filed a report to the Board at the 1953 summer meeting. The major recommendations that the Governors adopted at once were to 1) concentrate on first-year courses for able students, 2) articulate with physicists and engineers, and cultivate contacts in the social sciences, 3) avoid overemphasizing abstract topics, and 4) enlist college instructors in the process. Two future MAA presidents were instrumental in these activities: Albert Tucker (1961– 1962) and G. Baley Price (1957–1958). Duren had met Tucker during a sabbatical year at The Mathematical Association of America: Its First 100 Years 149 the Institute for Advanced Study in the mid-1930s, and credited him with getting the idea of replacing the college algebra course with calculus accepted: “Tucker gave it the prestige of Princeton, which was essential to its general acceptance” [6, p. 3]. Baley Price, as chair at Kansas, obtained a small grant and devoted space from his University to support CUP members to meet during the next summer to begin to carry out details. Up to this time it had been difficult for a national group to operate in this manner. As a result, in August 1954 CUP issued a revolutionary report urging the adoption of one universal freshman course. What set CUP apart from its predecessors was that it not only issued a set of recommendations for the course, but it produced source material for what it called the Universal Course. This material was tested at Tulane and several other institutions the next year. CUP reached a crossroads by 1958 when it realized that although its work had been restricted to mathematics in the freshman year, only the course designed for students in the social sciences (mostly under the direction of John Kemeny at Dartmouth) had found success. But the calculus part was generally unsuccessful and the overall program for mathematics majors remained unchanged from a half century earlier. CUP members felt strongly that the time was ripe for an expanded group to investigate the matter, resultingin a conference that November to examine its work over the prior five years and to assist in formulating plans and policies for its future work. Two of the reports from that conference described progress that had been made at the pre-college level on what came to be called the New Math. One was by Max Beberman, the head of the University of Illinois Committee on School Mathematics, which had been formed in 1951 to develop a four-year program of high-school mathematics based on conceptual understanding as well as manipulative skills. With the help of Bruce Meserve at Vermont, the group produced mimeographed notes that were distributed to high-school teachers throughout Illinois. Those notes were then revised based on feedback and redis- tributed the following year. The initial aim was to be experimental but by the 1960s this aspect got lost because those mimeographed notes were converted into the High School Mathematics textbook series published by Heath starting in 1964. The other report anticipating the New Math was given by Edward G. Begle, who had just been appointed as director of the School Mathematics Study Group (SMSG), an eight- person group created by the AMS. The launch of Sputnik in the fall of 1957 helped SMSG gain substantial support from NSF. However, the AMS distanced itself from the project’s further activities and neither the Bulletin nor the official book on the history of the So- ciety 1938–1988 (edited by Everett Pitcher) contains any mention whatsoever of SMSG. Nonetheless, efficient writing groups produced books that were adopted worldwideand developed ways of testing their success. Ultimately, however, parents and teachers joined in a massive public backlash against the New Math by the end of the 1960s and consequently SMSG fell out of favor. Five other reports from the November 1958 CUP conference indicate other types of activities that were of prime interest at the time: 1. Films and television for mathematical instruction 2. The role of numerical analysis in the undergraduate program 3. Undergraduate statistics in a mathematics department 150 A Century of Advancing Mathematics

4. The mathematical training of social scientists

5. The relation of mathematics to physics instruction. For our purposes, however, the most important report was given by W. L. Duren, who reviewed CUP accomplishments over the preceding six years. He noted that although CUP had made no specific curricular suggestions, its members felt strongly that they had completed their mission, yet they recommended that a much broader study be carried out by an expanded group.The name of the new group changed slightly,the Committee on the Under- graduate Program in Mathematics (CUPM), but it included most of the original members. R. Creighton Buck was appointed chair. CUPM set about expanding its investigation into the entire undergraduate program and five years later the Committee published its most important document. Over that period CUPM issued several vital reports that laid the foundations for the critical one in 1963. One of CUPM’s first official acts was to establish a Panel on Teacher Training under John Kemeny with a charge to prepare a set of recommendations of minimum standards for the training of teachers on all levels. Another initiative was to establish a Consultants Bureau to aid colleges and universities in upgrading and revising their present undergraduate mathematics offerings or with planning new curricula. CUPM also conducted a massive survey to learn about course offerings at smaller institutions; its findings were very reveal- ing and paint a telling picture of the mathematics undergraduate program circa 1960. In addition CUPM formed the Panel on Physical Sciences and Engineering to investigate the mathematics curriculum as it related to those clients. Arguably the most important group that CUPM assembled was the Panel on Pregradu- ate Training (PPT), initially chaired by Berkeley’s John C. Moore. PPT began in 1960 by constructing an idealized program suitable for honors students in mathematics. Two years later the Panel issued a report in the Monthly listing its recommendations for an honors program in mathematics that had received the approval of the full CUPM beforehand [14]. PPT recommended a modicum of the algebra of vector spaces as well as the introduction of appropriate geometric and topological concepts into the twelve-hour calculus sequence. Beyond that, the Panel endorsed a one-semester course on linear algebra at the sophomore level. Regarding upper-level courses, PPT listed five (or six) courses that every college should offer its majors: two in real analysis, one in each of abstract algebra, complex analysis, and geometry-topology, and either probability or mathematical physics. Of the four panels that performed a major portion of the CUPM work, PPT became the most influential. The critical CUPM report presented the PPT’s set of recommendations in 1963. The next year CUPM executive director A. B. Willcox wrote a rather extensive, though whimsical, summary for the Monthly. The crux of the recommendations was to bridge the gap between undergraduate instruction and contemporary mathematics by “an idealized program which, while keeping in touch with reality, would also help set the pace for curricular improvements for some years to come” [24, p. 1120]. Alan Tucker recently described how, in spite of all of CUPM’s work and its vetting process, the curriculum proposed in the 1963 report was soon seen to be overly ambitious[23]. To compensate, two years later CUPM issued a revision recommending a watered-down version. The full set of recommendations in the 1965 report was sent to all mathematics departments, with complimentary copies available for MAA members. CUPM provided The Mathematical Association of America: Its First 100 Years 151 course outlines as samples for all lower-division and most upper-division courses. In an attempt to gain grassroots support, the Committee asked MAA sections to evaluate the proposed curriculum and publish their findings in the Monthly. Many did, and almost every issue contained sectional reports on the CUPM recommendations over the next few years. Nonetheless, the actions taken by CUPM in its reports from 1963 and 1965 were mon- umental and allowed William Duren to answer the question, What was the raison d’être of the MAA? When the MAA celebrated its semicentennial he concluded (with brutal hon- esty), “I finally decided that [the] MAA existed to give comfort and status to college mathematicians ...This role of the MAA continues today [1967], but no longer as its only role” [5, p. 24]. The vetting process of the 1965 CUPM report by MAA sections and numerous colleges and universities over the next several years showed that it too was overly ambitious. Consequently the Committee proposed dramatic changes when it issued a new set of recommendations in 1972. This time the MAA issued a two-volume, 700-page publication that included a 64-page extensive revision of the earlier report called “Commentary on ‘A general curriculum in mathematics for colleges’.” It also contained a 32-page Basic Library List. The gist of the “Commentary” was a curriculum of twelve courses aimed at what the CUPM regarded as the three major problem areas: 1) the evolving nature of mathematics, 2) the service functions of mathematics departments, and 3) prerequisites for entry into the program and requirements for graduation. Regarding 1) the Commentary suggested using the broader term “mathematical sciences” to account for the subject’s growing intercon- nections with other human endeavors. Was calculus still the bedrock for the mathematical sciences? CUPM nodded in the affirmative. The report also recommended multiple curricula to account for 2). For 3), although CUPM did not go beyond recommending that the mathematics curriculum provide suitable points of entry for all students, the MAA ultimately became involved in constructing its own placement tests for use by colleges. The curriculum for mathematics majors that CUPM recommended in its 1972 report proved to be poised at the right level and was flexible enough to accommodate future changes. It has basically withstood the test of time over the past 42 years, with slight alter- ations since then to address ongoing changes in the nature of mathematics. For instance, the report recommended two linear algebra courses, one essentially dealing with Rn and the other proof theoretic (dealing with vector spaces over fields, triangular and Jordan forms of matrices, dual spaces and tensor products, bilinear forms, and inner product spaces). CUPM also advised that every college offer courses in abstract algebra and applied mathematics, though the latter proved to be problematic for many institutions. One other notable recommendation was that probability and statistics should be offered over two semesters and not combined into one. The flexibility built into the 1972 report allowed the mathematics major to seamlessly adjust to two developments since then. One was the “Introduction to Proofs” course aimed at easing the transition from computational to conceptual mathematics. The other was the reduction to one sophomore-level course in linear algebra that continues to have wide variation. Alan Tucker has singled out the period 1955–1974 as the Golden Age of Mathe- matics Majors, and CUPM played an important role in shaping the curriculum during that period [23]. 152 A Century of Advancing Mathematics

The 1972 report was not the last one issued by CUPM. A successor in 1981, for instance, recommended a multi-track structure to account for various alternatives within the mathematical sciences. Later activities brought CUPM into collaboration with NCTM in recommending sometimes controversial standards. Even now this influential Committee is preparing an updated report in 2015. The emphasis here has been on CUPM’s recommendations on the undergraduate program for mathematics majors but from its founding in 1958 the Committee also formed subcommittees on teacher training, applied mathematics, and statistics whose findings were instrumental in the lists of recommendations. Later CUPM added panels on two-year colleges, computer science, and minority participation to address the evolving nature of college mathematics and thosewho teach and study it. Most of these committees are no longer linked to CUPM. My history of the MAA EPADEL Section shows that up to the 1950s, its leaders were research mathematicians and its annual meetings generally featured lectures on cutting- edge advances. After that time the section began to expand horizons into various pursuits under the direction of a new generation of college professors. That account reads, “From 1956 to 1978 the character of the section changed from one devoted almost exclusively to the development of mathematics to one that sponsored a variety of activities on pedagogical and curricular themes” [27, p. 137]. These new activities included high-school contests, a newsletter, a panel on industrial opportunities, sessions of undergraduate speakers, competitions, films, presentations and workshops on curriculum and pedagogy, and special interest groups. The national MAA has been involved in all of these activities, and more, since that time and up to the present. One of the most successful was the result of an MAA Task Force formed in 1988 to address the issue of communities underrepresented in mathematics. Two years later the MAA established Strengthening Underrepresented Minority Math- ematics Achievement (SUMMA) to increase the representation of minorities and improve the education of minorities in mathematics. The initial full-time staff consisted of William Hawkins and Florence Fasanelli. During 1991–1997 SUMMA sponsored five conferences for project directors through its Consortium of Intervention Programs, two national conferences devoted to the issue of attracting minorities into teaching mathematics (with proceedings published by the MAA), and a survey of minority graduate students in the mathematical sciences. In 1994 the MAA began sponsoring the David Blackwell Lecture at its annual winter meeting, named in honor of the distinguished African-American mathematician; it has been sponsored solely by the National Association of Mathematicians at MathFests since 2006. In the second half of the 1990s SUMMA developed an archival record of minority PhDs; implemented calculator-based, technical-assistance projects; conducted calculator workshops for tribal college faculty; and published a poster on African and African-American pioneers in mathematics. Since 2000 SUMMA inaugurated the Na- tional Research Experiences for Undergraduates Program and developed the Tensor program designed to encourage the pursuit and enjoyment of mathematics among middle school students, high school students, and beginning college students from groups traditionally underrepresented in mathematics. Similarly, the Association of Women in Mathematics (AWM) was established to improve the status of women in mathematics, from changing attitudes about girls’ ability to The Mathematical Association of America: Its First 100 Years 153 learn mathematics as early as elementary school to ending discrimination against professional women in mathematics. Althoughthe MAA was not directly involved in the founding of AWM, in a speech at an MAA meeting in 1990 Judy Green noted: The [AWM] was formed in January 1971 with a goal of encouraging the participation of women in mathematics and, at the summer meeting that year, the MAA sponsored a panel entitled “Women in Mathematics.” In January 1974, the Board of Governors approved a recommendation “that the MAA participate in a joint committee with AMS in an investigation of the status of women in Mathematics.” That Joint Committee still exists and has been expanded to include representatives of the other mathematical organizations.... The MAA Committee on the Participation of Women was ... formed [in] 1987. The AWM inaugurated a series of lectures at the 1996 MathFest to enhance the attraction of women and minorities into scientific careers. These lectures became a joint effort of the AWM and the MAA eight years later, when the name was changed to Etta Z. Falconer Lectures. The Blackwell and Falconer lectures are but two of several that the MAA sponsors each year to honor a mathematician. The earliest were Hedrick Lectures, named after the first president when established in 1952; these three-lecture series have been held at summer meetings and MathFests. Two lecture series have been created in the last sixteen years. The Leitzel Lectures were established in 1998 for the improvement of mathematical sciences education to honorJames R. C. Leitzel for his efforts in improvingthat field. And the Porter Public Lectures, established in 2010 and sponsored jointlyby the MAA, AMS, and SIAM, deal with a mathematical topic accessible to the broader community; they honor former MAA treasurer and now retired University of Pennsylvania mathematician Gerald Porter and his wife Judith, a retired professor of sociology at Bryn Mawr College. Another successful MAA activity, launched in 1994, is Project NExT (New Experi- ences in Teaching), a professional development program for new and recent PhDs in the mathematical sciences that introduces them to senior established mathematicians, which had been very hard until then. The program helps to ease the graduate-student-to-faculty- member transition by addressing all aspects of an academic career: teaching, scholarship, and professional activities. Workshops, national meetings, and an electronic discussion list help Fellows develop a network of peers and mentors as they assume their new responsibilities. Almost 1500 Project NExT Fellows have been chosen to date. At national meetings they wear different colored dots on their badges to help them identify each other by year; thefirst ones wear red dotsand a recent Monthly author described himself as “a ‘brown-dot’ Project NExT fellow” [22, p. 915]. Many Fellows have become national leaders; for example, 1996 Fellow (blue dot) Francis Su is MAA President-Elect for 2014. The program was initially funded by the Exxon Education Foundation (now the ExxonMobil Foundation). The biggest current funder is the Mary P. Dolciani-Halloran Foundation. In 1999 an MAA Task Force headed by Ed Dubinsky and Ann Watkins reviewed the Association’s status heading into the 21st century. The most successful of the Task Force’s five priority action recommendations seems to have been the formation of Special Interest Groups—SIGMAAs. Designed to take advantage of affordable online communication, the first specially focused group was formed in January 2000—SIGMAA on RUME (Research in Undergraduate Mathematics Education). Two more SIGMAAs were formed over the 154 A Century of Advancing Mathematics next two years, whereupon the Task Force was disbanded and replaced by the standing Committee on SIGMAAs (now headed by Amy Shell-Gellasch). As of 2014 there were a dozen active SIGMAAs, with another in the works.

Prizes and awards In 1925 MAA president Julian L. Coolidge donated money to establish the MAA’s first award, the Chauvenet Prize for an outstanding expository article by an MAA member. The first recipient was G. A. Bliss. By the time the second Chauvenet Prize was given to T. H. Hildebrandt in 1929 the funding for the award had been supplemented by donations from the next two MAA presidents, Dunham Jackson (1926) and Walter B. Ford (1927). The Prize was awarded every three years from 1929 until 1963, when it began being awarded annually. The MAA has created eight other writingawards since then. The first was the Lester R. Ford Award, established in 1965 for expository articles published in either the Monthly or Mathematics Magazine. Six awards were given that year. In 1976 the Allendoerfer Award was created for articles in Mathematics Magazine. From that time onward Ford Awards were restricted to the Monthly, with up to five given each year. In 2012 the Board of Gov- ernors designated this the Halmos-Ford Award. The other writing prizes are (1) the Pólya Award, established in 1976 for expository articles published in The College Mathematics Journal, (2) the Beckenbach Book Prize, funded in 1986 as the successor to the MAA Book Prize, which had been created four years earlier, (3) the Hasse Prize, funded in 1986 by an anonymous donor for expository papers appearing in an Association publication, (4) the Evans Award, established by the Board of Governors in 1992 and first awarded in 1996, for articles in Math Horizons accessible to undergraduates, (5) the Robbins Prize, established in 2005 for outstanding papers in algebra, combinatorics, or discrete mathematics, and (6) the Euler Book Prize, established in 2005 with a gift from Virginia and Paul Halmos. The MAA did not create any awards after the Chauvenet Prize until 1961, when the Board of Governors established the Award for Distinguished Service to the Association. It was intended to be the MAA’s most prestigious honor for service; the first recipient (in 1962) was Mina Rees. The endowed successor to this award is the Gung and Hu Award for DistinguishedService to Mathematics, which was first presented in 1990 (to Leon Henkin). The MAA has created four other awards for service since the late 1970s. The first was the Certificate of Merit, given at irregular intervals for special work or service associated with the mathematical community. The first recipient was Henry M. Cox in 1977. Six years later the Board of Governors established a Certificate for Meritorious Service for service generally to an MAA section. The first certificates were presented in 1984. Four years later the Joint Policy Board for Mathematics created a CommunicationsAward to reward and encourage communicators for informingthe publicabout mathematical ideas. (The JPBM is a collaborative effort of the MAA with the AMS, SIAM, and the ASA.) Finally the Dolciani Award was established in 2012 for mathematicians who make distinguished contributions to the mathematical education of K–16 students in the United States or Canada. In additionto awards for writingand service, the MAA began to create prizes for teaching about 25 years ago. The first was the Award for Distinguished College or University The Mathematical Association of America: Its First 100 Years 155

Teaching of Mathematics established in 1991 to honor annually at most three extraordinarily successful teachers whose influence extended beyond their own institutions. Two years later it was renamed in honor of Deborah and Franklin Tepper Haimo; D. T. Haimo was MAA president 1991–1992. In 2003 the Alder Award for Distinguished Teaching by a Be- ginning College or University Mathematics Faculty Member was created for undergraduate mathematics teachers. A recent award combines teaching and research—the Selden Prize for Research in Undergraduate Mathematics Education, was established in 2004 for teachers with significant records of published research in undergraduate mathematics education. In 1926 Elizabeth Putnam created a trust fund to encourage team competition in college studies. Upon her death eight years later, the Putnam Competition assumed its present form and was administered by the MAA, which established an award for undergraduate students when the examination was first held in 1938. While that award is named for her husband, William Lowell Putnam, the Elizabeth Lowell Putnam Prize was established in 1992 for women with the highest scores. Since 1995 the Frank and Brennie Morgan Prize has been awarded jointly by the MAA, AMS, and SIAM to undergraduates for outstanding original work.

MAA headquarters

The MAA lived a vagabond existence over its first sixty years, with headquarters located at the home institution of its secretaries. Those officers passed along the massive records from one to another up to 1968, when the MAA set up its headquarters in Washington DC. Ten years later a permanent location was established when the Association purchased a three-building complex at 1529 Eighteenth Street NW in the same city. That location has served as the Association’s headquarters since then. The acquisition of the properties was aided by two anonymous members who pledged substantial amounts of money toward the project. In 2002 a $3 million donation from Paul and VirginiaHalmos restored one of the three buildings, the Carriage House, which is located behind the two buildings housing MAA headquarters and the Washington DC offices of the AMS. The Carriage House serves as a mathematical sciences conference center. The MAA used the gift to renovate the interior completely. The Carriage House was builtin 1892, eleven years before the buildinghousing MAA headquarters.

Conclusion

The MAA was founded in late 1915 by an exuberant group dedicated to the interests of collegiate mathematics. H. E. Slaught (the father of the Association) and E.R. Hedrick (the ﬁrst president) were at the forefront of this movement. With a large membership from the beginning, the Association engaged in many activities from the founding up to the mid- 1950s. Most of these activities were similar to AMS actions, especially meetings at the national and sectional level. The second half of the decade saw the MAA distinguish itself from the AMS under the leadership of E. J. McShane and W. L. Duren Jr. The formation of CUPM was, and remains, critical for deﬁning the undergraduate mathematics major. 156 A Century of Advancing Mathematics

Bibliography [1] Anon, Notes on the Columbus meeting, Amer. Math. Monthly 23 (1916) 29–30. [2] R. C. Archibald, A Semicentennial History of the American Mathematical Society 1888–1938. American Mathematical Society, New York, 1938. [3] A. A. Bennett, Brief history of the Mathematical Association of America before World War II, Amer. Math. Monthly 74 (1967; No. 1, Part 2, Fiftieth anniversary issue) 1–11. [4] W. D. Cairns, The Mathematical Association of America, Amer. Math. Monthly 23 (1916) 1–6. [5] W. L. Duren, Jr., CUPM, the history of an idea, Amer. Math. Monthly 74 (Part 2; 1967) 23–37. [6] ———, A career as a scientiﬁc generalist based in mathematics. www.wldurenjrmemorial.net/docs/ACareer.2.pdf. [7] E. T. Falconer, H .J. Walton, J. E. Wilkins, Jr., A. A. Shabazz, and S. T. Bozeman, A history of minority participation in the Southeastern Section, April 1995. This report, a supplement to “Threescore and ten: A history of the Southeastern Section of the Math- ematical Association of America 1922–1992,” sections.maa.org/southeastern/ minority/minority.html. [8] B. F. Finkel, The human aspect in the early history of the American Mathematical Monthly, Amer. Math. Monthly 38 (1931) 305–320. [9] H. M. Gehman, The fortieth annualmeeting of the Association, Amer. Math. Monthly 64 (1957) 212–215. [10] ———, The Washington conference, Amer. Math. Monthly 65 (1958) 575–586. [11] ———, Financial history, in [13, pp. 104–110]. [12] P. S. Jones,Historical background and founding of the Association, in [13, pp. 1–23]. [13] K. O. May (ed.), The Mathematical Association of America: Its First Fifty Years. MAA, Wash- ington, DC, 1972. [14] J. C. Moore, et al, Preliminary recommendations for an honors program, Amer. Math. Monthly 69 (1962) 976–979. [15] M. Morse, Report of the War Preparedness Committee, Amer. Math. Monthly 47 (1940) 500– 502. [16] E. J. Moulton, The unemploymentsituation for PhD’s in Mathematics, Amer. Math. Monthly 42 (1935) 143–144. [17] K. H. Parshall, The stratiﬁcation of the American mathematical community: The Mathematical Association of America and the American Mathematical Society, 1915–1925, in A Century of Advancing Mathematics, edited by S.F. Kennedy, MAA, Washington DC, 2015. 159–175. [18] E. Pitcher, A History of the Second Fifty Years: American Mathematical Society, 1939–1988. American Mathematical Society, Providence, RI, 1988. [19] F. A. N. Rosamond,A century of women’sparticipation in the MAA and other organizations,in Winning Women into Mathematics, edited by P.C. Kenschaft and S.Z. Keith, MAA, Washington DC, 1991. 31–53. This informative paper does not appear to be widely known today. [20] H. E. Slaught, The promotion of collegiate mathematics, Amer. Math. Monthly 22 (1915) 251– 253. [21] ———, The teaching of mathematics, Amer. Math. Monthly 22 (1915) 289–292. [22] W. Traves, From Pascal’s theorem to d-constructible curves, Amer. Math. Monthly 120 (2013) 901–915. [23] A. Tucker, The history of the undergraduate program in mathematics in the United States, in A Century of Advancing Mathematics, edited by S.F. Kennedy, MAA, Washington DC, 2015. 219–237. The Mathematical Association of America: Its First 100 Years 157

[24] A. B. Willcox, Pregraduate training in mathematics—A report of a CUPM panel, Amer. Math. Monthly 71 (1964) 1117–1129. [25] J. W. Young, The work of the National Committee on Mathematics Requirements, Mathematics Teacher 14 (1921) 5–15. [26] ———, Functions of the Mathematical Association of America, Amer. Math. Monthly 39 (1932) 6–15. [27] D. E. Zitarelli, EPADEL: A Semisesquicentennial History, 1926–2000. Raymond-Reese Book Co., Elkins Park, 2001. Available upon request from the author, and online at www.personal.psu.edu/ecb5/EPaDel/Zittarelli/EPL0_Intro.html.

3441 St. Louis Avenue Minneapolis, MN 55416 [email protected] The cover of the ﬁrst issue of The American Mathematical Monthly. The Stratification of the American Mathematical Community The Mathematical Association of America and the American Mathematical Society, 1915–1925

Karen Hunger Parshall University of Virginia

The Mathematical Association of America (MAA) officially came into existence over the course of a two-day-long meeting held on 30–31 December, 1915 in Columbus, Ohio. Some 450 peoplenationwidehad answered a writtencall in support of thecreation of a new organization that would specifically foster collegiate mathematics, and over 100 of them opted to spend their New Year’s Eve in Ohio in order to discuss and ratify a constitution and bylaws for the new society. There was, of course, already a national association for mathematicians. The American Mathematical Society (AMS) had been founded in 1888 as the New York Mathematical Society at what was then Columbia College; it had assumed the national mantle of “American” in 1894. This chapter examines the impetus behind the formation of the MAA, itsfoundingand first decade of activities, and the dynamics between the MAA and the AMS over the course of that first decade as each sought to define more clearly its place in the American mathematical landscape.

Impetus for change: The question of The American Mathematical Monthly In 1894, Benjamin Finkel, former secondary school mathematics teacher and then Profes- sor of Mathematics at Drury College in Kidder, Missouri, began the publication of The American Mathematical Monthly to ﬁll what he saw as the need to stimulate and encourage “teachers of mathematics in our high schools and academies and normal schools” [16, p. 310]. In Finkel’s view, “[m]ost of our existing journals deal almost exclusively with subjects beyond the reach of the average student or teacher of mathematics or at least with subjects unfamiliar to them, and little, if any, space is devoted to the solution of problems” [17, p. 1]. The thinkingthus went that the teachers, and by association their students, would beneﬁt from the challenges presented by a problems-and-solutions department as well as by articles of both a mathematical and an historical nature dealing with the subject matter presented in their classrooms. Despite this lofty goal, it soon became clear to Finkel and

159 160 A Century of Advancing Mathematics

Benjamin F. Finkel (1865–1947) others that the journal, “designed ... for the benefit of high school mathematics teachers particularly,” had become “occupied by a more virile race of mathematicians, namely the teachers of college and university mathematics, particularly the former” [16, p. 310].More- over, practical matters concerning the journal’s finances and its printer had arisen by 1912 that prompted Finkel to consult with friends at his alma mater, the University of Chicago, about the journal’s longer-term prospects. Herbert E. Slaught, Professor of Mathematics there, embarked on an active campaign to help Finkel secure broader institutional support for the Monthly. By 1913, his efforts had resulted in financial backing from the Universi- ties of Chicago, Illinois, Missouri, Minnesota, Nebraska, Kansas, Indiana, Iowa, Colorado, Michigan, Northwestern,and Washington(at St. Louis) as well as from Colorado and Ober- lin Colleges [26, p. 19]. The Monthly was, temporarily at least, on a firm financial footing. In June of that same year, Slaught, George A. Miller of the University of Illinois, and Earle R. Hedrick then of the University of Missouri, officially took over the editorship of the Monthly.The time seemed ripe for a rearticulationand refinement of the journal’sgoals. On behalf of his co-editors, Hedrick made those new goals explicit in the opening pages of its twentieth volume. “The American Mathematical Monthly, beginning with this issue,” he declared, “proposes to afford an opportunity for any discussions that seem valuable upon collegiate mathematics, and the editors invite contributions concerning the methods of in- structionas well as those that treat special topics or theorems” [20, p. 1 (my emphasis)]. He went on to explain, however, that this did not imply that the journal would become a venue in “the field of general pedagogy, nor [would] the Monthly entertain discussions that are concerned with research in general pedagogy, or that deal with new theories of pedagogy, however important these contributions may seem” [20, p. 4]. “What we do desire,” he continued, “is to inspire, not a discussion along these lines, but rather a discussion of definite mathematical problems. It is our hope ...[to] spur mathematicians to the realization of the new conditions that confront us, to the variety of problems that demand discussion, to the The Stratification of the American Mathematical Community 161 vital importance which their solution holds to a great body of intelligent people” [20, p. 4 (his emphasis)]. By fostering this sort of exchange of ideas, the Monthly sought “to awaken an interest ...even among those who now regard their own profession—theteaching of collegiate mathematics—with distrust as a possible field for that type of human thinking that is known as scientific research. In this distrust, we, the editors of the Monthly, emphatically announce that we do not share” [20, p. 5]. In short, the newly reconceived Monthly aimed to professionalize and more formally to legitimize the teaching of collegiate mathematics. A year later, Slaught assessed the Monthly’s progress toward another of its articulated goals, that of “occupy[ing] a unique position in this country” [34, p. 1]. As he noted, the Monthly had been careful to consider neither secondary school material nor the pedagogical issues associated with its instruction. Likewise, it had endeavored “not to encroach upon the field of the advanced scientific journals of mathematics” [34, p. 1]. At the same time, however, its editors were firm in their belief “that large numbers who would become active and effective in higher mathematical research are now lost to the cause simply by reason of the fact that there are no intermediate steps up which they can climb to these heights” [34, p. 1]. For that reason, they held “that the Monthly has a mission to perform in holding the interest of such persons by providing mathematical literature of a stimulating character that is within their range of comprehension, and by offering an appropriate medium for the publication of worthy papers which the more ambitious may produce” [34, p. 2]. In so doing, the Monthly served as both a resource and a stepping stone for those at that intermediate stage between student of higher-level mathematics and active mathematical researcher. What, though, should be its relationship to the American Mathematical Society, the nation’s one professional association for mathematicians? That was a questionthat Slaughtput before theChicago Section of the AMS at itsmeet- ing in April of 1914. As a result of his query, a committee consisting of Richard P. Baker of the University of Iowa, Hedrick, Miller, Edward B. Van Vleck of the University of Wis- consin, and Alexander Ziwet of the University of Michigan was charged with “report[ing] to the Section at the December meeting, concerning possible recommendations of the relation of the Society to the field now covered by the American Mathematical Monthly”[35, p. 450]. The report it presented carried the recommendation that those interested in the field of collegiate mathematics be eligible for associate membership in the AMS. Meanwhile, also at Slaught’s instigation, the Council of the AMS appointed its own committee of Hedrick, Slaught, and past AMS Presidents Henry Fine of Princeton, Thomas Fiske of Columbia, and William Osgood of Harvard to ponder and make recommendations on the same question.After much discussion,the latter committee voted three to two against a proposal that the AMS actually take over the publicationof the Monthly and thereby make collegiate mathematics an explicit part of its purview.1 In accepting the committee’s report, however, the AMS Council adopted the following resolution: It is deemed unwise for the American Mathematical Society to enter into the activities of the special field now covered by the American Mathematical Monthly; but the Council desires to express its realization of the importance of the work in this field and its value to mathematical science, and to say that should an organization be

1See [26, p. 20] and [3, p. 79]. It would be easy to assume that Hedrick and Slaught were the two votes in favor of the AMS taking overthe Monthly, but the sourcesare silent on this point. 162 A Century of Advancing Mathematics

formed to deal speciﬁcally with this work, the Society would entertain toward such an organization only feelings of hearty good will and encouragement [9, p. 482]. Article II of the Society’s constitution had read from the beginning that “[t]he object of the Society shall be to encourage and maintain an active interest in mathematical science” [12]. On the face of it, the charge was open to broad interpretation, but, in practice, the Society’s real aim had always been clear. It was to be, the initial call declared, a “mathematical society for the purpose of preserving, supplementing, and utilizing the results” of the mathematical studies of its members as well as for presenting those “original investigations to which the members may be led” [3, p. 4]. The Council’s action in 1915 made the emphasis on research explicit by effectively deﬁning—without, however, altering the language of—the second article of the Society’s constitution. The Society actually aimed to encourage and maintain an active interest in mathematical research as opposed to the teaching of mathematical science. The latter was important, the Council acknowledged, but the AMS needed to concentrate its efforts on research. A new organization that focused on teaching would, however, have the AMS’s blessing.

The founding of the Mathematical Association of America Over the course of the spring, summer, and fall of 1915, members of the American mathematical community, spurred by Herbert Slaught, debated the prudence of and the real need for founding a new mathematical association. Some, like Slaught, came to favor such a development on its own merits, or sided with the AMS Council in the belief that the AMS should center its attentions on research, even though teaching strength and expertise at the college level was critical to its goals. Some, like Osgood, were opposed, believing that the American mathematical community should remain united under one society that should foster both teaching and research. That was, after all, the dual mission of the majority of the members of that community.2 By December, a course of action seemed clear to Slaught and others, and they used the occasion of the thirty-sixth meeting of the Chicago Section of the AMS in Columbus, Ohio to come together to discuss it further. As early as 1897, mathematicians in the Midwest had formed the so-called Chicago Section of the AMS, the meetings of which became officially sanctioned by the AMS through the addition of bylaws to the Society’s constitution in that same year. The creation of this section acknowledged the facts that mathematical activities took place outside New York City,the official meeting site of the AMS, and that it was not always possible for mathematicians in the far-flung United States to travel east for the regularly scheduled So- ciety meetings.3 Indeed, the winter meeting of the Chicago Section in Columbus on 30–31 December—which was joint with a meeting of Section A of the American Association for the Advancement of Science (AAAS)—immediately followed the twenty-second annual

2See Herbert Slaught to Roland Richardson, 28 January, 1924, Container 12, Folder: “Slaught, H. E. 1923– 30,” in [2]. Deborah Kent is currently at work on a detailed account—provisionally entitled “The Genesis and Early Years of the MAA” [24]—of this debate and on the many nuancesof the arguments both pro and con. 3The Chicago Section was followed by San Francisco and Southwestern Sections in 1902 and 1906, respectively, as collegiate- and research-level mathematicians spread westward across the country. For more on the foundation of the Chicago Section and its historical importance, see [27]. The Stratification of the American Mathematical Community 163 meeting of the AMS in New York City on 27–28 December, and, perhaps not surprisingly, only three members of the AMS were in attendance at both.4 In addition to almost seventy AMS members, the Columbus gathering attracted, among others, at least nine women and more than thirty mathematics teachers from small colleges, high schools, normal schools, and academies. The attendees listened to some twenty-six talks ranging across a wide variety of topics from the calculus of variations to group theory to differential equations to geometry to real analysis, but many of them were also there in answer to Slaught’s call for the possible formation of a new mathematical society. As managing editor of the Monthly and representative of its Board of Editors, Slaught was asked to open a discussion of that topic on the afternoon of Thursday, 30 Decem- ber. He reiterated what he had said in print at the end of the first year of the then newly reconceived Monthly, but with a fresh twist. In its new incarnation, the Monthly aimed to advance “the interests of mathematics in the collegiate and advanced secondary fields” [5, p. 2]. Now, Slaught hoped, a “new organization might carry forward these aims with still greater effectiveness, coöperating, on the one hand, with the various well-organized secondary associations, and, on the other hand, with the American Mathematical Society in its chosen field of scientific research, but being careful to encroach upon neither of these fields” [5, p. 3 (my emphasis)]. From the very beginning, this new organization sought to carve out its own special niche—but without “encroachment”—in a professional, mathematical continuum that had previously been defined by the teachers of mathematics on the one hand and the research mathematicians on the other. Slaught and his colleagues had come to the meeting prepared with a draft constitution and bylaws, and those assembled constituted a committee of the whole to hammer out the details. In fairly short order, agreement was reached as to the substance of the document. Only a name for thenew organizationremained uncertain,but there were many suggestions. When the meeting reconvened the next morning, the Mathematical Association of America officially came into existence. Like that of the AMS, the MAA’s constitutionopened with a statement of purpose. “Its object shall be,” the writers declared, “to assist in promoting the interests of mathematics in America, especially in the collegiate field” [5, p. 3]. The wording was carefully chosen. The MAA would “assist” in the promotion of mathematics in the United States through its emphasis on “the collegiate field.” In this way, it would be a partner in common cause with the AMS at the same time that it explicitly took on an area of interest that the AMS had largely eschewed. In a shrewd business move, moreover, the MAA, as part of its constitution, would “publish an official journal,” namely, the Monthly, “which shall be sent free to all members of the Association” [5, p. 4]. The dues of the MAA membership would thus provide the assured financial support that the journal had previously lacked, and repeated negotiations for financial backing with selected universities would no longer be required. Also at the organizational meeting, the requests of three states—Kansas, Missouri, and Ohio—toform Sections of the Association were received, and a nominating committee was formed to put together the Association’s first slate of officers. On Friday, 31 December, Missouri’s Earle Hedrick became the MAA’s first President, with Harvard’s Edward Hunt- ington and Illinois’s George Miller as Vice Presidents and William Cairns of Oberlin as

4Compare the attendance lists given in [11, p.263]and[36, pp. 280–281]. 164 A Century of Advancing Mathematics

George A. Miller (1863–1951) Vice President of the MAA

Secretary-Treasurer. Twelve additional mathematicians were chosen to serve as members of the Executive Council, among them Eliakim Hastings Moore and Oswald Veblen, the former a past AMS President, leader of the research program at the University of Chicago, and arguably doyen of research mathematics in the United States and the latter a student of Moore, leader of the rising graduate program at Princeton, and soon-to-be force within the AMS and American mathematics as a whole.5 Slaught, as editor of the Monthly, was, quite naturally, appointed to the Committee on Publications, together with Robert Carmichael of the University of Illinois and William Bussey of the University of Minnesota. With all of these ofﬁcers in place, it was time for the new organization to begin the process of self- deﬁnition.

The MAA’s first decade Hedrick began to set the tone in an article in the February 1916 number of the Monthly. Entitled “A Tentative Platform for the Association,” it aimed to clarify the relationship between the MAA and other associations, especially the AMS, and to set at least a preliminary agenda for the new body. Echoing Slaught at the organizational meeting two months earlier in December, Hedrick proclaimed that “the chief motive” of the MAA “may well be said to be that of service to the whole body of teachers of mathematics in American colleges” and that “the Association will not stop at anything which will serve this body of men” [22, p. 31].6 That said, he immediately made an important qualiﬁcation that served both to acknowledge the stratiﬁcation that the MAA’s founding signaled within the broader

5For more on Veblen’s leadership role in the AMS in the 1920s and 1930s, see [28]. 6Although it was, indeed, true that most of the members of the MAA (and the AMS) were male (hence the turn of phrase “men” here), both organizations had signiﬁcant numbers of female members as their membership rosters attest. The Stratification of the American Mathematical Community 165

American mathematical community and to delineate the function of the MAA from that of the AMS. As Hedrick noted, “[t]he majority of those responsible for the new organization are themselves members of the American Mathematical Society. This older organization is itself bound by its constitution to promote the interests of mathematics in this country” [22, p. 31]. Still, the AMS’s Council had, in April 1915, chosen to circumscribe that broad mandate and “to restrict [the AMS’s] activities to the ﬁeld of pure research in mathematics, and to the promotion of those phases of mathematics which are commonly associated with that word” [22, pp. 31–32]. The AMS, in other words, would construe the term “research” as meaning “proving new theorems,” while the MAA would take a different, broader view. Hedrick, himself a leading member of the AMS who would, in 1921, assume the editorship of the Society’s Bulletin, put it this way: “[t]hose responsible for the new organization are by no means at variance withthis determination,and it is their aim to carry out in goodfaith the separation of ﬁelds of activity provided for by the action just mentioned” [22, p. 32]. That “separation,” then, involved a delineation between what the MAA would publish in its Monthly and what the AMS published in its Bulletin and Transactions. The Bul- letin contained brief announcements of new results, while the fully developed expositions of those and other results appeared in the Transactions. The Monthly, on the other hand, would not publish research, that is, new results in “the common acceptation of that word,” but rather articles “which nevertheless represent a great deal of labor of a purely investiga- tional sort which would seem worthy of being called research in a broader interpretation of that word” [22, p. 33]. Those articles might involve serious historical research or pene- trating inquiries into the subject matter of collegiate coursework at the elementary or more advanced levels or discussions of a pedagogical nature “in which a professional knowledge

William D. Cairns (1871–1955) Secretary Treasurer for the MAA’s first 27 years President from 1943–1944 166 A Century of Advancing Mathematics of the subject-matter is a necessary element toward the formation of any dignified conclusion” [22, p. 33 (his emphasis)]. In short, “[w]hile the new association will recognize fully the prior right of the American Mathematical Society in all questions which would ordi- narily be termed research under the common interpretation, the attitude of this Association will be to encourage and to dignify all investigations of the character which have here been called research in the broader sense” [22, p. 33]. Implicit in that separation, too, was the stratification already articulated by Slaught as early as 1914. By concentrating on collegiate mathematics, situated hierarchically as it is between elementary and secondary mathematics, on the one hand, and mathematics at the graduate and research levels, on the other, the MAA and its Monthly would provide the stepping stone that would allow collegiate mathematicians effectively to “climb to” the research “heights” [34, p. 2]. The Association and its publication would also provide the instructors of those collegians with the material and the inspiration to challenge their students in the classroom. With this sense of mission articulated and with the Monthly in Slaught’s capable hands as managing editor, the MAA held its first summer meeting on 1–2 September, 1916 at the Massachusetts Institute of Technology in Cambridge. Some 126 people attended. Of these, 111 were members of the MAA and 81 belonged to the AMS as well. At least thirteen women and ten secondary school teachers were also among the MAA or AMS members present [23, p. 273]. As Hedrick had foreseen, a majority of those participating in the MAA’s meeting were indeed also in the AMS. The MAA adopted a different format for its meetings from that of the AMS, however. Only nine talks were given over the course of two days, their speakers having been formally invited by the program committee chaired by Henry Fine of Princeton. Moreover, Vice President Edward Huntington’sopening lecture on “The Teaching of Elementary Dy- namics” was followed by a separately scheduled, open discussion moderated by Stanford’s Leander Hoskins. This formal program was supplemented, not surprisingly,by various business meetings. The Committee on Mathematical Requirements, chaired by John Young of Dartmouth, explored questions associated with mathematics in the collegiate curriculum: “I. What general educational values (utilitarian, disciplinary, cultural) can actually be secured by the study of mathematics? II. What should be the primary purposes of mathematical instruction? III. What topics and what treatment of these topics will best serve to realize the values and purposes under I. and II.? IV. How much of the content included under III should be required (a) of all students in the secondary schools; (b) for college entrance; (c) of all students in college? V. What should be the preparation of teachers in secondary schools and in colleges?” [23, p. 283]. The Committee on Bureau of Information discussed how best to provide accurate answers to mathematical questions that may arise in the course of secondary or collegiate instructionand arranged for a sort of mathematical clearinghouse to be headed by James Shaw of the University of Illinoiswiththe problem-solvingsupport of Ju- lian Coolidge at Harvard, Luther Eisenhart at Princeton, William Fite at Columbia, Mellon Haskell at Berkeley, and Wallie Hurwitz at Cornell. Delegates of eighteen of the fifty-three schools that had institutional memberships in the MAA also met to consider questions of an institutional nature that might make for valuable discussion in the future and to set up a program committee to organize their subsequent meetings. The Stratification of the American Mathematical Community 167

Finally, the MAA’s Council met to conduct the general business of the Association. In additionto settingthe place of the annual meeting and to electing new members and institu- tionsintotheMAA, it heard the reportof a committee chaired by E. H.Moorethat had been appointed in the spring “to consider ...the advisability of fostering the production and the publication of articles of an expository and historical nature” [23, p. 288]. Moore, together with his fellow committee members Raymond Archibald of Brown, Oswald Veblen, and Alexander Ziwet, devised a remarkable plan. The idea was to collaborate with the Annals of Mathematics, then underwritten by Princeton University and of which Veblen was on the editorial board. The MAA would provide an annual subvention of $300 to allow the Annals to increase its size from 200 to 300 pages annually, “[t]he added pages [being] devoted to expository articles of suitable character so far as these can be obtained.”7 As Veblen knew, the Annals already had a policy that did not debar the publicationof expository articles. The MAA’s move could thus be seen as encouragement for qualified authors to undertake such writing and thereby to reach a broader mathematical reading public than would be possible with a narrowly focused piece of specialized research. Moreover, the arrangement had the potential further to define the Monthly’s publication sphere, for if the agreement were not renewed at the end of the initial three-year period, “the Board of Editors of the Annals shall thereafter conduct the Annals as a journal devoted primarily to research, yielding the field of historical and expository articles (not necessarily absolutely but principally) to the publications of the Association” [23, p. 288]. The MAA was being true both to its spiritof cooperation and to its broader definition of research.

Oswald Veblen

7See [23, p. 288]. In 2013 U. S. dollars, this was equivalent in “economic status” (considering the money as income or wealth) or in “income value” (consideringthe moneyas a commodity) to an allocation of some $32,400 based on the nominal gross domestic product per capita. All currencyconversionsto follow will use this measure. For more, see www.measuringworth.com. 168 A Century of Advancing Mathematics

The matter of the delineation between the MAA and the AMS, however, was still on people’s minds at the MIT meeting in Cambridge in 1916. Among the speakers at the conference banquet, Thomas Fiske, seventh President of the AMS, held forth precisely on the topic of the “Relations between the Association and the Society.” After noting that the MAA in its eight months of existence boasted just over 1,000 members to the AMS’s not quite 800, Fiske laid out the two primary reasons why the “organizations must have intimate and friendly relations” [18, p. 296]. First of all, as was clear, “they have a large body of members in common” [18, p. 296]. Secondly, “they have, to a large extent, a common aim, or more accurately, closely related and mutually helpful aims” [18, p. 296]. In language that could easily have been adopted by the two bodies to clarify and delineate their statements of purpose, Fiske maintained that “[t]he primary aim of the American Mathe- matical Society is the advancement of mathematical science by the stimulation of research. . . . [T]he primary aim of the Mathematical Association of America is the advancement of mathematical science by the stimulation of teaching” [18, p. 296]. Given the close connection between these two goals, Fiske, for one, was “inclined to think that every American mathematician should wish to belong to both organizations” [18, p. 296]. By the time the Association held its second annual meeting in New York City three months later in December, the number of members had grown to 1,064; institutional memberships had increased to seventy-six; and a new Section for Maryland, the District of Columbia, and Virginia had been established [6, pp. 63–64]. A new Library Committee had also been formed with the broad charge of providing advice to the “mathematical libraries of our schools and colleges” as to, among other things, books “suitable for freshmen, sophomores, juniors, etc.” and reference works and books in foreign languages to procure [6, pp. 63–64]. Provisions were also made for new committees on sections and on membership. Over the course of the MAA’s first decade, meetings were held twice a year, in the summer and in December. Times and places were often coordinated with the AMS and/or with Section A of the AAAS in the ongoing spirit of collaboration between the organizations. The MAA also continued to expand its work and both to define and to refine its niche. At the second summer meeting held in Cleveland, Ohio in September of 1916, for example, MAA President Hedrick chose “The Significance of Mathematics” as the topic of his retiring address. In his view, the MAA had a large role to play in communicating the importance and relevance of mathematics not only in the college curriculum but also in society as a whole. As he put it,“[v]alues to theworld at large must be stated in terms of more concrete realities. Shall we hide the fact,” he queried rhetorically, “of the immense service of mathematics to society? To emphasize beauty and pleasure to the entire exclusion of the more convincing argument of benefit to mankind is as quixotic and short-sighted as is the corresponding formalization of our courses of instruction. To ignore the significance of our great subject is to spurn our birthright” [21, p. 404]. How best, then, could the MAA help the American mathematical community avoid these pitfalls? Hedrick had an idea. He rightlynoted that “[i]n America, up to recent years, the beauty and interest centering in pure mathematics has so absorbed all mathematical talent that we have almost if not quite neglected that other phase of mathematics in which the significance of all we do is so self-evident: applied mathematics” [21, pp. 404–405]. The Monthly, however, unlike the journals of the AMS, had already published “articles of research on The Stratification of the American Mathematical Community 169 topics in insurance, on mathematical history, on mechanics, and on other applied branches of mathematics” [21, p. 405]. “[B]y reorganizing our own instructionunder the auspices of this Association, and by the recognition and encouragement of workers in the various fields of applied mathematics,” Hedrick continued, “we may, and I think we should, increase the appreciation of the significance of mathematics among our students, among the public, and even among ourselves” [21, p. 405]. Hedrick had found a niche for the MAA that the AMS had largely never occupied, despite the applied mathematical bent of some of its early Presidents.8 Clearly another niche for the MAA was defined by its Committee on Mathematics Re- quirements. Under the guiding hand of John Young, that committee worked diligently over the course of the MAA’s first decade to prepare reports addressing a number of issues that confronted collegiate professors of mathematics as well as secondary school mathematics teachers in so far as they prepared students for entry into the collegiate mathematics classroom. By May of 1919, this committee had successfully interested the General Education Board (GEB) of the Rockefeller Foundation in its goals and had secured from the GEB a grant of $16,000 (or $1,130,000 in 2013 U. S. dollars) for “the appointment of one college man and one high school man to devote their full time to the work of the Committee, their salaries for one year to be paid out of the fund mentioned” [38]. Young himself was the “college man,” and J. A. Foberg of the Crane Technical High School of Chicago was the “high school man.” As Young expressed it in a subsequent report, “[t]wo specific problems face the Committee: (1) the revision of secondary school and college courses in mathematics; [and] (2) the revision of the college entrance requirements in mathematics” [37, p. 280]. In acknowledgment of the ongoing cooperation between the MAA and the AMS, Young was also careful to note that “[t]he latter problem has been referred to the Commit- tee not only by the Mathematical Association of America but also by the Council of the American Mathematical Society” [37, p. 280]. By 1923, this committee had produced The Reorganization of Mathematics in Secondary Education: A Report by the National Com- mittee on Mathematical Requirements under the Auspices of the Mathematical Association of America, Inc., a massive, 652-page report published by the MAA that made specific recommendations about the mathematical content to be taught grade by grade.9 The MAA also continued to support various publication initiatives throughout its first decade. For example, it actively considered the publication of a mathematical dictionary. Raised at the summer meeting in Cleveland in 1916 and further developed thanks to a committee initially chaired by Hedrick, the idea was to produce a two-volume “dictionary containing brief definitions of the words employed up to and including the end of collegiate work proper, or possibly the first graduate year” [7, p. 60]. By 1929, however, this project had still not materialized owing to the fact that the funding needed to realize it, estimated at $100,000 (or $6,180,000 in 2013 U. S. dollars), had not yet been secured [8, p. 123].The MAA had also renewed itscommitment to boththe Annals of Mathematics and

8This would certainly not be the last time that applied mathematics was viewed as a desideratum within the American mathematical community overthe course of the first half of the twentieth century, and especially in the 1930s and 1940s as the world once again erupted into war. In the ongoing process of professional delineation, however, neither the MAA nor the AMS ultimately provided the natural venue for the support of applied mathematics in the United States. For more on this issue, see [28] and [32]. On the early nineteenth-century applied work of American mathematicians, see [29, chapter 1]. 9For a brief accountof the report, see [4, pp. 26–27].For a review of the report, see [10]. 170 A Century of Advancing Mathematics thereby to the active encouragement of expository mathematical writing through its continued subvention to the Princeton journal, and began a new expository publication venture, the Carus Mathematical Monograph Series, thanks to the generous donation of Mrs. Mary Hegeler Carus of $1,500 (or $118,000 in 2013 U. S. dollars) a year for five years beginning in January, 1922 [33]. By 1925, the first in the series, The Calculus of Variations by Chicago’s Gilbert Bliss, had been published by The Open Court Press for the MAA; other volumes, Functions of a Complex Variable by Northwestern’s David Curtiss and Mathe- matical Statistics by Iowa’s Henry Rietz, followed in 1926 and 1927, respectively.10 All of these projects and initiatives attest to the fact that the MAA was actively engaged in its efforts to support mathematics at the collegiate level.

The MAA through AMS eyes

As Hedrick, Fiske, and others had acknowledged, the constituencies of the MAA and the AMS naturally and significantly overlapped. After all, the university mathematician had the dual mission of teaching and research, and some mathematicians at the colleges, having earned theirPhDs for a piece of originalresearch, continued to pursuea research agenda even though their primary institutional goal was to teach. Still, in the mathematical continuum, there were those who primarily wanted to teach and those who primarily wanted to be left to their research. As early as 1919, John Kline, a freshly minted PhD from the Uni- versity of Pennsylvania, had experienced firsthand this differing allegiance as an instructor at Yale during the 1918–1919 academic year. A student of the topologist,Robert L. Moore, Kline had thoroughlyimbued the research ethos that Moore had embraced during his own student days at the University of Chicago workingunder the guidance of E. H. Moore and in the vibrant research atmosphere that the elder Moore and his German colleagues, Heinrich Maschke and Oskar Bolza, had fostered there.11 Kline had committed himself to pure mathematical research, and he was surprised to find that his new colleagues at Yale did not all share that commitment. At a faculty meeting in February of 1919, the Yale senior faculty had lamented the fact that its graduate applicationshad dwindled to just one for the 1919–1920academic year and that itsgraduate program had largely dried up. Ernest Brown, then Chair of the Department, cited as part of the cause the fact that Yale “did not offer enough financial inducements” and that “the money [it] had to put out in fellowships and scholarships was very small as compared with Chicago, Harvard, and Princeton.”12 The senior faculty charged a committee of “the younger men” in the department to discuss more fully “the question of what was wrong and to make recommendations as to how to remedy the defects.” Kline was “furious” at the attitudes his fellow junior colleagues revealed. While, at first, they seemed supportive of pure mathematical research, they really believed “that most

10Mrs. Carus raised the amount of her gift to $2,500 (or $164,000in 2013 U. S. dollars) annually in 1927. See [4, p. 28]. 11On the Chicago programin which R. L. Mooredid his doctoral work, see [29, pp. 363–426]. 12J. R. Kline to Robert L. Moore, 9 February, 1919, Box 4RM74: Folder: Kline, John Robert (1918–1921),in [1]. The quotes that follow in this and the next paragraphare also from this letter. Kline refers to the “men” in the Yale department.Indeed, that department was all male as were most departments at the time. Women mathematicians were employed at the women’s colleges as well as at some land-grant and state-supported institutions. On this point, see [31, chapter 7, especially Tables 7.1 and 7.2 on pp. 170–173]and [19, pp. 74–75]. The Stratification of the American Mathematical Community 171 men ...are enthusiastic research men when in graduate school but when they got out into teaching and got away from this influence, they gradually returned to their normal selves and a correct balance of things.” In Kline’s view, this was characteristic of “the teaching gang.” They did “not think that [they could] teach and do research at the same time,” yet, to Kline’s way of thinking, that was precisely their mission. For him, the real problem at Yale in 1919 was that, of the younger men, he was only one of two who “had any interest in research.” Kline would remain a dyed-in-the-wool proponent of research in pure mathematics throughout what became a long career back at his alma mater, the University of Pennsylva- nia. He worked tirelessly both with his own graduate students and in support of the efforts of the AMS, serving as a member of the editorial boards of its Bulletin and Transactions and later as its Secretary from 1941 to 1950 during the tumultuous years of World War II and its aftermath. He recognized and respected the place of the MAA; he joined it after taking his position at the University of Pennsylvania in 1920. He clearly understood, however, a stratified American mathematical community that placed the researchers on top and valued a dual commitment to both teaching and research. His was not an isolated view. In 1924, during the second year of his two-year term as AMS President, Oswald Ve- blen reconsidered AMS-MAA relations in discussions with then AMS Secretary Roland Richardson of Brown University. The reappraisal, which took place at the same time that the AMS under Veblen’s guidance was trying to increase its revenues and establish a more viable business model, was prompted by the proposal that the MAA and the AMS consider the publication of a combined membership list. Veblen knew both organizations well, having served as Vice President of the AMS in 1915 and as Vice President of the MAA two years later in 1917 before becoming AMS President in 1923. He was, however, “dubious” about a combined membership list, despite the fact that he had “frequently thought about the possibility of amalgamating the two societies and having two classes of membership.”13 Richardson admitted in response that “[f]rom time to time this suggestion of combining the two organizations” had also occurred to him, but he was “not ready to either endorse or reject it.”14 In fact, he was not “even sure whether combining with the Association on the drives for membership would be advantageous,” a suggestion that had just been received in a letter to Richardson by then MAA President Herbert Slaught.15 To Richardson’s way of thinking, it would be purely a business decision, for “[t]here is no conflict at all in the aims of the two organizations.”16 “[W]hether they can be better run as one organization or as two cooperating,” was, however, “not at all clear to” him. Were they to combine, “there would have to be one group of men looking after the interests of research and one group

13Oswald Veblen to Roland Richardson, 25 January, 1924, Container 10, Folder: “Richardson,R. G. D. 1924,” in [2]. On Veblen’s efforts and motivations as AMS President to improve the finances of the AMS, see [15] and [28]. The “two classes of membership” to which Veblen referred here were “researchers” (the AMS members) and “teachers” (the MAA members). These were distinct from the categories of “sustaining” AMS members (of which the category of “patrons” was a subset) that Veblen and his colleagues actually did create during Veblen’s AMS presidency. 14Roland Richardson to Oswald Veblen, 28 January, 1924, Container 10, Folder: “Richardson,R. G. D. 1924,” in [2]. 15Herbert Slaught to Roland Richardson, 28 January, 1924, Container 12, Folder: “Slaught, H. E. 1923–30,”in [2]. 16Roland Richardson to Oswald Veblen, 28 January, 1924, Container 10, Folder: “Richardson,R. G. D. 1924,” in [2]. The quotes that follow in this paragraph are also from this letter. 172 A Century of Advancing Mathematics looking after those interests now coming under the head of the Association.” Moreover, the AMS, which, in Richardson’s view, would necessarily be the dominant force in such a combined society, “would have to create a new office for the man who stood at the head of the work now represented by the Association. There have been men president of the Association whom we should never want as president of a national organization which combined the two.” As Richardson explained to Slaught, he and Veblen intended to bring the matter of amalgamation up for consideration by the AMS Council at the time of its meeting at Columbia University in March 1924. Slaught politely but firmly objected to this course of action. First, he clarified that his suggestion had not been to “enter into any cooperative scheme at the present time” regarding the membership drive.17 Rather, what he “had in mind was for the future.” More significant, though,was the question of actually combining the two organizations.“My own judgment on this question,” he stated, “is that it would be unwise to try to make such an arrangement at the present time. As strongly as I felt at the beginning that such an arrangement was desirable I have become more and more convinced as time goes by that the wise thing after all was done and that both organizations have accomplished a whole lot more separately than they ever would have under a combined management. I do not believe that it is best now to try to make this change.” Ever cooperative, however, he made clear that he was “not adverse to considering such a proposition if it should be made,” even though he reiterated that “on the mutual relations of the two organizations, I am sure that everything is to be gained by very close cooperation. There is no question of the great stimulus to each organization by virtue of its relation with the other.” After further discussion with Richardson and others, Veblen made his decision. “I am quite convinced,” he wrote to Slaught, “that it would now be quite out of the question to attempt any amalgamation of the two mathematical organization[s]. I think we shall accomplish much more as formally distinct organizations which, however, coöperate very closely.”18 In particular, the AMS and the MAA did join forces in a membership drive beginning in 1924 that targeted, among others, teachers of mathematics at some 1200 schools in the United States and Canada. In a letter dated 16 December, 1924 and addressed to Veblen, Slaught, and Cairns, among others, Richardson effectively put an end to the discussion of amalgamation by emphasizing the spirit of cooperation between the two organizations. “I do feel,” he said, “that we have added to the solidarity of the mathematical teaching forces in the country and that the campaign is worth while [sic] from an educative standpoint.”19

Conclusion The AMS Council’s decision in April of 1915 marked an important moment in the history both of America’s oldest mathematical organization and in what by December of that year

17Herbert Slaught to Roland Richardson, 5 February, 1924, Container 12, Folder: “Slaught, H. E. 1923–30,”in [2]. The quotes that follow in this paragraph are from this letter. 18Oswald Veblen to Herbert Slaught, 14 February, 1924, Container 12, Folder: “Slaught, H. E. 1923–30,” in [2]. 19Roland Richardson to Oswald Veblen, George Birkhoff, William Cairns, [Abraham ?] Cohen, Arnold Dres- den, [Harry S. ?] Everett, William Roever, Clara E. Smith, and Herbert Slaught, 16 December, 1924, Container 10, Folder: “Richardson,R. G. D. 1924,” in [2]. The Stratification of the American Mathematical Community 173 had become itsnewest. By rejecting the proposal of Chicago’sHerbert Slaughtand others to assume official responsibility for the publication of The American Mathematical Monthly, the AMS set the wheels in motion for the foundation of the Mathematical Association of America. A line had been drawn. The AMS had clarified its mission; it served to foster the production of original mathematical research. The MAA would assume the responsibility of cultivating collegiate mathematics by providing stimulus for both college teachers and their students. Both organizations self-consciously defined groups of professionals: the research mathematicians in the case of the AMS and the teachers of collegiate mathematics in that of the MAA. While these two groups were not disjoint, they shared a clear sense of hierarchy that paralleled the educational hierarchy from undergraduate to graduate instruction and then to productive research mathematician. The creation of the MAA underscored that stratification; the AMS’s decision almost a decade later to remain separate and distinct from the MAA further reinforced it. The MAA’s success in its first decade as well as its ability to define its own agenda distinct from, yet complementary to, that of the AMS reflect, however, the development of an ever more complex mathematical community in the United States over the course of the first quarter of the twentieth century.20

Acknowledgments My thanks to the archivists at the Library of Congress, but most especially to Carol Mead, archivist of the Archives for American Mathematics at the University of Texas, Austin,for all of her help and encouragement duringmy tripsto workin her beau- tifullyorganized archives. I also owe a debt of gratitude to my friend, Albert Lewis, for his hospitality in Austin and elsewhere and for so many fruitful and inspiring conversations about the history of American mathematics.

Bibliography Unpublished and Archival Sources [1] R. L. Moore Papers, 1875, 1891–1975, Archives of American Mathematics, Dolph Briscoe Center for American History, The University of Texas at Austin. [2] The Papers of Oswald Veblen, Library of Congress Manuscripts Division, Washington, DC.

Published Sources [3] R. C. Archibald, A Semicentennial History of the American Mathematical Society. American Mathematical Society, New York, 1938. [4] C. B. Boyer, The ﬁrst twenty-ﬁve years, in [25], pp. 24–54. [5] W. D. Cairns, The Mathematical Association of America, Amer. Math. Monthly 23 (1916) 1–6 (see also [14, pp. 25–26] for an excerpt from this article). [6] ———, Second annual meeting of the Association, Amer. Math. Monthly 24 (1917) 49–64. [7] ———, Third annual meeting of the Mathematical Association of America, Amer. Math. Monthly 25 (1918) 45–68. [8] ———, Thirteenth annual meeting of the Association, Amer. Math. Monthly 36 (1929) 119– 131. [9] F. N. Cole, The April meeting of the Society in New York, Bull. AMS 21 (1915) 481–493.

20For a different case study in the stratiﬁcation of American mathematics, see [30]. 174 A Century of Advancing Mathematics

[10] A. R. Congdon, review of The Reorganization of Mathematics in Secondary Education by the Committee on Mathematics Requirements, Amer. Math. Monthly 31 (1924) 91–96. [11] ———, The twenty-second annual meeting of the American Mathematical Society, Bull. AMS 22 (1916) 263–280. [12] Constitution, Bull. AMS 1 (1892) 20. [13] A Century of Mathematics in America–Part II. Edited by P. L. Duren et al. American Mathe- matical Society, Providence, 1989. [14] A Century of Mathematics through the Eyes of the Monthly. Edited by J. Ewing. Mathematical Association of America, Washington, DC, 1994. [15] L. B. Feffer, Oswald Veblen and the capitalization of American mathematics: Raising money for research, 1923–1928. Isis 89 (1998) 474–497. [16] B. F. Finkel, The human aspect in the early history of the American Mathematical Monthly, Amer. Math. Monthly 38 (1931) 305–320 (see also [14, pp. 79-82] for an excerpt from this article). [17] B. F. Finkel and J. M. Colaw, Introduction, Amer. Math. Monthly 1 (1894) 1–2. [18] T. S. Fiske, Relations between the Association and the Society, Amer. Math. Monthly 23 (1916) 296–297. [19] J. Green and J. LaDuke, Pioneering Women in American Mathematics: The Pre-1940s PhDs, HMATH, Vol. 34, American Mathematical Society, Providence and London Mathematical So- ciety, London, 2009. [20] E. R. Hedrick, Foreword on behalf of the editors, Amer. Math. Monthly 20 (1913) 1–5 (see also [14, pp. 21–24] for an excerpt from this article). [21] ———, The significance of mathematics, Amer. Math. Monthly 24 (1917) 401–406 (see also [14, pp. 31–33] for a reproduction of part of this article). [22] ———, A tentative platform for the Association, Amer. Math. Monthly 23 (1916) 31-33 (see also [14, pp. 27-29] for an excerpt from this article). [23] E. R. Hedrick and W. D. Cairns, First summer meeting of the Association, Amer. Math. Monthly 23 (1916) 273–288. [24] D. Kent, The genesis and early years of the MAA, manuscript in preparation. [25] The Mathematical Association of America: Its First Fifty Years, Edited by K. O. May. The Mathematical Association of America, Washington, DC, 1972. [26] P. S. Jones,Historical background and founding of the Association, in [25], pp. 1–23. [27] K. H. Parshall, Eliakim Hastings Moore and the founding of a mathematical community in America, 1892–1902, Annals of Science 41 (1984) 313-333. Reprinted in [13, p. 157–177]. [28] ———, ‘A new era in the development of our science’: The American mathematical research community, 1920–1950, in A Delicate Balance: Global Perspectives on Innovation and Tra- dition in the History of Mathematics: A Festschrift in Honor of Joseph W. Dauben, Edited by D. E. Rowe and W.-S. Horng, Birkhäuser Verlag, Basel, 2014, to appear. [29] K. H. Parshall and D. E. Rowe, The Emergence of the American Mathematical Research Com- munity, 1876–1900: J. J. Sylvester, Felix Klein, and E. H. Moore, HMATH, Vol. 8, American Mathematical Society, Providence and London Mathematical Society, London, 1994. [30] D. L. Roberts, Albert Harry Wheeler(1873-1950): A casestudyin the stratification of American mathematical activity, Historia Mathematica 23 (1996) 269–287. [31] M. W. Rossiter, Women Scientists in America: Struggles and Strategies to 1940. Johns Hopkins University Press, Baltimore, 1982. The Stratification of the American Mathematical Community 175

[32] Siegmund-Schultze, R. The ideology of applied mathematics within mathematics in Germany and the U. S. until the end of World War II, LLULL 27 (2004) 791–811. [33] H. E. Slaught, The Carus mathematical monographs, Amer. Math. Monthly 30 (1923) 151–155. [34] ———, Retrospect and prospect, Amer. Math. Monthly 21 (1914) 1–3. [35] ———, Spring meeting of the Society at Chicago, Bull. AMS 20 (1914) 449–464. [36] ———, Winter meeting of the Society at Columbus, Bull. AMS 22 (1916) 280–295. [37] ———, The national committee on mathematical requirements, Amer.Math. Monthly 26 (1919) 279–280. [38] J. W. Young, Report of the national committee on mathematical requirements, Amer. Math. Monthly 26 (1919) 233–234.

Departments of History and Mathematics, University of Virginia, Charlottesville, Virginia 22904 [email protected] The cover of the ﬁrst issue of The American Mathematical Monthly after the MAA took it over. Time and Place: Sustaining the American Mathematical Community

Della Dumbaugh University of Richmond

Every human being feels the need of belonging to some sort of a group of people with whom he has common interests. Otherwise he becomes lonely, irresolute, and ineffective. The more one is a mathematician the more one tends to be unfit or unwilling to play a part in normal social groups. In most cases that I have observed, this is a necessary, though definitely not a sufficient, condition for doing mathematics. But it has made it necessary for mathematicians to group themselves together as mathematicians. The resultant organizations of various kinds have accomplished many importantthingsknown to us all. Of these accomplishments I am sure that the most important is the maintenance of a set of standards and traditions which enable us to preserve that coherent and growing something which we call Mathematics. — Oswald Veblen, Opening Address of the 1950 ICM, August 30, 1950, Harvard University [16].

Introduction

The “common interest” of mathematicians in the last quarter of the nineteenth century contributed to the emergence of an American mathematical community. What “standards and traditions” evolved as mathematicians strengthened this community in the opening decades of the twentieth century? The private correspondence of Leonard Dickson and Oswald Veblen provides a loose frame to explore the views of these leaders as they not only sustained, but also advanced this young community. It is almost an oxymoron to see the words “Leonard Dickson” and “correspondence” in the same sentence. The gruff Dickson worked tirelessly to advance mathematics and preserve his privacy, to the point of burning his papers when he retired. The far more charismatic and diplomatic Oswald Veblen, however, saved his correspondence throughout his career. Aside from one letter in 1910 and two letters in 1915, the known correspondence of twenty-ﬁve letters between the two essentially spans the decade from 1924–1933. This chapter begins with brief biographical introductions to Dickson and Veblen, and continues with a discussion of a few of the letters exchanged between Dickson and Veblen on issues related to publication. We then turn our attention to letters Dickson and Veblen

177 178 A Century of Advancing Mathematics exchanged with others about a mathematics institute in America. The chapter ends with some concluding remarks.

Leonard Eugene Dickson (1874–1954)

A quintessential American, Leonard Dickson drew his first breath on land claimed in Iowa by his father, Cam Dickson, a Civil War veteran (Union). Dickson’s family moved to Cle- burne, a small town in Texas, to take over a family-run hardware business, when he was four. His parents, particularly his father, were community activists before the American lexicon ever included that descriptive term. For the Dicksons, this meant a strong commitment to public service in their small town. They helped bring public water, schools and the railwayto Cleburne. This same commitment to America took root in what was then a young Dickson and later took shape in the form of his dedication to the American mathematical community [11]. Dickson attended the University of Texas for his undergraduate and master’s education. He arrived at the University of Chicago in 1894 and went on to earn one of the first doctorates awarded in the Mathematics Department under the direction of the influential E. H. Moore. In his thesis, “The Analytic Representation of Substitutionson a Power of a Prime Number of Letters with a Discussion of the Linear Group,” Dickson sought to extend the theory of finite fields and to establish further its connections with group theory. He followed his dissertation with his first book, Linear Groups with an Exposition of the Galois

L. E. Dickson was sketched during his 1920 International Congress lecture. Time and Place: Sustaining the American Mathematical Community 179

Field Theory, published in 1901. This text proved to be a primary source of information about finite simple groups for the next fifty years. Like many aspiring American mathematicians of his day, Dickson traveled to Europe for a year of study following the completion of his PhD. Given the focus of his dissertation, he chose, not surprisingly, to study with Sophus Lie in Leipzig and Camille Jordan, Emile´ Picard, and Charles Hermite in Paris. Once he returned to the States, he spent two years at the University of California and one at his undergraduate alma mater the Univer- sity of Texas, before he returned to the University of Chicago in 1900 to join his former professors as a colleague. In his forty-year tenure at the University of Chicago, he wrote more than 300 manuscripts and eighteen books, advised 67 doctoral students, served as President of the American Mathematical Society and edited the major journals. Early in his career, Dickson’s publication record established him as the most prolific of America’s researchers. Although group theory formed the focus of his dissertation and would remain among Dickson’s research interests throughout his career, he would add the theory of finite fields, invariant theory, the theory of algebras, and number theory to his repertoire of research interest. He was, in a word, dogged. An anecdote from his honeymoon captures this quality of Dickson in a more personal light. When a colleague asked, “how was your honeymoon?” Dickson replied, “it was great, except I only got 2 papers written” [14, p. 398]. This mathematical workhorse, who played billiards and bridge by day and did mathematics from 8:30 P.M. to 1:30 A.M. by night, interrupted—but did not stop—his pure mathematical researches to write a three-volume, 1602-page historical account of the theory of numbers. Why? As he explained it, he undertook this project because “it fitted with my convictionthat every person should aim toperform at some time in his life some serious useful work for which it is highly improbable that there will be any reward whatever other than his satisfaction therefrom” [8, p. xxi]. Although he viewed it as “highly improbable,” thisaltruisticmission paid handsome rewards for Dickson as thishistorical study ultimately formed the basis of his plenary address at the International Congress of Mathematicians in Strasbourg in 1920 and led to his celebrated work in the arithmetic of algebras. In particular, he defined a set of integral elements for an algebra that led to an arithmetic analogous to that of the ordinary integers and that paralleled Joseph H. M. Wedderburn’s structure theory of algebras. It would be Dickson’s 1923 text Algebras & Their Arithmetics, enlarged and translated into German in 1927, that would introduce the German number theorist Helmut Hasse to the work of the Americans on the theory of algebras. And it would be Hasse, only a few years later, who would feel a certain urgency to submit his work to the Transactions of the American Mathematical Society in order to reach an American audience [13]. Dickson’s motivation for undertaking what became his mammoth History of the Theory of Numbers provides particularly meaningful insight into broader developments in the community at large. It seems that nationalistic reasons, in part, compelled Dickson to pursue this historical project. Dickson, for example, initially sought out the Carnegie Institution of Washington, one of the new national agencies created to promote what we now call basic research, as a possible publisher of the project. From his perspective, “[i]t would seem desirable to have undertaken in this country something of the kind done by the British Association, the Deutsche Mathematiker Vereinigung, etc., in the preparation by specialists of note of extensive Reports each covering an important branch of science” [2, 11 February, 1911]. After describing his “ideal of a mathematical report” which would ap- 180 A Century of Advancing Mathematics peal both to specialists and non-specialists, Dickson admitted that he had “. . . already given a solid year’s work to such an expository Report on the theory of numbers (integral and algebraic),...” [ibid]. Thus the British and German “mathematical Report[s],” and, in particular, the lack of similar offerings in America, may have encouraged Dickson to write his own historical compendium on the subject of number theory. In the case of graduate training, it was not at all unusual for American mathematicians to look to Europeans for ideas. The initiative Dickson outlined in his letter to Carnegie President R. S. Woodward, however, required not only an acquaintance with the European literature but also an awareness of a perceived void in American publications. Moreover, the opening sentence of his letter seems to suggest that Dickson wanted to raise Ameri- can mathematics to the European standard in this particular realm. Throughout his career, Dickson remained avidly committed to establishing standards of excellence for and in the community of American mathematicians [10, 11], especially through his own publications and his role as an editor of the leading mathematics journals. Dickson retired from mathematics in 1940 without a backwards glance. He gave away his books and burned his papers before he returned to Texas for the final fifteen years of his life. In his publications, most notably in his History of the Theory of Numbers where he could have added commentary quite naturally, Dickson wrote with littlefeeling or emotion, hopeful that the lines between his texts and manuscripts contained no trace of his personality. “What is generally wanted is a full and correct statement of the facts,” as Dickson described it in the introduction to his History, “not an historian’s personal explanation of those facts. The more completely the historian remains in the background or the less conscious the reader is of the historian’s personality, the better” [8, vol. 2, p. xx]. His personal letters, however, were apparently not written with this seemingly self-imposed filter and, consequently, offer a rare glimpse into the more private Dickson. The letters he exchanged with Oswald Veblen, in particular, not only give an indication of the relevant topics of importance for these two leaders in American mathematics, but they also contrast the more private Dickson with the more public Veblen.

Oswald Veblen (1860–1960) Bornon June 24, 1880in Decorah, Iowa, OswaldVeblen seemed destined from birthto possess a pioneering spirit and an inclination for academic excellence that would later define his life. His father, Andrew Anderson Veblen, was a first generation American of Norwe- gian descent, who was raised in a Minnesota family of twelve children in the harsh conditions of early settlement in the Northwest [17]. Veblen’s father went on to teach physics at the University of Iowa, while his uncle, Thorstein Bunde Veblen, made a name for himself as a prominent sociologist and economist. Not surprisingly, Oswald Veblen blazed his own trail in academics, albeit at an altogether different time in higher education in America. Veblen graduated with a bachelor’s degree from the University of Iowa at the age of eighteen. He then earned a second undergraduate degree at Harvard University in 1900. That same year, Veblen traveled west to pursue graduate studies at the young but already prominent Universityof Chicago. Like Dickson,Veblen completed his doctorate at Chicago under the direction of E. H. Moore in 1903. By this time, however, Moore’s research interests had shifted towards questions in the foundations of mathematics and Veblen’s dis- Time and Place: Sustaining the American Mathematical Community 181 sertation on “A System of Axioms for Geometry” reflected this change [18]. During his time at Chicago, which included two additional years of post-doctoral study, Veblen not only met Dickson and G. D. Birkhoff, but he also met the Scottish mathematician Joseph Wedderburn when he visited on a Carnegie Scholarship in 1904–1905. In 1905, a new opportunityarose for Veblen when Woodrow Wilson, then president of Princeton, created a series of junior faculty positions called “preceptors” designed to em- ulate the British tradition and enhance the undergraduate teaching at the university. Henry Burchard Fine, chair of the Princeton mathematics department, selected Veblen as one of the first preceptors of mathematics. In this position, Veblen collaborated with fellow pre- ceptor John Wesley Young on a major study of projective geometry, which resulted in a book of the same title in 1910. That same year an external offer from Yale helped secure Veblen a promotion to full professor; thereafter he focused his attention on Poincaré’s con- ceptualization of algebraic topology, commonly referred to as analysis situs at the time. He continued to broaden his mathematical scope, visiting the great centers of European mathematics in 1913 and keenly observing their mathematical culture. The uninterrupted time for mathematical research in European institutions made a particular impression on him. He would not forget it. All the while, Veblen continued to focus his research on topology. His preeminence in the subject was confirmed by an invitationto showcase his work at the American Mathemat- ical Society Colloquium Lectures in 1916. Veblen’s lectures on analysis situs established his reputation as one of the country’s foremost mathematicians. Through the later publicationof his talks in 1922, after World War I, in his Analysis Situs [19], “an entire generation of mathematicians was given the opportunityto pursue the techniques originally conceived by Poincaré” [5, p. 106].1 But Veblen possessed more than mathematical prowess. His adept organizational and administrative skills would serve him equally well in the growing American mathematical community. Veblen found an especially significant outlet for these talents in 1917 when the United States entered the First World War. Veblen was commissioned a captain in the army reserve with command of the Office of Experimental Ballistics at Aberdeen Proving Ground in Maryland. Although the firing ranges of Aberdeen were far removed from the comfort of his Princeton classroom, Veblen immediately set to work converting his theoretical understanding of mathematics to a practical knowledge of the experimental work he would have to undertake in support of the war effort. Although Veblen’s service, along with nearly sixty other professional mathematicians who served with him, contributed to the American war effort, it came too late in the war to have any substantial impact on the eventual Allied victory [15]. At the completion of his military service in 1919, Veblen quickly became one of the most influential mathematicians in the country [9, p. 477]. Later that year, his stature was further enhanced by his election to the National Academy of Sciences. Combined with his previous military service, this appointment “brought Veblen into a circle of powerful and ambitious scientists centered in Washington” [9, p. 477]. Taking full advantage of his newfound prominence, Veblen began to advocate for the advancement of mathematical research within the United States. At the time, Veblen’s foresight was sorely needed by

1“This book was for several years,” as Raymond Clare Archibald described it, “the source from which mathematicians in all parts of the world learned Topology” [4, p. 207]. 182 A Century of Advancing Mathematics

Oswald Veblen (1880–1960) Veblen, who served at the Aberdeen Proving Grounds during World War I, in uniform. the American mathematical community, which, despite its wartime contributions, had been marginalized after the war while the more highly publicized and seemingly more practical fields of physics and chemistry enjoyed the political and financial largesse of newly created philanthropicand public organizations like the Rockefeller Foundation and the Na- tional Research Council [9, pp. 475–476]. By linking abstract mathematics with applied medicine through the scientific medium of physics and chemistry, Veblen emphasized the necessity of strong mathematics in order to further enhance the rest of the scientific community. Veblen’s strategy helped persuade the National Research Council to extend their postdoctoral fellowships to include pure mathematics. He also helped secure funds (albeit small) for AMS journals through a grant from the General Education Board of the Rocke- feller Foundation to the National Academy of Science [9, pp. 477, 483]. Having obtained this support for mathematics in the broader scientific sphere, Veblen was perfectly poised to serve as president of the American Mathematical Society (AMS) in 1924. In thisposition,Veblen put his superb organizationalskillsto the test with a campaign to enhance the society’s dwindling endowment. Faced with mounting costs to publish original American mathematical books and papers, the society needed to find a way to increase revenue without raising dues. Veblen’s response was to lead a nationwide campaign aimed at engineering and industrial corporations that relied heavily on mathematical expertise. The title of the pamphlet designed to advertise the campaign, “Our Debt to Mathematics,” captured the essence of the sales pitch. While idealistic in scope, Veblen’s campaign failed to win over substantial corporate support. As Feffer concluded, “[a]ppeals to ‘duty’ and ‘moral obligation’ in soliciting industrial support were not only ineffective but betrayed a remarkable arrogance and na¨ıvetéon the part of academic mathematicians in the AMS” [9, p. 487]. Having failed in this endeavor, Veblen found other ways to fund the society Time and Place: Sustaining the American Mathematical Community 183 through creating the new categories of sustaining member and patron, which granted special privileges to their holders for higher annual dues [4, p. 31]. In 1932, Veblen assumed a faculty position at the newly created Institute for Advanced Study. Not long after, as a member of the Emergency Committee in Aid of Displaced For- eign Scholars, Veblen’s efforts earned him the unofficial title of “statesman of mathematics,” and contributed significantly to his standing in the worldwide mathematical community [6, p. 616].

The Dickson-Veblen correspondence (1910–1924) Even with their decidedly different personal characteristics, Dickson and Veblen shared a common commitment to the advancement of American mathematics. The topics these two leaders in the American mathematical community took up in their twenty-five letters reflect broader concerns and provide key insights into the types of issues the community faced as it established a more cohesive structure. This correspondence included discussions, among other topics, of Thomas Grönwall’s election to the NAS, Joseph Wedderburn’s election to the Royal Society, and how to handle a paper submitted by Wedderburn to Crelle’s journal just before he had to be hospitalized for his mental health. It also called attention to the growing demand for the publication of American mathematics and the creation of a mathematics institute in America. Not long after Veblen completed his term as AMS President and brought the successful endowment campaign to a close, Dickson sought his advice on two manuscripts he had prepared for publication. The research needs of Dickson and his students in Algebra led him to prepare a book-length History of Algebraic Equations, for which he needed a publisher. Dickson requested that Veblen let him “know your advise [sic], in light of your recent attention to the whole subject of publication of advanced books in math.” “In particular,” Dickson inquired of Veblen, “would you advise trying to get Carnegie Institution to publish now the Hist[ory]Theory Algebraic equations and after a couple of years approach them on Vol. IV (about 250–300pp) on History of Theory of Numbers”, [2, 4 April, 1926]. Thus Dickson was thinkingabout the publicationof the fourth volume of his History of the Theory of Numbers in early 1926. This was, in fact, precisely the time Dickson’s graduate student Albert Everett Cooper completed his graduate work at Chicago. His thesis on “A Topical History on the Theory of Quadratic Residues” was originally intended to serve as a chapter in this fourth volume. Cooper (re)joined the University of Texas mathematics faculty after he completed his PhD at Chicago. At that time, Dickson appealed to the Pres- ident of the University of Texas, Dr. Harry Y. Benedict, a former classmate from Dickson’s undergraduate days at Texas, to finance the publication of the fourth volume. Just a month earlier, in the closing days of 1927, Dickson had negotiated the publication of “an entirely new work on the Theory of Numbers” with the Carnegie Institute of Washington (CIW) in exchange for “abandon[ing] the printing of Vol. IV” [2, 29 December, 1927]. Dickson’s letter to Veblen suggests that Dickson began to use the fourthvolume as a sort of bargaining chip as early as 1926 to secure the publicationof other texts. Interestingly,the new work on the theory of numbers that Dickson proposed to the CIW never appeared from their presses. AlthoughCooper appealed to the CIW in June, 1929 to publishthe fourthvolume, theCIW denied his request [12, pp. 624–625]. 184 A Century of Advancing Mathematics

Veblen replied with the kind of “advice” Dickson probably wanted to hear. “[Wedder- burn] and I,” Veblen wrote, “think that your History of the Theory of Algebraic Equations would be a suitable book to publish in the new series of Colloquium publications of the Mathematical Society” [3, 7 April 1929]. Veblen also suggested that it was “sensible” to let the “Carnegie people” publish the fourth volume of Dickson’s History. “I am filled with admiration,” Veblen concluded his letter, “at your ability to get so many things done at the same time.” For a man who had spent more than a decade informing the CIW of his hard work on his History, this remark must have pleased the diligent and determined Dickson. Dickson maintained a strong commitment to securing the publication of advanced level treatises in mathematics in general. His voluminous publication record made him well- positioned to understand the publication concerns within the mathematical community. When consulted about the possibilityof Duke University foundinga new mathematics journal in the late 1920s, for example, Dickson cautioned that “[a] new math. journal would have for considerable time a small circulation, practically nil in Europe. This would cause hesitancy of first rate articles.” Instead, Dickson suggested that Duke “undertake the publication of advanced math. books of real merit. Practically no such books have appeared in America since the war without being financed by some University or the Research Coun- cil.” To substantiatethis claim, Dickson offered his own analysis of the growth of American mathematics in 1927 when he wrote that “[t]here are a dozen readers of advanced math. now to every one of a decade ago—and 20 to 1 of 25 yrs. ago. The need of first class books is now very urgent.” The urgency for Dickson, however, was no longer tied to national interests (i.e., Americans need to write their own texts instead of relying on European ones). From Dickson’s point of view, increased time for research created a new role for advanced treatises. As he put it, “[a]t the centers, a man gives fewer lectures and more time to research and thesis direction. Instruction must be given in part by supplementary work in books” [2, 3 May, 1927]. For Dickson, then, advanced treatises in mathematics for students freed mathematicians to pursue their own research.

A mathematics institute and its young faculty While Dickson and Veblen did discuss issues related to publicationvenues for mathematics, it seems they did not discuss the seminal idea of a mathematics institute in America with each other. Apparently unbeknownst to each other, Dickson and Veblen each took up this possibility with other members of the mathematical community. In May, 1919, Robert S. Woodward, then President of the Carnegie Institute of Wash- ington and former President of the AMS, proposed the idea of a mathematics institute to “stimulate researches in the higher branches of mathematico-physical science in Amer- ica” [2, 19 May, 1919]. Dickson wholeheartedly embraced the idea and traveled back to Chicago to pursue the initiative further. In particular, Dickson collected information about libraries—the standard repository of publications—and emphasized to Woodward that a suitable location for an American mathematics institute would need to include access to existing collections of scientiﬁc journals and proceedings. Dickson concluded that current library holdings in America limited the possible siting of the mathematics institute to Boston, New York, Chicago, Washington or Pasadena. For Dickson, then, an institution devoted to mathematical research depended on access to publications. Time and Place: Sustaining the American Mathematical Community 185

With these details aside, Dickson took Woodward to the heart of the matter, to the significant, intangible benefits of this proposed mathematics institute. “[A]fter further reflection,” Dickson wrote to Woodward, “I am as fully convinced now, as at the moment when you outlined your plans of an institute,of the highly important role which such an institute would play in the development in America of pure and applied mathematics” [2, 16 May, 1919]. Dickson recognized Woodward’s proposed idea as one that furthered the development of American mathematics. Woodward could count on Dickson to cast his support for virtually any endeavor that promised potential for American mathematics. While Dickson had suggested a mathematics institute in his desiderata for the Chicago mathematics department in 1910, neither he, nor the department for that matter, were really in any position to see that idea become reality. With Woodward at the head of the CIW, Dickson seemed to recognize the very real possibility that this idea might actually reach fruition.But it was not to be. Woodward’s secretary replied to Dickson and indicated that “President Woodward left Washington yesterday for an absence of about ten days in order to obtain some much needed rest” [2, Barnum to Dickson, 27 May, 1919]. Apparently sensing a difficult situation, Dickson did not bring up the mathematics institute in his later correspondence, but, rather, focused on the positive press the first volume of his History had received. All was not lost, however. Five years later, Oswald Veblen, in his role as the Chairman of the Division of Physical Sciences of the National Research Council (NRC),wrote to Ver- non Kellogg, Permanent Secretary of the NRC with ideas to advance research mathematics. “My experience this year,” Veblen began,

has made me rather acutely conscious of the fact that the needs of mathematical research have not yet been brought to the attention of those whose position enables them to have a view of the strategy of Science. This, I think, is chieﬂy the fault of the mathematicians themselves, who have too easily assumed that an outside world which cannot understand the details of their work is not interested in its success. That such an idea is erroneous has been well illustrated by the generous action of the Rockefeller Foundation in providing funds for Research Fellowships in mathematics of the same type as for physics and chemistry. This was done immediately, and apparently as a matter of course, when the need for such fellowships was pointed out [1, 10 June, 1924].

Veblen’s term as president of the AMS and his appointment on the Division of Physical Sciences at the NRC had, apparently, made him mindful of the critical importance of gaining a foothold for mathematics within science as a whole. In his very diplomatic way, he blamed the mathematicians for creating the current separateness while he simultaneously implied that scientists—Kellogg in particular—would, of course, have an interest in mathematics. That Veblen could cite his recent success securing fellowships for mathematics only served to further substantiate his claim. ”This experience,” Veblen asserted, “as well as much evidence of a less tangible sort of friendly interest in mathematics,” led Veblen to approach Kellogg about the next “very important step” for mathematics. Veblen wanted to widen the wedge for mathematics within science that he had begun to make with the research fellowships. Just what did he have in mind? 186 A Century of Advancing Mathematics

Veblen described the next step to advance mathematics as “simply to give a number of men who have proved that they can do productivework in this field a chance to concentrate their efforts on it.” That is, Veblen hoped to stretch the idea of the newly acquired research fellowships for young scholars to “men who have already proved their ability” [1, 10 June, 1924]. To accomplish this goal, Veblen proposed what he called a “Mathematical Institute.” Veblen called attention to the “very simple” equipment required to set up an institute. For Veblen, “the main funds should be used for men or women whose business is mathematical research. Such an institute, in my opinion, could operate successfully wither [sic] in conjunction with a university or as an entirely separate institution. In either case it would treat mathematical research as a profession” [ibid.]. Thus Veblen hoped to create a mathematical institute for proven research mathematicians to pursue their studies unimpeded by teaching and other university work. Just thirty-two years had passed since Chicago initiated the first sustained commitment to teaching and research in mathematics. Before that, with the almost singular examples of the institution of Johns Hopkins University and the mathematician, Benjamin Pierce, most higher education in America meant teaching with no research. By 1924, Veblen argued that the community would benefit from opportunities for mathematicians to devote time to research alone. Encapsulating Veblen’s overall view of the intrinsic value of the mathematical community, he closed his letter to Kellogg by suggesting that “the mathematical institute has the advantage that it would provide a definite nucleus for mathematical research and foster cooperation in a subject that has been treated in the past in perhaps an unnecessarily individualisticway” [1, 10 June, 1924].Thus Veblen ended his letter where he began it—promoting the advantages of mathematics as a more collective enterprise, this time for more experienced researchers. Although apparently unable to convince the NRC to undertake a mathematical institute at this time, Veblen would see his idea take shape, almost exactly as he had articulated it to Kellogg, at the turn of the next decade, in the form of the Institute for Advanced Study. In the end, however, it wouldnot be the NRC or the CIW or some other seemingly natural source to fund this institute. Instead, the financial support would come in the summer of 1930 from department store magnates who had earned their fortune selling everything from bicycles to underwear. The wealthy New Jersey entrepreneurs Louis Bamberger and his sister Caroline Fuld donated five million dollars toward the creation of a center for research and scholarship [6, pp. 612–613]. Ultimately known as the Institute for Advanced Study, the center would be directed by Abraham Flexner and located in or near Newark, New Jersey, where Bamberger and Fuld had made their fortune. While not initially concerned with the intellectual focus of the institute, Flexner soon settled on mathematics, in part due to its low maintenance costs as compared to other, more expensive, laboratory sciences. Veblen was quick to congratulate Flexner on the institute and offer his assistance, recommending Princeton as a possible location for the institute’s campus. When Flexner initiated the faculty search for the Institutefor Advanced Study (IAS), he sounded out the best names in the country for potential professorships. During a meeting with David Eugene Smith, past vice president and long time librarian of the American Mathematical Society, Smith informed Flexner that Dickson was the only “mathematical genius” in the country, although Harvard’s G.D. Birkhoff was not far behind [5, p. 56]. With Dickson in his mid-fifties at the time, Flexner worried that he did not possess the potential to guide the institute for a long period of time. True to Flexner’s initial concern, Time and Place: Sustaining the American Mathematical Community 187

Dickson retired from Chicago in 1939, just nine years after the foundingof the Instituteand seven years after it opened its doors. When negotiationswith Birkhoff fell through, Flexner offered a mathematics professorship to Veblen, a role he assumed with enthusiasm in June of 1932. Finally, in October, 1933, the ﬁrst mathematics institute in America, an idea conceived as early as 1910 by Dickson, championed separately by Woodward, and articulated by Ve- blen in1924, had ﬁnallybecome a realityin 1933withthe help of a few criticalprotagonists outside of mathematics.

Conclusion: to belong Together, these seemingly disparate personalities of Dickson and Veblen represented two different yet mutually important sides to the success of the young American mathematical community. Dickson, with his impeccable standards and sheer mathematical output, needed colleagues like Veblen whose active role in the public sector led to the recognition and funding of key mathematical initiatives. Veblen needed the contributions of outstanding mathematicians like Dickson to realize his dream of an internationallyprominent and well- funded national research community that could train future generations in person and in print. At the time, the two were living the signal points Veblen would later articulate in his 1950 International Congress address about the importance of belonging to a community of people that share your common interests and maintaining high standards that preserve the ideals of that community. Dickson may have never realized (or admitted?) that he benefit- ted from belonging to a community, but Veblen understood this intrinsic characteristic of “human beings” and engaged Dickson all the same. Although Veblen possessed the same national commitment to mathematics, he viewed the advancement of research mathematics from a different perspective and focused his efforts more broadly. We might say that Dickson worked primarily from the “inside out,” writing and securing publication for advanced (and other) mathematical texts to train the next generation of American students and setting extraordinarily high standards for research in America. Following this analytical perspective, we might characterize Veblen as a member of this community who contributed from the “outside in” as he worked to build the structure to make these individual initiatives in mathematics not only possible, but also available at increasingly meaningful levels. Speaking to the larger issue of communities, and, in particular, sustaining them, Dick- son and Veblen demonstrate the myriad of individual personalities that combine together to advance a given discipline, mathematics in this case. Despite the different roles they played in the community, these two influential figures each arrived at the same “next step” for the mathematical research community in the early part of the twentieth-century. In that time and place the next step was time and place, specifically time for research and a place to pursue it. In some sense, the creation of a mathematics institutein the early 1930s announced, in a very tangible form, that a community of research mathematicians not only existed, but also flourished in this country. That community had evolved out of a collection of singular contributions of mathematicians in the late nineteenth and early twentieth centuries. The 188 A Century of Advancing Mathematics size and stature of the group had reached a point where “men who have proved they can do productive work in this field” would benefit from a designated place to “concentrate their efforts” on mathematics. This step alone signifies the community had attained a certain level of excellence. This thin sliver of correspondence exchanged between Dickson and Veblen provides a glimpse into both the professional and personal sides of American mathematics. Their individual ideas for building an infrastructure for an American mathematical community reflected their own personalities and strengths as members of that community. In particular, Dickson advocated for publication venues for mathematics and Veblen focused on creating spaces and opportunities to pursue mathematics. Their epistolary conversation represents a microcosm of the larger issues and events that shaped the American mathematical community, and the role two of America’s best mathematicians played in their unfolding.

Bibliography Unpublished and Archival Sources [1] Brown University Archives, American Mathematical Society Archives, Box 22, Folder 8. [2] Carnegie Institution Archives, Washington, D.C., Dickson Papers. [3] Library of Congress, Manuscript Division, Oswald Veblen Papers, Box 4, Dickson Folder.

Published Sources [4] R. C. Archibald, A Semicentennial History of the American Mathematical Society 1888–1938. New York: American Mathematical Society, 1938. [5] S. Batterson, Pursuit of Genius. Wellesley, Massachusetts: AK Peters, Ltd, 2006. [6] ———, The Vision, Insight, and Inﬂuence of Oswald Veblen, Notices of the American Mathe- matical Society 54 no. 5 (2007) 606–618. [7] ———, Institute for Advanced Study: Opening Day, October 2, 1933, The Institute for Ad- vanced Study Letter, (Fall 2008), pp. 6, 9. [8] Dickson, Leonard Eugene, History of the Theory of Numbers, 3 vols. New York: Chelsea Pub- lishing Company, 1919, 1920, 1923. [9] L. B. Feffer, Loren Butler, Oswald Veblen and the Capitalization of American Mathematics, Isis 89 no. 3 (1998) 474–497. [10] D. D. Fenster, Leonard Eugene Dickson and His Work in the Arithmetics of Algebras, Archive for History of Exact Sciences 5 (1998) 119–159. [11] ———, LeonardEugene Dickson(1874–1954): An American Legacy in Mathematics. A Cele- bration of the 125th Anniversary of his Birth, The Mathematical Intelligencer 21 (1999) 54–59. [12] ———, Why Dickson left Quadratic Reciprocity out of the History of the Theory of Numbers, The American Mathematical Monthly 106 (1999) 618–626. [13] D. D. Fensterand J. Schwermer, A Delicate Collaboration: A. Adrian Albert and Helmut Hasse and the Principal Theorem in Division Algebras in the Early 1930’s, Archive for History of Exact Sciences 59 (2005) 349–379. [14] J. A. Gallian, Joseph, Contemporary Abstract Algebra, seventh edition. Brooks/Cole, Belmont, CA 2010. [15] D. A. Grier, Dr. Veblen Takes a Uniform: Mathematics in the First World War, The American Mathematical Monthly 108 no. 10 (2001) 922–931. Time and Place: Sustaining the American Mathematical Community 189

[16] Harvard Mathematics Department, Veblen’s Opening Address to ICM 1950; available from www-history.mcs.st-andrews.ac.uk/Extras/Veblen_ICM_Address.html; accessed24 June 2008. [17] D. Montgomery, Oswald Veblen, Bulletin of the American Mathematical Society 69 (1963) 26– 36. [18] O. Veblen, A System of Axioms for Geometry, Transactions of the American Mathematical Society 5 (1904) 343–384. [19] ———, Analysis Situs. AMS Colloquium Publications, 1922, second ed., 1931.

Mathematics Department, University of Richmond, 28 Westhampton Way, Richmond, VA 23173 [email protected]

Abstract (Modern) Algebra in America 1870–1950: A Brief Account

Israel Kleiner York University

Introduction: Algebra before abstract algebra

Algebra in the US did not develop in isolation. American mathematicians were in contact with, and drew inspiration from, researchers in algebra in Germany, and to a lesser extent in England and France. As Eric Temple Bell notes, “the debt of American algebra to the Germany of the late 1880s and early 1890s is very great” [7, p. 3]; see also [59]. Very significant in the development of American algebra was also the seven-year stay at Johns Hopkins (1876–1883) of the distinguished English algebraist James Joseph Sylvester [59]. And “as late as 1904 ::: 20% of the members of the American Mathematical Society report having studied abroad” [37, p. 22]. In the early decades of the twentieth century, a number of prominent American mathematicians, among them the algebraists Garrett Birkhoff, Leonard Eugene Dickson, Saun- ders Mac Lane, and Eliakim Hastings Moore visited and studied at various German universities. (Dickson also traveled to Paris to see Camille Jordan, and Birkhoff went to Cam- bridge to study with Philip Hall.) What did their hosts offer? The answer, given in Sects. 2, 3, and 4 below, will provide a context for the consideration of abstract algebra in the US. First, some background. The Babylonians knew how to solve linear and quadratic equations about 3500 years ago. Muhammad al-Khwarizmi, dubbed by some “the Euclid of algebra,” systematized the study of such equations in his book Kitab al-jabrwa l-muqabalah (ca 830 AD), from which stems our word “algebra.” Girolamo Cardano (in 1545) and other Italian mathematicians of the Renaissance solved cubic and quartic equations by radicals, and François Viète (1590s) and RenéDescartes (1630s) supplied a symbolic notation, which enabled a theory of polynomial equations to emerge. This came to be known as “classical algebra” and was usually taught in the schools under the title “Theory of equations.” It is essentially pre-nineteenth- century mathematics. See [18]. A number of problems emerged in classical algebra, number theory, and geometry which were studied intensively in the nineteenth century but which would not yield to existing methods. Among these were solvability by radicals of polynomial equations of degree greater than four, solution of diophantineequations, such as the Fermat equation xn yn C D zn (n > 2), representation of integers by binary quadratic forms, n ax2 bxy cy2 D C C 191 192 A Century of Advancing Mathematics

(n and a, b, c integers), finding bases for invariants of various forms, and extension of the complex numbers to hypercomplex numbers. See [45]. To deal with these (and other) problems, mathematicians needed new tools (concepts, results, theories). To this end, they introduced what were at the time viewed as abstract notions, such as permutation groups, (groups of) equivalence classes of binary quadratic forms, rings of integers of algebraic number fields and algebraic function fields, ideals of such rings, rings of polynomials and their ideals, rings of invariants, and algebras (hypercomplex systems) of matrices and of n-tuples of real numbers (including quaternions and octonions). See [45] for details. All this occurred against the background of fundamental changes in mathematics in the nineteenth century. Mathematicians turned more and more for the genesis of their ideas from the sensory and empirical to the intellectual and abstract. Witness the introductionof non-commutative algebras, non-euclidean geometries, continuous nowhere differentiable functions, space-filling curves, n-dimensional spaces, and completed infinities of different sizes. Cantor’s dictum that “the essence of mathematics lies in its freedom” became a reality, though one to which many mathematicians took strong exception. Other pivotal changes were the emphasis on rigorous proof and the acceptance of non- constructiveexistence proofs,the focus on concepts rather than on formulas and algorithms, the stress on generality and abstraction, the resurrection of the axiomatic method, and the use of set-theoretic modes of thinking. It is perhaps not standard practice to focus on individuals when writing about history. But I believe that for the purposes of this account we are well served by highlighting the accomplishments of several mathematicians who were central players in the evolution of abstract algebra. We begin with Emmy Noether, the pivotal figure in the founding of that subject.

Emmy Noether (1882–1935) Noether was born in Erlangen, the German university town for which Klein’s Erlangen Program was named. (Max Noether, the well-known algebraic geometer, was her father.) In 1904 she registered at the University of Erlangen, studying mathematics, though enrollment of women at German universities was far from routine at that time. In 1908 she received her PhD degree, summa cum laude, having written a thesis on invariants under Paul Gordan. Between 1908 and 1915 she worked—without compensation—at the Uni- versity of Erlangen. In 1915 Hilbert and Klein invited her to Göttingen to help them with problems on differential invariants. Noether’s move to Göttingen was of singular importance. The university was at that time considered the world center of mathematics. With Gauss, Dirichlet, and Riemann as former professors, and with the contemporary faculty including Klein, Hilbert, Lan- dau, Minkowski, and Courant, and later Weyl, Bernays, and Neugebauer, Göttingen had become the “Mecca of Mathematics.” Noether thrived in these surroundings. The decade 1920–1930 was the decisive period in her mathematical life. This is when she made her groundbreaking contributions to abstract algebra. And she was then in her forties! Mac Lane asserted that “abstract algebra, as a conscious discipline1, starts with Noether’s

1 My emphasis. Abstract (Modern) Algebra in America 1870--1950: A Brief Account 193

Emmy Noether

1921 paper ‘Idea1 Theory in Rings’” [19, 49]. Kaplansky called her the “mother of modern algebra,” and Weyl claimed that she “changed the face of algebra by her work.” Accord- ing to van der Waerden, the essence of Noether’s mathematical creed is contained in the following statement [9, p. 42]: All relations between numbers, functions and operations become perspicuous, capable of generalization, and truly fruitful after being detached from speciﬁc examples, and traced back to conceptual connections. We identify these ideas with the abstract, axiomatic approach in mathematics. They sound commonplace to us. But they were not in Noether’s time. In fact, they are commonplace today in considerable part because of her work. For almost the entire nineteenth century algebra was concrete by our standards, connected for the most part, one way or another, with the real or complex numbers. For example, Dedekind’s rings and ideals were subsets of the complex numbers. Although work in algebra was important (see Sect. 1, pars. 4 and 5), it was viewed in the nineteenthcentury, in the overall mathematical scheme, as secondary. The primary subjects in that century were analysis (complex analysis, differential equations, real analysis) and geometry (projective, non-euclidean, differential, and algebraic). But after the work of Noether and others in the 1920s, algebra became central in mathematics. Noether contributed to the following major areas of algebra: invariant theory, commutative algebra, non-commutative algebra and representation theory, and applications of non-commutative to commutative algebra. More important than the subject matter was her approach, which initiated a new algebraic tradition—what has come to be known as “abstract algebra” or “modern algebra” (for the use of “modern” see Sect. 4). She attracted students and collaborators who promoted her view of the subject. The topologist P. S. Alexandroff highlighted the essential elements in her contribution to abstract algebraic thinking [3, p. 158]: 194 A Century of Advancing Mathematics

It was she whotaught us tothinkinterms of simpleand general algebraic concepts— homomorphic mappings, groups and rings with operators, ideals—and not in cumbersome algebraic computations, and [she] thereby opened up the path to finding algebraic principles in places where such principles had been obscured by some complicated special situation. Noether was of course not the only contributor to the abstract, axiomatic approach in algebra. Among her predecessors who advanced it were Dedekind, Frobenius, Weber, and Steinitz in Germany, and Wedderburn and Dickson in the US [45]. Among her contem- poraries, Dickson and Albert in the US and Artin and Brauer in Germany stand out. We will consider the contributions of Artin, Dixon, Wedderburn, and Albert (and several others) shortly (in Sects. 3, 8, 9, and 10, respectively), but first we recount Noether’s stay in the US. On January 31, 1933, Hitler assumed the office of Chancellor. On March 31 he announced the beginning of the Third Reich. On April 25 Noether was dismissed from her teaching position at Göttingen. The dismissal of Courant, Landau, and Bernays followed in short order. Courant was replaced as head of the Mathematics Institute by Neugebauer, who lasted one day in that position. He refused to sign the required declaration of loyalty [63]. Jobs at American universities were difficult to come by in those days of the Great Depression [39, p. 4], [63]. With Weyl’s assistance, Noether got a visiting position at Bryn Mawr, a women’s college in Pennsylvania. The transition might have been difficult but for the warm reception she received at Bryn Mawr and the mathematical contacts she made at the nearby Institute for Advanced Study in Princeton, recently established [33]. At Bryn Mawr she had her “Noether girls”—one doctoral and three postdoctoral students. The latter were Olga Taussky (later Taussky-Todd), Grace Shover (later Quran), and Marie Weiss. In a 1939 article in The American Mathematical Monthly Weiss outlined a course in abstract algebra that she had taught to juniors [70]. In 1949 she wrote a book on abstract algebra entitled Higher Algebra for the Undergraduate, which became quite influential [69]. Noether’s only PhD student at Bryn Mawr was Ruth Stauffer (later Mc- Kee). This foursome eagerly read the first volume of van der Waerden’s Moderne Algebra (Sect. 4). At Princeton Noether began in early 1934 to give weekly lectures on (abstract) algebra. Writing to Hasse about them, she said [20, pp. 81–82]: Princeton will receive its first algebraic treatment this winter, and a thorough one at that. My audience consists mostly of research fellows, besides Albert and Vandiver [who were faculty members]. I’m beginning to realize that I must be careful; after all, they are essentially used to explicit computation and I have already driven a few of them away with my approach! Between the University and the Institute, there are more than sixty professors and aspiring professors here. Among those who were not driven away were Albert, Brauer, Jacobson, Vandiver, and Zariski. In a book on Zariski, Carol Parikh pointed out that “Zariski’s contact with Noether was undoubtedly the single most important aspect of that year for him” [53, p. 74]. Jacob- son, who was completing his PhD dissertation at Princeton under Wedderburn, and who was to become one of the foremost algebraists of the era (Sect. 10), noted that his “contacts Abstract (Modern) Algebra in America 1870--1950: A Brief Account 195 with [Noether] were a memorable experience, as was attending her course” [39, p. 4]. In- cidentally, Jacobson’s first full-time teaching position was at Bryn Mawr, in 1935–36. He was hired to replace Noether following her death [40, p. 41]. The time she spent at Bryn Mawr and Princeton, Noether told Veblen before her death, was the happiest in her life. She was respected and appreciated as she had never been in her own country. But it was a brief, if happy, year and one-half. On April 10, 1935, she underwent an operation for a tumor. She was recovering well when, four days later, complications brought unexpected death. See [3, 20, 50, 52, 66] for further details on Noether’s life and work.

Emil Artin (1898–1962) Born in Vienna, Artin was brought up in Reichenberg, Bohemia. He was drafted into the army in late 1916(or early 1917) and entered the Universityof Leipzig in 1919, completing his PhD with Gustav Herglotz in 1921. He then went to Göttingen for a year, after which he obtained a positionat Hamburg University. He became full professor in 1921, at age 23. In 1937 Artin left Germany for the US, fearing for the fate of his Jewish wife. He spent one year at the University of Notre Dame, then eight years at Indiana University. In 1946 he moved to Princeton, where he stayed until 1958. He then returned to Hamburg, remaining there until his death of a heart attack in 1962 [25]. Artin was “one of the principal inspirationsof the movement for abstract algebra,” notes Mac Lane [49, p. 6]. He spread the gospel on the subject through outstanding publications, brilliant students, inspiring teaching, and influential books. We give several illustrations. Stimulated by Noether’s work (in 1921) on commutative rings with the ascending chain condition, now called Noetherian rings, Artin generalized (in 1927) Wedderburn’s structure theorem on algebras to non-commutative rings with the descending chain condition, now called Artinian rings (Sect. 9). Ring theory began to take its rightful place alongside the theories, by then well established, of groups and fields, as one of the pillars of abstract algebra. See Artin’s book Rings with Minimum Condition (1944). In 1927 Artin and Schreier defined the notion of a formally real field, one in which 1 is not a sum of squares. According to Bourbaki [15, p. 92]: One of the remarkable results of the Artin-Schreier theory of formally real fields is no doubt the discovery that the existence of an order relation on a field is linked to purely algebraic properties of the field. Specifically, a field can be ordered if and only if it is formally real. The theory of formally real fields, a “daring and successful construction of a bridge between algebra and analysis” [72, p. 4], enabled Artin to solve Hilbert’s seventh problem on the resolution of positive definite rational functions into sums of squares. Class field theory is the study of finite extensions of an algebraic number field having an abelian Galois group. It is a beautiful synthesis of algebraic, number-theoretic, and analytic ideas, in which Artin’s reciprocity law plays a central role. This law is a grand generalization (the grandest, it is said) of various reciprocity laws, investigated intensively throughout the nineteenth century, beginning in 1801 with Gauss’s renowned quadratic reciprocity law. See Artin’s books Algebraic Numbers and Algebraic Functions (1951) and 196 A Century of Advancing Mathematics

Emil Artin

Class Field Theory (1961),the latter co-authored with John Tate, one of Artin’soutstanding PhD students. Among his eleven doctoral students in Germany and twenty in the US were, in addition to Tate, such luminaries as Serge Lange, Timothy O’Meara, Richard Semple, Hans Zassenhaus, and Max Zorn. One of the fundamental results of Galois theory is the Primitive Element Theorem, which states that if E and F are fields with E a finite extension of F of characteristic zero, then E F.a/, for some a in E. This result was enunciated and proved by Galois D himself, and it became essential in all subsequent work in the subject. Artin bypassed it by “linearizing” Galois theory (1926), for he felt that the theorem was not intrinsic to the subject. In a 1950 talk he said [43, p. 144]: Since my mathematical youth I have been under the spell of the classical theory of Galois. This charm has forced me to return to it again and again, and try to find new ways to prove its fundamental theorems. Contemporary treatments of Galois theory usually follow Artin’s, as set out in his book Galois Theory (1942). Also worthwhile is his set of notes, “Contents and Methods of an Algebra Course,” intended for graduate students [4, pp. 539–546]. The editors of Artin’s Collected Papers, Serge Lange and John Tate, said the following about Artin as a teacher [4, p. x]: Artin loved teaching at all levels. Even though occupying research professorships, he never failed to give, regularly, courses in elementary Calculus. [See his book Cal- culus and Analytic Geometry (1957).] His Lectures and Seminars were renowned for their perfection and excitement. They contributed much toward spreading his point of view in algebra, for which van der Waerden’s text, derived from lectures by Artin and Emmy Noether, has been the fundamental reference for the past 30 years [see Sect. 4]. They also inspired his students, towards whom his generosity and affection were unsurpassed. Abstract (Modern) Algebra in America 1870--1950: A Brief Account 197

Hans Zassenhaus, one of Artin’s star PhD students, also comments on his teaching [72, pp. 1, 5, 6]:

Artin stands out as a great teacher. He tooka lively interest in theproblems of teaching mathematics at all levels. In Princeton Artin gave outstanding honors courses to mathematics freshmen beyond the call of duty. The 1957 book on geometric algebra is the most mature fruit of Artin’s ideas [on] how to modernize the teaching of mathematics from the high school level to the senior undergraduate level.

One of the giants of algebra, Artincontributed much to the subject in America—through his papers, his books, his teaching, and his students [25]. He was honored by many scientiﬁc societies. In 1932 he was awarded, jointlywith Noether, the Ackermann-Teubner Memorial Prize for the advancement of the mathematical sciences.

Van der Waerden’s Moderne Algebra (1931)

This book is a classic [68]. Published in German in 1931 (in two volumes), translated into English in 1949, it captured masterfully the essentials of a newly-emerging field— abstract algebra—and served as a model of what is important in algebra and how it should be presented and taught. Pointing to the book’s enduring merit, Dieudonnéobserved in 1971 that he was “often asked for advice on how to start out studying algebra, and to most people [he said]: First read van der Waerden, in spite of what has been done since” [16, p. 45]. A near-century after its publication, the book, though challenging, can still be read with profit and pleasure. Following its publication, and for several decades thereafter, the terms “abstract algebra” and “modern algebra” were used interchangeably. In 1959, in its seventh German edition, the word “Moderne” was dropped from the title. Van der Waerden’s book served, among other things, to spread awareness of the recent founding of a new subject, in which Noether and Artin had played major roles. In an acknowledgement on the title page, the author notes that the book is “in part a development from lectures by E. ARTIN and E. NOETHER.” In [67] he gives its major sources. “Without doubt,” claims the algebraist Robert Gilmer, “Moderne Algebra is the instrument through which Emmy Noether’s influence has been greatest” [16, p. 141]. Garrett Birkhoff expressed, in striking terms, van der Waerden’s brilliant achievement and far-reaching impact [10, p. 771]:

Even in 1929, its concepts and methods [i.e., those of “modern algebra”] were still considered to have marginal interest as compared with those of analysis in most universities, including Harvard. By exhibiting their mathematical and philosophical unity and by showing their power as developed by Emmy Noether and her other younger colleagues (most notably E. Artin, R. Brauer, and H. Hasse), van der Waerden made “modern algebra” suddenly seem central in mathematics. It is not too much to say that the freshness and enthusiasm of his exposition electriﬁed the mathematical world—especially mathematicians under 30 like myself.

And now we turn to discussing the beginnings of abstract algebra in America. 198 A Century of Advancing Mathematics

Benjamin Peirce (1809–1880) Peirce was the first American mathematician who did substantial work in pure mathematics. His singular achievement was a 150-page monograph of 1870 entitled Linear Associative Algebra, giving the structure and classification of all associative algebras (also known as hypercomplex number systems) of dimension lower than seven. He showed that there are over 150 such algebras, and gave their multiplication tables. Peirce was born in Salem, Massachusetts, enrolled at Harvard College in 1825, and graduated in 1829. He was appointed Professor of Mathematics at Harvard in 1833 and remained there until his death. Early in his career he published various textbooks, from which he taught undergraduate courses. Among the textbooks were An Elementary Treatise on Plane and Spherical Trigonometry, An Elementary Treatise on Curves, Functions, and Forces, and An Elementary Treatise on Algebra: To which are Added Exponential Equations and Logarithms. He also taught graduate courses—the first ever given in the US. These comprised the study of works of Cauchy, Gauss, Hamilton, Laplace, and Monge. Peirce was, however, a poor teacher and the books he used were too challenging for his students. Peirce was inspired in his work on Linear Associative Algebra principally by the publications of Hamilton on quaternions, but also by those of Cayley, Clifford, De Morgan, and Sylvester on various “number systems.” In fact, he taught Hamilton’s quaternions at Harvard as early as 1848, five years after their discovery (invention). The quaternions were the first example of a non-commutative number system, obeying all the (algebraic) laws of the real and complex numbers except for commutativity of multiplication. They were an example of what is now known as a division ring or a division algebra (also a skew field). The quaternions acted as a catalyst for the exploration of diverse number systems, with properties that departed in various ways from those of the real and complex numbers. Among such number systems, introduced around the mid-nineteenth century, are octonions (also known as Cayley numbers), group algebras, matrices, triple algebras, and biquater- nions [45]. Much more important than the major result of Peirce’s monograph—the classification of algebras of dimension less than seven—was the means used to obtain it. Here Peirce introduced concepts and derived results which proved fundamental for subsequent work in algebras. Among the conceptual advances were: (a) An abstract definition of a finite-dimensional associative algebra. Peirce defined such an algebra—he called it a “linear associative algebra”—as the totality of formal ex- n pressions of the form i 1 ai ei , where the ei are “basis elements.” Addition was D defined componentwise and multiplication by means of “structural constants” cijk; n P namely ei ej cijkek: Associativity under multiplication and distributivity D k 1 were postulated, butD not commutativity. This is likely the earliest explicit definition P of an associative algebra, though it is not the definition that we would give today (see Sect. 8).

(b) The use of complex coefficients. Peirce took the coefficients ai in the expressions ai ei to be complex numbers. This conscious broadening of the field of coefficients from the real to the complex numbers was an important conceptual advance on the road P to coefficients taken from an arbitrary field. It also necessitated consideration of zero divisors of an algebra. Abstract (Modern) Algebra in America 1870--1950: A Brief Account 199

(c) Relaxation of the requirement that an algebra have an identity. This too departed from past practice and indicated Peirce’s general, abstract approach. (d) Introductionof nilpotent and idempotent elements. These concepts proved basic for the subsequent study of algebras and, still later, of rings. Peirce proved the fundamental result that any algebra contains a nilpotent or an idempotent element. (e) The Peirce decomposition. He showed that if e is an idempotent of an algebra A then A eAe eB1 B2e B; where B1 x A xe 0 , B2 x A ex 0 D ˚ ˚ ˚ D f 2 j D g D f 2 j D g , and B B1 B2: This result is still a central tool in the study of rings and algebras D \ (see Sect. 9). Peirce’s work was well ahead of its time, and at first attracted little attention. Cay- ley, for example, who praised it in an address in 1883 to the British Association for the Advancement of Science, called it “outside of ordinary mathematics” [62, p. 548]. Even some of Peirce’s admirers in the United States characterized the work as “philosophy of mathematics” rather than mathematics proper. Peirce began the monograph with a definition of mathematics: “mathematics is the science which draws necessary conclusions” [61, p. 97]. This abstract definition was out of line with the view of mathematics at the time, more in tune with its conception in the early twentieth century. Following its elaboration [61, pp. 97–98], Peirce turned to algebra. He described it as “formal mathematics,” and distinguished between “logical algebra” and “arithmetical algebra” [61, p. 98]. “The language of algebra,” he observed, “has its alphabet, vocabulary, and grammar” [61, p. 99]. The next twenty pages or so expanded on what algebra is about. Little wonder such musings have been viewed as “philosophy.” Peirce was a devoutly religious man, and he saw his work as an affirmation of God’s design, so that he did not take these criticisms to heart. And in time, his Linear Associative Algebra was seen as “a pioneer work in American mathematics and in modern abstract algebra” [62, p. 551]. Its author, noted George David Birkhoff in 1938, “appears as a kindof father of pure mathematics in our country. In his deep appreciation of the elegant and abstract we may recognize a continuing characteristic of American mathematics” [11, p. 272]. The theory of associative algebras—the subject whose study he initiated—was an area in which American algebraists excelled during the next hundred or so years.

The Johns Hopkins University (founded 1876), and the University of Chicago (founded 1892) In a survey of higher mathematics education in the US undertaken in 1840, it was found that there were “a hundred and twenty colleges.... All teach mathematics, but where are [the] mathematicians?” [37, p. 10]. In fact, “in the first hundred years of therepublic[1776- 1876] ::: no American was an outstanding leader in world mathematics” [37, p. 9]. Ad- mittedly, Peirce’s Linear Associative Algebra of 1870 was a world-caliber work, but it was published only in 1881 (posthumously,in the American Journal of Mathematics), edited by his son Charles Saunders Peirce [61]. (In 1870 Benjamin Peirce made one hundred litho- graph copies of the monograph and distributed them to friends and colleagues, mostly in the US, some in Europe.) Yet, “by the end of the nineteenth century, the work of Americans 200 A Century of Advancing Mathematics was known and respected throughout the mathematical world” [37, p. 10]. Two distinctive events were largely responsible for this transformation—the founding of The Johns Hop- kins University (in 1876) and of the University of Chicago (in 1892). The former university was founded by Baltimore industrialist Johns Hopkins [36]. Some business leaders in the mid-nineteenth century began to recognize that scientific research was essential to the growth of American industry. Consequently, the mandate of Johns Hopkins was to focus on research—including mathematical research—as a major goal. In fact, the University was the first American institution of higher learning “consciously devoted to the pursuit of knowledge [for its own sake], the solution of problems, the critical appreciation of achievement, and the training of men at a really high level” [11, p. 273]. To put that vision into practice, the newly appointed president of the University, Daniel Coit Gilman, went in search of (among others) a distinguished mathematician. The sixty- one-year-old J. J. Sylvester, the eminent English mathematician who had retired in 1870, fit the bill admirably, and was given the task of buildinga department at Johns Hopkins focusing on mathematical research. He accomplished the mission remarkably well, remaining at Hopkins for seven years.

J. J. Sylvester

During that time he founded (with the assistance of Peirce and others) The American Journal of Mathematics, the ﬁrst research-oriented American mathematics periodical. This was an important outlet for European as well as American mathematics. For example, its ﬁrst issue had an article by Lipschitz from Bonn, two by Clifford from London, and three by Cayley from Cambridge. Early subscribers to the journal were Hermite, the University Library at Cambridge, and the library of the Ecole´ Polytechnique. Sylvester also directed eight PhD theses during his seven-year stay in the US, mainly in invariants and combinatorics. “The mathematical results which [Sylvester] and his students published [in the Abstract (Modern) Algebra in America 1870--1950: A Brief Account 201

American Journal of Mathematics] served to awaken Europe to America’s growing mathematical sophistication” [60, p. 10].See also [37, 59] for further details. Felix Klein was offered the position vacated by Sylvester in 1883, but he declined. However, his association with mathematics in the US was substantial. For example, he was the invited keynote speaker at the first International Congress of Mathematicians, held in Chicago in 1893. Followingthe Congress, he gave a series of lectures at Northwestern Uni- versity on recent developments in mathematics with which he had been closely associated. Given that he was an excellent teacher with a broad and deep knowledge of mathematics, he attracted a number of budding American mathematicians to Leipzig (in 1880–1885) and to Göttingen (in the following decade) for doctoral and postdoctoral studies. Among these were Osgood, Fine, Bolza, van Vleck, Cole, Haskell, Bôcher, and Maschke. See [65]. A very significant event in American mathematics was the foundingin 1892 of the Uni- versity of Chicago, financed by the tycoon John D. Rockefeller. Eliakim Hastings Moore, who had recently obtained his PhD at Yale, did postdoctoral work at Berlin with Kronecker and Weierstrass, and studied German at Göttingen, was picked as head of the department of mathematics. He appointed to the department two excellent mathematicians, Oskar Bolza and Heinrich Maschke, who taught both graduate and undergraduate students, with a focus on the former. “This team almost immediately made Chicago the leading department of mathematics in the United States” [48, p. 129]. See also [37, 54, 59, 60]. The University of Chicago awarded thirty-ninedoctorates in mathematics during 1892– 1910, among them to G. D. Birkhoff(analysis), L. E. Dickson (algebra and number theory), R. L. Moore (topology),and O. Veblen (geometry). All four were students of E. H. Moore, and were soon to become outstanding mathematicians in their own right. Mathematics came to flourish in the US and began to be respected internationally. The strongest area of mathematics at the University of Chicago during the first several decades following its founding was abstract algebra. Faculty members with an interest in algebraic research were E. H. Moore, Maschke, Dickson, and Wedderburn (the last visiting Chicago in 1904–1905). The following graduate algebra courses were offered in (for example) 1902: theory of equations (Bolza), finite groups (Dickson), and linear substitution groups (Maschke). (There was also a course in the history of mathematics (!), taught by Epsteen, a postdoctoral student.) “Algebra was there [at the University of Chicago] with a vengeance,” observed Mac Lane [48, p. 133]. We will discuss shortly the contributions to algebra of Dickson and Wedderburn, but first some comments on what came to be known as “postulational analysis,” an area of abstract algebra that attracted mathematicians in America.

Postulational analysis (1902–1908)

The axiomatic method, ably practiced in ancient Greece ca. 300 BC, fell into a deep slum- ber, with occasional awakening, during the next two millennia. It began to be used in the second half of the nineteenth century in geometry and in algebra [45]. In geometry the pioneers of modern axiomatics were Pasch, Peano, and Hilbert. (But note that Hilbert’s axiomatics is not Euclid’s, and that the use of axiomatics in geometry differs from its use in algebra [12].) Hilbert’sworkstands out as an example of depthand clarity. His Foundations of Geometry (1899) was, according to Birkhoff and Bennett, “the most inﬂuential book on 202 A Century of Advancing Mathematics geometry written in the ...[nineteenth] century” [12, p. 343]. Bell claimed: Hilbert’s classic inaugurated the abstract mathematics of the twentieth century. His great authority ﬁrmly established the postulational method, not only in the geometry of the twentieth century, but also in nearly all of mathematics since 1900 [12, p. 387]. In the US, E. H. Moore “strove unceasingly toward the utmost abstractness and generality obtainable” [7, p. 4]. He was taken with Hilbert’s masterpiece, in particular with his clear formulation of the notions of independence, completeness, and consistency of a set of postulates. In 1902, in an article titled “On the projective axioms of geometry,” which appeared in the recently founded Transactions of the American Mathematical Soci- ety, he showed that Hilbert’s axioms for Euclidean geometry were not independent. In the next ten years or so, E. H. Moore, and some of his students and colleagues at Chicago (and elsewhere), among them Dickson, Huntington,R. L. Moore, Veblen, and Wedderburn, published articles that analyzed postulates of various geometric and algebraic structures. This topic came to be known (in 1980) as “postulational analysis” [55, p. 221]. Aside from the technical problems that these authors were tackling, they promoted the rise of the axiomatic method in algebra (and also the establishment of model theory). It must be admitted, however, that these works did not make a great impression on European mathematicians. See [12, 19, 55] for further details on this section. In the next two sections we discuss some of the work of the two foremost American algebraists of the era—Dickson and Wedderburn. (The latter, though born in Scotland, spent almost all his academic life in the US) They made outstanding contributions to algebra in the early decades of the twentieth century. America began to appear on the world map of centers of algebraic excellence.

Leonard Eugene Dickson (1874–1954) Dickson was the most prominent American algebraist of his time. He was E. H. Moore’s first PhD student, gettinghisdoctorate in 1896. Followingpostdoctoralstudy in Leipzig and Paris (1896–97), he spent three years at the Universities of California and Texas, returning to Chicago in 1900, where he remained until his (formal) retirement in 1939, following which he became Professor Emeritus. Dickson was a prolific researcher and an inspiringteacher. He authored eighteen books and over three hundred research papers. Among the sixty-seven doctoral theses which he supervised, twelve were by women. It is noteworthy that “like his predecessors [Moore, Maschke, and Bolza], Dickson incorporated the most recent European mathematics into his graduate courses” [28, p. 316]. As for professional duties, he was editor of the Monthly (1902–1908) and the Transactions (1911–1916),and President of the American Mathemat- ical Society (1916–1918). Aside from his accomplishments in algebra, Dickson made significant contributions to number theory, which he viewed (with Gauss) as the queen of mathematics. Among his eighteen books, the following were especially valued: Linear Groups with an Exposition of the Galois Field Theory (1901) (later titled Linear Groups), Linear Algebras (1914), History of the Theory of Numbers, a prodigious three-volume work (1919, 1920, 1923), and Algebras and their Arithmetics (1923), likely his most influential work. Abstract (Modern) Algebra in America 1870--1950: A Brief Account 203

Linear groups, that is, groups of invertible matrices (or invertible linear transformations) were studied by Galois (1830s) and Jordan (1870s) over the field of integers modulo p: They were generalized by Dickson to linear groups over arbitrary finite fields. The groups yielded classes of finite simple groups, the building blocks of all finite groups. All this is discussed in the first half of Linear Groups with an Exposition of the Galois Field Theory. The second half “contains the most extensive and thoroughpresentation of the theory of Galois fields available in the literature” [22, p. v], noted the prominent algebraist Wilhelm Magnus in a 1958 introduction to the book, calling it “a milestone in the development of modern algebra,” and noting that “for many years Dickson’s book was the final word on the subject of linear groups in a Galois field” [22, p. vi]. It is noteworthy that E. H. Moore, Dickson’s mentor, defined the notion of an abstract field (as did Weber in Germany in the same year, 1893), and characterized finite fields, so-called Galois fields. Dickson’s Algebras and their Arithmetics (1923) was a very significant contribution to abstract algebra, and it attracted considerable attention abroad [21]. He presented this “new branch of number theory” at two International Congresses of Mathematicians, in Strasbourg (1920) and in Toronto (1924). The book inspired Artin and Hasse in their work in class-field theory [34, pp. 125–126]. Its general idea was to extend the factorization theory of integers in algebraic number fields, initiated by Gauss and developed by Kummer, Kronecker, and Dedekind, to hypercomplex number systems (associative algebras)—first over the rationals and then over an arbitrary field. Unsuccessful attempts in this direction had already been made in Europe—by Rudolf Lipschitz, by Adolf Hurwitz, and by Gustave Du Pasquier (Hurwitz’s doctoral student), although Hurwitz did succeed (in 1896) in defining “the integers” in the algebra of quaternions and proving uniqueness of factorization of such integers into primes. Dickson’s book was translated into German in 1927, with substantial revisions by the author and by Andreas Speiser. Noether mentioned the book in one of her papers, and Dickson’s notion of “crossed product” in her 1932 talk at the ICM in Zurich [46, p. 156]. Among other noteworthyfeatures, the book [21] included several firsts: (i) An abstract definition—in today’s style—of an associative algebra over an arbitrary field [p. 9]. (ii) A proof of the existence of finite-dimensional division algebras other than the quaternions or generalized quaternions, whose dimension over the ground field is greater than four [p. 65]. (iii) A solution of certain diophantine equations not previously amenable to a complete solution [p. 194]. (iv) An account in book form of Wedderburn’s structure theorems (see Sect. 9 below). In sum, the book “stimulated the great flowering of associative algebra theory of the 1930s” (Jacobson [41, p. 1076]). And it “was crucial not only for bringing together two different schools [the American and the German] but also for sparking the fruitful, mutual interaction that ultimately gave rise to new and powerful methods and to abstract theories...” (Frei [34, p. 142]). The book’sauthor won the American Mathematical Society’s Cole Prize in Algebra in 1928 (see Sect. 11). For further details on this Section see [1, 27, 28, 29, 30, 34, 46]. 204 A Century of Advancing Mathematics

Joseph Henry Maclagan Wedderburn (1882–1948) Wedderburn was born in Forfar, Scotland. In 1903 he completed a Masters degree at the University of Edinburgh with first-class honors in mathematics. He came under the influence of Peter Guthrie Tait’s work, which focused on the application of Hamilton’s quaternions to physics. In that spirit Wedderburn published four papers. Soon, however, his interest turned to a study of hypercomplex systems for their own sake, as algebraic entities. He might have been influenced in this choice by the work of William Burnside (who was also of Scottish ancestry). To broaden his horizons, he went for a year to Leipzig and Berlin to study, respectively, with Engel (a collaborator of Lie) and Frobenius. In 1904, Wedderburn moved for a year to Chicago as a Carnegie Fellow, to work in the congenial algebraic setting fostered by Moore and Dickson [57]. It is noteworthythat a gifted European mathematician now considered furtheringhis higher education in America. During 1905–1909 Wedderburn was a lecturer at Edinburgh, where he also completed his doctorate. In 1909 he got a positionat Princeton, taking early retirement in 1945, apparently due to long-lasting ill-health. Wedderburn published thirty-eightpapers and a textbook—Lectures on Matrices (1934). But his reputation rests largely on two results, the first proving that a finite division algebra is commutative, hence a field (“On finite division algebras,” Trans. Amer. Math. Soc. 6 (1905) 349–352), and the second establishing the structure of finite-dimensional associative algebras over an arbitrary field—what came to be known as the Wedderburn Structure Theorem (“On hypercomplex numbers,” Proc. Lond. Math. Soc. 6 (1907) 77–118). The theory of associative algebras (hypercomplex number systems) is an important branch of algebra, vigorouslypursued by American mathematicians since Benjamin Peirce’s work in 1870 (Sect. 5). And, as Artin pointed out, “from the very beginning the abstract point of view is dominant in American publications [on the subject] whereas for European mathematicians a system of hypercomplex numbers was by nature an extension of either the real or the complex field” [5, p. 65]. In one of the early results in the subject, Charles Saunders Peirce, who edited and published his father’s (Benjamin Peirce’s) Linear Associa- tive Algebra in 1881, showed that the only finite-dimensional associative division algebras over the reals are the reals, the complex numbers, and the quaternions [61]. (This result was established independently by Frobenius.) Wedderburn’s 1905 result, that a finite associative division algebra is a field, has “fascinated most algebraists to a very high degree...,” opined Artin [5, p. 71]. (Finite fields, we recall, were characterized by E. H. Moore, in 1893.) Moreover, that result is a “special case of a more general diophantine property of fields [as shown by Chevalley in 1935] and thus has opened an entirely new line of research” [5, p. 72]. In 1906 Wedderburn and Veblen used Wedderburn’s result to show that in a finite projective plane Desargues’s theorem implies Pappus’s theorem (Trans. Amer. Math. Soc. 8 (1907) 379–388). The algebraic proof of this geometric result is apparently the only one in existence. At the end of the nineteenth century the theory of finite-dimensional associative algebras had attained a degree of maturity. Connections had been made with Lie’s theory of continuous groups as well as with the theory of finite groups, via group representation theory. A major structure theorem for associative algebras (due independently to Elie Cartan, Georg Frobenius, and Theodor Molien) was also available. It stated that a semi-simple, finite-dimensional associative algebra over the real or complex numbers is a direct sum of Abstract (Modern) Algebra in America 1870--1950: A Brief Account 205 simple algebras, which in turn are matrix algebras over division algebras. Moreover, this representation is unique up to isomorphism [57]. The theory of finite-dimensional associative algebras thus became a distinct discipline for serious mathematical investigation. What was needed for further progress in the subject was a new departure, and this was provided by Wedderburn’s groundbreaking paper of 1907, “On hypercomplex numbers,” Proc. London Math. Soc. 6 (1907) 77–118. Wedderburn’s result “merely” extended the field of scalars of the algebra from the reals or complex numbers to an arbitrary field. This extension, however, necessitated a new approach to the subject—a rethinking and reformulation of its major concepts and results. Wedderburn’s approach to the study of the structure of finite-dimensional algebras was, unlike previous methods (due to the Europeans), conceptual rather than computational. “It is remarkable,” he wrote toward the end of the paper, “that the properties of a field with regard to division are not used in many of the theorems of the preceding sections.” Among the ideas which he either introduced for the first time or made central in the study of algebras, ideas now—a century later—still recognized by students of algebra as basic, are the notionsof ideal, quotient algebra, nilpotent algebra, radical, semi-simple and simple algebra, direct sum, and tensor product. His work “marked a veritable turning point in the theory of linear associative algebras,” wrote G. D. Birkhoff in 1938 [11, p. 287]. It served as an inspiration and a model for other ring-theoreticstructure theorems, for example those by Artin and Jacobson. See [5]. It has been said that “a good abstract theory” is one which summarizes and unifies previous results, placing them in a new perspective, and one which provides new directions for subsequent work in the field. Undoubtedly Wedderburn’s work qualifies for such a designation. See [5, 56, 57, 58] for further details on this section.

America ranks first in algebra (1930s and 1940s) Algebra in America flourished during this period. A second generation of outstanding algebraists, who got their education in the US, was joined by equally eminent colleagues fleeing Nazi Germany and Nazi-occupied Europe (there were about 140 mathematicians in the latter category [63]), to make theUS a center of research in algebra (and in mathematics as a whole) second to none. The center of gravity of algebraic excellence had shifted from Germany to America. In a speech in 1936 at Harvard, “Hardy stated that the United States had become number one in mathematics, ahead of Germany, France, or England” [9, p. 62]. The same can undoubtedly be said of algebra. Among the homegrown algebraists in this period we find such household names as Albert, Jacobson, Mac Lane, and Kaplansky; among the émigrés were luminaries such as Noether, Artin, R. Brauer, Baer, and Hochschild. The outstanding algebraist (and Bour- bakist) Claude Chevalley came as a visitor to the Institute at Princeton in early 1939 and stayed on in the US until 1955. We make very brief remarks about the works of some of these algebraists.

A. Adrian Albert (1905–1972) Albert was born and educated in Chicago. His doctoral-thesis advisor was Dickson, who steered him to investigate ﬁnite-dimensional associative algebras, a subject that occupied 206 A Century of Advancing Mathematics

Albert for the rest of his life. He also made important contributions to Lie and Jordan algebras and to Riemann matrices (the latter arise in the theory of complex manifolds) [27, 42]. His work on Jordan algebras, claimed Kaplansky, “created a whole subject” [42, p. 249]. And he resolved the major outstandingproblems in the theory of Riemann matrices [41]. Albert came to Princeton in 1928–29 to work with Wedderburn, who in his famous structure theorems of 1907 left unresolved the nature of division algebras over an arbitrary field. On the other hand, it was most important to determine all division algebras over the rationals—this was perhaps the major problem in the field of associative algebras. “Artin, Hasse, and Noether cherished the hope that [its solution] would give a clue as to how to extend class field theory to the non-commutative case” [34, pp. 117–118]. The problem of division algebras over the rationals was solved in 1931 by Brauer, Hasse, and Noether, although that of non-commutative class-field theory remained open. Albert worked very intensively on the rational division-algebras problem and was very close to a solution, but he “had stiff competition and was nosed out in a photo finish,” as Kaplansky put it [42, p.247].“In ajointpaper [that Albertand] Hasse publishedin 1932 the full history of the matter was set out, and one can see how close Albert came to winning” [42, p. 247].See also [31]. Albert published about 140 papers and had eighteen PhD students, among them Di- vinsky, Dubisch, Gerstenhaber, Rapoport, and Zelinsky. He was awarded the Cole Prize in Algebra in 1939. See [2, 27, 31, 41, 42] for further details on Albert.

Nathan Jacobson (1910–1999) Born in the Jewish ghetto of Warsaw, Jacobson emigrated with his family to the US when he was ﬁve. He was awarded the PhD at Princeton under Wedderburn in 1934. He taught at Bryn Mawr, North Carolina, and Johns Hopkins, before moving (in 1947) to Yale, where he remained for the rest of his life. Jacobson described the employment situation for academics during roughly the mid- 1930s to the mid-1940s as follows [39, p. 4]: In the depth of the Great Depression salaries declined in some instances and there were very few new positions. Moreover, for the new Jewish PhDs the situation was further aggravated by anti-Semitism that was prevalent, especially in the top universities—the only ones that had any interest in fostering research. Seligman, his colleague at Yale for many years, adds [8, p. 1062]: [T]he anti-Semitic barrier to senior appointments in the faculty of Yale College had fallen only in 1946, and there were still misgivings about that step in too many quarters; but the time had come when merit could prevail. Meritorious, indeed, Jacobson was “a giant of twentieth-century algebra,” according to Kaplansky [8, p. 1063]. Seligman elaborates [8, p. 1061]: His contributions have become a part of the daily vocabulary and working equipment of many of us.... His expository and research monographs and his ambitious textbooks have indebted a worldwide community to him for strong and articulate leadership. Abstract (Modern) Algebra in America 1870--1950: A Brief Account 207

Jake (as he was known) made outstanding contributions in all the three major areas of algebras: Lie, associative, and Jordan algebras. But perhaps most important was his work on the structure theory of rings without finiteness conditions, generalizing both Wedderburn’s and Artin’s contributions to this field. The Jacobson radical and the Jacobson-Chevalley density theorem are fundamental in this context. The above four subjects were enshrined in books which became classics and nourished generations of mathematicians worldwide [8, p. 1066]. Then there were two sets of challenging textbooks, perhaps suitable for graduate students: the three-volume Lectures in Abstract Algebra of 1975 and the two-volume Basic Algebra of 1980. Finally, Jacobson had over thirty PhD students, some of whom became first-rate algebraists [8, 39].

Saunders Mac Lane (1909–2005) Mac Lane was an undergraduate at Yale, and followinga year of graduate work at Chicago, went to study mathematical logic at Göttingen, where he earned his PhD (in 1933) under Bernays. He also imbibed the new algebra from Noether, and other subjects from Hilbert and Weyl. On his return to the US, he taught at Yale, Harvard, Cornell, and Chicago, before getting a permanent position (in 1938) as assistant professor at Harvard. In 1947 he left Harvard for Chicago, where he remained for the rest of his academic career. Mac Lane’s early papers were on valuation theory. In fact, his first doctoral student, Kaplansky, wrote a thesis entitled “Maximal Fields with Valuations.” But Mac Lane’s main claim to fame was the creation (in 1945), jointly with Eilenberg (another refugee from the Nazis), of category theory. The subject was not received well for about a decade after its introduction.Mac Lane related that the notions of category theory “were so general that they hardly seemed real mathematics,” and wondered: “would our mathematical colleagues accept them?” [45, p. 125]. In time category theory became central in mathematics—both as a language and as a subject in its own right.Some of itsmain concepts, noted Mac Lane, “are algebraic in character ... [and can be used to give] a fresh presentation of algebra” [51, p. v]. During his fourteen-year collaboration with Eilenberg, Mac Lane also had a hand in inventing homological algebra and cohomology theory. “In the hands of Eilenberg, Mac Lane, Grothendieck, and others [homological algebra] evolved into a new branch of algebra, embracing category theory and other new constructs” [5, p. 1480]. Mac Lane had broad mathematical interests, as reflected in (for example) his thirty-nine doctoral students. Among these were Kaplansky (in valuation theory, as noted), Solovay (in logic), Morley (in model theory), Eisenbud (in homological algebra), Thompson (in group theory), Szczarba (in algebraic topology),Ginali (in computer science), and Calinger (in history of mathematics). His book Mathematics: Form and Function stemmed from a longstanding interest in the philosophy of mathematics, and his book A Survey of Modern Algebra (joint with Garrett Birkhoff) was a result of his keen interest in teaching (see Sect. 12 below). See [47] for further details on this Section.

Gerhard Hochschild (1915–2010) Hochschild was born in Berlin. In 1933, soon after the Nazis came to power, he left for South Africa, where he got a BS degree in 1936 and an MS degree the following year. He then left for Princeton, completing his PhD in 1941 under Chevalley with a thesis entitled 208 A Century of Advancing Mathematics

“Semisimple Algebras and Generalized Derivations.” After serving in the army during the war, he got postdoctoral positions at Harvard and Princeton. In 1948 he was appointed to a tenure-track post at the University of Illinois as assistant professor and was promoted to full professor in 1952. In 1958 he moved to Berkeley. Hochschild was the first to introduce the cohomology groups of associative algebras and their modules, and to use cohomological tools to study local and global class-field theory. He also introduced (with Chevalley) the spectral sequence for the cohomology of a group extension. All these came to be basic tools in mathematics. His interests soon shifted to Lie algebras and Lie groups, their representations, and their cohomology. He spent the next twenty-five years making basic contributions(often in collaboration with George Daniel Mostow) to these topics. He was elected to the National Academy of Sciences and to the American Academy of Arts and Sciences. And he was awarded the American Mathematical Society’s Steele Prize. See [32] for details.

Irving Kaplansky (1917–2006)

Born in Toronto, he got his BA and MA at the University of Toronto in 1938 and 1940, respectively. He was on the winning team of the first Putnam competition, which resulted in a fellowship at Harvard (in 1940). He got his PhD the following year under Mac Lane, and moved to the University of Chicago in 1945, where he remained until his retirement in 1984. He was Director of the Mathematical Sciences Research Institute in Berkeley during 1984–1992. “For students interested in algebra, Kaplansky is virtually a household name,” asserted Hyman Bass [6, p. 1482]. He contributedto variousareas of the subject: topological algebra (including operator algebras and locally compact rings), commutative and homological algebra, non-commutative ring theory, Lie groups and Lie algebras, abelian group theory, and linear algebra; and he also advanced combinatorics, number theory, game theory, and probability and statistics. He had fifty-five doctoral students. Opined one of them, Richard Swan, “He was not only a fantastic mathematician but a marvelous lecturer, and he had a remarkable talent for getting the best out of students” [6, p. 1481]. In view of the contributionsof these five algebraists, as described in the preceding brief remarks, we can say that during the two decades under consideration (1930–1950) algebraic research in the US had not only become well established by assimilating, extending, and deepening the subject-matter of the Noether school—groups, rings, algebras, fields, and vector spaces/modules—but had branched out in new directions: homological algebra, category theory (algebraic aspects), and topological algebra. In brief, America turned into the world center of algebraic excellence.

Group theory (1886– )

In comparison with the very considerable achievements of American mathematicians in the theories of (associative, Lie, and Jordan) rings and algebras throughout the period under consideration in this article, their contributions in group theory were slight. But of course the subject is fundamental, and progress was made, so we give in thissection a brief outline Abstract (Modern) Algebra in America 1870--1950: A Brief Account 209 of some of what was done, focusingon finite groups. For highlightsin the historyof abelian groups see [35], for Lie groups consult [38], and for combinatorial group theory see [17] Groups in the guise of permutations were introduced by Galois in the 1830s in his work on the solvabilityof polynomial equations by radicals. These were finite groups, and throughout the history of group theory their investigation was a central problem. The quest turned to finding the finite simple groups, the “building blocks” of all finite groups [71]. This was a pursuit that came to fruition only about 150 years later. Gauss’s Disquisitiones Arithmeticae of 1801 was a source of abelian groups—finiteand implicit at this stage (e.g., nth rootsof unity,equivalence classes of binary quadratic forms). Infinite abelian groups showed up in algebraic number theory in the latter part of the nineteenth century (e.g., the units in the ring of integers of an algebraic number field). Infinite non-abelian groups were encountered as geometric transformations in Klein’s Erlangen Program (1872) and in Lie’s introduction,in connection with his study of differential equations, of continuous transformation groups, later called Lie groups. See [38, 71]. Frank Nelson Cole (1861–1926), famous for the Cole Prize in Algebra, was perhaps the first group theorist in America. He got his doctorate in 1886 under Klein, with a thesis titled “A Contribution to the Theory of the General Equation of the Sixth Degree,” and subsequently published about twenty-five papers. Noteworthy was his determination of all simple groups of order between 200 and 600 (Hölder had classified those of order up to 200). Also significant was his translation from the German of Netto’s 1882 book on group theory, The Theory of Substitutions and its Applications to Algebra (1892, reprinted in 1964). This was the first book on group theory in English, and it stimulated interest in the subject in the English-speaking world [71]. Cole taught at Harvard, Michigan, and Columbia. He was Secretary of the American Mathematical Society for over twenty years, and Editor-in-Chief of the Bulletin of the American Mathematical Society for about thirty years. One of Cole’s doctoral students was George Abram Miller (1863–1951). He spent two years in Europe, attendinglectures by Lie in Leipzig and Jordan in Paris. He publishedover 800 brief articles on group theory in about forty years. Of note was his proof of the simplicity of the four largest Mathieu groups. He collaborated with Hans Frederick Blichfeldt (1873–1945) and Dickson on a book on Theory and Applications of Finite Groups (1916). Blichfeldt was a group theorist who got his PhD in Leipzig, with Lie, writing a thesis “On a Certain Class of Groups of Transformations in Three-Dimensional Space.” He taught at Stanford for forty years. Dickson did significant work in group theory, as we have noted (Sect. 8). His remarkable book Linear Groups with an Exposition of the Galois Field Theory (1901) “contains virtually everything known at the time concerning the structure of classical groups over finite fields” [26, p. 43]. According to Feit, “the main contributors to the subject [of finite groups] during this period [1896–1911] were G. Frobenius and I. Schur in Germany, W. Burnside in England, and L. E. Dickson in the United States” [26, p. 42]. See also Sect. 8. Little of consequence in group theory was accomplished by American algebraists in the next two decades (ca. 1910–1930).But, with the influx of European mathematicians during the 1930s and 1940s (see Sect. 10), group theory in America came into its own. Promi- nent among algebraists who made outstanding contributions to the subject were Reinhold 210 A Century of Advancing Mathematics

Baer, who before coming under the influence of Noether studied mechanical engineering and philosophy; Richard Brauer, who was at the University of Toronto for thirteen years (1934–1947) and subsequently spent eighteen years at Harvard; Claude Chevalley, who studied under Artin for a year; Harold Scott MacDonald Coxeter, who came to Toronto from England in 1936; Marshall Hall, the only native in this group, whose book The The- ory of Groups (1959) was the first text in over twenty years devoted to finite groups; and Hans Zassenhaus, who came from Germany in 1949 to McGill, stayed there for a decade, and in 1965 went to Ohio State, from where he retired in 1982. This brilliant assemblage made many outstanding technical and conceptual advances in group theory. For example, Coxeter classified the finite groups generated by reflections, Brauer developed modular representations to study the structure of groups, and Chevalley introduced the powerful methods of the theory of Lie groups into the study of finite simple groups. However, “no method was as yet available to attack the structure of finite simple groups” [26, p. 52]—themajor outstandingproblem of finite group theory. “By 1961...the theory of finite simple groups had been transformed and the systematic study of finite groups had begun” [26, p. 53]. A landmark in the subject was a proof (in 1963) of the Feit-Thompson theorem (conjectured by Burnside in 1911). Also named the Odd Order theorem, it states that every finite group of odd order is solvable; equivalently, every non-abelian finite simple group has even order. The proof was over 250 pages long, exceeding by far the proof of any theorem in group theory up to that time. The classification of finite simple groups—thegrand prize—was accomplished in 1982, under the masterful guidance of Daniel Gorenstein [36].

Birkhoff and Mac Lane’s A Survey of Modern Algebra (1941)

It often takes decades from the time a subject is born and becomes established by researchers in the ﬁeld until it is expounded in textbooks and taught at universities. In the case of abstract algebra this time differential was relatively brief, perhaps because abstract algebra was “needed” by other disciplines, and likely also because it had exquisite promot- ers in (among others) Noether, Artin, and van der Waerden’s Moderne Algebra (see Sects. 2, 3, 4, and 13). Birkhoff and Mac Lane were taken with the abstract algebra emanating from Germany in the 1920s (see Sects. 2, 3, and 4). As young instructorsat Harvard in the mid-1930s, and following study visits to Germany (also to England in the case of Birkhoff), they taught graduate courses in abstract algebra, using van der Waerden’s Moderne Algebra as a text (Sect. 4). But the book was in German (an English translation became available only in 1949), and the material was rather challenging, even for graduate students. So Birkhoff and Mac Lane decided to write a more appropriate textbook, “intended to present this exciting new view of algebra to American undergraduate and beginning graduate students” [13, p. 26]. What was the state of undergraduate instruction in abstract algebra in the late 1920s and early 1930s? For the most part it was non-existent, principally because there were very few algebraists in the US at that time, and very few appropriate texts to teach from. Abstract (Modern) Algebra in America 1870--1950: A Brief Account 211

“Most beginning graduate students have not even the ghost of an idea as to what algebra is really about...,” wrote Robert Thrall in 1941 [65, p. 343]. Even at Harvard, Birkhoff confessed that as an undergraduate (1928–1932) he was “taught analysis and mechanics almost exclusively” [9, p. 54]. Up to that time, abstract algebra counted for littlewith most mathematicians when compared to analysis, geometry, or even topology. As for textbooks of the recently-emergent modern algebra, there were only three in the US in English prior to 1941: Dickson’s Modern Algebraic Theories (1926), Albert’s Modern Higher Algebra (1937), and MacDuffee’s An Introduction to Abstract Algebra (1940). None of these was suitable as a text for undergraduate students, for reasons of style, content, or level of presentation. Birkhoff and Mac Lane’s Survey, on the other hand, was admirably appropriate. Its preface is instructive (any edition will do). It points to desiderata for how (in the authors’ view) to structure a beginner’s course in abstract algebra (or other subjects, for that matter), and what a good textbookfor such a course should looklike. Here is an excerpt [14]: We have tried throughout to express the conceptual background of the various definitions used. We have done this by illustrating each new term by as many familiar examples as possible. . . . It serves to emphasize the fact that the abstract concepts all arise from the analysis of concrete situations. Modern algebra also enables one to reinterpret the results of classical algebra [essentially the theory of equations], giving them far greater unity and generality. Therefore, instead of omitting the results, we have attempted to incorporate them systematically within the framework of the ideas of modern algebra. Birkhoff and Mac Lane’s text was not well received at first. But after the war, with the great expansion of higher education thanks to the G.I. Bill of Rights, coupled with the growing recognition by the American mathematical community of the centrality of abstract algebra within mathematics (see Sect. 13), the Survey “became the text of choice for undergraduate courses [in abstract algebra]” [47, p. 82]. And it remained so for many years. Little wonder. “Our Survey in 1941,” note the two eminent authors in a 1992 article commemorating the fiftieth anniversary of its publication [13, p. 31], presented an exciting mix of classical and conceptual ideas about algebra. These ideas are still most relevant and worthy of enthusiastic presentation. They embody the elegance, precision, and generality which are the hallmarks of mathematics. [Some mathematicians might argue for removing the definite article in ‘the hallmarks.’] Instructors of algebra courses could do worse than to use Birkhoffand Mac Lane’s book as a text—this, seventy years after its publication.

The algebraization of mathematics A number of mathematicians and historians of mathematics have spoken of the “algebraization of mathematics” in the twentieth century—the penetration of abstract algebra into such areas as geometry, analysis, topology, number theory, logic, and combinatorics. As evidence, note the following ﬁelds, all having a substantial algebraic component: algebraic 212 A Century of Advancing Mathematics geometry, algebraic topology, algebraic number theory, algebraic logic, functional analysis, Banach algebras, von Neumann algebras, Lie groups, algebraic combinatorics, and normed rings. (Lie groups and algebraic number theory had their start towards the end of the nineteenth century.) Noether had no small share in the algebraization of several areas of mathematics, as she seemed to have acknowledged in a letter to Hasse in 1931: “My methods are really methods of working and thinking;this is why they have crept in everywhere anonymously” [20, p. 61]. Here is further evidence.

As is quite well known, it was Emmy Noether who persuaded Alexandroff and ...Hopf to introduce group theory into combinatorial topologyand to formulate the then existing simplicial homology theory in group theoretic terms in place of the more concrete setting of incidence matrices (Jacobson [40, p. v]). It was a pity that my Italian teachers never told me there was such a tremendous development of the algebra which is connected with algebraic geometry. I only discovered this much later, when I came to the United States (Zariski [53, pp. 36– 37]). When I came to G¨ottingen in 1924, a new world opened up before me. I learned from Emmy Noether that the tools [of abstract algebra] by which my questions [in algebraic geometry] could be handled had already been developed ...(Van der Waerden [67, p. 32]).

Kaplansky was another prominent contributor to algebraization, a pioneer in the rise of topological algebra. As he put it [6, p. 1477]: I like the algebraic way of looking at things. I’m additionally fascinated when the algebraic method is applied to infinite objects. The algebraization of mathematics is part of a broader process: the merger of two or more topics, methods, or fields into a single, powerful entity—a most important feature of twentieth-century mathematics, though it had occurred earlier. Dieudonnénotes its effectiveness [23, p. 537]:

Progress in mathematics results, most of the time, through the imaginative fusion of two or more apparently different topics.

Three far-reaching examples of the (non-algebraic) bonding of fields are differential topology, analytic number theory, and diophantine geometry. The abstract, axiomatic approach to algebra, which yielded such powerful results—in algebra and elsewhere—also had an impact on Bourbaki’s grand opus, The Elements of Mathematics, which aimed to give a rigorous, axiomatic foundation to all of mathematics: In some ways, van der Waerden’s Moderne Algebra (1931) served as a model for the highly influential work of Bourbaki (Dieudonné[16, p. 45]). Modern algebra ...was expanded by Bourbaki intoa new global view of mathematics (G. Birkhoff [9, p. 42]). Abstract (Modern) Algebra in America 1870--1950: A Brief Account 213

Postscript: what is algebra, anyway? Although the term “algebra” dates from the eighth century, the subject that has come to be known under that name—the study of solutions of polynomial equations—is over 3500 years old. We have come to call this subject classical algebra [18]. In thenineteenth century we witnessed the rise of topics—groups of permutations, rings of integers of algebraic number fields, hypercomplex systems of matrices and quaternions—which would in the first halfof thetwentiethcenturybe subsumed under what has come tobe knownas abstract or modern algebra: the study of groups, rings, fields, vector spaces, algebras, and modules (these were the initial objects of modern algebra) using the unifyingand abstracting notion of “the axiomatic method.” Abstract/modern algebra thrived during roughly 1930–1960. But it ceased to be “modern” in the later 1950s. For example, the title of van der Waerden’s Moderne Algebra became Algebra in its seventh German edition, in 1959, and that of Dickson’s Modern Al- gebraic Theories turned into Algebraic Theories in 1955. The subject also ceased to be abstract (relatively speaking). Compare, for example, Birkhoff and Mac Lane’s A Survey of Modern Algebra of 1941 [13] withMac Lane and Birkhoff’s Algebra of 1967 [51]. The latter text tried to promote a new view of algebra, with a focus on categories and morphisms. The former text was no longer abstract when compared with the latter. So abstract algebra has evolved: on the one hand towards more abstraction—much of that due to the rise of category theory—and on the other hand towards less abstraction, in large part due to the influence of the computer (which has of course revolutionizedmuch of mathematics). Birkhoff describes various computer-influenced areas of algebra which have emerged in the 1970s: numerical linear and multilinear algebra, combinatorial algebra, theory of automata, algebraic coding theory, and computational complexity in algebra [9, 10]. Algebra has also been substantially influenced by its interaction with other branches of mathematics (see Sect. 13). And even a traditional area of algebra such as group theory can no longer always fit the mold of an axiomatic structure. For example, the proof of the classification of finite simple groups required the assistance of a computer [36]. Given these various developments of the subject, is there a definition or description of algebra that will capture “the essence of algebra,” the multitude of algebraic expression described above? Well, yes: Algebra is what algebraists do or define to be Algebra, and algebraists are people doing Algebra or declaring themselves to be algebraists (Tamari, 1978, [19, p. 402]).

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34 Harrington Crescent, Toronto, ON M2M 2Y5 [email protected] Part III Pedagogical Developments

The History of the Undergraduate Program in Mathematics in the United States

Alan Tucker Stony Brook University

The undergraduate mathematics program in America has had a punctuated evolution. The Mathematical Association of America was organized 100 years ago, at the end of a period of dramatic rethinking of American education at all levels, one product of which was the introduction of academic majors. The mathematics major was static in its ﬁrst forty years, followed by great changes from 1955 to 1975, and then a period of relative stability to the present. The educational concerns of the Mathematical Association of America also changed in the 1950s. Initially, its educational recommendations focused on preparing high school students for college mathematics, but starting in 1958, the MAA’s Committee on the Un- dergraduate Program in Mathematics (CUPM) played a leading role in promoting a major reworking of the mathematics major.

Early history Mathematics had a greater role in the college curriculum in America in the 1700s and 1800s, when it was studied as a classical training of the mind instead of as the language of science and engineering it is today. The first colleges in the colonies were modeled on English colleges, whose curriculum was largely prescribed, focused on Latin, Greek, Hebrew, and mathematics, and whose original purpose was to train ministers. The faculty had attended college, but nothing more, and the education was dry and involved mostly rote learning. College mathematics initially consisted of euclidean geometry and some algebra. In the 1700s, there were few high schools and students (mostly children of the well-to- do) entered college at age fifteen or sixteen with a couple of years of education, typically from tutors, beyond primary school. (For readers wantingmore information about thisearly history, see Cajori [7], Smith and Ginsberg [33], the first chapter of Hofstadter [20], and the beginning of Duren [14]. These references were the primary sources of this section. Also see Cohen [8] and Hofstadter [19].) In the nineteenth century, colleges welcomed middle-class students, and the growing number of private academies and public “Latin” high schools in larger cities gave college-

219 220 A Century of Advancing Mathematics preparatory instruction that caused the age of freshman to rise to today’s 18 years. Admis- sion requirements in mathematics also rose over the century to include euclidean geometry and a year of algebra. Until the late 1800s, school teachers rarely had a college degree and generally attended normal schools, which were high schools that specialized in teacher training. As students arrived knowing more mathematics, the level of mathematics taught in colleges rose slowly through the nineteenth century. In the early 1800s, calculus had not yet entered the regular US college curriculum, although a few institutions, e.g., Yale and West Point, offered classes about “fluxions.” In the second half of the 1800s, the typical college curriculum for freshmen in mathematics was algebra and geometry; sophomores took more algebra and trigonometry. Technically-oriented students continued with a junior year of analytic geometry; possibly calculus started then or else in the senior year. Well- prepared students at better colleges took calculus in the sophomore year after a year of college algebra and trigonometry. In the late 1800s, mathematics offerings at the University of Michigan (a university with strong academic standards then as now) included algebra (several courses), determinants, euclidean geometry, trigonometry, axiomatic geometry, analytic geometry, calculus, differential equations, calculus of variations, quaternions, and elliptic functions. At this time, Harvard offered a similar curriculum, supplemented by additional courses in theory of functions, higher plane curves, mechanics, and Fourier series. Before 1850, all engineers were trained at the US Military Academy or at Rensselaer Polytechnic Institute, but by the second half of the nineteenth century there was a growing demand for technically-trained college graduates who required a more practical training than the classical curriculum offered. Scientific schools offering BS degrees were established within Harvard, Yale, and other leading colleges. The 1862 Morrill Act created land grant public universities, whose primary purpose was to “teach such branches of learning as are related to agriculture and the mechanic art” [37]. However, the level of mathematics was not high in BS programs; engineers used mostly finite differences instead of calculus. Even when PhD programs were instituted—the first mathematics PhD was awarded by Yale in 1862—nearly all faculty teaching in these programs had only a BA or possibly an MA degree. Mathematics PhD production increased from one or two per year in the 1860s and 1870s, to ten per year by 1900 (see Richardson [30] for more early PhD data).

A period of rethinking American education at all levels The end of the nineteenth century and beginning of the twentieth century was a time of dramatic change in US education. The number of students going to college had started declining in 1840. In reaction to students’ unhappiness with the classical curriculum, and seeking to have a more open “democratic” college experience, Harvard president Charles Eliot changed his college to an all-elective curriculum in the mid-1880s. The model for this change was the all-elective curriculum in German universities. By the early 1900s, most US institutions had changed to an elective curriculum. The program of study tended to be very practical, in keeping with the spirit of the country. The reaction to these changes was dramatic: College enrollments soared from 150,000 in 1890 to 350,000 in 1910 [20, p. 31]. This increase was also fueled by the growth of public high schools, even though they still were serving fewer than 10% of eligible students in 1910 (US Census [36]). The History of the Undergraduate Program in Mathematics in the United States 221

The elective system led to a dramatic decline in the study of mathematics. What saved collegiate mathematics from being totally decimated was the growing need for engineers and technically-oriented professionals in industry and agriculture. Thus, it came to be that while mathematics was previously associated with a classical education, henceforth it became associated with science and engineering. However, mathematicians continued to hold to the classical view that the study of mathematics is valuable as a general training of the mind. The decline in mathematics instruction then came to high schools. While college preparation curriculum standards had long been set by the colleges, K–12 education came under the control of education specialists in the early twentieth century, led by John Dewey, who argued that the primary purpose of school education was social development and personal fulﬁllment. An inﬂuential 1918 National Education Association report [26] had fourteen subject-area subcommittees, but mathematics was not one of them. Mathematics became an elective subject in many high schools. (It is worth noting that in Europe at this time, calculus was being made a mandatory subject in college-preparatory high schools.) The impact of this movement on college mathematics instruction was a substantial increase in precollege (remedial) mathematics courses in arithmetic and beginning algebra.

The first forty years of the MAA, 1915–1954 The ﬁrst MAA president, E. R. Hedrick, laid out a platform for the MAA and its Amer- ican Mathematical Monthly [17, p. 28]: “the great majority of the work fostered by the Association will be on questions directly affecting collegiate courses in mathematics.” This included articles and talks on ways to improve instruction in introductory, as well as advanced, courses. Mathematics preparation for college was also a concern of his, given the situation just described. A broader concern was that the elective system in colleges was being abused, with many students graduating with little more than a collection of freshmen- level courses. This depressing situation gave rise to a counter-movement in higher education. The change was again led by Harvard, whose president Lawrence Lowell instituted a system of required academic majors in 1910. A few years later, Woodrow Wilson, the president of Princeton, expanded on Lowell’s efforts by adding a core curriculum, what we now call general education requirements. The academic major and general education requirements were soon widely adopted, althoughthe number of courses required in a major was initially modest. (Note that England and many European countries had by this time moved to a system where students applied to study an academic discipline upon admission to college and virtually all their studies were related to that discipline.) The social sciences and humanities, which had the majority of college majors, led an effort to update the old classical curriculum with what became known as a liberal arts curriculum. This curriculum was driven by the mission of training a mind to think and explore knowledge as a foundation for future on-the-job learning. Mathematics was frequently a required subject at liberal arts colleges. At this time, there were also calls to raise the standards for freshman instruction. A leading American mathematician, E. B. Wilson [41], called for requiring all freshmen to take a yearlong mathematics course that would be half calculus and half what he called 222 A Century of Advancing Mathematics

‘choice and chance’—much of what later came to be called finite mathematics. Some selective institutions, led by Harvard, soon made calculus the second semester of freshman mathematics, with preparatory instruction available. Nonetheless, for most college students, there were no longer any mathematics requirements. For general education purposes, mathematics was grouped with the natural sciences, so that science courses could be taken in place of any mathematics. When students needed to take college-level mathematics for their major, they often first needed extensive precollege work. Students who had a good high-school mathematics preparation usually started college mathematics with freshmen study of college algebra and trigonometry,followed by analytic geometry and calculus in the sophomore year. The educational reforms advocated by Dewey also called for extending mandatory schooling through twelfth grade (motivated as much by keeping students out of gruel- ing factories as by greater learning). This recommendation was widely implemented after World War I. From 1910–19 to 1920–29, college graduation rates increased by over 150% (for such data see US Census [36]) to meet the demand for high school teachers (high school graduation rates grew even faster) as well as for the growing number of engineers and other professionals required for the commerce, infrastructure, and technology of the twentieth century world. Colleges, eager for these larger enrollments, had to lower standards to accommodate many of the applicants. Precollege mathematics enrollments grew even more. The overall enrollment growth was accompanied by an increased production of mathematics PhDs—from about 25 per year in the 1910s to about 50 per year by 1925 [30]. Still, typical tenured college/university mathematics faculty had at most a Master’s degree, since most mathematics instruction was at a low level. However, the Monthly had several articles in the first third of the twentieth century discussing how to ensure that new mathematics PhDs would be good teachers; for example, see Slaught [32]. These articles highlight the fact that, despite the advent of doctoral programs in many university mathematics departments, teaching was still viewed as the primary duty of university mathematics faculty. Beginning mathematics faculty at leading doctoral mathematics departments were typically teaching twelve to fifteen hours per week up through the 1940s. While early MAA educational activities focused on secondary-school preparation in mathematics, a 1928 MAA report [22] addressed complaints about the first two years of college mathematics; also see Schaaf [31]. It acknowledged calls to offer a survey course of mathematical ideas with historical aspects for non-technical students. The report made recommendations for enrichment readings to address these concerns, but in the end it defended the mental discipline developed by traditional drill and problem-solving courses. Until World War II, a typical student who chose to major in mathematics had in mind becoming either a high school teacher or an actuary. This was a pragmatic career-oriented major, although professors also lectured about general theory and the beauty of mathematics. It should be noted that actuary work was an offshoot of broader statistical concerns. Measuring and counting things had interested business-minded Americans from the repub- lic’s founding. For example, the American Statistical Association was founded in 1839, half a century before the American Mathematical Society. Nonetheless, despite the practical values of American society, there was always a small number of college students who became fascinated with mathematics. Although the under- The History of the Undergraduate Program in Mathematics in the United States 223 graduate mathematics offerings at most colleges and universities were limited in the first half of the twentieth century, the higher level of mathematical knowledge among faculty with PhD and MS degrees created opportunities for undergraduates to learn more mathematics. There were topic seminars, reading tutorials, and better mathematical libraries. During the 1920s and 1930s, US institutionseducated a generation of mathematicians who went on to become world leaders in their fields by the 1950s. More broadly, during this period American universities evolved from places that passed along knowledge to places that also created knowledge. While mathematics instruction did not change much in the 1930s and early 1940s, major world events during this period had a dramatic influence on higher education. En- rollments plunged during the Great Depression and some institutions closed. These hard times heightened the focus on career training in mathematics and other disciplines. Dur- ing World War II, most colleges and universities ran accelerated specialized programs for selected soldiers; some allowed women to enroll. Some closed for the duration of the war.

Mathematics offerings at some institutions To look more closely at the mathematics undergraduate program during the first 40 years of the Association’s existence, we present a sampling of offerings and requirements for the mathematics major at several institutions, gleaned from their old college catalogs. In 1920 (soon after the institution of academic majors) the University of Pennsylvania [39] required just six courses for an academic major, but most mathematics majors took more. All Pennsylvaniastudents had to take two courses in the physical sciences, choosing among offerings in chemistry, mathematics, and physics. Math majors were expected to complete at least one semester of calculus by the end of their sophomore year. Low-level offerings: Math 1–Solid Geometry, Math 2–Plane Trigonometry, Math 3–Inter- mediate Algebra, and Math 5–College Algebra. Intermediate-level offerings: Math 6–Analytic Geometry, Math 7–Elements of Analytic Geometry and Calculus (not open to math majors), Math 8–Differential and Integral Calculus, Math 9–Advanced Plane Trigonometry and Spherical Trigonometry, Math 10–History of Mathematics, Math 11–Determinants and the Theory of Equations, Math 12–Determinants and Elimination, Math 13–Solid Analytic Geometry, Math 14–Infinite Series and Products. Advanced offerings (not all offered annually): Math 15–Projective Geometry, Math 16– Modern Analytic Geometry, Math 18–Quaternions and Vector Methods, Math 19– Differential Equations, Math 20–Advanced Calculus, Math 21–Foundations of Ge- ometry, Math 25–Mathematical Analysis (a deeper look at series, representations of functions, and other classical topics), and Math 26–Functionsof a Complex Variable. We note that Advanced Calculus in those days covered basics of multivariable calculus, some special functions, some functions of a complex variable (up to Cauchy’s theorem), and possibly some partial differential equations; e.g., see Woods [42]. At Pennsylvania, there were few changes over the next thirty years. By 1930, new courses in statistics and projective geometry became available, but the complex variables course was dropped. By the late 1930s, Intermediate Algebra was dropped but was reinstated in the late 1940s for returning GIs. Around 1940, a topics course was added, along 224 A Century of Advancing Mathematics with a new three-semester calculus sequence specifically for engineers. In the late 1940s, a course in abstract algebra was added. Throughout this period, calculus was a sophomore- year course for mathematics, natural science, and engineering majors. In contrast to the fairly static undergraduate program at Penn, the graduate mathematics offerings expanded extensively. The University of Nebraska’s mathematics offerings [38] over this period were very similar to Penn’s. Nebraska also required six courses for an academic major. Nebraska required all students to have two majors, or else a major and two minors. In mathematics, one minor had to be in the sciences and the other in French, German, Latin, or philosophy. The Nebraska mathematics department offered a number of courses in secondary school mathematics and in insurance mathematics (for actuarial training); Penn was unusual in its paucity of such courses. In 1920, Nebraska had an advanced course in functions of a real variable that was dropped within a decade. Nebraska did not yet offer abstract algebra in 1950. As another example, consider the University of Rochester [40], then a technical institution (heavily influenced by the major employer in Rochester, Eastman Kodak). It initially had just five majors: arts, chemical engineering, mechanical engineering, chemistry, and economics. The limited mathematics offerings were mostly courses for engineers, up through advanced calculus. In 1920 Rochester, like Harvard, offered only one course below the level of calculus, indicating highly selective admission requirements. While there was still no mathematics major in 1940, there were by then increased elective offerings for engineers and scientists, including a rudimentary abstract algebra course and a course in complex variables, a course not then offered by Pennsylvania and Nebraska. A mathematics major was finally offered shortly after World War II, but few new courses were added. Note that by 1960, Rochester had a highlyregarded research faculty and a doctoral program in mathematics. We turn now to some liberal arts colleges. Colgate[9] required two mathematics courses of all students. A preamble to the mathematics offerings in the old Colgate catalogs reflected a liberal-arts focus independent of career training: The goal of mathematics instruction was “to form habits of accurate and precise expression and to develop the power of independent and logical thinking.” In 1920, there were seven courses below calculus, two sophomore calculus courses, and only occasional mathematics electives beyond that. The pre-calculus courses decreased over time and were eliminated by 1940. Business mathematics, investment mathematics, and statistics (for actuaries) were added in the late 1920s, along with more electives (not taught every year) in number theory, axiomatic geometry, projective geometry, and mathematics for teachers. We see here that Colgate had felt the pressure to offer a career-oriented curriculum side-by-side with liberal-arts offerings. By 1940, decreased enrollments forced junior-senior courses to be reclassified as seminars that were offered when there was adequate demand. By 1950, the electives were almost all gone and the pre-calculus courses were also gone. At smaller Macalester College [21], mathematical offerings in 1920 consisted of four pre-calculus courses and two calculus courses. By 1940, there were four semesters of calculus with differential equations; other additions were an investment mathematics course and a seminar in higher mathematics. By 1950, there were also courses in advanced calculus, mathematical statistics, and engineering mechanics, and, unlike Colgate, there were still four pre-calculus courses. The History of the Undergraduate Program in Mathematics in the United States 225

So what sequence of courses did a strong mathematics major take during this period? An example close to home comes from my mother’s college transcript of her 1934–38 mathematics studies at Northwestern University. She went to a highly-regarded highschool, came from a mathematical family (her father was a Northwestern mathematics professor), and was accepted for graduate study in mathematics at Radcliffe upon graduation. She took courses in College Algebra and Analytic Geometry in her freshman year, in Differential and Integral Calculus in her sophomore year, and in her junior year, she took courses in Differential Equations, Advanced Calculus, and Theory of Equations. In her senior year, she took Higher Geometry, Functions of a Real Variable, and Honors Seminar. Note that these ﬁrst two years of study were littlechanged from what a well-prepared student took in the late 1800s.

Impact of World War II Two dramatic changes in the stature of American mathematics in the 1930s and 1940s foreshadowed major changes in the mathematics major in the 1950s. First, the immigration of great European mathematicians fleeing Nazi Germany catalyzed America’s emergence as the world’s leading center of mathematics research. Second, mathematicians showed how mathematics could make major contributions to a wide range of allied efforts during the war, including aerial combat, fluid mechanics, shock fronts, code breaking, logistics, and designing the atomic bomb (for more, see Rees [29]). Significantly, these mathematicians mostly had backgrounds in pure mathematics, yet proved extremely adept at solving a range of important practical problems. John Von Neumann epitomized this versatility with his seminal work in the theory of games, neural networks, design of programmable computers, and numerical analysis, as well as applications of partial differential equations. After the war, mathematical and statistical models became an integral part of the study of engineering, economics, and quantifiable aspects of commerce, as well as the physical sciences. By the end of World War II, industry came to value mathematics majors almost as much as engineers. Since pure mathematicians had been responsible for the success of mathematical models, the study of core mathematics was seen as an ideal combination of the classical values of intellectual training along with a preparation to solve a range of problems of importance to industry. This led to an increase in the proportion of college students interested in mathematics, as well as the sciences and engineering. After the war, returning soldiersusing the GI Bill to attend college for free caused a substantial increase in the overall college population.All these factors led to a growth in mathematics enrollments at all levels. On the other hand, as noted above, mathematics offerings at most institutions still had changed little in 1950 from what they were in 1920. However, incoming students planning on scientific and engineering majors were better prepared. Many of these freshmen were ready to study calculus, as by 1950 the level of high school mathematics for such students had come to include two years of algebra, euclidean geometry, and a “precalculus” course of analytic geometry and trigonometry. To accommodate the growing demand for high school teachers, regional teachers’ colleges expanded from training primary teachers to educating secondary school teachers, but 226 A Century of Advancing Mathematics their curriculum for prospective highschool mathematics teachers was limited and typically did not extend beyond one year of calculus. At the other extreme, some leading universities were offering modern mathematics courses such as point-set topology, abstract algebra, and logic. Most mathematics faculty still did not have PhDs, and in the early 1950s faculty at many leading research departments still saw teaching as their primary mission. Even senior administrators often taught two courses per semester. When my father, A. W. Tucker, was chair of the Princeton mathematics department in the 1950s, not only did he have the same teaching load as other senior faculty, but every other semester he was also in charge of the freshman calculus course taken by almost all students. When I questioned him years later why he tookon this huge extra obligation,he responded, “The most important thingthat the Princeton Mathematics Department did was teach freshman calculus and so it was obvious that as chair, I should lead that effort.”

The golden age of the mathematics major, 1955–1974 This was the only time of substantial change in the 100-year history of the mathematics major in America. The period was marked by a growing demand for mathematics instruction and mathematics graduates at all levels. This demand was driven by a quadrupling of college enrollments from 1950 to 1970 (US Census [36]) and an increasingly quantitative world, highlighted by the growing use of computers. The launching of Sputnik in 1957, in the larger context of the Cold War competition with the Soviet Union, made mathematicians, scientists, and engineers the country’s Cold War heroes. These twenty years saw the largest fraction of incoming students interested in a mathematics major—5% in the early 1960s—and largest fraction of Bachelor’s degrees in mathematics—around 4% in the late 1960s [10]. Mathematics’ appeal among the brightest students was even greater. For example, in 1960 half my freshmen class at Harvard stated an intention to major in mathematics or physics. While many of these bright students ended up with successful careers in engineering, economics, and computer sciences, a number of them, e.g., several Economics Nobel Laureates, first earned a Bachelor’s or even a PhD in mathematics. In 1953, the MAA formed the Committee on the Undergraduate Program (CUP), later renamed the Committee on the Undergraduate Program in Mathematics (CUPM). Its initial focus was a common first-year mathematics syllabus for all students, paralleling the common syllabus for introductory courses in the natural and social sciences. The course, called Universal Mathematics, consisted of one semester of functions and limits, the real number system, Cartesian coordinates, functions (with focus on exp.x/ and log.x/), limits, and elements of derivatives and integrals, followed by one semester of mathematics of sets, logic, counting, and probability. Note that this syllabus was very similar to the first-year mathematics syllabus proposed by E. B. Wilson in 1913. The second-semester component was based on the expectation that newer areas of applied mathematics, such as statistics and operations research that were so useful in the war effort, would become a major part of the mathematics used in many disciplines. There was also a proposed “technical laboratory” for engineers and physicistswith more extensive work in calculus. The CUP-prepared 1954 text [23] for the functions half of Universal Mathematics offered a highly theoretical approach on odd-numbered pages (e.g., replacing sequences by filters) along with a traditional The History of the Undergraduate Program in Mathematics in the United States 227 approach on even-numbered pages. The theoretical approach was laying the foundation for a more theoretical mathematics major to prepare students for graduate study. See Duren [13] for more about the early history of CUP/CUPM. CUP’s proposal, with textbook, for an important new college course appears to have been withoutprecedent. Professional academic organizations make reports about highschool preparation but normally stay clear of telling their members what to teach, much less propose a major organization of the introductory course in the discipline. However, the Asso- ciation for Computing Machinery followed the CUPM example in 1968, with recommendations for a curriculum to define the newly-emerging computer science major. Universal Mathematics was doomed, no matter what its reception might have been by mathematicians, by the decision of physicists in the mid-1950s to teach calculus in their freshman physics course for engineers. These physics courses started with supplementary workshops on basic calculus formulas. Mathematics departments, wary of losing calculus instruction,quickly made a full-year course in calculus the standard freshman sequence for engineers and scientists. As calculus came to be appreciated as a foundationfor the modern world and one of science’s greatest intellectual achievements, a number of selective private institutionsinstituted a general education requirement that all students take three semesters of calculus or else, for the math-phobic, three semesters of a foreign language. Calculus, or preparation to take calculus, was thus firmly established as the primary focus of first-year college mathematics. Moving in another direction, John Kemeny and colleagues at Dartmouth reworked the mathematics-of-sets part of the Universal Mathematics course into their 1956 textbook, In- troduction to Finite Mathematics. Along with some logic, discrete probability (with some combinatorics), and matrix basics, they chose two modern topics, Markov chains and linear programming, that were accessible and had interesting applications. This course, with gentler textbooks, is still widely taught to business and social science majors. (Interestingly, the mathematics requirement for many business majors evolved to cover finite mathematics and a gentle introduction to calculus, essentially the curriculum of Universal Mathematics and that advocated by E.B. Wilson in 1913.) Over the next twenty years, other alternatives to precalculus and calculus were developed, including introductory statistics, mathematical modeling, and surveys of mathematical ideas. As an aside, we note that in the mid- 1950s Kemeny also anticipated that personal computing would soon be used extensively in mathematics instruction (it was 25 years before the IBM PC). His group created the easy-to-use programming language BASIC and developed the country’s first (non-military) time-sharing system supporting terminals in every Dartmouth dormitory. At the high-school level, mathematicians led a Commission on Mathematics of the Col- lege Board that made recommendations for upgrading the school curriculum, including set theory and inequalities, topics that were introduced for their role in newer areas such as statistics and linear programming. The College Board forced schools to adopt this curriculum by including these new topics on the SAT tests. Around the same time, a Ford Foundation report led to the establishment of Advanced Placement exams to give college credit for college-level courses taken in high school, although initially the program was active only at selective private schools. Calculus was one of the first subjects with an AP exam. Good data about mathematics undergraduate degrees is not available before 1960, but it is estimated that the number of Bachelor’s degrees in math at least doubled from the 228 A Century of Advancing Mathematics early 1950s to 1960, with 13,000 math degrees. In 1966, there were 20,000 math degrees. In 1970, the number had grown to 27,000 per year, over 4% of all college graduates [27].

Changing the mathematics major The success of pure mathematicians in war efforts supported efforts to move the mathematics major away from an explicit career focus (high-school math teacher or actuary) to a major that included more theoretical courses, but was also believed to be a good preparation for industrial careers such as computer programming. The result was a dramatic change in the undergraduate program in mathematics that began in doctoral institutions and leading private colleges during the 1950s. In 1960 at the University of Pennsylvania, ten courses were required for the math major and there were now only two pre-calculus courses, one a liberal arts survey course. There was a sophomore-level “introduction to proofs” course. There were now upper-level courses in real variables, abstract and linear algebra, number theory, topology, differential geometry, Fourier analysis, numerical analysis, and axiomatic set theory. At Nebraska, the major in 1960 required five courses beyond Calculus III. There were still five courses before calculus, in algebra, trigonometry, and analytic geometry, along with a new Elements of Statistics course, but there was also now a three-semester honors calculus sequence. Many new upper-level courses were similar to Pennsylvania’s. Colgate had a similar expansion, although not to as many new courses as the universities. On the other hand, smaller colleges like Macalester were slower to change. In 1960, Macalester still had four pre-calculus courses. A two-semester abstract algebra course was added, but only two other post-calculus courses were offered, advanced calculus and honors seminar. Another sign of the slow rate of change was experienced by this writer, who took an AP calculus course in 1960–61 at a well-regarded prep school, using the text by Granville, Smith, and Longley, first published in 1904. The addition of more modern, theoretical courses and honors sequences at leading in- stitutionswas motivated in part by a desire to prepare more mathematics majors to continue on to doctoral study in mathematics. In 1962, the President’s Science Advisory Committee optimistically estimated [16] that there would be a need for seven times as many new PhDs annually by the 1970s, and indeed the number of PhDs grew from about 300 per year in the late 1950s to 1200 per year in 1970 (a level not matched again for 30 years); see AMS [2] for yearly PhD data. This demand for mathematics PhDs was driven by: (i) growingcollege enrollments in the 1960s as baby boomers entered college and the percentage of students going to college was rising; (ii) replacing retiring faculty hired during the last expansion in college enrollments in the 1920s; and (iii) the Cold War-focus on science, mathematics, and engineering. In the late 1950s, many able students were still going to graduate school from mathematics programs that did not teach the Riemann integral. They dropped out of graduate study when faced with first-year courses in modern topics like Lebesgue integration. CUPM took the lead in addressing this problem with its first comprehensive curriculum report, published in 1963, Pregraduate Preparation of Research Mathematicians. The report’s preface notes that the recommendations were “idealized” and intended for an honors program. The four-semester sequence for the first two years covered calculus and differential equations, along with some real analysis (results about compactness, absolute con- The History of the Undergraduate Program in Mathematics in the United States 229 tinuity, Riemann integration, Ascoli’s theorem), linear algebra, and other topics such as differential p-forms. For juniors and seniors, yearlong courses were recommended in real analysis, complex analysis, abstract algebra, geometry-topology, and either probability or mathematical physics. Real analysis went up to Banach spaces and spectral resolution of self-adjoint operators. Complex analysis went up to harmonic measures and Hardy spaces. Needless to say, this report was swinging the pendulum too far in the other direction, with a curriculum that was over the head of most freshmen who considered themselves talented in mathematics. At the same time, the MAA and AMS presidents, in coordinationwith the NCTM president, created the School Mathematics Study Group (SMSG), the most prominent of the “New Math” efforts, to produce textbooks for a more abstract school mathematics curriculum for stronger students. The CUPM Pregraduate Preparation report mentioned the SMSG curriculum as a valuable preparation for its recommended major. SMSG was led by mathematicians who got carried away with an axiomatic approach to school mathematics, e.g., the field axioms were discussed in its junior high school texts, and produced in time a massive public backlash.

CUPM’s General Curriculum in Mathematics for Colleges A follow-up 1965 CUPM report, the General Curriculum in Mathematics for Colleges (GCMC), presented recommendations for a mainstream mathematics major that were a watered-down version of the Pregraduate Preparation recommendations. This curriculum of 11 courses was aimed at four-year colleges. However, this program was still too ambitious for many institutionsand as a result, CUPM produced a gentler version of GCMC in 1972. That GCMC recommended a beginning core of six courses: calculus I, calculus II (with some differential equations), linear algebra, multivariable calculus I, abstract algebra, and linear algebra. Suggested additional courses were multivariable calculus II, real variables, probability and statistics, numerical analysis, and applied mathematics. The GCMC was widely adopted and formed the basis of the mathematics major by the mid-1970s. Vir- tually all college mathematics departments also retained their separate course in differential equations. The one new course that GCMC missed was the Introduction to Proofs course that many colleges and universities felt was needed to help math majors make the transition from calculus to more theoretical upper-division courses, such as abstract algebra. Possibly the most signiﬁcant curricular change arising from GCMC was the introduction of a sophomore-year linear algebra course; it was adopted widely at universities and colleges. Linear algebra had entered the undergraduate curriculum in the 1950s as part of a yearlong upper-division abstract algebra course. It is somewhat ironic that this sophomore linear algebra course evolved from Kemeny et al.’s Introduction to Finite Mathematics text. In 1959, Kemeny’s group produced a variation of this text for mathematics majors called Finite Mathematical Structures, with a major part of the text devoted to linear algebra. CUPM issued other reports about teacher preparation, statistics, and computational/ applied mathematics in the late 1960s and early 1970s, but none had the impact of the 1972 GCMC report. While the primary change for the mathematics major was a more theoretical approach, other additions to the undergraduate program involved new areas of applied mathematics such as computer science, numerical analysis, mathematical modeling, and operations research. 230 A Century of Advancing Mathematics

As undergraduate enrollments quadrupled from 1950 to 1970, the fraction of mathematics enrollments in precollege (remedial) mathematics and pre-calculus courses actually dropped substantially while the percentage of enrollments in calculus grew. This trend would seem to reﬂect better mathematics instruction in the schools. The largest percentage growth, 150%, in below-calculus instruction was in elementary statistics courses, whose enrollment of 140,000 in fall 1975 matched precollege enrollments (many additional students were taking statistics courses in other departments) [10].

Statistics and computer science spin off Statistics flourished after World War II based on its wide use during the war effort. Spe- cialization in statistics initially involved graduate-level education. About 50 statistics departments split off from mathematics departments in the 1950s and 1960s, and most started their own undergraduate majors. To this day, however, there are a modest number of statistics majors, probably because most mathematically-oriented students are not ready to specialize in a particular mathematical science at the time they must pick a major. On the other hand, a good fraction of mathematics departments began to offer the popular option of a concentration in statistics. The other major discipline to split off from mathematics was computer science. Aca- demic study related to machine computation initially focused on numerical analysis and logic topics such as recursive function theory. So computer science started primarily as an area of mathematics, taught at the graduate level, although in some institutions it was associated with electrical engineering. In the 1960s there were few computer undergraduate degrees, but student demand and the growth of computing in industry soon created pressure for more undergraduate majors. Computer scientists did not have a good idea of what courses should be created for an undergraduate program in computer science. Copying CUPM, the Association for Computing Machinery (ACM) issued a report for an undergraduate curriculum in 1968 [3], although it was almost as ambitious as CUPM’s 1963 Pregraduate Preparation report, i.e., the initial textbooks for the proposed sophomore-year Discrete Structures course were actually aimed at a graduate audience. As with GCMC, two more reports were needed to settle on a workable undergraduate program. There were 2,000 computer science graduates in 1970, 5,000 in 1975, 11,000 in 1980, and 50,000 in 1985 [27]. This growth is one of the reasons that math major enrollments declined after 1970. A limiting factor in the growth of computer science programs was the shortage of computer science faculty, since industry was hiring most new PhDs at high salaries. To help address this shortage, mathematics faculty with computing interests were encouraged, with financial incentives and summer training programs, to move into computer science. This was essentially the only option to staff computer science programs at liberal arts colleges. Even today, many colleges have combined mathematics and computer science departments, reflecting this history of shared faculty.

University faculty turn focus to research While in the early 1950s most faculty at doctoral institutionsstill saw undergraduate teaching as their primary mission, by 1970 that mission had changed, with research becoming the primary focus. One consequence was that freshman-level mathematics instruction at The History of the Undergraduate Program in Mathematics in the United States 231 public universities now typically involved graduate student instructors or large lectures by faculty with supporting TA-led recitations. During this period, a change in values discouraged most faculty in doctoral departments from taking a serious interest in undergraduate education. To lure star research faculty to Berkeley from Ivy League institutions in order to raise Berkeley’s reputation to their level, University of California Chancellor Clark Kerr made offers to star faculty that included no undergraduate teaching and a light graduate teaching load. I remember my father lamenting that there was no way Princeton could counter such offers, since Princeton believed that all faculty should teach at least one undergraduate course a semester and no one taught fewer than two courses a semester. Kerr succeeded in attracting Ivy League star faculty to Berkeley, but as a whole, academia suffered. Other public universities building up their doctoral programs copied Berkeley, and eventually the Ivy League universities also reduced their teaching loads. The Berkeley model for recruiting star faculty backﬁred when regular faculty started pressing for reduced teaching loads and minimal undergraduate teaching, as a sign that they too were elite researchers. We note that the supposed incompatibility of research and undergraduate teaching was refuted by a 1957 National Research Council study [1] that found littleimpact on research productivity for teaching loads of up to 15 hours per week (a common teaching load for junior faculty in top departments before World War II). This NRC study also found an important beneﬁt of teaching-oriented liberal arts colleges. They produced a disproportionate number of future mathematics PhDs, despite the attraction that future mathematics PhDs should have found in undergraduate study at universities with their distinguished research faculty. In the 1970s, the attractiveness of the mathematics major declined dramatically. The percentage of freshman interested in a mathematics major dropped from 4.5% in 1966 to 3.2% in 1970 to 1.1% in 1975, and never rose above 1.5% again (data here come from CBMS [10] and NSF [27]). From 27,000 math degrees awarded in 1970, the number declined to 18,000 per year in 1975 (a 1970 CBMS estimate had projected 50,000 per year in 1975) and 11,000 per year in 1980. The 1975 CBMS enrollment survey revealed a shift of student interest away from pure mathematics courses, a number of which saw 70% enrollment declines from 1970 to 1975, while applied mathematics courses registered modest declines. Internally, CUPM’s more theoretical GCMC curriculum may have overwhelmed some students at the same time that the new computer science major offered an attractive, career-oriented alternative. Externally, the Vietnam War was turning students off to working for the “military-industrial” complex, a major employer of mathematics majors. Also, as baby boomers started graduating from high school, the demand for high-school mathematics teachers dropped. In short, the mathematics major in 1970s experienced a tidal wave of problems, causing a sharp decline in its enrollments and general appeal.

The last forty years, 1975–2015 The decline in interest in the mathematics major stabilized, and the number of majors re- bounded a little in the 1980s. Initially, there was a tension between students who wanted more career-oriented coursework, often in newer applied areas like statistics and operations research, and mathematics faculty who wanted the major to maintain its focus on the beauty and intellectual depth of core mathematical theory. In 1981, CUPM published 232 A Century of Advancing Mathematics

Recommendations for a General Mathematical Sciences Major to address this tension. The report provided a rationale for a mathematics major with a broader scope than the GCMC. The guiding philosophy of these recommendations was that a mathematical sciences major should develop attitudes of mind and analytical skills required for the efficient use and understanding of mathematics, and should be designed around the abilities and needs of the average major (with supplementary work to attract and challenge talented students.) This philosophy led to the recommendation that in-depth study in one of the diverse areas of the mathematical sciences was a valid substitute for core courses in real variables and abstract algebra. While the Mathematical Sciences report was geared toward a single flexible major in a mathematics department, it actually helped catalyze the development of multiple tracks in the mathematics major, even at small colleges. Often, the applied tracks still required one or both of real variables and abstract algebra, but then subsequent electives could be applied. This compromise worked for both students and faculty. The mathematics major today has the same multi-track structure. One important area that the Mathematical Sciences report neglected was the preparation of future high-school mathematics teachers. Specifically, the report did not question the prevailing thinking that a standard mathematics major, with the new flexibility of the Mathematical Sciences recommendations, should develop a deep mastery of high school mathematics. Although this was a convenient assumption for mathematics faculty, a 1965 studyby Begle [4](anda later1994studybyMonk[25]) challenged this assumption: Begle actually found a slightlynegative correlation between the number of upper-level mathematics courses taken by high-school mathematics teachers and the test performance of students of these teachers. In the late 1990s, an awareness developed among some mathematicians that substantial mathematics in the K–12 curriculum is worthy of study in college courses, starting with place value in early grades. But it was not until 2012 that a CBMS teacher education report [11] recommended that a mathematics major for future high-school teachers should include three courses in high-school mathematics from an advanced perspective, along with modifications to core courses such as emphasizing polynomial rings and fields in abstract algebra. While there have been no significant national changes in the mathematics major in the years since the CUPM Mathematical Sciences report, there have been differing levels of interest in mathematics. Namely, liberal arts colleges have had a higher percentage of graduates majoring in mathematics than research universities, mirroring the 1950s Albert report [1] finding about the disproportionate success of liberal arts colleges in producing future PhD mathematicians. Some liberal arts colleges have had a far larger percentage of mathematics graduates than the national average of 1%. A common theme at successful math major programs is the deep involvement of faculty who personally engage students in introductory courses. Arnie Ostebee, then mathematics chairman at St. Olaf College, where up to 10% of degrees have been in mathematics, gave a sense of what drives such success when he told me, “calculus is the favorite course of our faculty because that is where we can change students’ lives; teaching courses like real analysis is just preaching to the converted.”

Enrollment trends and two-year colleges Over the last forty years, the annual number of mathematics Bachelor’s degrees has remained fairlyconstant in the range of 10,000–15,000(about 50% of the high-watermark of The History of the Undergraduate Program in Mathematics in the United States 233

27,000 in 1970),whileoverall college graduationrates increased by over 50%,to 1,500,000. For comparison, engineering graduation rates are up by 75% to 75,000, and physical sciences rates have stayed ﬂat at about 15,000 (they had grown only modestly in the 1960s). While the number of mathematics graduates has stayed fairly steady, there continued to be a decline in advanced mathematics enrollments, down 30% from 1985 to 1995, probably because many math majors were pursuing outside minors or double majors. Data in this subsection comes from CBMS [10]. Total mathematics enrollments in four-year colleges and universities have also stayed fairly constant over this period, at about 1,600,000 per semester, while overall enrollments doubled. This discrepancy is likely explained by two factors: (i) The growth of overall enrollments has occurred in departments with light mathematics requirements; and (ii) there has been no growth in precollege mathematics enrollments. (As noted below, a huge increase in precollege mathematics instruction has occurred at two-year colleges.) The overall division of mathematics enrollments has also remained fairly constant, with precollege courses around 12%, courses below calculus around 45%, calculus courses around 40%, and post-calculus courses at 4–6%. The growing role of mathematics in science and engineering resulted in an increase in mathematics courses being offered in other departments. While the large enrollments in statistics courses in the social sciences and business were well-known, a study by Garfunkel and Young [15] found that, in 1990, there were about 170,000 annual enrollments in post- calculus mathematical sciences courses in other departments, as opposed to about 120,000 annual enrollments in post-calculus courses in mathematics departments. A major feature of higher education in the last forty years has been the growth of two-year college enrollments, from about 1,000,000 students in 1965, to around 6,000,000 students in 2000. Mathematics enrollments in two-year colleges have grown apace, but the majority were precollege mathematics enrollments, which grew from 200,000 in 1970 to 1,100,000 in 2010. At the same time, their enrollments in college-level mathematics only doubled. As a result of these surging precollege enrollments, current mathematics enrollments of 1,800,000 per semester in two-year colleges now exceed the mathematics enrollments in four-year colleges and universities, even though colleges and universities have almost three times as many students as two-year colleges. Initially, two-year colleges focused on preparation for technician-type careers that did not require a four-year degree, but they have increasingly become a starting place for Bach- elor’s degree study and have been offering the standard lower-level mathematics courses for mathematics majors: single and multivariable calculus, differential equations, and linear algebra. Many college graduates now complete all their mathematical coursework in two-year colleges. In particular, it is estimated that close to half of all elementary school teachers complete their mathematics requirements in two-year colleges.

Articulation with high schools and calculus instruction Arguably, the greatest area of change and concern in the past forty years has been the articulation between high school and college mathematics. It used to be that freshmen were placed in mathematics courses based on which high-school mathematics courses they took. However, by the early 1980s, mathematics faculty were dealing with large numbers of stu- 234 A Century of Advancing Mathematics dents in freshman courses who showed limited knowledge of needed algebra skills. States had been raising high school mathematics requirements, but courses were watered down to make these requirements workable. Further, cramming for tests became a widely-accepted substitute for course-long learning. In response, mathematics departments instituted placement tests required for all students, except those with AP calculus credit, to determine which mathematics courses they were eligible to take. Another concern has been the unintended consequences of the AP calculus program. In time, large numbers of high school students were taking AP calculus classes and then repeating first-semester calculus in college. A survey by Bressoud and colleagues [6] found that 70% of calculus students at universities and colleges had taken calculus in high school, and 25% received a 3 or better on the AP Calculus AB exam (the numbers were lower at Master’s and Bachelor’s institutions). Half of all calculus students believed that one must take calculus in high school to succeed in college calculus. However, the effect was that many students coasted through most of the first semester of calculus and failed the second semester, i.e., second-semester calculus failure rates came to exceed first-semester failure rates at many institutions. Other students who rushed through the standard high school mathematics curriculum to take AP calculus (but not take the AP test), did so poorly on math placement tests that they had to start college with a college algebra course. Equally worrisome was the fact that by 2000, a number of the students who earned AP Calculus credit in high school never took a mathematics course in college. There has also been a problem with superficial learning at the college level in calculus. As a broader audience of students from computer science, biology, business, and economics started taking calculus—partly as a screening device—the contents got watered down. Many students reacted by learning calculus in a superficial way to pass tests, never expecting to see calculus again. A common horror story was that when students were asked after taking a calculus course, what the derivative of a function is, they would answer, “for x2, it is 2x.” A prominent computer scientist led a well-publicized campaign in the early 1980s for computer science departments to stop requiring their majors to take any calculus because of these problems. In reaction, dozens of efforts were undertaken to re-invigorate calculus instruction under the banner of “lean and lively calculus” [12]. Greater attention was given to understanding the derivative as a rate of change with graphs, numerical examples (using technology), and applications. Less attention was given to techniques of integration, epsilon-delta limit proofs, and series of constants. Another approach focused on more student-active modes of instruction, such as cooperative learning. This general initiative was provocatively labeled Calculus Reform. (See Paul Zorn’s contributionto thisvolume [pp. 275–281].) While many of the changes were not controversial, there were aspects, especially cooperative learning, that gave rise to heated criticism for making calculus too “fuzzy.” While only one calculus reform textbook did become a commercial success, the less controversial aspects of calculus reform were incorporated into leading calculus texts.

Conclusion This paper has tried to provide a chronicle of the evolution of the undergraduate program in mathematics in American colleges and universities. It also sketched out major trends in The History of the Undergraduate Program in Mathematics in the United States 235 higher education that affected mathematics instruction. While the undergraduate program in mathematics has been relatively stable for the past forty years, the same was true for the undergraduate program in the ﬁrst forty years of the Association’s history. Who knows if this stability will soon be followed by dramatic changes comparable to those around the beginning and middle of the twentieth century?

Acknowledgments I wish to acknowledge the assistance I received in collecting information for this article from Doug Baldwin, David Bressoud, Gail Burrill, Christa Nilsen, Arnold Ostebee, Arnie Pizer, Gerry Porter, James Tatersall, Tom Tucker, and H. H. Wu. Dozens of other mathematicians helped me develop my personal knowledge of this subject over the forty years of my educational efforts with the MAA. I also want to thank the two excellent reviewers of this paper for their corrections and improvements.

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[16] Meeting manpower needs in science and technology,Edited by E. Gilliland, Graduate Training in Engineering,Mathematics, and Physical Sciences, President’s Science Advisory Committee, The White House, Washington, DC, 1962. [17] E. R. Hedrick, A tentative platform of the Association, Amer. Math. Monthly 23 (1916) 31–33, available at 10.2307/2972334. [18] R. Hofstadter, Education in a democracy, Anti-Intellectualism in American Life, Vintage Press, New York, 1962. [19] R. Hofstadter, C. D. Hardy, The Development and Scope of Higher Education in the United States, Vintage Press, New York, 1952. [20] American Higher Education, a Documentary History, Edited by R. Hofstadter and W. Smith, University of Chicago Press, Chicago, 1961. [21] Macalester College, College Catalogs from 1920, 1940, 1950, 1960. [22] Mathematical Association of America, Report of committee on assigned collateral reading in mathematics, Amer. Math. Monthly 35 (1928) 221–228. [23] ———, Committee on the Undergraduate Program, Universal Mathematics, Washington, DC, 1954. [24] ———, Reports of Committee on the Undergraduate Program in Mathematics: Pregraduate Preparation for Research Mathematicians, 1963; A General Curriculum in Mathematics for Colleges, 1965 and 1972; Recommendations for a General Mathematical Science Program, 1981. [25] D. A. Monk, Subject area preparation of secondary mathematics and science teachers and student achievement, Economics of Education Review 13 (1994) 125–145, available at 10.1016/0272-7757(94)90003-5. [26] National Education Association, Cardinal Principles of Secondary Education, United States Bureau of Education, Washington, DC, 1918. [27] National Science Foundation, S& E Degrees: 1966–2006, Report 08-321, Washington, DC, 2008. [28] K. Parshall, D. Rowe, American mathematics comes of age: 1875–1900, A Century of Mathe- matics in America, Vol. 3, American Mathematical Society, Providence, 1989. [29] M. Rees, The mathematical sciences and World War II, A Century of Mathematics in America, Vol. 1, American Mathematical Society, Providence, 1989. [30] R. Richardson, The PhD degree and mathematical research, Amer. Math. Monthly 43 (1936) 199–211, available at 10.2307/2300615. [31] W. L. Schaaf, Required mathematics in a liberal arts college, Amer. Math. Monthly 44 (1937) 445–448, available at 10.2307/2301130. [32] H. Slaught, The teaching of mathematics in the colleges, Amer. Math. Monthly 16 (1909) 173– 177, available at 10.2307/2969511. [33] D. Smith, J. Ginzburg, A History of Mathematics in America Before 1900, Carus Monograph Vol. 2 , Mathematical Association of America, Washington, DC, 1934. [34] State University of New York College at Geneseo,College Catalogs from 1920, 1940,and 1950. [35] A. W. Tucker, personal communication. [36] United States Bureau of the Census, Education Summary: High School Graduates, and College Enrollment and Degrees: 1900 to 2001, No. HS-21. Washington, DC, 2008. [37] United States Code, Article 304. [38] University of Nebraska, College Catalogs from 1920, 1930, 1940, 1950, 1960. The History of the Undergraduate Program in Mathematics in the United States 237

[39] University of Pennsylvania,College Catalogs from 1920, 1930, 1940, 1950, 1960. [40] University of Rochester, College Catalogs from 1920, 1930, 1940, 1950. [41] E. B. Wilson, Let us have calculus early, Bull. Amer. Math. Soc. 20 (1913) 30–36, available at 10.1090/S0002-9904-1913-02435-2. [42] F. Woods, Advanced Calculus, Ginn, Boston, 1926. [43] R. Woods, How can interest in calculus be increased, Amer. Math. Monthly 36 (1929) 28–32, available at 10.2307/2300174.

Department of Applied Mathematics and Statistics, Stony Brook University, Stony Brook, NY 11794 [email protected]

Inquiry-Based Learning Through the Life of the MAA

Michael Starbird University of Texas

A mathematical awakening Robert Lee Moore was one of the most famous mathematics teachers in the mid-twentieth century. He conducted his classes in a distinctive manner. He never lectured. Instead, he posed questions and his students were required to discover answers to these difficult questions independently without any outside help from books, teachers, or each other. In class, Professor Moore started with the student he considered the least able and asked, “Mr. can you answer the next question?” If that student could not answer the question, Profes- sor Moore asked the same question to the next weakest student and so on until he found a student who claimed to be able to answer the question.That student would go to the blackboard and present his or her independently created solution. If the other students found no flaw in the answer, then Professor Moore would simply move on to the next question. Sam Young was a student in a mathematics course taught by R. L. Moore and found himself with the dubious distinction of being considered the weakest student in the class. He never answered a questioncorrectly duringthe entire fall. His daily answer of “No, sir, I can’t answer the question.” became somewhat amusing to the whole class—even to him. He was convinced that the other students must have some previous mathematical knowledge that he lacked. At that time, the first semester did not end until several weeks after the winter break. Duringthat winter vacation, Sam went with his parents to Big Lake, Texas. His only chance to pass this class was to somehow find a way to answer some of those difficult questions. He isolated himself in his room and thought. He returned to the first questions of the semester. By looking back at the beginning of the term, the mathematics began to make sense. He saw why the first theorems were true and why no previous mathematical knowledge was necessary to prove them. It was a revelation to him. He built up from the ground floor and question-by-question solidly constructed answers for himself that taught him the material in a deep and meaningful way. He eventually found himself tackling the questions that had not yet been discussed in class. “The time came to go back to Austin and a great fear came over me as I visualized the inevitable scene that would unfold in the classroom. I was ready and confident with proofs of several of the upcoming theorems but I was scared to death. I had

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never “been to the board.” When the time came, Dr. Moore looked at me and asked the usual “Mister Young, can you prove Theorem 26?” I am pretty sure that my exact answer was “yes sir, I can.” I had learned that he did not like for anyone to say “yes, I think I can”. The fear that I felt at that moment is indescribable but I want to repeat that I was confident that my proof was correct. The fact that I knew I had it, the fact that I was confident in spite of the awful fear of the moment is a tribute to the Moore method of teaching. I went to the board and presented my proof. He asked if anyone had any questions. There were none and I took my seat. “My friends who have had courses from Dr. Moore are always surprised when I tell them what happened next. He never called upon the same person who had just presented a proof. It was just not done. He would turn the pressure on someone else. But I distinctlyremember that he looked around the room as if tryingto decide whom to call upon. His eyes fell upon me again and he asked “Mister Young, can you prove Theorem 27?” I said that I could and went to the board again. When I finished I sat down again and again he looked around pretending to decide and said “Mister Young, can you prove Theorem 28?” This impish routine continued for an hour and started again at thenext class meeting. In fact, no one else was called upon for the remainder of the term. I proved all the theorems that we had time for.” (From “Christmas in Big Lake” by Sam W. Young [16].)

At Big Lake, Mr. Young had become a mathematician. He went on to earn a PhD in mathematics and Dr. Young had a long and successful career as a professor of mathematics at Auburn University. Modern implementations of Inquiry-Based Learning (IBL) don’t usually feature such overt exposure of weak students, but this story of intellectual awakening illustrates a transformation that thousands of students have experienced through IBL. Today guided Inquiry-Based Learning methods of mathematics instruction are common and growing in use across the country. Educational research and teaching practice are reflecting the reality that students learn mathematical skills and attitudes through their individual effort and that student-centered instructional methods that call for students to work on and present their own mathematical ideas are effective strategies for getting students to become producers of knowledge and become more effective thinkers. The evolution of student-centered teaching strategies in mathematics during the last hundred years includes all the features of a robust drama including challenges to tradition and complex characters. The story of Inquiry-Based Learning begins with the very origins of the MAA. The creation of the MAA was a response to an interest in educational issues including methods of experiential learning. The history of Inquiry-Based Learning during the last 100years presents us witha parable about the powerof persistence and the influence of individualeffort, and this story suggests lessons for us to embrace for the next 100 years. Before telling some of the evolution of Inquiry-Based Learning, let’s take a moment to describe what it is. These days when you talk about Inquiry-Based Learning (often abbrevi- ated IBL), you are referring to any one of a suite of student-centered instructionalmethods that have in common that during a typical IBL class session, students do most of the talking, and they are working on or explaining their own mathematical work. Recently, studies of teaching methods have been conducted about IBL, and one of the objective measures Inquiry-Based Learning Through the Life of the MAA 241 these studies report is the percentage of class time spent in various activities: instructor explaining, instructor answering questions, students presenting their own work, students working in groups, students discussing ideas. The most obvious and most easily measurable distinction between IBL instructional methods compared with standard lecture methods of teaching is the percentage of time during the class session in which the teacher is speaking. In classes using the standard lecture method of instruction, the teacher typically explains mathematical ideas or speaks while doing examples more than 80% of the time. In Inquiry-Based Learning classes, the teacher typically speaks less than 30% of the time— and commonly less than 10% of the time would be described as having the teacher giving didactic explanations. In Inquiry-Based Learning classrooms the students spend most of the class time either making presentations of their own work or working in groups or individually on problems or proofs of theorems while the instructor provides some commentary and assistance. Most of the remaining class time is generally spent in student discussion and critique. Inquiry-Based Learning techniques vary widely. Some variations involve no group work and some consist almost exclusively of group work. Some IBL classes require a great deal of written work that is critiqued by peers; other IBL classes have little or no written work critiqued by peers. Recently, new online-based methods of class discussion are being developed including having the students asynchronously edit others’ proofs and problem solutionsbetween class sessions. Some IBL classes include the production of a textbook of student produced and class-edited polished solutions to problems and proofs of theorems. For the purposes of this essay, all these techniques where the main experience of students in the class is their producing and evaluating student-created proofs of theorems and student-created solutions to problems and exercises will be included under the umbrella of Inquiry-Based Learning. Here at the end of this century of development, many experiments about and variations of Inquiry-Based Learning methods are being conducted and developed. New users of IBL methods, including some Project NExT fellows, are being introduced to the techniques through organized workshops, mentoring, and informal influences. New IBL materials are being written and produced both for use by students in classes and by instructors as they learn the methods. The MAA has recently created a subseries of its Textbook Series called Mathematics Through Inquiry for IBL materials. Inquiry-Based Learning methods of instruction are influencing the vision of what many departments consider essential experiences for an undergraduate mathematics major. As colleges and universities respond to the challenges posed by online materials and presentations, Inquiry-Based Learning presents a strong example of an educational experience that has a significant impact on students and that is not easily duplicated through an online medium. The Mathematical Association of America, the National Science Foundation, the Ed- ucational Advancement Foundation, the Howard Hughes Medical Institute and other pro- moters of educational improvement are all supporting the further development of Inquiry- Based Learning methods of instruction. This broad support indicates that IBL methods are becoming mainstream strategies of instruction at colleges and in some pre-college education across the country. Let us now look at the historical development of Inquiry-Based Learning methods of instructionin mathematics. We will focus on the last 100 years, but let’s start with Socrates. 242 A Century of Advancing Mathematics

Ancient history The history of Inquiry-Based Learning started millenniaago. The Socratic method provides a clear historical precedent for using inquiry methods during instruction. Of course, not all Athenian leaders gave a warm reception to Socrates’s encouragement of the youth of Athens to think independently. Dangers aside, let’s begin our investigation of the early uses of Inquiry-Based Learning methods by looking at an example of teaching mathematics using IBL methods as recorded in Plato’s Socratic dialogue Meno—although Socrates may not have referred to the method as IBL. In the Meno, Socrates attempts to demonstrate that people have knowledge of mathematics before they are born. Frankly, the demonstration of pre-birth knowledge is not too persuasive, but the dialogue is an early illustrationof a guided discovery method of instruction. In this scene, Socrates takes a slave boy and through an inquisitiongets the slave boy to prove a theorem in geometry. His goal is to prove an instance of the Pythagorean Theo- rem, although he does not call it by that name. He begins with a 2 2 square and seeks to prove that the square on the diagonal has area 8. Here is an excerpt of the instruction as it unfolds in Plato’s Meno. J H G

C D E

A B F

First, Socrates creates a picture consisting of four copies of the 2 2 square that is of interest. So AF GJ is a 4 4 square. Socrates then proceeds to extract the proof from the Boy. SOCRATES: And how many times the size of the ﬁrst square (ABCD) is the whole (AF GJ )? BOY: Four times. SOCRATES: Now do these lines going from corner to corner cut each of the four squares in half? BOY: Yes. SOCRATES: And these are four equal lines enclosing this area? (BEHD.) BOY: They are. SOCRATES: Now think. How big is this area? BOY: I don’t understand. SOCRATES: Here are four squares. Has not each line cut off the inner half of each of them? BOY: Yes. SOCRATES: And how many such halves are there in this ﬁgure? (BEHD.) BOY: Four. SOCRATES: And how many in this one? (ABCD.) Inquiry-Based Learning Through the Life of the MAA 243

BOY: Two. SOCRATES: And what is the relation of four to two? BOY: Double. SOCRATES: How big is this figure (BEHD) then? BOY: Eight [square] feet. SOCRATES: On what base? BOY: This one (BD). SOCRATES: The line that goes from corner to corner of the square of four [square] feet? BOY: Yes. SOCRATES: The technical name for it is ‘diagonal’; so if we use that name, it is your personal opinion that the square on the diagonal of the original square is double its area. BOY: That is so, Socrates. [Here Socrates and Meno discuss the issue of whether the Boy had knowledge of geometry before birth.] SOCRATES: At present these opinions, being newly aroused, have a dream-like quality. But if the same questions are put to him on many occasions and in different ways, you can see that in the end he will have a knowledge on the subject as accurate as anybody’s. MENO: Probably. SOCRATES: This knowledge will not come from teaching but from questioning. He will recover it for himself. MENO: Yes. [] SOCRATES: I shouldn’t like to take my oath on the whole story, but one thing I am ready to fight for as long as I can, in word and act: that is, that we shall be better, braver and more active [people] if we believe it right to look for what we don’t know than if we believe there is no point in looking because what we don’t know we can never discover. MENO: There too I am sure you are right. Few of us would accept Socrates’s conclusion that the slave boy knew geometry before he was born; however, we would probably have to admit that the slave boy learned a geometrical insight through this experience. The trouble with this dialogue is that the slave boy did not really come up with the salient ideas on his own. Socrates’s questions were so leading that it was clearly Socrates who knew each step of the argument. The degree to which the questions in an IBL experience guide the student to discover ideas is one of the choices that lead to a range of possible IBL methods. Inquiry-Based Learning techniques guidestudents to focus in directions that predictably lead to specific mathematical knowledge. A lecture that includes all the details of a proof is at one end of the spectrum of possibleinvolvement ofstudentsin figuringout themathemat- ics on their own. At the other end of the spectrum would be a blank piece of paper saying, “What can you say about a triangle?” That question is so open-ended that it might lead to no results or might lead in directions that the instructor could not guess in advance. Most Inquiry-Based Learning methods of instruction include strong guidance through a carefully arranged sequence of challenges so that the students discover specific mathematical concepts, methods of proof, and proofsof theorems. Some IBL courses include asking students 244 A Century of Advancing Mathematics to make conjectures as well. IBL is a training step in the direction of pure mathematical research. The teacher of an IBL course must respond appropriately to mathematical arguments that the students produce. So the teachers must have a sufficiently sophisticated understanding of the mathematics that they can recognize and deal with lines of reasoning that may be different from the standard approaches. The teacher of an IBL course should model the view that new ideas are on the horizon and can be discovered and explored by clear thinkers. Socrates’s dialogue ends with the provocative insight that people can discover new insights by intentionallyturning their minds to do so. One of the features of Inquiry-Based Learning is that students learn the attitude of expecting to discover new insights through their own thought. The record of research success on the part of students who learned mathematics partly using IBL techniques provides persuasive evidence of the potential impact of IBL on the research abilities of students.

Eliakim Hastings Moore E. H. Moore was a professor of mathematics at the University of Chicago in the early part of the twentieth century. He was a prominent research mathematician who served as Pres- ident of the American Mathematical Society and was a member of the National Academy of Sciences. As a prominent and influential figure both in research and teaching, he viewed those two activities as interconnected. His interest in novel methods of mathematical instruction played a part in the development of IBL methods during the last century as well as the connection between the MAA and IBL methods. E. H. Moore was strongly influenced by the nineteenth-century success of science and science instruction using laboratory methods. So he created a method of mathematical instruction that he described as a mathematical laboratory. During class periods, instead of presenting a lecture, he had students work together with him on unsolved problems. He asked them to work out mathematical ideas on their own as the main experience of those classes. His interest in this style of pedagogy was part of his motivation for becoming one of the founders of the MAA. In 1913, he and others proposed the creation of an organization somewhat parallel to the American Mathematical Society, but an organization that would encourage educational advances. Of course, that new organization became the Mathematical Association of Amer- ica. During the years when E. H. Moore was using his laboratory method of instruction at the Universityof Chicago, one ofhis graduate students was Robert Lee Moore. E. H. Moore and R. L. Moore shared a common last name; they were seventh cousins but only knew each other as professor and student. Duringthe 20thcentury, R. L. Moore became byfar the most well-known developer and user of inquiry methods of instruction in mathematics.

Robert Lee Moore The most colorfulcharacter and themost influential individualin the story of Inquiry-Based Learning during the last century was Robert Lee Moore. R. L. Moore was a distinguished Inquiry-Based Learning Through the Life of the MAA 245 research mathematician. His research in topology gave him an international reputation and accolades including membership in the National Academy of Sciences and Presidency of the AMS. Like E. H. Moore, R. L. Moore combined his interests in research and teaching and viewed both parts of the profession as intertwined and mutually supportive of the other. R. L. Moore is best remembered for creating an extremely distinctive student-centered teaching method, from which modern Inquiry-Based Learning methods of instruction are evolving. In fact, Moore was so prominently identified with his teaching style, that his inquiry-based methods were, and sometimes even now are, referred to as the Moore Method or the Texas Method of instruction. As we have seen, and as is invariably the case in history, R.L. Moore’s methods of inquiry-basedteaching had their originsin ideas and practices that preceded him. One of the preceding influences was the laboratory method of instruction that E. H. Moore practiced at Chicago. An even earlier influence on R. L. Moore’s mathematical research and teaching came from George Bruce Halsted when Moore was a student at the University of Texas previous to Moore’s attending graduate school at the University of Chicago. But R. L. Moore’s methods of instruction were certainly distinctive. The opening story presented a vignetteof an R. L. Moore class. We will now describe his method of instructiona bit more thoroughly and then relate some of his personal history.

R. L. Moore’s method of instruction Moore’s method of instruction in his smaller upper-division topology classes was unusual: he would never explain mathematics to the students; he would not allow students to consult books to learn the mathematics of the course; he would not allow them to ask other professors for help with the mathematics; and the students were not allowed to discuss the mathematics with one another. Instead, Moore would pose mathematical questions to the students and their daily job was to resolve the questions on their own, thus developing the concepts for themselves. In other words, every mathematical idea that each student learned in the course was expected tocome ideallyfrom his orher ownthinkingabout a challenging question. The questions Moore posed were carefully scripted in such a way that they elicited the concepts and theorem-proving techniques that Moore wanted the students to discover and learn. Most important of all was the technique of individual effort and self-confidence, which attitudes were some of the enduring legacies of the Moore experience for many of his students. As we described in the opening scenario, during class Moore would systematically ask students whether they had been able to resolve the challenges he had posed. When one of the students claimed to be able to settle the question, either by presenting a proof or by constructing a counterexample, that student would present his or her own work to the other students in the class. The other students had one job, and that was to find flaws in the reasoning. If students from the class noticed a mistake or a gap, they would point out the flaw, but not correct the mistake. If the student at the board was unable to correct the mistake, then the student at the board would sit down and the question would either be deferred until the next class period or another student might attempt it. 246 A Century of Advancing Mathematics

In this format of teaching, the daily experience of the students was to do and evaluate mathematics independently. The challenge questions were organized in such a way that the students naturally constructed a whole body of mathematical ideas for themselves, beginning with easier fundamentals and increasingly becoming deeper and often more difﬁcult. Students regularly felt the intoxicating thrill of discovery and satisfaction of personal victory over the unknown. It turns out that Moore was far ahead of his time from the point of view of pedagogical techniques. Now educational researchers basically unanimously emphasize the importance of having students actively engage in struggling through ideas to make the students learn concepts in a meaningful way and having the learning make a permanent effect on the students. There are many Inquiry-Based Learning methods of instruction,usually less extreme than Moore’s strategy; however, the fundamental ingredient is the students’ engagement in doing mathematics for themselves.

The story of Robert Lee Moore R. L. Moore earned his PhD from the University of Chicago in 1905. After his graduation, he took a series of short positions at Chicago, Northwestern, and Tennessee. During the period 1911–1920, he was a professor at the University of Pennsylvania, where he began developing his famous teaching method. But he and his teaching method are most closely identified with the University of Texas where he was a professor from 1920 to 1969. R. L. Moore was definitely a character. Perhaps it was necessary for a person who was breaking truly new ground in educational practice to be an extremely independent thinker. So some of his idiosyncrasies of character may be a necessary part of the make up of anybody who is devising such a radical departure from the standard lecture method of instruction. Moore was turning the educational practice on its head. He was having the students produce the knowledge and present the instruction. In his early years, his methods of teaching were new to the mathematics community. His methods had neither a history of common practice nor any organized evaluation or support from educational research. Under those circumstances, it certainly required a person of great personal self-confidence and strong opinions to undertake such a fundamentally different practice of instruction. R. L. Moore’s personal characteristics probably arose from his own social background and time. Moore was born in 1881 in Texas, just 16 years after the conclusion of the Civil War. So Moore had been inculcated into the opinions of the south and certainly his given names Robert Lee (after Robert E. Lee who commanded the Confederate Army) suggest the strength of opinion of his family in their sympathies in regard to the Civil War. So it is not surprising that R. L. Moore had social opinions consistent with those of his family and local culture. Some of those values were constructive and clearly informed some of the central parts of Moore’s teaching methods. For example, the traditional Texas values of personal responsibility and fierce independence clearly influenced Moore’s teaching methods. The values of independence and focus on individual opinions encouraged Moore to develop teaching methods that were his own and clearly distinctive from others around him. His cultural roots perhaps encouraged him to create methods that promoted those same values—individual responsibility and independence—in his students. Unfortunately, some of the opinions that Moore absorbed from the culture of his childhood time and place included racial bias. However, Moore’s promotion of independent Inquiry-Based Learning Through the Life of the MAA 247 thinking made his own students independent as well. Independence of thought has a self- correcting effect. Students of Moore who were most successful did not follow Moore in his political opinions. They learned independence. So even though Moore himself never allowed a black student in his classes, Moore’s first PhD student (at Penn), J. R. Kline, was the PhD adviser for two of the first three mathematics PhDs awarded to African Americans in the United States. In one of those delightful ironies of history, some of the recent studies of Inquiry-Based Learning methods suggest the possibility that the student-centered methods that Moore pioneered may be especially effective in nurturing the mathematical potential of those same groups that Moore himself was biased against. R. L. Moore was quite assertive in his opinions. He sometimes threatened people with whom he was having some dispute with a fistfight. He sometimes brought a gun to class and wouldtake it out and put it on the table presumably indicatingsome seriousness of purpose in the classroom. He had arguments withsome of the other professors in the university to such an extent that for several decades the University of Texas mathematics department itself was split into two different departments—the pure mathematics department and the applied department. It is said that for thirty years Moore did not speak to some of his colleagues whom he disliked.But he also had allies in his department and in other departments includingH. S. Wall and H. J. Ettlingerin mathematics and R. N. Littlein physics who used methods of teaching compatible with Moore’s methods of instruction. During the early days of the University of Texas, registration was held in the gymna- sium. During those years, Moore would sit at a table in Gregory Gym and students would walk up to the table to ask to be registered in mathematics classes. Moore would begin the process by quizzing students by giving them various puzzles or other somewhat mathematical challenges. Based on the responses during his discussions with the students, he would decide whether to put the students in his own classes or into somebody else’s classes. One of the features that Moore was looking for was ignorance of the mathematics associated with any class that the student was to be in. He did not want the student to know too much in a particular class because then that student could spoil the newness of the mathematics for the other students. But he also wanted students who were bright and able to deal with abstractions to consider becoming mathematics majors. Folklore has it that one of the kinds of questions that he would ask would be the following, “Suppose cows had six legs, then how many pairs of legs would a cow have?” This kind of a question would give an understanding of whether the student was able to deal with abstractions and hypotheticals.

Moore’s students As you can imagine Moore’s method of instruction was quite an intense experience for his students. His students often came away with a profound sense that something of great importance had happened in their educational careers. The entire responsibility for understanding the mathematics rested on the shoulders of the individual students. So the impact of this experience was predictable. The students would come to have the sense that they could personally resolve difficult mathematical questionson the basis of their own ability to investigate those questions. Students who were successful in Moore’s classes often gained a tremendous self-confidence in their ability to resolve difficult problems—mathematical 248 A Century of Advancing Mathematics problems or problems well beyond the academic sphere. Many experienced an epiphany of intellectual awakening. Many of Moore’s students used that confidence and the skills that they gained in his classes to great effect. The record of Moore’s PhD students in their own research, teaching, and service careers was extremely impressive. Three of Moore’s PhD students became members of the National Academy of Sciences. Many of his students went on to produce many PhD students of their own, with their students continuing the pattern. In fact, to date more than 3,000 PhDs are mathematical descendants of R. L. Moore. Perhaps because of the distinctiveness of the teaching methods that Moore’s students had experienced in his classes, many of his students were active in educational issues. Specifically, many of them became leaders in the MAA, participating actively in the educational issues of their day. Five of Moore’s students became President of the MAA: R. H. Bing, R. D. Anderson, E. E. Moise, G. S. Young, Jr., and R. L. Wilder. Lida Barrett was another MAA President who could be partly added to the list since she studied with Moore although her PhD was finished under the direction of Moore’s students Kline and Ander- son. Three of Moore’s students became President of the AMS: R. L. Wilder, R. H. Bing, and G. T. Whyburn. Many more of Moore’s students also served in other offices in the MAA and in the AMS. Some of his students chose different career paths other than mathematics or academia and many of them viewed their experiences in Moore’s classes as important contributors to their success. They learned the skills of clear thinking, perseverance, innovation, and self-confidence that laid the foundation for success in any walk of life. Moore’s method of instruction was not successful for all students, of course. Some students in his courses were not able to prove the theorems on their own. Some of his students were no doubt discouraged by their lack of success. However, recent educational research suggests that inquiry methods of instruction may actually help weak students to an even greater degree than their relative impact on stronger students.

The middle years of IBL Moore did not himself actively promote his teaching methods. From the 1930s into the 1960s, much of the teaching inthe style of R. L. Moore occurred from Moore’s studentsand theirstudents. But dissemination of hismethods of instructionwas of interest to many of his students and to others who came to appreciate the method. R. H. Bing encouraged Moore to participate in the creation of the 1967 MAA film titled ‘Challenge in the Classroom.’ Moore was somewhat grudging in his participation in the film; however, the film was duly created, and it shows Moore in his mid-80s talking about his teaching methods and showing some footage from his actual classrooms. It is definitely worth viewing. Moore created and used his distinctive style of teaching mathematics, but he was not an educational theorist. He tended to denigrate people who studied instruction. His reluc- tance to promote his method of instruction extended even to his own students. After their graduation, he did not ask them whether they used his method—he viewed such a question as somehow not within the rules of the game. However, several of his students wrote articles about the method includingLucille Whyburn, F. Burton Jones, R. L. Wilder, and E. E. Moise. Inquiry-Based Learning Through the Life of the MAA 249

Many of Moore’s students used his method of instruction in their own classes and over the years modified that method in many important ways. Some professors beyond Moore’s personal influence learned about and were strong advocates for Inquiry-Based Learning as an effective method of instruction. One notable example of such a proponent of IBL methods was Paul Halmos, who became convinced of the efficacy of inquiry methods of instruction and wrote about his perspective: Some say that the only possible effect of the Moore Method is to produce research mathematicians—but I don’t agree. The Moore Method is, I am convinced, the right way to teach anything and everything—it produces students who can understand and use what they have learned. It does, to be sure, instill the research attitude in the student—the attitude of questioning everything and wanting to learn answers actively—but that’s a good thing in every human endeavor, not only in mathematical research. Not all of Moore’s students liked or used inquiry methods of instruction and certainly many or most of them modified the method they experienced with Moore to accommodate the method to their own circumstances and personalities. But many of Moore’s PhD students and their PhD students in turn used or use Inquiry-Based Learning methods of instruction. Many of Moore’s descendants are leaders today in the continuing development and dissemination of IBL.

Recent developments of IBL Recently the Inquiry-Based Learning movement has gained considerable impact and range well beyond R. L. Moore’s students and their mathematical progeny. Moore’s strategy of instruction was the primary origin for the burgeoning of recent IBL methods in mathematics instruction. But new developments in teaching methods in the sciences have made inquiry methods of instruction gain considerable attention and positive reputation throughout the STEM disciplines. Eric Mazur, a professor of physics at Harvard University, has successfully promoted student-centered methods in the teaching of physics. Carl Wieman, a Nobel-prize winning physicist, has done extensive research showing the value of student- centered methods. Bruce Alberts, former President of the National Academy of Sciences, has promoted inquiry methods of teaching and learning. The National Science Foundation has given significant grant support to developing and disseminating more student-oriented methods of instruction in all the STEM fields, basing its support on a growing body of educational research. The MAA has played no small part in recent developments of IBL through the MAA’s support and involvement in at least five significant areas that foster the evolution and dissemination of IBL. The MAA has sponsored or co-sponsored IBL workshops,conferences, sessions at meetings, Project NExT events, and publications. For example, in 2010 the MAA began co-sponsoring the annual Legacy of R. L. Moore Conference, which dis- seminates IBL methods and innovations. Several MAA PREP workshops have focused on disseminating IBL methods of instruction, some have been co-sponsored with the Na- tional Science Foundation and many have been significantly supported by the Educational Advancement Foundation and its principal, Harry Lucas, Jr., who was himself a student 250 A Century of Advancing Mathematics of R. L. Moore and who attributes a significant amount of his success in business to the lessons he learned in R. L. Moore’s classes. To no small extent, the recent increase in Inquiry-Based Learning in mathematics has arisen from Harry Lucas’s persistence and encouragement, both of which attributes were among the lessons learned in Moore’s classes. Now a growing team of IBL practitioners, some from the Moore school and many newly introduced to IBL methods are actively developing and disseminating IBL methods nationally. Many students who experience IBL methods look upon those experiences as formative in their own lives. Their support, both personal and financial, of the further development and dissemination of IBL reflects the impact that they feel the method had on their lives.

Current status and assessment of Inquiry-Based Learning and future directions

IBL methods are becoming increasingly more standard across the county. Recent educational research has helped to confirm the benefits that an IBL experience frequently has on students. Sandra Laursen et al. ([7, 8]) have done and published extensive studies on the efficacy of IBL instruction to thousands of students and have described the effects of IBL methods in mathematics. And recent work on the Calculus Concept Inventory by Jerome Epstein ([3]) also indicates that IBL methods of instruction have special value for mathematics students. Many people are involved in systematically developing and disseminating effective IBL methods of teaching nationally. The IBL movement includes IBL Centers at the University of Texas at Austin, the University of Chicago, the University of California, Santa Bar- bara, and the University of Michigan. These Centers actively disseminate IBL methods on their own campuses and nationally. In addition, the Academy of Inquiry-Based Learning (AIBL) supports many small grants to faculty members who are doing experiments and implementing IBL in classes at a broad range of colleges and universities nationwide. Regular workshops, conferences, and sessions at national meetings are systematically bringing IBL expertise into the standard practice of mathematics faculty nationally. The very fabric of education is being challenged today through technological developments and through the challenge of educating a far broader range of students who seek college-level education. In the past, teaching procedures for doing arithmetic, algebra, and calculus formed a significant part of the mathematical education of most students. Now, many of the more mechanical aspects of this staple of mathematical education are readily available online. Not only in mathematics, but in every part of higher education, colleges are strugglingto define the special contributionsthat the experience of personally attending a college can provide. In thissetting, Inquiry-Based Learning may well provide a potent example of the special experiences a student can have in a classroom that truly can transform lives. The MAA continues to play a central role in the national discussion of the best practices and future of education in mathematics. On a practical level, PREP workshops, Project NExT programs, and meeting sessions often feature discussions of IBL methods and developments. On a philosophical and education research level, the IBL perspective may Inquiry-Based Learning Through the Life of the MAA 251 well contribute to reasoned decisions about how to create increasingly effective methods of instruction. Today many fundamental questions about mathematics education confront us. This moment in the joint history of both IBL and the MAA is definitely a moment of enthusiastic and quick evolution in educational practice. Student-centered methods of instruction are becoming sufficiently prevalent that conscientious teachers must actively consider alternatives to pure lecture instruction even if they do not personally embrace student-centered methods. A century from now, the discussions about how people learn may well be based on a far more sophisticated understanding of human learning than we currently possess; however, current developments in educational practice and theory suggest that IBL and the MAA will continue to be closely connected as central players in the evolution of education in mathematics during the next century.

Acknowledgement I want to thank Albert Lewis for many helpful suggestions on all aspects of this article. He certainly improved it enormously, and I appreciate his kind contributions.

Some further reading on the history of IBL [1] B. Alberts, Prioritizing science education, Science 340 (2013) 249. 10.1126/ science.1239041. [2] C. A. Coppin, W. T. Mahavier, E. L. May, G. E. Parker, The Moore Method: A Pathway to Learner-Centered Instruction. Mathematical Association of America, Washington, DC, 2009. Additional information at www.LegacyRLMoore.org. [3] J. Epstein, The Calculus Concept Inventory—measurementof the effect of teaching methodology in mathematics, Notices of the AMS 60 (2013) 1013–1026. [4] P. R. Halmos, I Want to be a Mathematician. Springer-Verlag, New York, 1985. [5] ———, What is teaching?, Amer. Math. Monthly 101 (1994) 848-855. [6] F. B. Jones, The Moore Method, Amer. Math. Monthly 84 (1977) 273–277. [7] S. L. Laursen, From innovation to implementation: multi-institution pedagogical reform in undergraduate mathematics, in D. King, B. Loch, L. Rylands (Eds.), Proceedings of the 9th DELTA conference on the teaching and learning of undergraduate mathematics and statistics, Kiama, New South Wales, Australia, 24–29 November 2013, Sydney: University of Western Sydney, School of Computing, Engineering and Mathematics, 2013, available at www.colorado.edu/eer/research/steminquiry.html. [8] S. L. Laursen, M.-L. Hassi, M. Kogan, T.J. Weston, Beneﬁts for women and men of inquiry- based learning in college mathematics: A multi-institution study, Journal of Research in Math- ematics Education 45 (2014) 406–418. [9] The Mathematical Association of America: Its First Fifty Years, edited by K. O. May. Mathe- matical Association of America, Washington, DC, 1972. [10] J. Mervis, Transformation is possible if a university really cares, Science 340 (2013) 292–296. 10.1126/science.340.6130.292. [11] E. E. Moise, Activity and motivation in mathematics, Amer. Math. Monthly 72 (1965) 407–412. [12] ———, On the foundations of mathematics, (retiring address as president of the American Mathematical Society), Bulletin of the AMS 9 (1903) 402-424; also in Science (2) 17 (1903) 401–416. 252 A Century of Advancing Mathematics

[13] J. Parker, R. L. Moore: Mathematician and Teacher. Mathematical Association of America, Washington, DC, 2005. [14] L. S. Whyburn, Student-oriented teaching—the Moore method, Amer. Math. Monthly 77 (1970) 351–359. [15] R. L. Wilder, Robert Lee Moore 1882–1974, Bull. AMS 82 (1976) 417–427. [16] S. W. Young, Christmas in Big Lake (1998), available at legacyrlmoore.org/ reference/young.htm. [17] D. Zitarelli, Towering ﬁgures in American mathematics, Amer. Math. Monthly 108 (2001) 606– 635.

University of Texas, Department of Mathematics , 2515 Speedway Stop C1200, Austin, TX 78712 [email protected] A Passport to Pleasure

Bob Kaplan and Ellen Kaplan The Math Circle

Math cuts you no slack. You can argue the merits of Crime and Punishment with experts because beauty is in the eye of the beholder, but doubt what V E F add up to and C proof will send you sprawling. It is this absolutismthat by turns appeals and appalls. It can make a lifelong devotee like Bertrand Russell conclude that what he thought would be as beautiful as Dante’s Paradiso was no more than a series of hollow tautologies. But it can set each of us dancing at 2 AM when the doors to this paradise all at once swing open. Our inner GPS faithfullypoints us toward the intersection of “if” with “then”—so why do we shout angrily back at its impersonal voice when it tells us it is recalculating? Because it is the pits of our stomachs that tell us how stark math is, since that’s where our architectural instinct lurks—the instinct that lets us explore, colonize and understand the world. That’s why a defeat in it is so crushing and a victory so exhilarating; it is our very selves, our sense of worth, that is being nibbled away. One glimpse of the timeless struts of how things have to be, and our summer dream beneath the tamarind tree marks us no longer as childlike but childish. “Only if” always trumps “if only.” How are we to persuade our friends, how are we to raise our children,that they too may savor this architectural grandeur and relish the efforts and contortions that slowly lift arch and beam into place? Not through calculational calisthenics, nor repetitions that would take the edge off the keenest delight, but by setting an intriguing mystery just at their farthest reach. Ask some four-year-olds if there are numbers between numbers, and the rest, as they say, is analytic. Draw for a class of eight-year-olds the triangle at whose vertices live the As, Bs and C s and ask them where best to locate the ﬁre hydrant to serve them equally. The engines of deduction rev up, the power of transitivitydawns, and in time the stars of incenter, centroid and orthocenter too will glimmer in their cranial sky. Our Math Circle’s motto is “Tell me and I forget. Ask me and I discover.” This has evolved from the Nufﬁeld Foundation’s adoption of the Chinese proverb “Tell me and I forget; teach me and I may remember; involve me and I will learn.” Let a small group of age-mates sit with you as if it were a congenial dinner and tug at such a mystery together. Your guests are any who want to be there—the pernicious myth of talent has been left at the door to the last century. Chest-thumping and put-downs are out of place here: this is a cohort of friends. No tests, no grades, no hints, neither praise nor blame, are on the menu; as their host you no more than nudge the conversation as unobtrusively as possible toward likely paths. If it takes a turn in an unexpected but interesting direction, follow along and discover with them.

253 254 A Century of Advancing Mathematics

These Math Circle conversations set out from mystery and tack through turbulent wa- ters to insight—and then comes the new sailing toward proof. What a strange passage this is: what we have discovered—out there, independent of us, timeless, unpersoned—we now want to fair into each of our own personalities, with their different points of access. Different—yet with some sort of shared understanding and assumptions, Mind emerging through minds. We tinker, discard, invent, refine. Language, which up to now had been the invisible air our thinking breathed in, takes on a texture of its own, liable to twist or splinter when we work it—butequally likely to gleam and lead us along its grain to shapes that disclose deeper structure. Discovery has become invention—in new idioms, with fresh tropes—but of that unchanging abstract connection you saw, fleshed out now in your own persona. Is this what Kant meant by calling mathematics synthetic a priori: made, but with the tools that make the tools? The Math Circle aims to open its students’ eyes to these accessible wonders, and give the lightest of shaping touches to their exploring: encouragement, comradeship, focus, a toolkit with IKEA directions (this end is the handle). For that its leaders need good cheer, patience, an experience of wonder and its resolution themselves—but above all, they have to remember having been novices once. Your first absolute value sign might have stopped you dead in your tracks; that the converse of a true statement may not also be true can rank with The Greater Mysteries. Constructivists aren’t alone in balking at proofs by contradiction; is x an unknown, a place-holder or a variable? Is induction diabolic sleight-of-hand? To recall the queasiness you felt when you first encountered it, think about transfinite induction now. It helps too if “No” is replaced by “What does that imply?” Like the Duke of Plaza-Toro, we lead best from behind, empowering the climbers. We have had more than two thousand students in our Math Circle at Harvard over the past twenty years—but generations of people the world over are growing up fearing, hating or being bored by what is the greatest of the arts, and our widest gateway to meaning. How would you spread literacy in a country? Legislate that whoever learns to read must teach two others (figure out how soon 0% becomes 100%) . Why could numeracy not spread as quickly, by a similar initiative? With this in mind, we wrote Out of the Labyrinth: Setting Mathematics Free [1] in 2007. One upshot of it was being invited to Brazil in the summer of 2013, to open a thousand Math Circles there. They are set up in the poorest sections of major cities. This pilot program, massively funded by the Italian telecom company TIM, doubled in its expansion phase in 2014. The teachers we train there in our approach are enthusiastic, optimistic, passionate, imaginative. They haven’t, by and large, graduate-school experience of math, but what counts is their eagerness to learn, along with those in their care. It is as lazy to think that you have to be born a Math Circle leader as to think that some emerge from the womb as mathematicians. Imagination, and its guardian angel, a sense of humor, may not come when called, but can be persistently wooed. After thirteen years of our Math Circle at Harvard, we began as well, in 2008, a Math Circle Summer Teacher Training Institute at Notre Dame. We start the week with a Math Circle class on a topic the members are unlikelyto know, so that they can remember what Monday mornings feel like. The conversation is a cross-ruff between the math and the unfolding of it: “Why did you pause just there?” “Why did you answer that question with a question, and why that question?” Then the members each teach a practice class to local children, with the A Passport to Pleasure 255 rest watching and afterwards going over what happened or didn’t happen or should have happened. Sessions each afternoon on the ins and out of running a Math Circle—choice of topics, unrulychildren, the silent, theshy, the sly, obtrusiveparents, funding,the beginnings and ends of a class, your ignorance exposed (that means the pleasure of exploring too!), resources, shared ideas, a support network. Evenings are devoted to talking about math. We have had people coming from across the country and around the world, a department chair from Argentina sittingnext to a kindergarten teacher from a farm community. We find it immensely encouraging. A Math Circle leader once asked us, how do you know if a class went well? We answered: were you having a good time? We are there for one another’s pleasure. And Russell’s complaint that math turned out to be a series of hollow tautologies? Had he been last year in one of our classes with ten-year-olds, he would have plunged with them into this prologue to tackling the Wallace-Bolyai-Gerwien Theorem (equi-areal polygons are equi-decomposable): can you turn a triangle into a rectangle? It doesn’t look very promising, does it? Gross cuts, fine cuts, clever cuts, awkward cuts: a litter of disjecta membra spills over the table. “What’s a triangle’s area?” Adam asks. “Half the base times the height,” answers Sarah. “So that’s : : : 1=2 b h;” says the formalist Tom. Not much help. “Or b=2 h;”offers Eden, and Sarah cuts her new trianglealong a perpendicular at the base’s mid-point, moves the quadrilateral and right triangle around—nothing doing. “Or b h=2;” Tom finishes the triple tautology. Adam draws an altitude, bisects it, cuts along his two perpendicular lines, swivels the top two triangles left and right—a rectangle! Some tautologies are useless, some misleading—and some unlock the gate, since mathematics is a series of tautologies! Plato said that as playthings of the gods we should play the noblest games. Have you any doubt that he was thinking of mathematics? Our love of music ever reminds us that we are joined at the mind to the dance of sheer form.

Bibliography [1] B. Kaplan,E. Kaplan, Out of the Labyrinth: Setting Mathematics Free, Oxford University Press, New York, 2007; second ed., Bloomsbury Press, New York, 2013.

The Math Circle, 27 East Street, Southampton, MA 01073 [email protected]

Strength in Numbers Broadening the View of the Mathematics Major

Rhonda Hughes Bryn Mawr College

The prevailing ethos in many mathematics departments is that only the “best and brightest” should be encouraged to major in mathematics. Many students are quickly dismissed as not good enough or not cut out for mathematics. The conventional view of potential majors is based on stereotypes that rely on inaccurate and narrow views about who can and should do mathematics. These assessments are subjective and often based on expectations derived from one’s own educational and cultural experiences. The February 2012 Presidential Report Engage to Excel: Producing One Million Ad- ditional College Graduates with Degrees in Science, Technology, Engineering, and Math- ematics identified recruitment and retention of undergraduate STEM students as an urgent national challenge [5]. In its analysis the report states, . . . many students, and particularly members of groups underrepresented in STEM fields, cite an unwelcoming atmosphere from faculty in STEM courses as a reason for their departure. Given the current emphasis on STEM training, it is in the best interest of themathemat- ics community as well as the nation that mathematicians reevaluate the notionof their ideal student. Data presented in the Presidential Report revealed that mathematics consistently has the least number of majors among STEM fields. In order to ensure that more interested students pursue and thrive in the major, we must recognize that some students develop more slowly than others, have weaker preparation, or lack confidence in their abilities. The Presidential call for action presents an enormous opportunity for the mathematics community. The community’s response will determine whether or not mathematics will emerge as an important voice in the discussion and implementation of educational innovation and national achievement in STEM fields. When I arrived at Bryn Mawr College in 1980, there were four mathematics majors. Bryn Mawr is a selective liberal arts college for women with an esteemed mathematical heritage. Prominent Bryn Mawr professors Charlotte Angus Scott, Anna Pell Wheeler, Emmy Noether, John Oxtoby, and Frederic Cunningham, Jr. shaped its legacy as a premier institution for the study of mathematics. They inspired studentsto strive for excellence and pursue graduate work. There are many advantages and strengths to such a program; nevertheless, I was repeatedly disappointed by how few students chose to major in mathematics. Early

257 258 A Century of Advancing Mathematics efforts to address the low numbers produced inconsistent results. A charismatic calculus teacher could go a long way in moving the needle, but unless changes were institution- alized, progress was fleeting. One year there were seventeen majors, the next year five. There were also legitimate concerns: would more majors dilute the effectiveness and reputation of the program? How could we recruit more students to major in mathematics while preserving the challenge for the, so-called, best students? Through a series of efforts, some failed, the department eventually began to consistently attract students to the major. We learned that in order to attract more students to mathematics, a department must ensure that the courses in the first two years be engaging and demonstrate to students the possibilities available to them with a major in mathematics. We need to provide learning experiences that allow our students to witness their own strengths and potential, and to overcome temporary setbacks. Most Bryn Mawr math majors did not intend to major in mathematics at first, but rather were influenced to consider mathematics as a major because of their experiences in first- year calculus. Introductory courses should not present barriers to advancement, but rather opportunities for recruitment. There are several concrete steps that can be taken and individual departments will undoubtedly have their own paradigms for success. What is most important, and less tangible, is the change in attitude required to reimagine the cohort of math majors. Deploy the best and most charismatic teachers to teach introductory courses. These teachers should embrace the notion of inclusiveness. They may be tenured professors or adjuncts, but it is preferable to have faculty with a stake in the future of the program teaching these courses. Ensure that students understand the sequence of courses open to them should they wish to major or minor in mathematics, or any STEM field. Most students have no idea what linear algebra is, but once persuaded of its widespread usefulness, its ease of entry (there are really no concrete prerequisites for linear algebra other than an ability to add and multiply and the elusive “mathematical sophistication”), its refreshing intellectual challenge, and its welcome break from calculus, they flock to the course and often end up majoring in mathematics. Invite faculty who will be teaching the next course in the curriculum to come to your class to pitch it. We need to market our product, not protect it from unwanted outsiders. Employ students as graders and peer leaders. Our calculus students are kept engaged by becoming graders for their previous classes. This work provides students with a stake in the department and useful reinforcement of the material. Provide early mentoring and research experiences for students. If a student gets an impressive score on an exam, seek her out and congratulate her. Encourage students to attend REUs and other summer research programs. Involve students in departmental activities as early as possible. Such involvement increases their interest in the major and helps them see the department as a welcoming environment. Many of my colleagues give bonus points for attendance at departmental functions. The points are rarely enough to make any real difference, but highly- motivated students will attend and guarantee a large and enthusiastic audience. Strength in Numbers: Broadening the View of the Mathematics Major 259

Take steps to ensure that assessment actually reflects what has been taught in a course and is not being used to find out which students are “stars” or to “thin the herd.” An exam on which the average is 30% should generate self-reflection on the part of the instructor, rather than a lament about the decline of the quality of students.

If possible, hire a long-term faculty member whose position is devoted to excellence in teaching and the recruitment and retention of students. The Bryn Mawr Mathemat- ics Department could not have enjoyed its success without the appointment of Mary Louise Cookson, a dedicated and charismatic teacher who consistently attracted and encouraged students for nineteen years. The mathematics major provides excellent preparation for a wide variety of careers, many of them unexpected. In addition to research and teaching in math and science, a number of our majors have remarkably successful careers in finance, industry, medicine, law, epidemiology, environmental studies, and criminology. Others have found their lives’ work in dance, choreography, music, theater, television, film, or fashion. With such a diverse cohort of alumnae, I can enter old age with a trusted financial advisor, top-notch medical and dental specialists, a crack team of lawyers, a kind and gentle veterinarian, and a fashion consultant, all of whom were former math majors. Moreover, this diversity promotes extremely vibrant discussions and perspectives in advanced courses. One might ask whether the strongest students are well served by such an approach. As evidenced by our math alumnae completing their graduate work at Harvard, Princeton, Stanford, Johns Hopkins, and Wharton, our students are indeed able to succeed at the highest levels. While not all remain in mathematics, they will all be successful and contribute to society as leaders and innovators. Moreover, our students enter the world with a deep sense of optimism and altruism. An exceedingly selective and unforgiving approach to the major is counterproductive. What successful enterprise discourages customers? Mathematics is already an equalizer, and has its ownpower todemoralize and defeat. It is ourresponsibilityto coach ourstudents through and past their setbacks, rather than persuade them that a low grade is irrefutable proof that they should pursue a more suitable major. Walking the halls of Park Science Center, one sees a diverse and vibrant population of students excited about mathematics. Witnessing my former calculus students, who intend to be dentists or surgeons, standing at the board in a colleague’s office discussing Reidemeister moves fills me with pride and joy. A robust body of research in social psychology supports the tenets on which this approach is based. There is a range of qualities that contribute to student success: perseverance, a capacity to rebound from setbacks, a strong work ethic, creativity, and grit. Angela Duckworth has found that grit, the disposition to pursue especially challenging goals with perseverance and passion, predicts objective indicators of achievement over and beyond measures such as IQ, SAT, and standardized achievement test scores [1, 2]. A host of other researchers have demonstrated that achievement and success are not necessarily dependent on the usual predictors of intelligence, academic sophistication, or preparedness [1, 2]. These findings have spurred an upsurge of interest in identifying non-cognitive criteria that predict academic and professional achievement. Educators and social psychologists are beginning to look beyond traditional measures of intelligence and are searching for more creative and effective ways to evaluate student potential. For example, participation in sports 260 A Century of Advancing Mathematics or music, leadership, faith-based or community activities, volunteering, or paid work might be better indicators of potential than the ones universities typically use to evaluate students. The research on grit and related topics, such as growth mindset, belonging uncertainty, self-affirmation, and stereotype threat, steadily provide evidence that achievement is more malleable and subject to outside influences than we had thought. People can be encouraged to achieve beyond their own expectations and to overcome feelings of inadequacy, uncertainty, or self-doubt. With persistence and hard work, achievement is not only possible, but likely. Recent rigorous randomized experiments have shown that how students interpret the adversities that accompany academic transitions can determine whether students respond resiliently or whether they disengage. (See [4] for a review.) Changing the ways that students interpret setbacks can lead to new and improved academic trajectories [4]. Finally, numerous experiments have pointed to the fact that the educational environment frequently exerts effects on student achievement through students’ psychology and that interventions have effects on persistence even without changing the objective environment [3, 4]. Trying to predict mathematical or any other success can be humbling. Students have surprised and delighted me with their achievements; I would have regretted passing judgment too soon. The doors of mathematics should be wide open. There are many students who come from less rigorous backgrounds, yet may have been the strongest mathematics student in their school. They feel discouraged and demoralized when they come up short against the better-prepared competition. There should be mechanisms to identify such students early and provide them with the support they need to catch up and take their place in the mathematics curriculum. I recognize that this approach is not for everyone. For me, it has resulted in a rewarding career that never grew tiresome. Students with diverse backgrounds bring energy and fresh insights to the study of mathematics. They are bright, hard-working, and fully capable of the same levels of achievement as students from more elite backgrounds. Rather than discourage the aspiring major we do not deem to be a good fit, let’s give everyone who is interested an opportunityto study mathematics as a major in college.

Bryn Mawr College Mathematics Majors Years Average Number of Majors 1972–76 2.4 1977–81 4.0 1982–86 10.6 1987–91 9.6 1992–96 11.8 1997–2001 19.2 2002–06 29.8 2007–11 28.4 2012–14 35.0

Acknowledgements I would like to express my gratitude to Dr. Angela Duckworth for her generous willingness to discuss her work and for including me in her series Talks for Teachers in 2013–14. In addition, I am extremely grateful to Sarah Patrick for helping me Strength in Numbers: Broadening the View of the Mathematics Major 261 understand this work and write the paragraphs on grit and growth mindset, and to Alyssa Matteucci for her expert and generous editing of this manuscript.

Bibliography [1] A. L. Duckworth, C. Peterson, M. D. Matthews, and D. R. Kelly, Grit: Perseverance and passion for long-term goals, Journal of Personality and Social Psychology 92 no. 6 (2007) 1087–1101. [2] L. Eskreis-Winkler, A. L. Duckworth, E. Shulman, and S. Beal, The grit effect: Predicting retention in the military, the workplace, school and marriage, Frontiers in Personality Science and Individual Differences (2014). [3] D. S. Yaeger and C. S. Dweck, Mindsets that promote resilience: When students believe that personal characteristics can be developed, Educational Psychologist 47 (2014) 1–13. [4] D. S. Yaeger and G. Walton, Social-psychologicalinterventions in education: They’re not magic, Review of Educational Research 81 (2011) 267–301. [5] President’s Council of Advisors on Science and Technology, Report to the President: En- gage to excel: Producing one million additional college graduates with degrees in science, technology, engineering, and mathematics (2012) available at www.whitehouse.gov/ administration/eop/ostp/pcast.

Department of Mathematics, Bryn Mawr College, 101 North Merion Avenue, Bryn Mawr, PA 19010 [email protected]

A History of Undergraduate Research in Mathematics

Joseph A. Gallian University of Minnesota Duluth

Introduction Research in mathematics by undergraduates is now commonplace. Summer undergraduate research programs abound. Many institutions fund undergraduate student research. Senior theses routinelyincludeoriginal results. There are numerous conferences where the focus is on presentations by undergraduates. There are mathematics journals that specialize in publishing papers with undergraduate authors. Research experience is expected for admission to leading graduate programs. The annual Joint Mathematics Meetings (JMM) of the Amer- ican Mathematical Society (AMS) and the Mathematical Association of American (MAA) and the annual summer MAA MathFest are attended by large numbers of undergraduates. These are recent developments. In this article we identify the key events that have led to the current widespread acceptance of the importance of opportunitiesfor undergraduates to engage in research in mathematics.

URP programs

The ﬁrst national effort to promote research by undergraduatesbegan in 1959 when the Na- tional Science Foundation funded Undergraduate Research Participation (URP) programs. In 1961 the NSF promoted the URP program by sponsoring a ﬁve-day conference at Car- leton College at which 75 mathematics professors representing 70 colleges and universities from across the United States gathered “to discuss certain fundamental questions regarding undergraduate research in mathematics [4].” The questions were: 1. Is research desirable at the undergraduate level?

2. Is there a role for research in the basic undergraduate curriculum?

3. What are the aims of undergraduate research?

4. What are the criteria for undergraduate research?

5. Why is it that so few of the colleges and universities have undergraduate programs in mathematical research?

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Following is the content of a letter I received from Jerry Alexanderson in 2011 com- menting on his recollections of the era and the Carleton conference.

From my own experience an introduction of undergraduates to research began seriously in the late 50s. There was lots of opposition to the idea among faculty. They thought that it made sense in the other sciences—test tubes needed to be washed, white rats had to be tended to, ::: there were lots of things that undergraduates could do to help in research. But it was widely viewed as unworkablein mathematics. Undergraduates just didn’t know enough to do research. So it was about that time that I came to Santa Clara (1958) and Abe Hillman ::: was very keen on the idea of undergraduate research and he pulled me into the effort locally. There were some good supporters in the mathematical community: Arnold Ross, R. L. Wilder, former president of both the AMS and MAA, and Ken May at Carleton. But it was a hard idea to sell. The great step forward was the organization of a week-long conference on undergraduate research in mathematics by Seymour Schuster and Ken May, held at Car- leton College in the summer of 1961. That same year NSF started funding “undergraduate research projects in mathematics” and we got a grant for that first year and continued with them, for summers and academic years, for 20 years. But in my view, it all started at Carleton. And the projects certainly did not consist of the mathematical equivalent of test-tube washing. One of my students (now on the faculty at the University of Edinburgh)was the first to break into the AMS Proceedings with a nice paper. Other projects led to publications in the Pacific Journal of Math- ematics, Linear and MultilinearAlgebra, Journal of Mathematical Analysis and Its Applications, Semigroup Forum, Proceedings of the Edinburgh Society, The Amer- ican Mathematical Monthly, and on and on. Of course at some point funding was discontinued at NSF, but then it was reinstated with the introduction of the REUs.

Although the URP program had noticeable success, when it ended—in 1981 when the Reagan administration greatly reduced the NSF education budget—research by undergraduates in mathematics was still uncommon. The attitude of the math community regarding research by undergraduates is well illustrated by the following statement made to the Na- tional Science Board by Lynn Steen, President of the Mathematical Association of Amer- ica, in November 1985 [5].

Research in mathematics is not like research in the laboratory sciences. Whereas undergraduate research can thrive in most chemistry, biology, or physics research laboratories, research in mathematics is so far removed from the undergraduate curriculum that little if any immediate beneﬁt to the undergraduate program ever trick- les down from standard NSF research grants. Publication patterns provide vivid proof: hardly ever does one see papers in mathematics jointly authored with students, either graduate or undergraduate.

For the historical record, I have included in the appendix a complete list of the NSF- funded URPs between 1964–1966 as well as some other URP programs I was able to identify. A History of Undergraduate Research in Mathematics 265

REU programs Although the killing off of the NSF URP program in 1981 was a temporary setback, the first group of REU programs, in 1987, began a new era when the URP program was resur- rected under the NSF Research Division. This came at the recommendation of the National Science Board, the policy-making arm of the NSF. That year eight mathematics sites were funded. The 1987 NSF Division of Mathemati- cal Sciences (DMS) budget for REU sites and NSF grant supplements was $380,000. The annual math REU budget for 1988–1990 was $500,000. In 1991 it rose to $750,000. In 1987 and 1988 REUs ran four to six weeks with four to twelve participants. By 1991 seventeen of the eighteen REU sites ran from eight to ten weeks with six to ten participants. These numbers have been the norm ever since. “Paradigm shift” is a term that is overused but I think it is appropriate to describe the change in attitude that has occurred in the mathematics community regarding undergraduate research since the beginning of the REU program. A dramatic way to see thischange in mindset in the math community regarding research by undergraduates is to compare the expectations stated in the NSF announcement of the REU program in 1987 with the role that undergraduates played at the Joint Mathematics Meetings in 2012. Here is the announcement in 1987 in the Notices of the American Mathematical Society of the first REU program sponsored by the NSF [1]. To clarify the range of activities eligible for support under this program, the DMS has formulated the following examples. Direct involvement of a student in a research project operating in an experimental mode, e.g., generating data or working out examples in order to develop conjectures. Independent study activities where the student is expected to carry out literature searches that indicate the development over time of the area under study, possibly working through the details in seminal papers. Depth and difficulty of the material could be adjusted to meet the student’s background. Today most people who run REUs would consider these activities as the starting point, not the end product of an REU. In contrast to low expectations for the role of research by undergraduates in the 1987 NSF announcement, in 2012 948 undergraduates attended the Joint Mathematics Meetings (15.4% of the registered mathematicians) with 152 giving talks. The undergraduate poster session had 310 entries representing the research of 525 students. (See Table 1.) Here is an except from the REU announcement that appeared in the AMS Notices [2] in 1988 that addressed the problem that the REU program was meant to rectify. The decline in recent years in the number of mathematics doctorates has been a continuing source of concern within the mathematical community. Many believe that the root of the problem lies at the undergraduate level, where studentsare rarely exposed to the excitement of mathematical research when they are in the process of choosing a field of study. In the laboratory sciences, students can participate in research in a variety of ways, and even simply observing the laboratory environment 266 A Century of Advancing Mathematics

can demonstrate how research is conducted in such ﬁelds. But in mathematics, it is more difﬁcult to convey the nature of the research process, and the lack of technical background can be a barrier. A major goal of the program is to involve students who come from institutions where research programs are limited.

Diversity of REU programs One of the noteworthy strengths of REU programs is their great diversity in structure and target audiences. Although most REUs have students do group work in one or two areas under the supervision of a few faculty advisers, some institutionsmatch studentsone on one with a faculty adviser. Examples include Indiana University, the University of Michigan, Kansas State, the University of Tennessee, and a number of schools that had NSF VIGRE grants. Less common is for an REU to be run by a singlefaculty member. This is the model used at the University of Minnesota Duluth by the author, by Anant Godbole at Michigan Technological University and East Tennessee State, and by Charles Johnson at the College of William and Mary. For long-running REUs involving multiple faculty members it is common for the faculty and the program leadership to change over the years. Some REUs are designed for students who are from institutions that offer limited opportunities for research while others are aimed at students from leading PhD-granting institutions who have already done successful research. Some programs—particularly those in locations that have a large pool of students from underrepresented groups and possessing faculty from those groups—have had outstanding success at attracting those students. Other programs have had little success in this regard. There have been three large-scale REUs. Williams College has about twenty participants per year, Iowa State University has had thirteen to twenty-seven students, DIMACS and RutgersUniversity administer four REU programs each year that run concurrently with up to thirty students in total. Since it began in 1990, more than 450 undergraduates have participated in the National Security Agency’s Director’s Summer Program. Beginning with three sites in 2004, the MAA National REU Program (NREUP) has had funding from the NSF and NSA to support faculty and local minority undergraduates to engage in research at their own campuses during the summer. In 2012 twelve NREUP sites were funded, each with a minimum of four students. Since 1997 Worcester Polytechnic Institutehas operated an REU that focuses on industrial mathematics and statistics. The program partners with industries to identify problems of both industrial and mathematical signiﬁcance. Students, faculty and industrial liaisons work together on these problems. By 2012 twelve REUs had run for 20 years or more. They are listed below with the names of the initial directors, years of operation, and when available, the number of participants, PhDs produced, and papers published in refereed journals. Long-running REUs Oregon State University, Robert Robson, 1987–2012, 252 participants, 57 PhDs. Universityof Minnesota Duluth,Joseph Gallian, 1987–2012,191 participants, 106 PhDs, 189 publications, 8 AMS Fellows. A History of Undergraduate Research in Mathematics 267

Rose-Hulman Institute of Technology, Gary Sherman, 1988–2012, 182 participants, 39 PhDs from 1995 on, 23 publications. University of Washington, James Morrow and Charles Curtis, 1988–2012. Mt. Holyoke College, Donal O’Shea, 1988–2012 (missed 2003). Williams College, Frank Morgan, 1989–2012, 500 participants, 20 PhDs, 200 publications (all numbers are estimates). College of William and Mary, Charles Johnson, 1990–2012. NSA Director’s Summer Program, 1990–2012. Indiana University, Daniel Maki, 1991–2012 (missed a few years). Michigan Technological University, Anant Godbole, 1991–1999 moved to East Ten- nessee State University 2000–2012 (missed 2005 and 2009). Hope College, David Carothers, 1991–2012 (missed 1997 and 2007), 118 participants, 30 PhDs, 22 publications. DIMACS Rutgers University, 1992–2012. Louisiana State University, Neal Stofzfus and William Hoffman, 1993–2012. Highlights in Undergraduate Research in the REU Era In this section we document the growth in interest in fostering research in mathematics by undergraduates in the ﬁrst quarter-century of the REU era by providing some of the important events along the way. 1987 The ﬁrst NSF math REU site awards were granted to Harvey Mudd College, the Uni- versity of Colorado, Oklahoma State University, the University of Minnesota Duluth, Oregon State University, the University of Tennessee, Rice University and the Univer- sity of Utah. In addition to REU sites, individuals with standard NSF research grants could request a supplement to support a few undergraduate students [2]. The MAA sponsors an “experimental” contributed paper session for undergraduates at the summer meeting that has four speakers. Thereafter, the session becomes an annual event. Pi Mu Epsilon sponsors a contributed paper session for undergraduates at the MAA summer meeting that attracts thirty speakers. 1988 The MAA forms a subcommittee on Research by Undergraduates. This committee is responsible for many of the advances that appear on this list. NSF funds fourteen REUs; among them are Worcester Polytechnic Institute and Indiana University. 1990 The NSA Director’s Summer Program is established. It provides a twelve-week REU- like program for top-level undergraduates. 268 A Century of Advancing Mathematics

The Council of Undergraduate Research (CUR) establishes a division for mathematics and computer science. The Association for Women in Mathematics Alice Schafer Prize is established. The award was not created to recognize outstanding research but over the years research has become the decisive factor in the selection of the winner. 1991

The MAA committee on Research by Undergraduates sponsors a panel on “Models for Undergraduate Research” at the joint meetings. The ﬁrst MAA poster session is held at the Joint Math Meetings. There were twelve posters. The AMS and the MAA jointly sponsor a three-part special session for research papers by undergraduates that features twenty-two talks representing the work of ﬁfty-four students. 1994

The MAA sponsors the ﬁrst minicourse on undergraduate research at the Joint Meetings. The MAA and CUR sponsor the second poster session on undergrad research with nineteen students participating. The poster session now becomes an annual event. The AMS sponsors a four-part special session for research by undergraduates at the Joint Meetings with thirty-eight talks representing the work of sixty-eight students. 1995

Aparna Higgins gives the ﬁrst Project NExT course on undergraduate research. This course becomes an annual offering and that has thirty or so participants each year. The MAA, AMS and SIAM jointlyestablish the Morgan Prize for Outstanding Research by an Undergraduate. 1996

The Notices of the American Mathematical Society begins identifying talks by undergraduates at the Joint Meetings in the program announcement. There was not a special session for research by undergraduates but six undergraduates spoke at contributed paper sessions at the Joint Meetings. 1997

The MAA sponsors a minicourse on undergraduate research at the Joint Meetings. Fifteen papers are presented in a contributed paper session for faculty on establishing and maintaining undergraduate research programs in mathematics. 1999

The number of students involved in the poster session at the JMM is a record sixty-eight. A History of Undergraduate Research in Mathematics 269

The National Security Agency and the AMS sponsor a three-day conference on summer research programs in mathematics for undergraduates. The proceedings are published by the AMS [6]. The MAA minicourse on getting undergraduates involved in research at the Joint Meet- ings becomes an annual event. Registration varies between 35–50 per year. 2000 One hundred and forty students are involved in the poster session at Washington D.C. In many subsequent years the number is limited by the size of the room as demand exceeds the space available. The poster session annually becomes one of the best attended events at the Joint Meetings. The AMS sponsors a four-part special session on research by undergraduate students with thirty-eight talks representing seventy students at the joint meetings. 2001 The MAA includes in its Mission Statement: “We support research, scholarship, and its exposition at all appropriate levels and venues, including research by undergraduates.” 2002 A special session for research by undergraduates becomes standard at all subsequent annual Joint Mathematics Meetings. 2003 The number of undergraduate posters at the Joint Meetings reaches 200 for the ﬁrst time. 2004 The MAA initiates a national REU with emphasis on providing opportunities for underrepresented groups. With support from the NSA and the NSF, the program continues in subsequent years. 2006 Of 377 undergraduates attending the annual Joint Meetings, forty-four give talks. Michael Dorff from BYU receives $1.28 million from the NSF to provide twelve to ﬁfteen mini-grants to fund faculty each working with two to four undergraduate students during the academic year on research. The MAA sponsors its twentiethannual contributedpaper session at MathFest in Knoxville. Sixty students give talks in six sessions. The NSA and the AMS sponsor a three-day conference on promoting undergraduate research in mathematics. The proceedings are published by the AMS [7]. 2008 The Joint Mathematics Meetings has 527 undergraduates in attendance (11.1% of the mathematicians registered). 270 A Century of Advancing Mathematics

Undergraduates at the Joint Meetings At MathFest Number Talks Posters Number Year Registered Given Presented Registered 1991 - - 12 - 1992 - - - - 1993 71 - - - 1994 153 - 19 - 1995 125 - 13 - 1996 141 6 32 - 1997 109 23 13 - 1998 176 22 36 - 1999 236 23 68 - 2000 275 35 140 96 2001 276 15 148 122 2002 300 38 185 149 2003 377 50 200 123 2004 292 38 - 184 2005 361 51 120 166 2006 377 43 129 158 2007 476 64 140 255 2008 527 62 150 269 2009 650 94 210 247 2010 683 95 241 336 2011 759 137 265 279 2012 948 152 310 235

Table 1. Undergraduate Participation at National Meetings 2012 The NSF funds sixty-four math REUs. The number of undergraduates at the JMM exceeds the number of graduate students in any previous year. The NSA, NSF and the MAA sponsor a three-day conference on trends in undergraduate research in mathematics.

Reasons for growth in undergraduate research Although there are many reasons for the dramatic rise in the number of undergraduates doing research in mathematics, here are the ones I feel are the most signiﬁcant. NSF and NSA funding By a far margin, the generous support from the NSF and the NSA for summer REUs and REU-like programs has been the most important impetus for nearly all other efforts. A History of Undergraduate Research in Mathematics 271

The Council on Undergraduate Research The lobbyingefforts by the Council on Undergraduate Research were largely responsible for the NSF creating the REU programs. MAA minicourses Between 1994 and 2012 enrollment in the sixteen minicourses at the Joint Meetings on how to involve undergraduates in research was approximately 600. Most of these were offered jointlyby Aparna Higgins and the author. MAA poster session The MAA poster session at the Joint Meetings has become a showcase event for undergraduate students to exhibit their work. Project NExT Over 500 new faculty have taken Aparna Higgins’s Project NExT workshop on getting undergraduates involved in research. MAA undergraduate conferences Between 2003 and 2012 the MAA regional conferences, funded by the NSF, featured over 5500 talks by undergraduates and over 17,000 participants. Pi Mu Epsilon For many years Pi Mu Epsilon has sponsored regional conferences and paper sessions at MathFest dedicated to undergraduate research. MAA paper session at MathFest The MAA has sponsored a student paper session at MathFest for twenty-six years. NSA Director’s Summer Program The NSA Director’s Summer Program is one of the largest research programs for undergraduates in existence. Moreover, it is one of the few that providesa research opportunity for students in the summer prior to their entering graduate school. VIGRE programs Several major PhD-granting institutions had REU-like programs as part of their VIGRE program. Pipeline effect REUs have been around long enough that REU alumni are now providing research opportunitiesfor their own students. Deans are demanding it More and more deans are demanding that faculty in all disciplines provide research opportunities to undergraduates. This is even the case at many schools where faculty have very high teaching loads. Not long ago it was commonplace for math departments to be exempt from this pressure but this is becoming less the case. With Lynn Steen’s statement made in 1985 given earlier in this article in mind, I wish to close this essay with my own made at the 2006 AMS-NSA Conference on Promoting Un- dergraduate Research in Mathematics. 272 A Century of Advancing Mathematics

“Under the right circumstances, undergraduates CAN participate in mathematics research.”

Bibliography [1] New program for undergraduates, Notices Amer. Math. Soc. 34 (1987) 297–298. [2] Undergraduate students participate in research, Notices Amer. Math. Soc. 34 (1987) 911–912. [3] A. Jackson, Research Experiences For Undergraduates: NSF program stimulates student’s interest in mathematical research, Notices Amer. Math. Soc. 35 (1988) 686–689. [4] Undergraduate Research in Mathematics: Report of a Conference Held at Carleton College, Northﬁeld, Minnesota June 19–23, 1961, Edited by K.O. May and S. Schuster, Carleton, North- ﬁeld, MN, 1961. [5] L. A. Steen, Restoring scholarship to collegiate mathematics, MAA FOCUS 6:1 (1986) 1–2, 7. [6] Proceedings of the Conference on Summer Undergraduate Mathematics Research Program, Edited by J. A. Gallian, American Mathematical Society, Providence, RI, 2000, available at www.ams.org/employment/REUproceedings.pdf. [7] Proceedings of the Conference on Promoting Undergraduate Research in Mathematics, Edited by J. A. Gallian, American Mathematical Society, Providence, RI, 2007, available at www.ams.org/programs/edu-support/undergrad-research.

APPENDIX I URP Sites NSF URPs 1964--1966 Harvey Mudd College, John Greever University of Santa Clara, A. P. Hillman Wesleyan University, Robert Singleton, James Cronin Florida Presbyterian College, Forrest Dristy Emory University, John Neuberger Illinois Institute of Technology, Lennert Pearson University of Illinois, Urbana-Champaign, Hiram Paley Mount Saint Scholastica College [Kansas], Sr. H. Sullivan University of Kansas, William Scott Bowdoin College, David Christie University of Maryland, College Park, John Brace College of the Holy Cross, Patrick Shanahan Michigan State University, C. P. Wells University of Michigan, Ann Arbor, Nicholas Kazarinoff Jackson State College, Benjamin McLemore University of New Hampshire, Evans Munroe Princeton University, Ross Finney Rutgers University, Hyman Zimmerberg Stevens Institute of Technology, Henry Polowy A History of Undergraduate Research in Mathematics 273

New Mexico State University, John Giever Brooklyn College of SUNY, Meyer Jordan City College New York of SUNY, Gerald Freilich Fordham University, Henry DeBaggis New York University, Albert A. Blank Polytechnic of Brooklyn, Robert N. Dheedene Rosary Hill College, Sr. Marion Beiter SUNY of Buffalo, Frank Olson Syracuse University, Paul Gilbert North Carolina College, Durham, Marjorie Brown Case Institute of Technology, Fred C. Leone Kenyon College, Daniel T. Finkbeiner University of Oklahoma, Gene Levy Reed College, Thomas P. Dennehy University of Oregon, Andrew F. Moursund Bucknell University, Harvey Arnold Lehigh University, Everett Pitcher University of Pennsylvania, George Schweigert University of the South, Stephen Puckette Vanderbilt University, Donald Coleman Washington State University, Donald Bushaw

Other URP programs I was able to track down include: Indiana University, George Springer (late 1960s); Wesleyan University (1961); Clemson University,Bill Hare and John Kenelly, (1967, 1968, 1970); Duke University, Bill Hare and Joe Kitchen (early 1960s); Caltech, Gary Lordon (1971, 1972); Reed College; St. Olaf College, Lynn Steen, Arthur Seebach and Loren Larson; University of Wisconsin at Madison, Don Crowe; Macalester College, Wayne Roberts (1970s);University of MinnesotaDuluth, Joe Gallian (1977, 1979– 1981). Department of Mathematics and Statistics, University of Minnesota Duluth, Duluth, MN 55812 [email protected]

The Calculus Reform Movement A Personal Account

Paul Zorn St. Olaf College

What follows is a limited, personal, and opinionated account, not a comprehensive history, of some aspects of the calculus reform movement of the 1980s and 1990s. The author was himself closely involved, and therefore by no means disinterested. He thanks his friend Deborah Hughes Hallett for useful conversation and ideas, but all opinions and errors here are his.

Early days The era of calculus reform began “officially” in early 1986, when the Tulane Conference on Calculus Instruction, organized by Ronald Douglas, met in New Orleans. The conversation caught on. A follow-up gathering in 1987 attracted over 600 mathematicians and scientists to a colloquium in Washington, DC, titled Calculus for a New Century and sponsored by the National Academy of Sciences and the National Academy of Engineering. It was at this gathering, apparently, that the goal (and memorable metaphor) of making calculus “a pump, not a filter” was first enunciated, by Robert M. White, then President of the National Academy of Engineering. For much more information about these conferences and their direct products, see the MAA Notes volumes Toward a Lean and Lively Calculus and Calculus for a New Century [2, 3]. The importance of calculus teaching and learning and the desire to improve it were not suddenly discovered in the 1980s; they began at least 150 years earlier. In a delightful Mathematics Magazine article [1], “The Lengthening Shadow: The Story of Related Rates,” Bill Austin, Don Barry, and David Berman find roots of calculus reform in 1830s England, beginning with the Rev. William Ritchie, Professor of Natural Philosophy at Lon- don University. (Ritchie’s better-known colleague, Augustus de Morgan, was also a serious calculus reformer in the then-contemporary sense.) A remark by the Rev. Ritchie from the preface to his own 1836 calculus textbook, quoted by Austin, Barry, and Berman, has a modern ring, pre-Victorian diction notwithstanding: The Fluxionary or Differential and Integral Calculus has within these few years become almost entirely a science of symbols and mere algebraic formulae, with scarcely any illustration or practical application. Clothed as it is in a transcendental dress, the ordinary student is afraid to approach it; and ...many [students] ...do

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not appear to derive all the advantages which might be expected from the study of this interesting branch of mathematical science. With computing technology still far in the future, Rev. Ritchie recommended motivating the calculus by linking it to kinematics and other presumably intuitive physical applications, including what we know now as related rates problems. Curiously enough, related rates problems now play at best a minor role in typical calculus course, reformed or not. What particularly spurred the 1980s version of calculus reform? One factor was high failure rates: At many universities the DFW rate for calculus students was around 50%. A corollary result was that too few students continued successfully either to further mathematics or to other areas of study requiring calculus. Another impetus for change was a renewed version of the Rev. Ritchie’s lament: the perception that calculus courses, although ostensibly about beautiful, exciting, and powerful ideas, often boiled down to dull performance of a compendium of symbolic algorithms. Even a student who calculated antiderivatives adeptly and carried out series convergence tests successfully might have little idea what she was doing, or why, and therefore little chance of applying these methods in novel situations. A third driving force for calculus reform was the growing availability of mathematical computing, including both graphing calculators and computer algebra systems (CAS) that could “do” a lot of calculus, and thus arguably undercut the logic of standard courses. The late Herbert Wilf posed this issue memorably in a 1982 Monthly article [7], “The Disk with the College Education.” The eponymous disk was a ﬁve-inch ﬂoppy, and the “education” was the early computer algebra system muMATH. The package was then sold by Microsoft Consumer Products for about $75—not cheap back then, but surely a bargain by college education standards. And the price differential has only increased since 1982: Wolfram Alpha is now free, while college education ...not so much.

Support from the National Science Foundation The calculus reform movement enjoyed generous—and crucial—support from the National Science Foundation. Over the period 1987–95 the NSF invested around $43 million in calculus reform efforts, through various programs in the Divisions of Undergraduate Education (DUE) and Mathematical Sciences (DMS).1 That’s a lot of money. But as William Haver, a DUE program director during those years, observes in [5], amortizing this substantial investment over millions of present and future enrollments in calculus makes the cost per student negligible. Two key features of the NSF’s funding strategy deserve special mention: (i) support for development and testing of curricular materials, including full-fledged textbooks for eventual commercial publication; and (ii) emphasis on scaling up local improvement efforts. See [5] for much more information about the NSF’s role in calculus reform, and the panoply of projects supported over a decade. Suffice it to say here that the calculus reform movement would likely not have had anything like the influence it did without NSF support.

Goals and strategies of calculus reform Calculus reformers aimed, of course, to address the course-speciﬁc problems mentioned above. Reformers also tried to envision generally how the teaching and learning of un- 1The author of this piece received NSF support for textbook developmentefforts. The Calculus Reform Movement: A Personal Account 277 dergraduate mathematics, and the role of calculus courses therein, could be improved. A handful of common ideas and strategies emerged and appeared as themes in many calculus reform projects and reform-driven textbooks that appeared (often with signiﬁcant NSF support) over in the next decade.

The rule of three (or more) If “unreformed” calculus focuses too narrowly on symbolic operations on symbolically presented elementary functions, then a broader view is called for. The objects and processes of calculus—functions, derivatives, integration, summation, etc.—can be viewed from numerical, graphical, and sometimes verbal perspectives. Requir- ing students to negotiate and use all of these forms can help focus attention on mathematical structures and properties that persist across a variety of representations. A function f is more than a formula, for example, and the derived functions f 0 and f represent more than arcane symbolic recipes. Presenting functions graphically or tabularly in examples and R exercises (and exams) became standard operating procedure in reformed approaches. The study of inﬁnite series presents similar challenges in a deeper context. Performing symbolic tests (e.g., the ratio test) may help students label series as convergent or divergent, but not necessarily shed much light on what either label means. With a little computing power, students can both experience some basic numerics, hands on, and ideally develop intuition for what convergence and divergence actually mean.

More modeling and data That students should see in action the power of calculus in modeling and predicting phenomena of change is hardly a “reformed” idea: optimization and related rates problems are old, sturdy, and worthwhile standards. What may be new is applying calculus ideas and methods in real-world contexts, often described by tabulated data. For example, students might explore the Gini coefﬁcient, which measures inequality in resource distribution. Doing so depends on access to real-world data, and links nicely to numerical approximation of integrals.

Less weight on symbolics Time and energy for new calculus course ingredients needs to be found somewhere. This was typically done by deleting or de-emphasizing symbolic manipulation topics deemed dispensable. Among methods of antidifferentiation, for instance, candidates for trimming or outright deletion might include trigonometric substitution, the method of partial fractions, and techniques for handling rational functions of sines and cosines. The standard combinatorial rules for differentiation generally survived the cut, but time formerly spent buildingpaper-and-pencil facility with baroque symbolic combinations was often judged better spent elsewhere.

More (but cautious) use of technology The calculus reform movement coincided in time, as noted above, with new availability and user-friendliness of mathematical computing tools. Graphing calculators, especially, were becoming cheaper and more able, and many reformers advocated using them and other technologies to do calculus operations and illuminateideas. An old standard problem genre had students plot elementary functions, locating and labeling extrema, inflection points, and asymptotes. In their reformed (and more interesting) incarnations, such problems asked students to analyze geometric features that, with technology available, were likely to be evident from the outset. Numerically-focused 278 A Century of Advancing Mathematics exercises might ask students not just to label convergence or divergence of series, but to use technology to find or estimate numerical limits. With access to computer algebra systems (Maple, Mathematica, muMath, and Derive were just starting to appear in those days), students might not only handle more, and more complicated, symbolic expressions, but also use examples to discover symbolic patterns and explore structure in new ways. They might be surprised to find, for instance, that antidifferentiation of rational functions can produce transcendental results, like logarithms and arctangents.

Conceptual understanding (but not necessarily formal rigor) Calculus reformers talked (and still talk) freely and often about promoting conceptual understanding: students’ deeper engagement with and ownership of calculus ideas. Varying modes of representation was a common strategy: reformers believed that seeing and studying things graphically and numerically, as well as symbolically, could deepen students’ encounters with calculus objects. Mathematical computing, including in then-new forms, played an important role in this conceptual refocusing. Computing was employed not only, or even mainly, to speed up or automate routine calculations. More interesting, computing could make new and different problems and modes of representation readily accessible. For example, zooming in on a graph to illustrate local linearity (or non-linearity)is easy with technology, but hard to manage with paper and pencil alone. Plotting software makes this easy, of course, but further possibilities exist.

Example: On beyond slope If f .x/ sin.x/, then f .x/ cos.x/ and f .1/ D 0 D 0 D cos.1/. This is all symbolically innocuous, readily understood in terms of slope, and not obviously of much interest. But there’s more to “see.” The fact that f .1/ 0:54 has several useful and accessi- 0 ble meanings, touching on slope, linear approximation, magnification, rate of change, and more. Graphical zooming is helpful, but so is “numerical zooming”: x 0:95 0:96 0:97 0:98 0:99 1: 1:01 1:02 1:03 1:04 1:05 f .x/ 0:8134 0:8192 0:8249 0:8305 0:8361 0:8415 0:8468 0:8521 0:8573 0:8624 0:8674 A little staring at the second row reveals a lot: (i) gaps between adjacent entries are roughly equal, so f is roughly linear; (ii) comparing first-row gaps (all 0:01) to second-row gaps (all near 0:0054) gives a rate of change near 0:54, as expected; (iii) the input interval Œ:95; 1:05 has length 0:1, while the output interval Œ0:8134; 0:8674 has length 0:054, so f “magnifies” small intervals by a factor around 0:54.

Resequencing calculus Some resorting of standard calculus topics was tried. One approach was to merge single- and multi-variable incarnations of important ideas, such as the derivative and the integral. Another new strategy was to introduce differential equations much earlier than in typical earlier treatments. One reason for doing so was the possibility, with help from technology, of solving ODEs and even PDEs numerically; no longer need one wait to develop a varied kit of sophisticated symbolic tools. Another advantage of early DEs was “linguistic.” DEs offer a powerful “language” for mathematical modeling at its best: using DEs students could describe varying quantities and their rates of variation, and (often with help from technology) draw meaningful conclusions from those models. The Calculus Reform Movement: A Personal Account 279

What happened? Which changes stuck? Which didn’t? How does the calculus landscape look almost thirty years after the Tulane Conference, and twenty years after NSF’s reform funding period? Has a durable “reformation” in calculus teaching and learning taken hold, or have inertia, concerns about dumbing down, uneven access to technology, and other forces of “counter-reformation” won the day? I see generally positive but mixed developments, including some signiﬁcant improvements and some surprising disappointments. A few are reviewed in what follows.

A changing student population Any assessment of calculus reform’s effects, especially at the college level, should reckon with changing “academic demographics.” That more and more students now take calculus of some sort in high school means that fewer highly talented and motivated students take calculus courses, or as many courses, at the college level. While the beneﬁts to students of taking more mathematics earlier seem clear in theory, the situation is complicated in practice. The MAA’s study of Characteristics of Successful Pro- grams in College Calculus (CSPCC)[4] has revealed, for instance, that a distressing number of students who take some calculus in high school need to start all over, often with precalculus, in college. Moreover, students who succeed in relatively challenging and rigorous high school calculus courses, and proceed to more advanced courses in college, will likely have taken the Advanced Placement AB or BC courses, which have (for good and obvious reasons) relatively inﬂexible and nationally determined curricula.

Varied representations This good idea took firm hold, especially as regards emphasis on graphical presentations. Modern calculus textbooks, both reformed and mainstream, are now full of examples and exercises that feature functions presented graphically, with tables of data, or in words. A problem asking students to deduce properties of a function f from a graph of the derivative f 0 was once an edgy thing; now it’s entirely standard. Graph-plotting(on paper, by hand) was a significant part of a Calculus 1 course around 1970; in those days plotting was seen as an attractive application of the derivative. So it was, but nowadays accurately plotted curves are cheap, and can be seen as representing derivatives. We may spare students the hard work of plottingcurves, but there is still plenty to do with analyzing curves carefully. Following is a perhaps surprisingly difficult example from a “reformed” text [6] of that period. Nothing more esoteric than the chain rule is involved. Can you do it?

Example: The chain rule on graphical steroids Functions f and g are the functions deﬁned by the graphs below. Graph of f Graph of g 7 5

–3 –5 –5 0 5 –5 0 5 280 A Century of Advancing Mathematics

Let h.x/ .f g/.x/. D ı (a) Evaluate h. 2/, h.1/, and h.3/. (b) Estimate h . 2/, h .1/, and h .3/. 0 0 0 (c) Is h increasing at x 1? Why or why not? D (d) What values of x correspond to stationary points of h (where h .x/ 0)? 0 D A tech revolution? Hardly. Mathematical technologies have indeed somewhat changed the teaching and learning of calculus, especially in boosting graphical and geometric views of functions, derivatives, and integrals. Graphing calculators and similar technologies are now frequently permitted or even required—but almost as frequently forbidden. Problems and exercises are now routinelyposed in graphical forms, and textbooksare full of pictures. While the picture is mixed, technology has by no means revolutionizedcalculus courses fundamentally. Despite Wilf’s pointer (he called it a “distant early warning”), high-level mathematical packages like Mathematica have had surprisingly—and disappointingly— little effect on calculus teaching and learning. The advent of “CAS for the masses,” in such forms as Sage and Wolfram Alpha, may ﬁnally change our practice, but that remains to be seen. Meanwhile, and despite some notable exceptions, standard calculus courses are only cautiously dipping toes into the water of high-level mathematical computing. The possibilities of exploiting, rather than merely coping with, Mathematica and its relatives remain largely unexplored.

Harder or easier? Reformers’ push for deeper conceptual understanding was neither a back-to-rigorous-basics movement—epsilonics had long since vanished from mainstream calculus courses—nor a move to dumbed-down or “fuzzy” mathematics. The latter charge was sometimes advanced, but usually incorrectly: both faculty and students, for different reasons, usually experienced reformed calculus courses as more, not less, intellectually demanding than what they replaced. Whether today’s typical college calculus course, if there is such a thing, is harder or easier than its counterpart twenty years ago is hard to tell, especially given changes in student populations and preparation. But reform efforts have, on balance, steered calculus students’ efforts in more useful and productive directions than would have been the case without reform.

Openness to new pedagogy Calculus reform was all about rethinking the content and pedagogy of calculus courses, but the effects have spread well beyond calculus courses themselves. Flipped classrooms, projects, group work, use of historical sources, and other innovations were all tried and developed partly as elements of calculus reform, and their effects have spread. Calculus reform was not, as a matter of history, principally or patently focused on these techniques, but calculus reform helped contribute to a new and surviving spirit among mathematics faculty of willingness to experiment with new pedagogies. An analogy to the Chinese revolutionary ideal of “let[ting] a hundred flowers bloom” may be overdrawn in this context—five or six flowers might be closer to the truth. But it’s fair to say that calculus courses no longer constitute an academic monoculture. (It may be worth mentioning here that the Hundred Flowers campaign of 1956 ended soon, and abruptly.) The Calculus Reform Movement: A Personal Account 281

Efforts to “lively up” calculus teaching and learning, often with fundamental help from technology, have spread widely among other undergraduate courses. Courses in ordinary differential equations, for instance, now routinely include topics and viewpoints, such as those associated with dynamical systems, that would make little sense without technology, and draw essentially on qualitative viewpoints. And, not least, these courses appear to be a lot more fun than the one the present author took well back in an earlier millennium.

A bottom line The calculus reform movement has usefully focused attention on calculus teaching in particular and on mathematical pedagogy in general. Valuable as these improvements are, nothinglike nirvana has been attained in calculus classrooms. Much to the contrary, the large-scale CSPCC study has revealed that “Calculus I, as taught in our colleges and universities, is extremely efficient at lowering student confidence, enjoyment of mathematics, and desire to continue in a field that requires further mathematics” [4]. Other elements of the CSPCC findings are less bleak, and point encouragingly to recommended classroom and pedagogy practices that have been shown to produce good results at a broad range of institutions. Has the calculus reform movement, then, repaid its investments? I think so. Problems with calculus teaching and learning, some new and some old, remain unsolved. But useful rethinking, productive strategies, and a measure of openness to change have taken hold, endured, and begun to be institutionalizedin mainstream textbooks,courses, and classroom practice. That’s progress.

Bibliography [1] B. Austin, D. Barry, D. Berman, The Lengthening Shadow: The Story of Related Rates, Mathe- matics Magazine 73 No. 1 (Feb., 2000), pp. 3–12. [2] Toward a Lean and Lively Calculus, MAA Notes #6. Edited by R. G. Douglas. Mathematical Association of America, Washington DC, 1987. [3] Calculus for a New Century: A Pump, not a Filter, MAA Notes #8. Edited by L. A. Steen. Mathematical Association of America, Washington DC, 1987. [4] Characteristics of Successful Programs in College Calculus: A Report of the MAA National Study of Calculus I, MAA Notes. Edited by D. Bressoud, V. Mesa, C. Rasmussen.Mathematical Association of America, Washington DC, to appear. [5] Calculus: Catalyzing a National Community for Reform, Awards 1987–1995. Edited by W. E. Haver. MAA/NSF, 1999. A PDF summary is available at www.maa.org/sites/default/files/pdf/CUPM/pdf/Haver.pdf. [6] A. Ostebee, P. Zorn, Calculus from Graphical, Numerical, and Symbolic Points of View. W. H. Freeman, New York, 2008, p. 176. [7] H. Wilf, The Disk with the College Education, The American Mathematical Monthly, 89, No. 10 (Dec., 1982), pp. 801–808.

Introducing ex

Gilbert Strang Massachusetts Institute of Technology

Introduction The day when ex appears is important in teaching and learning calculus. This is the great new function—but how to present it? The presentation decides whether the chance to connect with key ideas (past and future) is taken or missed. Textbooks offer four main approaches to ex : 1. Use the derivative of xn=nŠ Add those terms to match dy=dx with y: 2. Take the nth power of .1 x=n/ as in compound interest. Let n approach infinity. C 3. The slopeof bx is C times bx: Choose e as the value of b that makes C 1: D 4. Construct x ln y by integration. Invert this function to find y ex: D D We have a favorite and we explain why. The second crucial step is to find ex times eX . The exponential function y ex is one of the great creations of calculus. Algebra is all D we need for x; x2; : : : ; xn. Trigonometry leads us to sin x and cos x. But ex, the last in this short list of all-important functions, cannot come so directly. That is because ex requires us, at one point or another, to take a limit. The most important function of calculus depends on the central idea of the whole subject—perfect for every teacher. Still a very big question remains. How do we approach ex? That limiting step can come in many places, sometimes openly and sometimes hidden. At the end of this note we mention several of these approaches (the reader may know others). My chief purpose is to advocate the choice that seems most direct and straightforward. This choice builds on what we know (the derivative of xn), it goes immediately to the two key properties of ex, and it brings out the central goal of calculus: to connect functions with their rates of change.

n n 1 What we know : The derivative of cx is ncx x X x X Propertywewant: Theproductof e and e is e C Connection we need : The derivative of ex is ex Calculus is about pairs of functions. Function 1 (the distance we travel or the height we climb) is changing. Function 2 (the velocity df=dt or the slope dy=dx) tells the rate of change. From one of those functions, we ﬁnd the other. This is theheart of calculus, and we must not let studentslose sightof it.The relationof Function 1 to Function 2 is learned by examples more than by deﬁnitions, and these great

283 284 A Century of Advancing Mathematics functions are the right ones to remember:

y xn y sin x and y cos x y ex and y ecx D D D D D With ex as our goal, let me suggest that we go straight there. If we hide its best property, students won’t ﬁnd it (and won’t feel it). What makes this function special? The slope of ex is ex Function 1 equals Function 2 y ex solves the differential equation dy=dx y: D D Differential equations are laws of change. The whole purpose of calculus is to understand change. It is wonderful to see the most important differential equation so early, and doubly wonderful to solve it. One more requirement will eliminate solutions like y 2ex and y 8ex (the 2 and 8 D D will appear on both sides of dy=dx y, so the equation still holds). At x 0, e0 will be D D the “zeroth power” of the positive number e. All zeroth powers are 1. So we want y ex D to equal 1 when x 0: D dy y ex is the solution of y that starts from y 1 at x 0: D dx D D D Before that solution, draw what it means to have y dy=dx. The slope at x 0 will D D be dy=dx 1 (since y 1). So the curve starts upward, along the line y 1 x. But as D D D C y increases, its slope increases. So the graph goes up faster (and then faster). “Exponential growth” means that the function and its slope stay proportional. The time you give to that graph is well spent. Once formulas arrive, they tend to take over. The formulas are exactly right, and the graph is only approximately right. But the graph also shows e x 1=ex; rapidly approaching but never touching y 0: D D This introduction ends here, before ex is formally presented. But a wise reader knows that we all pay closer attention when we are convinced that a new person or a new function is important. I hope you will allow me to present ex partly as if to a class, and partly as a suggestion to all of us who teach calculus. At the very end, I will add a small observation that you might like.

Constructing y D ex I will take three steps, in this order: 1. Construct the function y.x/ that has dy=dx y. D 2. Identify the number e 2:718 : : : which is y.1/. D 3. Verify that the function has the property exeX ex X . D C The function will be built from the powers xn. The quickest way isto write down thepower series in equation (1) below. The derivative of each term is the previous term. This is so simple and even wonderful, we must not lose it (even if the derivative of an inﬁnite series makes us pause to think). Introducing ex 285

I like to build that series for ex step by step, partly because the ﬁrst terms are so important for any function. Here we produce 1 x 1 x2 directly from dy=dx y. Start from C C 2 D y 1, which also means dy=dx 1: D D y 1 y 1 x dy y 1 x Start D Update y D C Update D C dy=dx 1 dy=dx 1 dx dy=dx 1 x D D D C After the ﬁrst update, y 1 x has the correct derivative dy=dx 1: But then dy=dx D C D has to change to stay equal to y: And I can’t stop there:

y 1 1 x 1 x 1 x2 cubic C C C 2 equals #%#%#%# dy=dx 1 1 x 1 x 1 x2 cubic C C C 2 1 2 3 The extra 2 x gives the correct x in the slope. Now we need a new term x =6 with this 1 2 4 derivative 2 x : After that comes x divided by 24 : x4 x4 x4 4x3 x3 x3 has derivative : 24 D .4/.3/.2/.1/ D 4Š .4/.3/.2/.1/ D .3/.2/.1/ D 3Š The pattern becomes clear. Each xn is divided by n factorial, which is nŠ .n/.n D 1/ : : : .1/. The derivative of that term xn=n! is the previous term xn 1=.n 1/! because n from the derivative cancels n from the factorial. As long as we don’t stop, this sum of infinitely many terms does achieve dy=dx y: D 1 1 1 y.x/ ex 1 x x2 x3 xn (1) D D C C 2 C 6 C C nŠ C Here is the function we want. Take the derivative of every term and this series appears again. If we substitute x 10 into this series, do the infinitely many terms add to a finite D number e10? Yes. The numbers nŠ grow much faster than 10n. At step n, we multiply the 1 previous term by 10=n. As soon as n > 20 the terms are decreasing by factors below 2 . So the terms 10n=nŠ in this “exponential series” become extremely small as n . Analysis ! 1 shows that the sum of the series (which is y ex) does achieve dy=dx y. D D Note 1 Mathematics has two supremely important infinite series. The exponential series in (1 ) solves dy=dx y (a linear equation). The other is the geometric series, which solves D dy=dx y2 (a nonlinear equation): D 1 y.x/ 1 x x2 x3 equals : D C C C C 1 x The derivative is 1=.1 x/2 y2. At x 0 we again have y 1. But this series runs D D D into a problem at x 1 : the sum y 1 1 1 1 is infinite. The nonlinearity D D C C C C produces a blowup, where ex is safe—because the powers xn are divided by the rapidly growing numbers nŠ. Every term xn=nŠ is the previous term multiplied by x=n: Those multipliers approach zero and the limit step succeeds (theinfiniteseries has a finite sum). This is a great example to meet, long before you learn more about convergence and divergence. 286 A Century of Advancing Mathematics

Note 2 The derivative of dy=dx y is d 2y=dx2 dy=dx. Then this second derivative D D and all derivatives are again equal to y. At the starting point x 0, we have y 1 and D D dy=dx 1 and all derivatives equal to 1. D This gives another way to look at that series for ex. Take n derivatives of xn. First get nxn 1 and then n.n 1/xn 2. Finally the nth derivative is n.n 1/.n 2/ : : : .1/x0, which is n factorial.When we divide by that number, the nth derivative of xn=nŠ is equal to 1. Now look at ex. All its derivatives are still ex; so they also equal 1 at x 0: The series D is matching every derivative (they all equal one) at the starting point x 0. It is a Taylor D series.

Note 3 Set x 1 in the exponential series. This produces the amazing number e1 e: D D 1 1 1 1 e 1 1 2:71828:::: D C C 2 C 6 C 24 C 120 C D The first three terms add to 2:5. The first five terms almost reach 2:71. We never reach 2:72. It is certain that e is not a fraction. It never appears in algebra, but it is the key number for calculus.

Multiplying by adding exponents Is it true that e times e equals e2? Up to now, e and e2 came separately. We substitute x 1 and then x 2 in the inﬁnite series. The wonderful fact is that for every x, the D D series produces the “xth power of the number e:” When x 1, we get e 1 which is 1=e: D 1 1 1 1 1 Set x 1 e 1 1 1 : D D e D C 2 6 C 24 120 C If we multiply that series for 1=e by the series for e, we get 1. We want to multiplyany ex and eX . The rule of adding exponents says that the answer x X 0 is e C . The series must say this too. When x 1 and X 1, this rule produces e 1 1 D D from e times e . Add the exponents .ex/.eX / ex X : D C (2) We only know ex and eX from the inﬁnite series. For this all-important rule, we can mul- x X tiply those series and recognize the answer as the series for e C . Make a start:

1 1 ex 1 x x2 x3 Multiply each term D C C 2 C 6 C 1 1 ex times eX eX 1 X X2 X3 D C C 2 C 6 C

x X 1 2 1 2 Hoping for ex X .e /.e / 1 x X x xX X . (3) C D C C C 2 C C 2 C

1 x X is the right start for ex X . Then comes 1 .x X/2: C C C 2 C 1 1 1 .x X/2 x2 xX X2 matches the “second degree” terms in (3). 2 C D 2 C C 2 Introducing ex 287

The step to third degree takes a little longer, but it also succeeds: 1 1 3 3 1 .x X/3 x3 x2X xX2 X3 matches the next terms in (3). 6 C D 6 C 6 C 6 C 6 For highpowers of x X we need the binomial theorem (or a healthy trust that mathematics C comes out right). When ex multiplies eX ; this produces all the products of .xn=nŠ/ times m x X .X =mŠ/: Now look for that same term inside the series for e C : .x X/n m xnXm .n m/Š xnXm Inside C C is times C which gives : .n m/Š .n m/Š nŠ mŠ nŠ mŠ C Â C Ã Â Ã When n m 2, the number 4Š=2Š2Š gives 6 ways to choose 2 aces out of 4 aces. This D D number 24=.2/.2/ 6 will be the coefﬁcient of x2X2 in .x X/4: All terms are correct, D C but we are not going there—we accept .ex /.eX / ex X as now conﬁrmed. D C Second approach to exeX A different way to see thisrule isbased on dy=dx y. Start D from y 1 at x 0. At the point x, you reach y ex: Now go an additional distance X D x X D D to arrive at e C : Notice that the additional part starts from ex (instead of starting from 1). That starting x X x X x X value e will multiply e in the additional part. So e times e must be the same as e C : This is a “differential equations proof” that the exponents are added.

Third approach to exeX Here is another neat way to use the equation dy=dx y.x/. D For any fixed number s, take the derivative of the function y.x/ y.s x/ using the product rule: d dy dy Œy.x/y.s x/ .x/ y .s x/ y.x/ .s x/ dx D dx C dx y.x/ y.s x/ y.x/ y.s x/ 0: D D Since its derivative is zero, the product y.x/ y.s x/ is a constant. At x 0, this product D equals y.s/ because y.0/ 1. Now set s x X: D D C y.x/ y.s x/ y.s/ becomes y.x/ y.X/ y.x X/: D D C That is the desired rule exeX ex X . This quick proof from [5] uses the most basic of all D C differential equations: If the slope stays at zero, the function stays constant. The rule immediately gives ex times ex: The answer is ex x e2x. If we multiply C D again by ex, we find .ex /3. This is equal to e2x x e3x. We are finding a rule for all C D powers .ex /n .ex /.ex / .ex /: D Multiply exponents .ex /n enx : D This is true for n 1; 2; 3 and then n 1; 2; 3 and eventually all numbers x and n. D D That last sentence about “all numbers” is important. Calculus cannot develop properly without working with all exponents (not just whole numbers or fractions). The infinite series (1) defines ex for every x and we are on our way. The graph shows Function 1 D Function 2 ex exp.x/. D D 288 A Century of Advancing Mathematics

d f

x X x X .e /.e / D e C .ex/n D enx c eln y D y b g h a x −2 −1 0 e 2 The exponentials 2x and bx

We know that 23 8 and 24 16: But what is the meaning of 2 ? One way to get close D D to that number is to replace by 3:14 which is 314=100. As long as we have a fraction in the exponent, we can live without calculus:

Fractional power 2314=100 314th power of the 100th root 21=100: D But this is only “close” to 2 . And in calculus, we will want the exact slope of the curve y 2x . The good way is to connect 2x with ex, whose slope we know (it is ex again). So D we need to connect 2 with e. The key number is the logarithmof 2. This is written “ln 2”and it isthe powerof e that produces 2. It is specially marked on the graph of ex:

Natural logarithm of 2 eln 2 2: D This number ln 2 is about 7=10. A calculator knows it with much higher accuracy. In the graph of y ex, the number ln 2 on the x-axis produces y 2 on the y-axis. D D This is an example where we want the output y 2 and we ask for the input x D D ln 2: That is the opposite of knowing x and asking for y. “The logarithm x ln y is D the inverse of the exponential y ex.” This idea is explained in two video lectures on D ocw.mit.edu—inverse functions are not always simple. When we have the number ln 2, meeting the requirement 2 eln 2, we can take the xth D power of both sides:

Powers of 2 from powers of e 2x ex ln 2: D All powers of e are deﬁned by the inﬁnite series. The new function 2x also grows exponentially, but not as fast as ex (because 2 is smaller than e). Probably y 2x could have the D same graph as ex, if I stretched out the x-axis. That stretching multiplies the slope by the constant factor ln 2. Here is the algebra:

d d Slope of y 2x 2x ex ln 2 .ln 2/ ex ln 2 .ln 2/2x: D dx D dx D D Introducing ex 289

For any positive number b, the same approach leads to the function y bx : First, ﬁnd D the natural logarithm ln b: This is the number (positive or negative) so that b eln b: Then D take the xth power of both sides: d Connect b to e b elnb and bx ex ln b and bx .ln b/bx : D D dx D When b is e (the perfect choice), ln b ln e 1. When b is en, then ln b ln en n. The D D D D logarithm is the exponent. Thanks to the series that deﬁnes ex for every x, that exponent can be any number at all. Allow me to mention Euler’s great formula eix cos x i sin x. The exponent ix has D C become an imaginarynumber. (You know that i 2 1.) If we faithfullyuse cos x i sin x D C at 90 and 180 (where x =2 and x ), we arrive at these wonderful facts: ı ı D D Imaginary exponents ei=2 i and ei 1: D D Those equations are not imaginary, they come from the great series for ex.

Continuous compounding of interest There is a different and important way to reach e and ex (not by an inﬁnite series). We solve the key equation dy=dx y in small steps. As these steps x approach zero (a limit is D always involved!) the small-step solution Y becomes the exact y ex. D I can explain this idea in two different languages. Each step multiplies Y by 1 x : C 1. Compound interest. After each step x, the “interest” Yx is added to the “principal” Y . Then the next step begins with a larger amount .1 x/Y . C 2. Finite differences. The continuous dy=dx is replaced by small steps Y=x: dy Y.x x/ Y.x/ y changes to C Y.x/ still with Y.0/ 1: dx D x D D Let me compute compound interest when 1 year is divided into 12 months. The interest rate is 100% and you start with Y.0/ $1. If interest is added once, at the end of the year, D then Y.1/ $2: D 1 If interest is added every month, you now get 12 of 100% each time (12 times). So Y is multiplied each month by 1 1 . (The bank adds 1 for every 1 you have.) Do this 12 C 12 12 times and the ﬁnal value $2 is improved to $2:61: 1 12 After 12 months Y.1/ 1 $2:61: D C 12 D Â Ã Now add interest every day. Y.0/ $1 is multiplied 365 times by 1 1 : D C 365 1 365 After 365 days Y.1/ 1 $2:71 (close to e): D C 365 D Â Ã Very few banks use minutes, and nobody divides the year into N 31;536;000 seconds. D It would add less than a penny to $2.71. But many banks are willing to use continuous compounding,the limit as N : After one year you have $e: ! 1 1 N Another limit gives e 1 e 2:718 : : : as N : C N ! D ! 1 Â Ã 290 A Century of Advancing Mathematics

This is the same number e as 1 1 1 1 from the approach that I prefer. C C 2 C 6 C To match this continuous compounding with ex; invest at the 100% rate for x years. Now each of the N steps is x=N years. Again the bank multiplies at every step by 1 x : The C N 1 keeps what you have, the x=N adds the interest in that step. After N steps you are close to ex : x N Second construction of ex 1 ex as N : C N ! ! 1 Á Comment. I would allow this second approach into my classroom, since everything about ex is so important. But I wouldn’t prove that it gives the same ex as the equation dy=dx y: Of course this is quite reasonable, since the derivative of .1 x /N is .1 D C N C x /N 1 . And equally reasonable to expect the difference equation Y=x Y to stay N D close to dy=dx y. D N Hairer and Wanner [4] have compared the product Y 1 1 to the partial sum D C N S 1 1 1 1=N Š of the series. Quite a difference: D C C 2 C C N 1 Y 2:000 S 2 D D D 2 2:250 2:5 3 2:370 2:67 4 2:441 2:708 5 2:488 2:7166 6 2:522 2:71805 7 2:546 2:718253 8 2:566 2:7182787 9 2:581 2:71828152 10 2:594 2:718281801 11 2:581 2:7182818261 12 2:594 2:71828182828

One column shows the slow convergence of the discrete Y=x Y to the continuous D dy=dx y. The error y Y is of order x 1=N . (This “Euler method” is still chosen D D for difﬁcult problems, but here it is not at all impressive.) The errors of order 1=N Š in the second column look more like a modern “spectral method.” Euler himself had seen this contrast before 1748, the date of his great textbook [2]. Johann Bernoulli connected logarithms to exponential series in 1697 [1]. And by 1751, Euler could resolve a hot debate between Bernoulli and Leibniz about the logarithm of a negative number [3]. The key was his wonderful formula eix cos x i sin x: D C Third construction of ex Authors frequently produce ex by starting with 2x and 3x : Those curves have slopes proportional to 2x and 3x: The slope of any function bx is proportional to that function:

bx h bx bh 1 slope of bx limit of C bx times limit of bx times C: D h D h ! D Introducing ex 291

That limiting number C is smaller than 1 for b 2. It is larger than 1 for b 3. Some- D D where between 2 and 3, there must be a number for which C 1. This reasoning produces D a number e for which the slope of ex is ex. It is not right to criticize this approach on mathematical grounds. Pedagogically, I don’t see how a student can buildon it. To me, the steps from 1 to 1 x to 1 x 1 x2 are going C C C 2 somewhere. We are seeing central ideas of calculus, the tangent line y 1 x that gives D C linear approximation and the tangent parabola that gives quadratic approximation.The motivation is clear and the correctness can be seen term by term, by using (and reinforcing) the derivative of xn: An inﬁniteseries is still a big jump. But it is good to show studentswhere we are going, by an example that we really need and use.

The equation dy=dx D ay The “use” of calculus is to understand change. The first step is from y to dy=dx (Function 1 to Function 2). The next step reaches d 2y=dx2 and its meaning and importance (this second derivative can be Function 3). There is one more absolutely crucial step, to connect those functionsby equations like dy=dx y and d 2y=dx2 y. These are fundamental D D equations of nature and why wouldn’t we solve them? Yes, nonlinear problems can wait for that future course on differential equations. But the essential points are clearest for three linear equations with constant coefficients: dy dy dy y ay ay s: dx D dx D dx D C The solution to the first also solves the second, after a scale change on the x-axis : dy Change the interest rate to a ay is solved by y.x/ eax: dx D D The series for eax is 1 ax 1 .ax/2 so its derivative begins with a: C C 2 C d .eax/ a a2x a.1 ax / aeax: dx D C C D C C D The derivative of eax contains the extra factor a. Thus y eax solves dy=dx ay. D D This soon becomes a key example of the chain rule. And dy=dx ay s has a D C constant solution s=a to add to the exponentials Ceax. Fourth construction by inverse functions Instead of constructing y ex, we could start with the inverse function x ln y. Either D D way willyield all pairs .x; y/, and the natural logarithmneeds only an ordinary integration: dy dx 1 1 Invert y to and then x dy: dx D dy D y D y Z Starting that integration at y 1 gives the correct value x ln 1 0. After inversion this D D D y is y e0 1. And introducing t as a dummy variable leaves x ln y 1 dt. D D D D 1 t R 292 A Century of Advancing Mathematics

The “limiting step” that ex always needs is now in the definition of the integral. The key property exeX ex X becomes ln.yY / ln y ln Y . This is proved directly from D C D C the integral. This fourth approach has its attractions. But look for the ideas that need to be understood first : 1. The meaning of an inverse function 2. The definition of an integral 3. The chain rule for x f 1.y/ that gave .dx=dy/.dy=dx/ 1. D D Maybe there is a way to escape that chain rule, but not the others. So ex would have to come long after the derivative of xn. “Early Transcendentals” will be impossible this way, and the ideas themselves seem much more subtle. Explicit constructions are my choices as winners. You can say, “Here is the function.”

A small observation The construction of ex is further developed in the textbooks [6] and [7]. Here is a question that occurred to me after completing those books. I am happy that this paper gives me a chance to ask it: For a given x, which is the largest term in the series for ex? For x < 1, certainly the constant term 1 is the largest. At x 1, that constant is overtaken D by the linear term x. All other terms xn=nŠ will be smaller until x reaches the second crossover point. That second point is reached when x2=2 catches up to x at x 2. This is D typical—the crossovers are at the integers.

xn 1 xn equals when x n: .n 1/Š nŠ D Thus the largest term is the one with n Œx, the greatest integer x. D Ä The largest terms in en are nn=nŠ and nn 1=.n 1/! This will remind you of Stirling’s formula nŠ p2n nn=en. The p2n comes because other terms in the series—not only the two dominant terms—contribute to en. May I emphasize: I am certainly not advocating an ambitious study of inﬁnite series in general! Much much better to study the exponential series in particular.

Bibliography [1] Joh. Bernoulli, Principia calculi exponentialium, Opera 1 (1697) 179–187. [2] L. Euler, Introductio in analysin infinitorum, Opera 8 (1748). [3] ———, De la contreverse entre Mrs. Leibniz and Bernoulli sur les logarithmes des nombres négatifs et imaginaires, Mém. Acad. Sc. Berlin 5 (1751) 139–179; Opera 17 195–232. (Note that Mrs.DMessieurs) [4] E. Hairer and G. Wanner, Analysis by Its History, Springer (2008). [5] P. Lax, S. Burstein, and A. Lax, Calculus with Applications and Computing, Volume 1, Springer (1976). Introducing ex 293

[6] G. Strang, Calculus (2nd edition), Wellesley-Cambridge Press (2010). [7] ———, Differential Equations and Linear Algebra, Wellesley-Cambridge Press (2014).

Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA 02139 [email protected]

Part IV Computational Developments

Computational Experiences in the Pre-Electronic Days

Philip J. Davis Brown University

The events recorded here took place almost seventy years ago. This article is therefore what Benjamin Disraeli, novelist and twice Prime Minister of England, has called an instance of one’s “anecdotage.” My first knowledge of the details of scientific computationcame from a book discarded by the MIT Library and brought home by my elder brother, then an undergraduate at MIT [1]. This book derived from the computation laboratory of the University of Edinburgh run by Sir Edmund Whittaker (1873–1956). It is interesting and amusing to read how the individual computer’s desks were outfitted. (Incidentally, in those years a computer was not an instrument but a person and a computation laboratory was a rarity on university campuses.)

Each desk contains a locker in which computing paper can be kept without being folded. Each desk is supplied witha copy of Barlow’s tables (which give the square, square root , cube and cube root,and the reciprocal of all the numbers up to 10,000) and with tables giving the values of trigonometric functions and logarithms. These may, of course be supplemented by a slide rule or any of the various calculating machines now in use.

In the pre-electronic days, then, scientiﬁc computations was carried out by a variety of means. Pencil and paper employing the rules of arithmetic taught in elementary school. There were slide rules of the ten-inch, twenty-inch, and circular varieties. There were special purpose slide rules adapted to special technologies. There were electro-mechanical computing machines such as the Marchant or the Friden. There were mathematical tables of logarithms, exponentials, and special functions such as the Bessel functions. (One of the very ﬁrst jobs that the electric relay computers carried out circa 1944 was to compute tables of the Bessel and related functions.) There were sets of French curves used for interpolation, approximation or smoothing. (Smoothing was often called fairing.) Very large French curves were employed in both auto and ship design. There were nomograms galore [2, 3]. There were planimeters; simple ones that were used to get areas and more complicated ones that would also yield moments. Computational mechanisms and devices have a very long history [4].

297 298 A Century of Advancing Mathematics

In spring 1944, a number of young mathematics or physics majors and I were recommended for and took jobs at the laboratories of the NACA (now NASA) at Langley Field, Hampton, Virginia. This was during World War II. Some months later, we were inducted into the US Air Force, placed on reserve status and given the equivalent GS rank of second lieutenant. I was employed in the Aircraft Loads Division of the NACA. My immediate boss was Henry Pearson, the head of the Division was Richard Rhode. They were “old- timers” at the NACA and aerodynamic experts, both theoretical and experimental. I will now mention three particular jobs I was put on and how I was instructed to carry them out. Of course, all the above mentioned computational devices were at my disposal. Job 1 Experimental airplanes were instrumented up to determine the in-flight wing pres- sures along the cross-section of the wing at perhaps a dozen equally spaced positions. This discrete data was recorded and then faired using French curves. Its area was then determined with a planimeter to obtain lift. This was the standard practice. It occurred to me that well-known numerical quadratures would do the job just as accurately, but in the middle of a war I didn’t think I had the option of suggesting as much to my superiors. Nonetheless, this was the seed from which, some years later when both Philip Rabinowitz and I were working at the National Bureau of Standards in Washington, DC, my book with Philip grew [5]. The planimeters were beautiful instruments of shiny metal and kept in velvet-lined dust-proof cases. They were of German make. In great demand by various sections of the NACA, and with importation blocked by World War II, the Instrument Shop at the NACA undertook to make a number of them on their own, at the cost, it was rumored, of $5,000 each. Job 2 Determine the theoretical pressure distribution over a given airfoil profile assuming plane potential flow. This had been worked out in an early NACA report [6]. If I recall correctly, their algorithm was numericized by harmonic analysis. There were stencils (Sch- ablonen) available for this purpose, made in the early 1900s by the German mathematician Carl Runge [7] which we copied with blueprints. To take advantage of the symmetries in- herent in the relevant sines and cosines, the stencils were of highly factorable numbers of points such as eighteen, twenty-four or thirty-six. This process is now called the FFT or DFT (the Fast or Discrete Fourier Transform). It took several hours to fill out a twenty- four point stencil and today this computation is performed in microseconds or even more rapidly. A detailed description of these stencils can be found on pp. 234–247 of [4]. Job 3 Determine theoretically the dynamic load on the vertical tail of a flying boat due to an in-flight rudder deflection and compare with measured experimental results. Deter- mine theoretically the tail loads due to abrupt and to sinusoidal rudder deflections. This work, carried out with my NACA colleague John Boshar, led to my first published paper [8]. As might be inferred from the title, a tremendous amount of simplification was necessary. As we wrote, “The amount of labor involved in effecting the computation has made impracticable the application of the complete theory to the tail loads problem.” The simplified theory led us to a second-order linear (ordinary) differential equation with constant coefficients and with a given right-hand forcing function recorded graphically. The coefficients were combinations of aerodynamic parameters, many obtained from wind tunnel data. I do not now remember how the solution was computed and the paper Computational Experiences in the Pre-Electronic Days 299 gives no indication of this. Apparently, numerical algorithms were taken for granted while the emphasis was placed on the aerodynamic details. It should be remembered that the strategies of numerical computation became a subject in their own right and were chris- tened “numerical analysis” and “computer science” by George Forsythe only in the early 1950s. Computations were often done by women (who were thought to be more accurate than men). My wife, Hadassah F. Davis, was so employed in the structural divisionof the NACA where she worked under the supervision of Bernard Budiansky. Later, in 1950, Budiansky took a PhD in Applied Mathematics at Brown University and became a distinguished professor at Harvard. Sixty-five years have now elapsed since the events recorded. The computations that requiredhours oreven days can now be done ina flash withthe scientific software available. Simplified theories can now often be replaced by “full” theories. Yet, lest my readers be too proud of today’s “cutting-edge” facilities, note that the first supersonic plane, the Bell X-1, was designed in 1945 using the computational resources described here. I doubt, though, whether we could have gotten to the Moon using a slide rule and an adding machine.

Bibliography [1] David Gibb, A Course in Interpolation and Numerical Integration for the Mathematical Labora- tory, (Edinburgh Mathematical Tracts, No. 2) G. Bell and Sons, London, 1915. [2] Maurice d’Ocagne, Trait´e de nomographie: Th´eorie des abaques. Applications prat´ıques, Gauthier-Villars, Paris, 1899. [3] Maurice Kraitchik, Alignment Charts, Construction and Use, D. Van Nostrand, New York, 1946. [4] E. M. Horsburgh, Handbook of the Napier Tercentenary Celebration or Modern Instruments and Methods of Calculation, G. Bell and Sons, London,1914. New Edition and Introduction by M.R. Williams, E. Tomash, MIT Press, 1984. [5] Philip J. Davis and Philip Rabinowitz, Numerical Integration, Blaisdell Publishers, 1967. Elabo- rated as Methods of Numerical Integration, Academic Press, 1975. [6] T. Theodorsen and I. E. Garrick, General potential theory of arbitrary wing sections, NACA Report 452, 1934. [7] Paul Terebesi, Rechenschlablonen fur harmonische analyse und synthese nach C. Runge, Verlag Julius Springer, Berlin, 1930. [8] John Boshar and Philip J. Davis, Consideration of Dynamic Loads on the Vertical Tail by the Theory of Flat Yawing Maneuvers, NACA Report 838, 1946.

Division of Applied Mathematics, Brown University, Providence, RI 02912 [email protected]

A Century of Visualization One Geometer’s View

Thomas F. Banchoff Brown University

Here is a collection of stories about the dramatic changes in the role of visualization in mathematics, particularly in geometry and topology, during the first hundred years of the MAA. I have lived three-quarters of that century and I have been teaching in colleges and universities for the past fifty years. I myself am a very visual person and I have met a number of other mathematicians with that proclivity over the years. Of course I have met a number of mathematicians who don’t see a need to accompany arguments with picture or models, and some who completely disdain “visual crutches” as being signs of weakness or even distractions from formal mathematical activity. In some cases, a visual approach is the only available one and progress in that area can prepare the way for a later formal argument. In other cases a formal argument proves something that is so counterintuitive and so much against common expectation that mathematicians seek for a number of ways to visualize what is going on. Whatever preferences one has, the facts are very clear—things have changed with respect to the role of visualization in mathematics teaching and research and they will continue to do so. The first story starts slightlybefore the foundingof the MAA, in 1901, with the discovery of the first immersion of the real projective plane into three-space by Werner Boy in Göttingen [5]. The folk tale is that his thesis advisor, David Hilbert,had encouraged him to show that it was not possible to find an immersion of that object, i.e., a locally one-to-one mapping into three-space, and that he was surprised when his student actually accomplished this seemingly impossible task. He did it without formulas, describing his construction in pictorial terms in a way that was totally convincing, using techniques of slicing by a family of horizontal planes to obtain a one-parameter family of curves that generated a surface in space. The surface was self-intersecting so some of the curves had double points, and one curve had a triple point. There were exactly three positions where the curve had a special critical point,the maximum and minimum positions where the slice was a single point and one other position where the slice had a “saddle point” like a mountain pass. This third “critical point” was unusual. In the case of an embedding with no self-intersections, the slices not containing critical points are themselves embeddings, and it is a theorem that the number of level curves changes by one or minus one as we pass a saddle point. In the case of the real projective plane, Werner Boy found a sequence where the level curves were immersed, not embedded, and there was only one curve before the saddle point and only one

301 302 A Century of Advancing Mathematics curve afterward. This gave a function on the surface with one maximum, one minimum, and one saddle point. Now here is aremarkable fact: Werner Boyused what is now knownas Morse Theory to prove that he had actually constructed an immersion of the real projective plane because he assumed that his readers knew that a surface is completely characterized by its Euler char-

Figure 1. Illustrations from Werner Boy’s 1901 doctoral thesis. A Century of Visualization: One Geometer’s View 303 acteristic and whether or not it contains any Möbius band. The real projective plane is the only connected surface with Euler characteristic one. Furthermore the Euler characteristic could be computed by calculating the number of maxima of a function plus the number of minima minus the number of (ordinary) saddle points. Werner Boy had constructed a surface with a height function that had one maximum, one minimum, and one ordinary saddle so it had Euler characteristic one. That was enough. He got his PhD, taught for a while in a secondary school, and disappeared from history’s view. His construction was modeled in plaster, where the triple point is clear. He described his construction in various ways, one of which has three-fold symmetry about an axis in three-space. Slices perpendicular to that axis produce one minimum, three maxima, and three saddle points. In the nineteenth century, mathematicians already knew two different ways of constructing representations of the real projective plane in three-space but both of them involved “pinch points” where the mapping was not locally one-to-one. The “cross-cap” had two pinch points, one at the top and one in the center. Steiner’s Roman surface, named because he was in Rome when he discovered it, has six pinch points and one triple point which is the image of three different points on the abstract surface; the cross-cap has two pinch points and no triple points. Werner Boy’s immersion had no pinch points and, as previously noted, one triple point. In the 1960s a very different problem began to focus attention back on Boy’s surface: Stephen Smale’s surprising proof of the fact that it was possible to “turn a sphere inside out” [16]. To understand what this means, mathematicians invariably would cite an earlier theorem of Hassler Whitney and Caspar Graustein which showed that the analogous result for a circle is impossible. Any closed curve in the plane with a well-defined unit tangent vector at each point has a well-defined unit normal vector obtained by rotating the unit tangent by a quarter-turn in the counterclockwise direction. For a circle traversed in a counterclockwise manner, the normal vector points toward the inside and if all the normal vectors are collected at the origin, they trace out the circle once in a counterclockwise way. If we trace out the circle in the clockwise direction, then the normal vectors point outward, and they trace out the unit circle in a clockwise direction. There is no way to deform the clockwise circle to the counterclockwise circle in a continuous way without introducing singularities, like cusps, so we say that a circle cannot be “turned inside out” in a singularity-free way. But, Smale asserted, exactly this is possible in the case of a sphere. One can deform a sphere with unit normals pointing outward to a sphere with unit normals pointing inward without introducing any singularities whatsoever. At each stage of the deformation, there is a well-defined tangent plane at every point and those tangent planes move continuously as the sphere “turned inside out.” The trouble was that the proof was indirect and reportedly Smale’s thesis advisor Raoul Bott doubted that the result could be true. Many other mathematicians were suspicious, and they wanted to see how this could be done. Then Arnold Shapiro came up with a way to exhibit the required deformation. If a sphere could be deformed into a double covering of Boy’s surface, then it would be possible to continue deforming it back the other way to the sphere turned inside out. (A description of Shapiro’s eversion of the sphere was developed by Bernard Morin and George Francis, the master of geometric and topological drawings, in The Mathematical Intelligencer in 1980 [8]). But how could this deformation to Boy’s 304 A Century of Advancing Mathematics immersion of the real projective plane be exhibited? The person who carried out this task was a graduate student at Princeton, Anthony Phillips, and his drawings made the cover of the Scientific American in 1966 [15]. It was a triumph of visualization in mathematics. Note that this explicit exhibition of a so-called sphere eversion was constructed without the aid of computers since there was no algebraic description of Boy’s surface. Fortunately an alternative description of a sphere eversion appeared that turned out to be easier to understand, described by Bernard Morin [14], where at each stage of the

Figure 2. Cover illustration of sphere eversion article by Anthony Phillips. A Century of Visualization: One Geometer’s View 305 deformation the only double points were situated on curves, rather that the double covering of Boy’ssurface where all pointsof thesurface were doublepoints.The central figure inthis sphere eversion had one curve of double points that went through a particular “quadruple point” four times. Once again this central stage in the eversion was not represented by an algebraic equation so it was difficult to see how to describe the deformation explicitly. In particular, there was not a direct way to make a computer graphics film of the eversion. Fortunately, Nelson Max, at Case Western Reserve University was able to create just such a film. He started out by creating films to illustratea famous theorem of Hassler Whit- ney, which asserted that one curve in the plane could be deformed explicitly into another curve if and onlyif the“normal degrees” of the curve were the same, so that theunitnormal vectors collected at the origin went around the unit circle the same number of times, and in the same direction. In the meantime, Charles Pugh, a young professor at the University of California, Berkeley, had constructed physical models out of chicken wire showing the various stages in the deformation from a sphere to the central surface with the single quadruple point and these were hung from the ceiling in the common room of the Mathematics Department. Nelson Max was able to measure the coordinates of each of the points of these models and thereby construct a computer graphics representation of the various stages and make films of the transitions between the stages. Supported by a substantial grant from the National Science Foundation and using the resources of a number of computer facilities throughout the US, he was able to produce a spectacular film, Turning a Sphere Inside Out [13]. It was fortunate that he had preserved images of all the intermediate chicken-wire models since inexplicably someone stole the originals! Using pictures from the film, Banchoff and Max [2] showed that any general sphere eversion must have an odd number of quadruple points, and John Hughes [10] then produced a formal proof of the theorem using techniques from algebraic topology. The story does not end here. In 1978 Bernard Morin produced equations for the surfaces in Nelson Max’s film [13]. Later in the 1980s François Apéry [1] in France and John Hughes [11] independently found algebraic equations for Boy’s surface. George Francis produced remarkable drawings of a number of sphere eversions in his A Topological Pic- turebook [7]. Robert Bryant was able to describe the central surface of the Morin eversion in terms of algebraic equations for surfaces that originated in four-dimensional space [6]. Two other films of sphere eversions followed, one by William Thurston and colleagues at the Geometry Center called Outside In [12] and another by John Sullivan and colleagues at the Technical University of Berlin called The Optiverse, utilizing relaxation methods from Kenneth Brakke’s Surface Evolver [9]. All in all, visualization had come of age in response to the challenge of these two problems, finding and describing the immersion of the real projective plane and creating an explicit deformation turning a sphere inside out.

Computer animation and interactive graphics The second story starts fifty years ago at the beginning of my interaction with computer graphics from the point of view of the technology involved in creating visual materials for research and teaching. During the last hundred years communication and visualization technology has made incredible progress. In 1915, radio was still in its infancy and television (black and white) 306 A Century of Advancing Mathematics was only gaining popularity thirty years later. Computers existed during World War II, but they were at least the size of a large auditorium and no one at that time could imagine that their use would become widespread. (In 1953, Thomas J. Watson Jr. estimated that there would be twenty potential users for their new IBM machine, one of the few predictions that actually came true.) About this time the invention of transistors changed the scale, and subsequently microchip and integrated circuit technology changed the scale once more. While computer technology was changing, so was the delivery system for audio and video. Vinyl records in three different sizes gave way to audiotapes in monophonic and stereophonic sound, followed by compact disc technology. Photography progressed from black and white snapshots to sophisticated cameras for color prints and Polaroid technology, ultimately to digital cameras, now available on cell phones. When I came to Brown University in 1967, one of the first people I met was Charles Strauss, the first PhD student in computer graphics at Brown. I was told he had developed a three-dimensional blackboard and that is something I had always wanted. His doctoral thesis involved providing computer graphics for the design and maintenance of a large piping network in a factory, but it was expensive to operate—it would do the work of eleven engineers but the costs were more than the salaries of eleven engineers. Charles had great techniques and he needed new problems. I arrived with great problems involving

Figure 3. Charles Strauss and Thomas Banchoff in 1980. A Century of Visualization: One Geometer’s View 307 visualization of transformations of surfaces in spaces of three and four dimensions and I needed new techniques to visualize them. It was a perfect collaboration opportunityand we worked together for the next twelve years, producing a sequence of animations that were very well received by mathematical audiences. At the beginning, things were slow. It took a minute to produce an image on a mono- chrome cathode ray tube of a wire-frame surface with a few dozen vertices and segments joiningpairs of vertices. Charles programmed using Fortran or Cobol or machine language, whatever was convenient, producing stacks of IBM cards that were fed pneumatically into a machine that would compile the program overnight. The single images were exciting, but we wanted to see the transformations in motion, and that required animation techniques. At first we had to stay up all night in a dark room with an animation camera pointed to the computer screen, waiting a minute or more for the machine to produce a picture. We would then take two shots of the picture, go on to the next frame in the programmed sequence, and repeat the process. After a night’s work we had a small reel of 16-mm film, enough for a short film to be displayed at thirtyframes per second. We sent it on a bus to a studio in Boston and the next day we got back about a half-minute of developed film. The modern language department had a 16-mm projector where we could view our work and plan the next step. We started with an ambitious project, to film a torus in the three-dimensional sphere projected stereographically into three-space as it rotated in four-space. The individual frames were already surprising and impressive, but the film sequences themselves were spectacular. I decided on the spot that I never wanted to do geometry any other way. Within a month we had a film that repeated the basic sequence of transformations and then played some additional scenes to rotate objects in three-dimensional space. We exhibited all of the surfaces in three-space that had a certain curvature property and I proudly showed the film as the climax when I gave a research talk. We never should have been so successful with that first film. It took us another five years to come up with film sequences that were as good, but by that time we had things down pretty well. We produced films of graphs in real four-space of complex squaring and cubing functions as well as the complex exponential function. We were delighted that having the graph of a function automatically gave us the graph of the inverse function—we just had to turn the graph around in four-dimensional space before we projected it into three-space. We also were able to exhibit the Veronese surface, one of the basic objects of study in my topology courses. Our films showed unexpected projections, for example a multiply covered equilateral triangle. (It took a while to see why this should have been predicted by Kostant’s Convexity Theorem.) When my PhD advisor Shiing-Shen Chern saw the films in 1976 he suggested that it would be a good time to have them exhibited at the International Congress of Mathemati- cians in Helsinki in 1978. I was invited to show them to Armand Borel at the Institute for Advanced Study in Princeton and several dozen people came to a showing the next day. As a result of the positive reception of the films, I was invited to give a 45-minute talk in the Section on History and Pedagogy at the ICM [4]. The auditorium was full, and I was asked to give a second presentation of the films three days later. Part of the reason for the success of the presentation was that many of the participants in the audience were very familiar with classical engineering drawing (Darstellende Geometrie) and they appreciated 308 A Century of Advancing Mathematics the power of the graphics computer to produce these visualizations. The films also included Cusps of Gauss Mappings, and Focal Surfaces of an Ellipsoid with Unequal Axes. By this time, ten years after Charles and I had started collaborating, we had automated the process so that we did not have to advance the film ourselves. In fact our film The Hypercube: Projections and Slicing used four colors, obtained by making each frame a multiple exposure through four different filters on a color wheel, with the amount of exposure depending on the color of the filter. We could make color slides using a similar technique. We could also make short videotapes of transformations in real time using a variety of tape widths. Soon the VHS format won out over Betamax so our films were all transferred to VHS. In the early 1980s, when we had our first raster graphics machines with color monitors, we produced slides and videotapes in full color, still using fairly primitive scripting techniques so that much of the film took shape in post production, resulting in a nine-minute film with original music composed by Gerald Shapiro of the Music Department to celebrate the dedication of the Thomas J. Watson Jr. Computer Science Building. Ultimately everything was transferred to DVD and CD format, a long process leading to present-day standards.

Technology and theory developing together The third story is the relationship between the development of visualization technology and the response of mathematicians. In the late 1960s, when the images were all spider- web frameworks, the features that were easiest to see were the singularities,cusps and folds of surfaces projected to the plane. A typical problem made for this technology is a result of RenéThom, namely that if a surface is projected generically to the plane, then the number of cusps is congruent modulo 2 to the Euler characteristic. For example, we can see four cusp pointson some projectionsof a torusof revolutionprojected into the plane while some images of the real projective plane would have three cusp points. That was all we had for the first fifteen years of our collaboration but that changed in the early 1980s, when raster graphics arrived, providing color, shading, highlighting and textures. Now the same projections to planes exhibited double points where surfaces in- tersected themselves, and generically there would be a finite number of triple points where three planes crossed similar to the intersection of three coordinate planes at the origin.From this point onward, the major theorem has to do with triple points of immersions, namely that the number of triple points is congruent modulo 2 to the Euler characteristic of the surface. Therefore, a surface embedded with no triple points must have even Euler characteristic; Werner Boy’s immersion of the real projective plane with exactly one triple point reflects the odd Euler characteristic of the real projective plane. A poster prepared for the InternationalConference of Mathematical Software,a satellite conference of the ICM in Beijing in 2002, exhibits both the wire-frame models of the 1978 ICM and the raster graphics images available now. [3]

The Geometry Center The second-to-last story is about the Geometry Center. During the 1980s up through the middle of the 1990s, this institute was a great resource available to mathematicians who A Century of Visualization: One Geometer’s View 309

Figure 4. Poster comparing wire-frame images from ICM 1978 and colored and shaded images 2002. wanted to use visualizationin the most up-to-date forms. It was housed at the University of Minnesota and supported by the National Science Foundation. It functioned as a clearinghouse for the most current techniques, still in the process of development. My PhD student Davide Cervone spent three years there as a postdoctoralfellow; when I arrived on my sabbatical during his ﬁnal year in 1995, I received a crash course in what was to be a vision of the future of geometric visualization. The Center was capable of producing videos of the highest quality, for example Not Knot, William Thurston’s Outside In, and The Shape of Space based on the book by Jeff Weeks. I recall discussing a proposal for a new project with Stuart Levy and he said, “Let’s go over and see what it looks like.” I asked, “You mean just go over to a machine and expect it to respond just like that?” “Watch,” he said, and he produced the effect on the ﬂy. “It’s called Mosaic,” he explained. The next year it was called by its new name, the internet. Unfortunately for various reasons the Geometry Center survived less than two years more and a great resource for visualization disappeared. Many efforts continue to extend the power of the internet for interactive visualization, inviting all users to participate in the process of investigation and exploration. The last twenty years have seen great advances, as well as a few setbacks when software suddenly starts responding in new ways, depending on security or technological requirements. Still there is no turning back. Visualization is a major part of any mathematics that involves geometry, and that encompasses a very large area indeed. 310 A Century of Advancing Mathematics

Communications in Visual Mathematics The final visualization story features the MAA. It was at the summer MathFest in Burling- ton Vermont in 1995 that I first came up withthe idea of a journal for the MAA that would be devoted to a totally new kind of mathematical article, one that was illustrated not only by single pictures but also by computer-generated pictures that could be manipulated and modified. If there was an experiment described in the article, then the reader could not only look at a report of an investigationbut could also participate in the exploration by changing parameters in an expression or by considering a totally different expression in the same context. The articles to be published in this new journal would have the characteristic that they could not be published in any traditional journal. I discussed this idea with publications officers and more particularlywith Davide Cervone and we came up with a prototype, which we decided to call Communications in Visual Mathematics,CVM for short.The year was 1996, the year I received the Haimo Award.

Figure 5. Table of Contents of prototype of Communications in Visual Mathematics, 1998, www.maa.org/external_archive/CVM/1998/01/welcome.html.

The 1998 version of that journal featured several important entries. For one thing, it included an important interactive article by Frank Farris, later to become editor of Math- ematics Magazine and a member of the MAA Executive Committee. Also there is a ﬁne example of an article written by Davide Cervone for this new type of journal, providing a proof of a conjecture that had been unsolved for decades, never before published. The prototype had the potential to change the face of mathematics publishing. We haven’t realized that potential; it was far ahead of its time. This 1998 article is listed as Volume 0 for the Journal of Online Mathematics and Applications, founded by Gerald Porter, which published from 2002 to 2008. We are still waiting for new journals to take up the call to publish in this way. In conclusion, the ﬁrst century of the MAA has seen a great many changes that affect our mission, none more dramatic than the stories of developments and challenges of mathematical visualization. I would like to acknowledge the contributions of many colleagues A Century of Visualization: One Geometer’s View 311 to the stories, especially the late Charles Strauss (1938–2013) and the many students who have worked with me on teaching and research projects over the years. We all look forward to what lies ahead.

Bibliography [1] F. Apéry, Models of the Real ProjectivePlane: Computer Graphics of Steiner and Boy Surfaces. Vieweg, Braunschweig, Germany, 1987. [2] T. Banchoff, N. Max, Every sphere eversion has a quadruple point, Contributions to Analysis and Geometry, Johns Hopkins Univ. Press (1981) 191–209. [3] T. Banchoff, Computergraphics in mathematicalresearch,from ICM 1978to ICM 2002:A personal reflection, in Proceedings of the First International Congress on Mathematical Software, Beijing, China, World Scientific, River Edge, NJ, 2002. 180–189. [4] T. Banchoff, Computer animation and the geometry of surfaces in 3- and 4-space, in Proceed- ings of the International Congress of Mathematicians, 1978, Helsinki, ICM, Helsinki, 1980. 1005–1013. [5] W. Boy, Ueber die Curvatura Integra und die Topologie Geschlossener Flaechen, Dissertation Göttingen, Math. Ann. 57 (1903) 151–184. [6] R. Bryant, A duality theorem for Willmore Surfaces, J. Diff. Geom. 20 (1984) 23–53. [7] G. Francis, A Topological Picturebook. Springer-Verlag, New York, 1987. [8] G. Francis, B. Morin, Arnold Shapiro’s eversion of the sphere, Math. Intelligencer 2 (1980) 200–203. [9] G. Francis, J. Sullivan, R. Kusner, K. Brakke, C. Hartman, G. Chappell, The minimax sphere eversion, Visualization in Mathematics, Springer-Verlag (1996) 3–20. [10] J. F. Hughes,Another proof that every sphereeversion has a quadruple point, Amer. J. Math 107 no. 2 (1985) 501–505. [11] J.F.Hughes, PolynomialModels of Smooth Surfaces. Computer Graphics in Pure Mathematics, Univ. of Iowa, 1990. [12] S. Levy, D. Maxwell, T. Munzner, Outside In, The Geometry Center, Minneapolis, MN, 1994. [13] N. Max, Turning a SphereInside Out, International Film Bureau, 1976. [14] B. Morin, Equationsdu retournement de la sphère, Comptes Rendus Acad. Sci. Paris 287 (1978) 879–882. [15] A. Phillips, Turning a sphere inside out, Sci. Amer. 214 (1966) 112–120. [16] S. Smale, A classification of immersions of the two-sphere, Trans. Amer. Math. Soc. 90 (1959) 281–290.

Department of Mathematics, Brown University, Providence, RI 02912 [email protected]

The Future of Mathematics 1965 to 2065

Jonathan M. Borwein University of Newcastle

William Gibson, the science fiction writer, who coined the term cyberspace well before he purchased his first personal computer has commented that the future is already here but that it is very poorly distributed.Mindful of the dangers of futurology, I shall look forward and back fifty years while where possible eschewing the unknowable.

The bigger picture

It’s generally the way with progress that it looks much greater than it really is. (Ludwig Wittgenstein, 1889–1951, “whereof one cannot speak, thereof one must be silent”) The world will change. It will probably change for the better. It won’t seem better to me.... There was no respect for youthwhen I was young,and now that I am old, there is no respect for age. I missed it coming and going. (J. B. Priestley, 1894– 1984)

I was asked to take on a daunting and futile task—that of talking about the future of our discipline. I negotiated myself back to the current title. At least that way, I can be demonstrably wrong about past events even as I fail miserably when looking at the future. Or I could take the coward’s way out as Ken Davidson and I did when we agreed to write Mathematics in Canada. The Future of Mathematics in Canada 50 years later. [10]. We fairly accurately reprised the present (1995) and then made milquetoast predictions about the very proximate future. These were so conservative that I shuddered when I reread the article—just fifteen years later—in preparation for my current remit. “Futurology” is the domain of fools and flim-flam artists. It can be very profitable. I prefer science fiction like Daniel H. Wilson’s Robocalypse (2001) to fiction masquerading as science—my view of Ray Kurzweil’s The Singularity is Near (2005). The further out one looks the less one can say, and the loonier the results are likely to be in retrospect. An astonishing example is a recent attempt by Scientific American to look at computing

Research supported by the Australian Research Council.

313 314 A Century of Advancing Mathematics in 2165; that’s right 150 years away! 1;2 A fine reprise of the errors of my fore-runners is Leon Kappelman’s 2001 article The Future is Ours [28]. He wrote Predicting the future is an activity fraught with error. Wilbur Wright, co-inventor of the motorized airplane that successfully completed the first manned flight in 1903, seems to have learned this lesson when he noted: “In 1901, I said to my brother Orville that man would not fly for 50 years. Ever since I have ... avoided predictions.” Despite the admonition of Wright, faulty future forecasting seems a favored human pastime, especially among those who would presumably avoid opportunities to so easily put their feet in their mouths. Some of the quotes in [28] were somewhat or very plausible when uttered: This ‘telephone’ has too many shortcomings to be seriously considered as a means of communication. The device is inherently of no value to us. — Western Union internal memo, 1876. Folks, the Mac platform is through — totally. — John C. Dvorak, PC Magazine, 1998. By the turn of this century, we will live in a paperless society. — Roger Smith, chair of General Motors, 1986. Some were folly. As defined by Barbara Tuchman [41, p. 5], folly is error “perceived as counter-productive in its own time.” The abdomen, the chest, and the brain will forever be shut from the intrusion of the wise and humane surgeon. — Sir John Eric Erichsen, British surgeon to Queen Victoria, 1873. Radio has no future. Heavier than air flying machines are impossible. X-rays will prove to be a hoax. — William Thomson (Lord Kelvin), English physicist and inventor, 1899. Credit reports are particularly vulnerable ... [as] are billing, payroll, accounting, pension and profit-sharing programs. — Leon A. Kappelman on likely Y2K problems, 1999. And some were (unintentionally) correct in part or in whole: The problem with television is that people must sit and keep their eyes glued on a screen; the average American family hasn’t time for it. — New York Times, 1949. Where ... the ENIAC is equipped with 18,000 vacuum tubes and weighs 30 tons, computers in the future may have only 1,000 vacuum tubes and weigh only 1.5 tons. — Popular Mechanics, 1949. That said some things are clear. Yet unimagined advances in information and communication technology will have radically reworked much of what we do. This, of course, assumes a certain level of optimism—that the ravages of global warming, and concomitant competition for increasingly scarce resources, or horrifying as-yet-unknown pandemics (or a wandering meteorite) have not overwhelmed our race.

1See www.scientificamerican.com/article.cfm?id=computing-in-2165. 2A more useful Scientiﬁc American article discusses assessments made in 1913 of the ten most important inventions of the prior quarter century, see www.scientificamerican.com/article.cfm?id= inventions-what-are-the-10-greatest-of-our-time&WT.mc_id=SA_DD_20131021. The Future of Mathematics: 1965 to 2065 315

Some probable verities We mathematicians are not entirely free agents We beneﬁt from societal advances (Moore’s law3 and ﬁber-optics,the abolitionof mandatory retirement in most ofthe English- speaking world) and are constrained by societal missteps (current intellectual property and copyright law). We swim in the sea of the other sciences [30]. All professions lookbad in themovies ... why should scientists expect to be treated differently? —Michael Crichton4 Further professionalization of university administration will continue and we will be bombarded with new initiatives. We may look at Khan Academy and MOOCs as new phe- nomena5. But for many decades—in the USA and elsewhere—spending on private-sector in-house tertiary education has been much greater than the total university budget. Yet, it would be wrong to count the university out. As Giamatti [22] and others have noted it is, with all respect to the Vatican, the last surviving successful medieval institution.

We are not the least popular academic profession Roughly twenty-ﬁve years ago, my brother and frequent collaborator Peter surveyed other academic disciplines. He discovered that students who complain mightily about calculus professors still prefer the relative certainty of what we teach and assess to the subjectivity of a creative writing course or the rigors of a physics or chemistry laboratory course.

Mathematics will continue to be important Unlike Geography or Latin departments, as long as there are universities [22] in some form or another there will be mathematics departments. What is less clear is whether there will be funds for research in mathematics for its own sake.6 Mathematics as the language of high technology will thrive (at least in the medium term), but the fate of curiosity-driven research is less clear. One can argue that only for thirty-ﬁve years (1945–1980) was mathematical research funded as a good in its own right. It started with Vannevar Bush7 and ended with the Reagan revolution.

Some things will not change much Certainly there are areas being deeply studied that were not in 1965 but in broad brush— despite G¨odel and Turing—the nature of mathematical research has changed surprisingly little over the last half-century [24].8 We do collaborate more, but most researchers take stunningly little advantage even of tools like Skype. ... Like Ol’ Man River, mathematics just keeps rolling along and produces at an accelerating rate “200,000 mathematical theorems of the traditional handcrafted

3See www.scientificamerican.com/article.cfm?id=moores-law-found-to-apply -beyond-transistors 4Addressing the 1999 AAAS Meetings, as quoted in Science of Feb. 19, 1999,p. 1111. 5See www.newrepublic.com/article/112731/moocs-will-online-education-ruin- university-experience. 6See Keith Devlin’s gloomy assessment regarding the death of mathematics in edge.org/response- detail/23783. 7Bush who directed the war-time Ofﬁce of Scientiﬁc Research and Development is inarguably the father of NSF, see en.wikipedia.org/wiki/Vannevar_Bush#National_Science_Foundation. 8Much of this subsectionis taken from [30]. 316 A Century of Advancing Mathematics

variety ... annually.” Although sometimes proofs can be mistaken—sometimes spectacularly—and it is a matter of contention as to what exactly a “proof” is— there is absolutely no doubt that the bulk of this output is correct (though probably uninteresting) mathematics. — Richard C. Brown [15, p. 239]

Pressure to publish This is unlikely to abate, and qualitative/quantitative measurements of performance9 are for the most part fairer than leaving everything to the whim of one’s Head of Department. Thirty years ago my career review consisted of a two-line mimeo “your salary for next year will be ... ” with the relevant number written in by hand. At the same time, it is a great shame that mathematicians have a hard time ﬁnding funds to go to conferences just to listen and interact. Cs¨ıkszentmih´alyi [18] writes:

[C]reativity results from the interaction of a system composed of three elements: a culture that contains symbolic rules, a person who brings novelty into the symbolic domain, and a field of experts who recognize and validate the innovation. All three are necessary for a creative idea, product, or discovery to take place. —Mihály Cs´ıkszentmihályi

We have not paid enough attention to what creativity is and how it is nurtured. Confer- ences need audiences, and researchers need feedback other than the mandatory “nice talk” at the end of a special session. We have all heard distinguished colleagues mutter a stream of criticism during a plenary lecture only to proffer “I really enjoyed that”as they pass the lecturer on the way out. A communal view of creativity requires more of the audience. The technologies now available—from Skype to MOOCs10—can both improve and exacerbate the situation. Most of this relies on organization and motivation not machinery and cash, see [12]. Though typically, a lot more cash is also needed than senior academic administrators are wont to assert. None of it entirely removes the need for physical removal to conferences and retreats. I began university in 1967. My academic life started in the short but wonderful period of a vast infusion of resources for science and mathematics “after sputnik,” see [15]. It now includes the iPad Kindle app (on which I am listening11 to a fascinating recent biography of the Defense Advanced Research Projects Agency, DARPA). The tyranny of a Bourbaki-dominated curriculum has been largely replaced by the scary grey-literature world of Wikipedia and Google Scholar. And pure research has already been kicked pretty far out of the playing ﬁeld.

The robustness of who does what and where The impact of taste and bias will not diminish: Some subjects can be roughly associated with geographic locations: graph theory is a Canadian subject, singular integrals is an Argentine subject, class ﬁeld theory an Austrian subject, algebraic topology an American subject, algebraic geometry an

9For an incisive analysis of citation metrics in mathematics I thoroughly recommend the relatively recent IMU report and responses at: openaccess.eprints.org/index.php?/archives/417- Citation-Statistics-International-Mathematical-Union-Report.html. 10See http://en.wikipedia.org/wiki/Massive_open_online_course. 11It will read to you in a friendly if unnatural voice. The Future of Mathematics: 1965 to 2065 317

Italian subject, special functions a Wisconsin subject, point-set topologya Southern subject, probabilitya Russian subject. —Gian-Carlo Rota [32, p. 216]. This is quite akin to the salt water-fresh water divide in economics12 in which a University of Chicago centered supply-side prejudice has overwhelmed reasonable Keynesianism for decades. It also shows in the selection of ﬁelds represented at Ivy League schools and in those in which one can win Fields medals (most strikingly in 2002 when only two were awarded).

The need to make our subject accessible This is more not less pressing than 50 years ago [15]. Rota [32, p. 216] captures some of the problems: It takes an effort that is likely to go unrewarded and unappreciated to write an interesting exposition for the lay public at the cutting edge of mathematics. Most mathematicians (self-destructive and ungrateful wretches that they are, always ready to bite the hand that feeds them) turn their noses at the very thought. Little do they realize that in our science-eat-science world such expositions are the lifeline of mathematics. —Gian-Carlo Rota

The Matthew effect or principle This is sociologist of science Robert Merton’s apt dictum that “to those who have shall be given”13. And yet the rise of the blogospheremakes it all the easier to focus on a few star names. Thus, Polymath quickly becomes synonymous with “Gowers and Tao” (who are two of the really smart and really good guys—those two qualities are not tightly coupled, something our discipline often neglects).

Dieudonné-like shortsightedness A 1970 survey of the Bourbaki project includes [19, p.140]: Let us now see what is excluded [from the corpus]. The theory of ordinals and cardinals, universal algebra (you know very well what that is), lattices, non-associative algebra, most general topology,most of topologicalvector spaces, most of the group theory (finite groups),most of number theory (analytical number theory, among others). The processes of summation and everything that can be called hard analysis— trigonometrical series, interpolation, series of polynomials, etc.; there are many things here; and finally, of course, all applied mathematics. —Jean Dieudonné. Rota again is spot-on about this sort of attitude in his review of Paul Halmos’s mathematical autobiography [33, p.701]: Take, for example, the turningpointof the author’scareer, theincident of hisleaving the University of Chicago. ... Whatever his other merits, Halmos is now regarded as the best expositor of mathematics of his time. His textbooks have had an immense influence on the development of mathematics since the fifties, especially by their influence on mathematicians in their formative years. Halmos’s glamor would have been a far sounder asset

12See en.wikipedia.org/wiki/Saltwater_and_freshwater_economics. 13See en.wikipedia.org/wiki/Matthew_effect_(sociology) 318 A Century of Advancing Mathematics

to the University of Chicago than the deep but dull results of an array of skillfular- tisans. What triumphed at the time is an idea that still holds sway in mathematics departments today, namely, the simplistic view of mathematics as a linear progression of problems solved and theorems proved, in which any other function that may contribute to the well-being of the field (most significantly, that of exposition) is to be valued roughlyon a par with that of a janitor. I have a first-rate younger colleague at a major state university who is a driving force in the intelligent use of computers in mathematics. His software alone should be enough to have him tenured and celebrated. But he has been told that it will count for (less than) nothing in all academic assessments. For his local academic judges he is most assuredly a janitor.

Some things have changed From the perspective of pure intellectual progress, I offer eight results we could only have dreamt of having achieved in 1960.

A few good theorems I suggest that the following are fine candidates: 1. Independence of Continuum Hypothesis (Cohen 1963) 2. Luzin Conjecture: a.e. Fourier convergence in L1 (Carleson 1966) 3. Four Color Theorem (Appel-Haken 1976, Robertson-Seymour-Thomas 1997) 4. Classification of Finite Simple Groups (Feit-Thompson, Gorenstein, et al. 1955–1995? [39]) 5. Fermat’s Last Theorem (Wiles-Taylor 1993–94) 6. PoincaréConjecture (Hamilton-Perelman 2004, ...) 7. Primes in long arithmetic progressions (Green-Tao 2008) 8. The Most StrikingResult in Your Own Area At the other end of the spectrum, I list a sampler of “digital assistance” tools [3] that have radically changed the prospects for howwe will do mathematics in the next fifty years.

A few tools: good, bad and ugly The existence of the current tools, starting with TEX, and platforms (including tablets, smart phones, Google glasses) is the start not the end of a changing praxis (see Section 21). Here are some of the many tools I use or personally avoid—like Facebook and Twitter.14 1. Use of Modern Mathematical Computer Packages: Symbolic, Numeric, Geometric, Graphical, Statistical, . . . 2. Use of More Specialist Packages or General Purpose Languages:Fortran, C++, Python, CPLEX, GAP, PARI, MAGMA, Cinderella, Blender, ... 3. Use of Web Applications: Sloane’s Encyclopedia, Inverse Symbolic Calculator, Fractal Explorer, Euclid in Java, Weeks’ Topological Games, MathOverﬂow, Polymath, . . .

14URLs for the bulk of these are most easily found by a web search. The Future of Mathematics: 1965 to 2065 319

Figure 1. A walk on 200 billion bits of . Pi day 2015 is special: 3.1415.

4. Use of Web Databases: Google, MathSciNet, ArXiv, JSTOR, Wikipedia, MathWorld, PlanetMath, DLMF, MacTutor, Amazon, Kindle Reader, Nook, Wolfram Alpha, ... All entail data-mining—“exploratoryexperimentation” and “widening technology”[3]— as is occurring in a big way in pharmacology, astrophysics, biotechnology,.... An example of large-scale data mining in mathematics [1] is given in Figure 1. Clearly the boundaries are vague and getting even vaguer, while judgments of a given tool’s quality vary and are context dependent. For instance, NIST’s wonderful Digital Library of Mathematical Functions15 is fundamentally still a nineteenth century handbook in twenty-ﬁrst century garb. By contrast INRIA’s prototype Dynamic Dictionary of Mathematical Functions16 points to a future in which mathematical knowledge is generated algorithmically and extensibly, rather than from lookup tables.

15dlmf.nist.gov 16ddmf.msr-inria.inria.fr/ 320 A Century of Advancing Mathematics

A few things are certain First, I am willing to predict that—as both Felix Browder (in his final address as AMS President) and Tim Gowers (at a meeting of Fields medalists) have said—the future of mathematics is intimately coupled to computing. (But then, so is everything else [34].) We should proudly consider ourselves one of The Sciences of the Artificial [36]. As a reminder of the youth of computer science as a discipline17, I recall that the Uni- versity of Toronto undergraduate computer science (CS) program18 was founded as late as 1971. In the case of mathematical research, this tight coupling with computational—as opposed to computer—science presages more emphasis on algorithms and constructive methods [8], visualization [23], aesthetics [37] and the like; and less focus on abstraction for its own sake. These are trends we can already observe. The increasing role of collaborative research [9, 12], which can be exaggerated (mathematicians have always needed to talk to each other [29]) may help make the subject more attractive as a career for women, by keeping them in the STEM stream long enough to develop a passion for science [42]. As it stands why would bright and articulate young women (including my own three adult daughters and my brother’s three) with many op- tions be likelyto opt for a STEM career? Second, we shall assuredly know vastly more about the working of the brain and in particular neurology [17] of mathematical creativity [25, 35] but whether that will resolve the mind-bodyproblem to the satisfaction of philosopherslikeThomas Nagel [30] only time will tell. In consequence, one may also hope that “evidence-based” mathematical education will become the rule not the exception. Finally, as the economic transformation of South America, Asia, and Africa accelerates, the geographic dispersal of mathematical research is certain to grow; we are a pretty cheap science. As any editor can attest the sheer quantity of submissions originating in Asia is now stunning. One may hope a corresponding increase in quality is not far behind. The rapid development of first-rate universities in places such Hong Kong, Singapore, Korea, and China is a positive sign. But building robust cultural traditions of academic enquiry is a long and difficult job.

The proximate future Watson for mathematics: intelligent assistance Since “academic plans” and “technology road-maps” are good for at most about ﬁve years, let me reprise one I was asked to write in 2011 under the title“If I had a blank cheque for mathematics” in a series in The Conversation on what one could do with inﬁnite resources in the sciences.19

Project: Retool IBM Watson for mathematics. Cost: $500 million. Time frame: 5 years.

17If it is one rather than an amalgam of mathematics, engineering,library science, business and other subjects. 18Computer Science as an academic entity dates from 1964 at Toronto, 1965 at Stanford and Carnegie-Mellon, while EE became EECS in 1975 at MIT. 19See theconversation.edu.au/if-i-had-a-blank-cheque-id-turn-ibms-watson- into-a-maths-genius-1213. The Future of Mathematics: 1965 to 2065 321

Mathematics has many grand challenge problems, but none that can potentially be settled by pouring in more money—unlike the case of the Large Hadron Collider, the Square Kilometre Array or other such projects. Math is a different beast. But, of course, you’re offering me unlimited, free dosh, so I should really think of something. Grand challenges in mathematics In his famous 1900 speech The Problems of Mathe- matics David Hilbert listed 23 problems that set the stage for 20th century mathematics. It was a speech full of optimism for mathematics in the coming century and Hilbert felt open (or unsolved) problems were a sign of vitality: The great importance of definite problems for the progress of mathematical science in general ... is undeniable ... [for] as long as a branch of knowledge supplies a surplus of such problems, it maintains its vitality... every mathematician certainly shares . . . the conviction that every mathematical problem is necessarily capable of strict resolution... we hear withinourselves the constant cry: There is the problem, seek the solution.You can find it through pure thought ... Hilbert’s problems included the continuum hypothesis, the “well-ordering” of the reals, Goldbach’s conjecture, the transcendence of powers of algebraic numbers, the Riemann hypothesis, the extension of Dirichlet’s principle and many more. Many were solved in subsequent decades, and each time it was a major event for mathematics. The Riemann hypothesis (which deals with the distributionof prime numbers) remains on a list of seven “third millennium” problems. For the solution of each millennium problem, the Clay Mathematics Institute offers—in the spirit of the times—a one-million- dollar prize. This prize has already been awarded and refused by Perelman for resolving the Poincaré conjecture. The solution also merited Science’s “Breakthrough of the Year” in 2009, the first time mathematics had been so honored.

And their limitations Certainly, given unlimited moolah, learned groups could be gathered to attack each problem and assisted in various material ways. But targeted research in mathematics has even less history of success than in the other sciences . . . which is saying something. Doron Zeilberger famously said that the Riemann hypothesis is the only piece of mathematics whose proof (i.e., certainty of knowledge) merits $10 billion being spent. As John McCarthy wrote in Science in 1997: In 1965 the Russian mathematician Alexander Konrod said ‘Chess is the Drosophila [a type of fruit fly] of artificial intelligence.’ But computer chess has developed as genetics might have if the geneticists had concentrated their efforts, starting in 1910, on breeding racing Drosophila. We would have some science, but mainly we would have very fast fruit flies. What a fine illustration of the likely unintended consequences of certain, perhaps most, scientific grand challenges. Unfortunately, the so-called “curse of exponentiality”—whereby the more difficult a mathematical problem becomes, the challenge of solving it increases exponentially—pervades all computing, and especially mathematics. As a result, many problems—such as most cases of Ramsey’s Theorem—will likely be impossible to solve by computer brute force, 322 A Century of Advancing Mathematics regardless of advances in technology. Except, just possibly by an unconventional mode of computing such as quantum computing.

Money for nothing But, of course, I must get to the point. You’re offering me a blank check, so what would I do? A holiday in Greece for two? No, not this time. Here’s my manifesto: Google has transformed mathematical life (as it has all aspects of life) but is not very good at answering mathematical questions, even if one knows precisely the question to ask and it involves no symbols. In February 2011, IBM’s Watson computer walloped the best human Jeopardy! players in one of the most impressive displays of natural language competence by a machine. I would pour money into developing an enhanced Watson for mathematics and would acquire the whole corpus of math for its database.20 Math ages very well and I am certain we would discover a treasure trove. Since it’s hard to tell where mathematics ends and physics, computer science, and other subjects begin, I would be catholic in my acquisitions.Since I am as rich as Croesus, claim no rights, and can buy my way out of trouble, I will not suffer the same court challenges Google Books has faced in the past few years. I should also pay to develop a comprehensive computation and publishing system with features that allow one to manipulate mathematics while reading it and which ensures published mathematics is rich and multi-textured, allowing for reading at a variety of levels, see [11]. Since I am still in a spending mood, I would endow a mathematical research institute with great collaboration tools [9] for roughly each ten million people on Earth. Such institutes have greatly enhanced research in the countries that can afford and chose to fund them. Content with my work, I would then rest. But soon I would realize how much I had left uncovered. For instance, ﬁnding the quote by Rota about special functions and Wisconsin. One wants to ask who wrote “Spe- cial functions [is] a Wisconsin subject”? With quotes and without the “is” it was easy. Without quotes or with the “is” I labored for many hours. I asked a knowledgeable friend who assured me it was probably not a Rotaquote. We are a long way from being able to get satisfactory answers from “What is known about something like the following ...?” questions. In a 2013 paper [5] the authors describe a variety of promising if limited attempts to build “ﬁngerprints” for extracting knowledge about theorems in a given area.

Reliability and reproducibility Conventionalwisdom sees computingas the thirdleg of science, complementing theory and experiment. That metaphor is out-dated. Computing now pervades all of science. Massive computation is often required to reduce and analyze data; simulations are employed in ﬁelds as diverse as climate modeling and astrophysics. Unfortunately, scientiﬁc computing culture has not kept pace. Experimental researchers are taught early to keep notebooks or computer logs of every work detail—design, procedures, equipment, raw results, processing techniques, statistical

20Sadly, IBM has so far only shown interest in more mundane if lucrative things like “Watson for oncology.” The Future of Mathematics: 1965 to 2065 323 methods of analysis, etc. In contrast, few computational experiments are performed with such care. Typically, there is no record of work-ﬂow, computer hardware and software con- ﬁguration, or parameter settings. Often source code is lost. While crippling reproducibility of results, these practices ultimately impede the researchers’ own productivity.

The state of experimental and computational mathematics Experimental mathematics —application of high-performance computing technology to research questions in pure and applied mathematics, including automatic theorem proving—raises numerous issues of computational reproducibility [3, 8]. It often pushes the bounds in very high precision computation (hundreds or thousands of digits), symbolic computation, graphics, and parallel computation. As with all computational science, one should carefully document algorithms, implementation, computer environments, experiments, and results.

Mathematics is special Even more emphasis needs to be placed on unique aspects of the discipline: a) Are precision levels (hundreds or thousands of digits) adequate? b) What independent consistency checks were employed to validate results? c) If symbolic manipulation software was employed (e.g., Mathematica or Maple), which version was used? What precise functions were called, what parameter values and environmental settings? d) Have numeric spot-checks been performed for derived identities etc.? e) Have symbolic manipulations been validated, say using two different packages? Such checks are crucial, because even the best symbolic and numeric computation packages have bugs and limitations—often exhibited only during hard computations. It is worth emphasizing that it is impossible to be expert with all of the algorithms now embedded in a computer algebra system. Hence, most of us are prone to oscillating between expert and ingenue as we move through different parts of our computationally-assisted research, see [2]. Automatic theorem proving has now achieved some truly impressive results such as fully formalized proofs of the four color theorem and the prime number theorem. While such tools currently require great effort, one can anticipate a time in the distant future (by 2065?) when all truly consequential results are so validated.

Interactive theorem proving for mathematics and computation “Interactive theorem proving,” a method of formal veriﬁcation, uses computational proof assistants to construct formal axiomatic proofs of mathematical claims. Examples as of now include coq, Mizar, HOL4, HOL Light, ProofPowerHOL, Isabelle, ACL2, Nuprl, Veritas, and PVS. Notable theorems such as the four color theorem have been veriﬁed in this way, and Thomas Hales’s Flyspeck project, using HOL Light and Isabelle, aims to obtain a formal proof of the Kepler conjecture. Each one of these projects produces machine-readable and exchangeable code that can be integrated into other programs. For instance, each formula in the web version of NIST’s authoritative Digital Library of Mathematical Functions may be downloaded in TeX or MathML (or indeed as a PNG image) and the fragment directly 324 A Century of Advancing Mathematics embedded in an article or other code. This dramatically reduces chances of transcription error and other infelicities being introduced.

Reproducibility in computational and experimental mathematics Motivated by such concerns, a workshop on the topic was held in late 2012 at the Institute for Computa- tional and Experimental Research in Mathematics at Brown University. Participants included computer scientists, mathematicians, computational physicists, legal scholars, journal editors, and funding agency ofﬁcials, representing academia, government labs, industry research, and all points in between. While different types and degrees of reproducible research were discussed, an overwhelming majority agreed the community must move to “open research”: research using accessible software tools to permit (a) “auditing” computational procedures, (b) replication and independent veriﬁcation of results, and (c) extending results or applying methods to new problems Of course, the level of validation should be proportional to the importance of the research and strength of claims made.

Three principal recommendations Coming out of the ICERM workshop, these were: 1. First, researchers need persuasion that efforts to ensure reproducibility are worthwhile, leading to increased productivity, less time wasted recovering data or code, and more reliable conversion of results from data files to published papers. 2. Second, the research system must offer institutional rewards at every level from departmental decisions to grant funding and journal publication. The current academic and industrial research system places primary emphasis on publication and project results and little on reproducibility. It penalizes those devoting time to developing or just following community standards. It is regrettable that software development is often discounted and becomes one of Rota’s janitorial tasks. It is typically compared, say, to constructing a telescope, rather than doing real science. Thus, scientists are discouraged from spending time writing or testing code. Sadly, NSF-funded web projects remain accessible only about a year after funding stops. Researchers are busy running new projects without time or money to preserve the old. Given the ever-increasing importance of computation and software, such attitudes and practices must change. 3. Finally, standards for peer review must be strengthened. Editors and reviewers must insist on rigorous verification and validity testing, along with full disclosure of computational details [21]. Some details might be relegated to a website, with assurances this information will persist and remain accessible. Exceptions exist, such as where proprietary, medical, or other confidentiality issues arise, but authors need to state this upon submission, and reviewers and editors must agree such exceptions are reasonable.

Help is already here Many extant tools help in replicating past results (by the researcher or others). Some ease literate programming and publishing computer code, either as commented code or notebooks. Others capture provenance of a computation or the complete The Future of Mathematics: 1965 to 2065 325 software environment. Version control systems are not new, but current tools facilitate use for collaboration and archiving complete project histories. Numerical reproducibility itself is a major issue, as is hardware reliability. For some applications, even rare interactions of circuitry with stray subatomic particles matter. The enormous scale, however, of state-of-the-art scientiﬁc computations, using tens or hundreds of thousands of processors, presents unprecedented challenges. Not to mention what the future will bring. Forty years ago, concern about proof by appeal to authority, “von Neumann says,” was a commonplace philosophical response to the four color theorem proof’s re- liance on computation. Without adoption of the sort of recommendations above we will have to settle for “Mathematica says we have a proof.”

Open source By 2013, the US had followed the UK, Australia, and others in mandating some form of public release of publicly funded research, including data [31]. I hope this brings a cultural change in favor of consistently reproducible computational research. This is further discussed in the ICERM workshop report [40] and Wiki [27] and in a 2013sum- mary of articles on the “[T]ectonic shifts [that] are happening in the way scientiﬁc research is done” commissioned by Nature.21 Yet, I remain unconvinced that mathematics, with its “long-tail” use of the literature and very large unfunded publication base, will ﬂourish in a regime driven by the needs only of the “big” sciences.

The far future By this I count things more than fifteen years out. For fifteen years it is fair to extrapolate. We will stillhave to fill out toomany pointlessforms. We willhave goodverbal and gestural control of our computers. The computers will look a lot different. We shall certainly have natural computer voices by then—they already exist in the research lab. If you want your Mom or Tom Cruise to read to you that will be fine. Computer algebra systems will be much better and way faster than now. The 3D Maple graphics will look pretty damn good in my Google glasses. And so on. Further out isguess-work. As David Bailey trenchantlywrote to me whileI was drafting this piece: How could anyone, even ten years ago, have predicted that by now almost everyone above the age of eight would now be on Facebook for minutes (or hours!) of every day?22 How could anyone have predicted twenty years ago that almost every person above the age of eight would have their own personal supercomputer, systems far more powerful (and useful!) than the Cray supercomputers of the time? Who could have foreseen, even five years ago, that toddlers would take to smart- phones and tablets with such aplomb that concerned parents would find it necessary to ration the time kids spend on these devices? For that matter, how could anyone have anticipated that CO2-induced global warming would loom as an existential threat to civilization? 21See theconversation.com/open-publishing-is-happening-the-only-question- is-how-13100. 22Care to predict if Facebookwill be a powerhouseor a footnote in 2025? 326 A Century of Advancing Mathematics

What will universities look like? My own guess is: much like now only different. I recently read some science ﬁction by Connie Willis about time-travelling historians based in Oxford of the 2060s. The history I think was pretty good, but the future had pre-mobile phones, and Oxford seemed entirely unchanged from my stint there over forty years ago. By 2030,willthe immanence of global warming23 lead toa Nixonian“war onwarming” with a corresponding gutting of pure research and the demise of mathematics as a ding-an- sich, or will it spur another sputnik-likegolden period? Make both predictions and one will likely be right. As my brother has observed, 50% is a great average in the prognostication business. Will there be general-purpose quantum computers? I have no basis for well-founded insights, but my gut feeling is “no.” Largely, because there are very few free lunches. In this—and this alone—I agree with Milton Friedman. As Cris Calude has pointed out, my gut feeling is in accord with the more reasoned analysis (some of it mathematical) of the quantum computing community. But it would be great to be wrong, probably. Working out what are the implications of quantum computing for mathematics is an exercise I propose to undertake, probably.24 I suggest, perhaps rather hope, that by 2065, the working philosophy of mathematics will have evolved from its current inchoate Platonism [8] to something a little more nuanced. This is not entirely an issue for philosophers. As Edwards [20] has pointed out the distinction between inventions (patentable) and discoveries (not protectable) is basic to current intellectual property law. One of the epochal events of my childhood as a faculty brat in St. Andrews, Scotland was when C. P. Snow (1905–1980) delivered an immediately controversial 1959 Rede Lec- ture in Cambridge entitled The Two Cultures.25 Snow argued that the breakdown of communication between the“two cultures”of modern society—thesciences and the humanities— was a major obstacle to solving the world’s problems—and he had never heard of global warming. In particular, he noted the quality of education was everywhere on the decline. Instancing that many scientists had never read Dickens, while those in the humanities were equally non-conversant with science, he wrote:

A good many times I have been present at gatherings of people who, by the standards of the traditional culture, are thought highly educated and who have with considerable gusto been expressing their incredulity at the illiteracy of scientists. Once or twice I have been provoked and have asked the company how many of them could describe the Second Law of Thermodynamics, the law of entropy. The response was cold: it was also negative. Yet I was asking something which is about the scientiﬁc equivalent of: ‘Have you read a work of Shakespeare’s?’

I hope that by 2065, scientiﬁc literacy will be ubiquitous and that C. P. Snow’s two cultures will both still be thriving, and will be well-aware and appreciative of each other; though I honestly do not expect that to happen.

23Storms of our grandchildren [26] are too distant to have any political impact. 24Interesting work in this direction by Calude and Tadaki can be found at www.cs.auckland.ac.nz/ research/groups/CDMTCS//researchreports/434CT.pdf. 25Subsequently republished in [38]. The Future of Mathematics: 1965 to 2065 327

Conclusion After 60 years with really only two input modalities: ﬁrst via keyboard and command line computing; and then thirty years later with Apple’s adoption of Douglas Engelbart’s mouse26 along with iconic graphical user interfaces (GUI), we are now in a period of rapid change. Speech, touch, gesture, and direct mental control are all either realized or in prospect. As noted, the neurology of the brain has developed in twenty-ﬁve years from ignorance to a substantial corpus. It is barely twenty years since the emergence of the World Wide Web27 and it wouldbe futile to imagine what interfaces will look like in another twenty.28 We are still exploring the possibilities suggested by Vannevar Bush in his seminal 1945 essay As We May Think29 and some parts of Leibniz’ dream [16] still seem very distant. In any event, in most of the futures, mathematics will remain important and useful, but those of us who love the subject for its own sake will have to be nimble. We cannot risk leaving the task of looking after the health of our beautiful discipline to others.

Acknowledgements These are due to many colleagues (including Sara Billey, Richard Brent, Neil Calkin, Cris Calude, Greg Chaitin and Veso Jungic) for feedback provided. Especial thanks is due to the coauthors of [40], and in particular Victoria Stodden and David Bailey who coauthored [4] from which Section 21 is largely taken. The cartoon is published by permission of the author, Simon Roy, and Veselin Jungic.

Bibliography [1] F. Aragon, D. H. Bailey, J. M. Borwein and P. B. Borwein, Walking on real numbers, Mathematical Intelligencer 35 (1) (2013), 42–60. dx.doi.org/10.1007/ s00283-012-9340-x. [2] David Bailey, Roberto Barrio and Jonathan Borwein, High-Precision Computation: Mathemat- ical Physics and Dynamics, Applied Mathematics and Computation 218 (2012), 10106–0121. dx.doi.org/10.1016/j.amc.2012.03.087 [3] D. H. Bailey and J. M. Borwein, Exploratory Experimentation and Computation, Notices of the Amer. Math. Soc. 58 (10) (2011), 1410–1419. E-published October 13, 2011 at www.ams.org/notices/201110/rtx111001410p.pdf. Published in translation in Mathematical Advances in Translation (Chinese Academy of Science), June 2012. [4] D. H. Bailey, J. M. Borwein and Victoria Stodden, Set the default to “open,” Notices of the Amer. Math. Soc. 60 (6) 2013, 679–80. [5] S. Billey and B. Ragnarsson,Fingerprint Databases for Theorems, Preprint April 2013. [6] J. M. Borwein, Exploratory Experimentation: Digitally-assisted Discovery and Proof, Chapter in ICMI Study 19: On Proof and Proving in Mathematics Education. G. Hanna and M. de

26See sloan.stanford.edu/mousesite/1968Demo.html. Note that William Gibson was right— the future was already there for Steve Jobs to distribute. 27On a slow news day in 2013, The Washington Post reposted a 1995 CNN report www.washingtonpost.com/blogs/wonkblog/wp/2013/03/29/what-the-internet- looked-like-in-1995/?tid=pm_business_pop. 28A 2013 summary of applets useful in taming scientiﬁc literature can be read at blogs.scientificamerican.com/information-culture/2013/03/26/mobile-apps-for -searching-the-scientific-literature/?WT_mc_id=SA_DD_20130326. 29See en.wikipedia.org/wiki/As_We_May_Think. 328 A Century of Advancing Mathematics

Villiers (Eds.) The 19th ICMI Study, New ICMI Study Series, 15, Springer-Verlag. ISBN 978- 94-007-2128-9, March 2012. (Invited). [7] J. M. Borwein, Who we are and howwe got thatway,in Mathematicians on Creativity. MAA, Washington, DC 2014. [8] JonathanM. Borwein and David H. Bailey, Mathematics by Experiment,Edition 2, AK Peters, 2008. [9] Jonathan Borwein, Peter Borwein and Veselin Jungic, Remote Collaboration: Six Years of the Coast-To-Coast Seminar Series, Science Communication, 34 (3) (2012), 419–428.E-published at scx.sagepub.com/content/early/2012/03/26/1075547012443020. [10] J. M. Borwein and K. R. Davidson, Mathematics in Canada. The Future of Mathematics in Canada 50 years later, pp. 231–248 (pp. 249–268 francais) in Canadian Mathematical Society 50th Anniversary Volume 1, CMS Publications, 1995 (Invited). [11] Jonathan Borwein and Veselin Jungic, Organic Mathematics then and now, Notices of the Amer. Math. Soc., 59 (2012), 416–419. See www.ams.org/notices/201203/ rtx120300416p.pdf. [12] JonathanM. Borwein, EugenioM. Rochaand JoseF. Rodrigues, Communicating Mathematics in the Digital Era, AK Peters, 2008. [13] Jonathan Borwein and Brailey Sims, The Douglas-Rachford algorithm in the absence of convexity, Chapter 6, pp. 93–109 in Fixed-Point Algorithms for Inverse Problems in Science and Engineering in Springer Optimization and Its Applications, volume 49, 2011. [14] Peter B. Borwein,Peter Liljedahland Helen Zhai, Mathematicians on Creativity. MAA, Wash- ington, DC 2013. [15] Richard C. Brown, Are Science and Mathematics Socially Constructed? World Scientific, 2009. [16] Cristian S. Calude (editor), Randomness And Complexity, From Leibniz To Chaitin, World Scientific Press, 2007. [17] Paul Churchland, Neurophilosophy at Work, Cambridge Univ Press, 2007. [18] Mihály Cs´ıkszentmihályi, Creativity: Flow and the Psychology of Discovery and Invention, Harper Collins, 1997. [19] Jean A. Dieudonné, The work of Nicholas Bourbaki, Amer. Math. Monthly 77(2) (1970), 134– 145. [20] David A. Edwards, Platonism is the law of the Land, Notices of the Amer. Math Soc, 60 (4) (2013), 475–478. Available at www.ams.org/notices/201304/rnoti-p475.pdf. [21] Daniele Fanelli, Redefine misconduct as distorted reporting, Nature, 13 Feb 2013, available at www.nature.com/news/redefine-misconduct-as-distorted-reporting -1.12411. [22] A. Bartlett Giamatti, A Free and Ordered Space: The Real World of the University. Norton, 1990. [23] Marco Giaquinto, Visual Thinking in Mathematics. An Epistemological Study, Oxford Univ. Press, 2007. [24] Bonnie Gold and RogerSimons (editors), Proof and Other Dilemmas. MAA, Washington, DC, 2008. [25] Jacques Hadamard, The Mathematician’s Mind: The Psychology of Invention in the Mathe- matical Field, Notable Centenary Titles, Princeton University Press, 1996. First edition, 1945. [26] James Hansen, Storms of My Grandchildren: The Truth About the Coming Climate Catastro- phe and Our Last Chance to Save Humanity. Bloomsbury USA, New York, 2009. The Future of Mathematics: 1965 to 2065 329

[27] ICERM Reproducibility Workshop Wiki: wiki.stodden.net/ICERM_ Reproducibility_in_Computational_and_Experimental_Mathematics: _Readings_and_References. [28] Leon A Kappelman, The Future is Ours, Communications of the ACM, March 2001, p. 46. [29] J. E. Littlewood, Littlewood’s Miscellany, Cambridge University Press, 1953. Revised edition, 1986. [30] Thomas Newman, Mind and Cosmos: Why the Materialist Neo-Darwinian Conception of Nature Is Almost Certainly False, Oxford University Press, 2012. [31] OSTP, Expanding Public Access to the Results of Federally Funded Research, 22 Feb 2013. Available at www.whitehouse.gov/blog/2013/02/22/expanding- public-access-results-federally-funded-research. [32] Gian-Carlo Rota, Indiscrete thoughts, Birkhäuser Boston, Inc., Boston MA, 1997. [33] ———, Reviews: I Want to Be a Mathematician: An Automathography, Amer. Math. Monthly 94(7) (1987), 700–702. [34] Kerry Rowe et al., Engines of Discovery: The 21st Century Revolution. The Long Range Plan for HPC in Canada, NRC Press, 2005, revised 2007. [35] Bertrand Russell,The Study ofMathematics in Mysticism and Logic: AndOther Essays. Long- man, 1919. [36] Herbert A. Simon, The Sciences of the Artificial. MIT Press, 1996. [37] Nathalie Sinclair, David Pimm and William Higginson (Eds.), Mathematics and the Aesthetic: New Approaches to an Ancient Affinity. CMS Books in Mathematics. Springer-Verlag, 2007. [38] C. P. Snow, The Two Cultures and the Scientific Revolution, Cambridge Univ Press 1993. First published 1959. [39] R. Solomon, A brief history of the classification of the finite simple groups, Bull. Amer. Math. Soc. (N.S.) 38(3) (2001), 315–352. [40] V. Stodden, D. H. Bailey, J. Borwein, R. J. LeVeque, W. Rider and W. Stein, Setting the De- fault to Reproducible:Reproducibility in Computationaland Experimental Mathematics, 2 Feb 2013, http://www.davidhbailey.com/dhbpapers/icerm-report.pdf. [41] Barbara Tuchman, The March of Folly: From Troy to Vietnam, Ballantyne Books (1984). [42] Lewis Wolpert and Alison Richards (Eds). A Passion for Science, Oxford University Press, 1989.

Centre for Computer Assisted Mathematics and its Applications (CARMA), School of Mathematical and Physical Sciences, University of Newcastle, NSW, Australia [email protected]

Part V Culture and Communities

Philosophy of Mathematics What Has Happened Since G¨odel’s Results?

Bonnie Gold Monmouth University

Introduction

When the MAA was founded in 1915, the mathematical community was in the midst of its “foundational crisis.” Cantor had recently developed his theory of transfinite numbers (which led to assorted paradoxes). Frege’s attempt via logicism to put mathematics on a sound foundation had foundered on the Russell paradox. Brouwer’s intuitionism had introduced a completely different approach to mathematics and philosophy of mathematics. And Hilbert was about to begin (in 1917) his program to set mathematics on a sound foundation via formalism and finitism, by finding a finitary consistency proof for arithmetic and analysis. There are those who say that “crisis” is too strong a word: most mathematicians continued to develop their mathematics despite the foundational concerns. However, it was certainly a crisis in the philosophy of mathematics, and essentially all work in the philosophy of mathematics until at least the 1950s turned to foundational issues. Although mathematicians continued to work on mathematical problems, largely ignoring foundational concerns in their own work, they were very aware of these discussions. Yet even today many mathematicians are unaware of the turn, after Gödel’s theorems, by philosophers of mathematics back to more traditional philosophical questions. They still believe that foundations of mathematics and philosophy of mathematics are identical. In this article I will survey the philosophy of mathematics since Gödel, including, at the end, a survey of some of the work in foundations that has substantial philosophical import. I will not, however, try to survey the remainder of the work in foundations, which is quite substantial. One good place to learn more about many of the topics mentioned in this essay is the online Stanford Encyclopedia of Philosophy [34]. This resource is overseen by an editorial board consisting of respectable philosophers, and the articles are written by philosophers working in the field they are writing about. Information on the authors is available via a link at the top of each article.

The end of foundationalism as the dominant direction

With the publication of G¨odel’s incompleteness theorems, interest among mathematicians in the three foundationalschools declined signiﬁcantly. Logicism appeared to have foundered

333 334 A Century of Advancing Mathematics on the set-theoretic paradoxes. Attempts to fix it involved Russell-Whitehead type theory — sets are built on different levels, avoiding the possibility of the set of all sets. In their system, however, the same set could be built at one level and at each later level. One didn’t want these to be different sets, but an axiom saying they were the same was clearly not simply a logical truth. Further, there is no logical property that implies the existence of an infinite set. So this work did not succeed at the aim of logicism, reducing mathematics to logic. Intuitionism had been developed to the point that it was clear that it involves a complete revision of mathematics. For most mathematicians, giving up completed infinities and the law of the excluded middle results in a very bizarre real number system (using “free choice sequences” to build non-definable real numbers) and throws out too many theorems we like, such as the intermediate value theorem. Finally, Gödel’s incompleteness theorems showed that one cannot give a finitistic, metamathematical proof that a contradiction cannot be derived from the axioms for number theory/analysis, putting an end to Hilbert’s program. Meanwhile, no new contradictions in mathematics had arisen since the early 1900s, giving mathematicians more confidence that, as long as we stay away from enormous objects such as the set of all sets, mathematics is unlikely to lead to further contradictions. As I will discuss toward the end of the article, work in foundations continued, much of it now mathematically quite technical, but some of it has significant philosophical implications. However, gradually interest among philosophers of mathematics turned to more traditional philosophical questions.

Benacerraf’s articles Much of the work in post-foundational philosophy of mathematics can be traced back to two seminal papers by Paul Benacerraf, “What Numbers Could Not Be” in 1965 [4] and “Mathematical Truth” in 1973 [5]. In the first of these papers, he observed that there are several different ways to identify the natural numbers (including 0) with sets, for example: 1. For any natural number n, identify it with the set consisting of ¿ inside n nested pairs of set brackets. So 0 is identified with ¿, 1 is identified with ¿ , 2 with ¿ , and f g ff gg so on. To get the successor of a natural number you simply put set brackets around it: S.n/ n . To determine whether m < n, simply see if you find m somewhere inside D f g the nested set brackets of n. 2. For any natural number n, identify it with the set of all previous numbers, after identifying 0 with ¿. So 1 is identified with ¿ (just as before), but now 2 is identified with f g 0; 1 , that is, with ¿; ¿ , 3 with 0; 1; 2 , that is, with ¿; ¿ ; ¿; ¿ , and so f g f f gg f g f f g f f ggg on. To determine whether m < n, simply see if m n. 2 He makes several observations: first, the “facts” about numbers are different in the different representations: in the first, 17 is a set with just one object, in the second, with 17 objects. In the first, 13 is not an element of 17; in the second it is. Second, since there is no good reason to prefer one of these representations over the other, “numbers could not be objects at all; for there is no more reason to identify any individual number with any particular object than with any other (not already known to be a number)” [4, pp. 290–291]. For the only issue of interest is the relations they bear to one another. To be 3 is no more or less than to be greater than 2 and less than 4. “Any object can play therole of 3” [4, p. 291]. He Philosophy of Mathematics: What Has Happened Since Gödel’s Results? 335 concludes, “There are no such things as numbers; which is not to say that there are not at least two prime numbers between 15 and 20” [4, p. 294]. Among other things, this paper is saying that, while reducing all of mathematics to set theory might have been valuable in the foundational attempt to restore confidence in the consistency of mathematics, there are problems with this approach from the perspective of other philosophical issues. In the second paper, he observed that there are two concerns motivating accounts of mathematical truth: “(1) the concern for having a homogeneous semantical theory in which the semantics for the propositions of mathematics parallel the semantics for the rest of the language and (2) the concern that the account of mathematical truthmesh witha reasonable epistemology. It will be my general thesis that almost all accounts of the concept of mathematical truth can be identified with serving one or another of these masters at the expense of the other” [5, p. 403]. Since both are necessary for an adequate account of mathematical truth, none is satisfactory. This has come to be known as “Benacerraf’s dilemma.” The next section will illustrate this.

Platonism (realism) versus nominalism These two articles by Benacerraf started a discussion among philosophers of mathematics that is ongoing. The basic philosophical problem is, despite the paradoxes in set theory, mathematical knowledge nonetheless appears to be the most certain knowledge we have beyond very basic knowledge such as that of nearby medium-sized physical objects. Many mathematical theorems have outlasted virtually all other scientific claims. Yet philosophers have not been able to account for mathematics in a way that is coherent with their understanding of the physical world or their other accounts of knowledge. The issues are: 1. Ontological: are there mathematical objects? If there are, what is their nature? Are mathematical statements correctly classifiable as “true” or “false”? Most people cat- egorize mathematical objects, if they exist, as some kind of abstract object. What’s striking about mathematical objects is that they seem to have objective properties — we have to learn about a given object, but then, once we understand what it is, we all agree about it. This is unlike most abstract objects such as “beauty.” 2. Epistemological: is there such a thing as mathematical knowledge? Assuming there is, explain, in a way consistent with our understanding of how we acquire other kinds of knowledge, how human beings acquire it. It is, of course, obvious to mathematicians that there is mathematical knowledge — after all, that’s what our research is all about! Philosophers who want to hold views that deny that there is mathematical knowledge have to explain how it appears that there is mathematical knowledge and why, if it doesn’t exist, mathematics is so useful. 3. Semantic: how should we interpret mathematical statements, in a way consistent with everyday language? “There are four chairs in my office” means that there are four physical objects that fit the definition of chairs, and they are located in a spatial location that has been assigned to me as my office, or some such. Now interpret “there are four primes between 40 and 55” in a manner consistent with the previous. A subset of this issue, the “problem of reference,” is, I can refer to physical objects by pointing them 336 A Century of Advancing Mathematics

out, touching them, or via accounts that go back to similar acts. How can we refer to mathematical objects, which are not in physical space with us?

4. Applicability: how can we account for the usefulness of mathematics in explaining phenomena in the physical world? Most of the work outside foundations in the philosophy of mathematics since these papers by Benacerraf has been an attempt to address these issues, while avoiding Benacerraf’s dilemma. There have been two primary approaches to these questions; I will look at how each responds to them.

Platonism or realism The first, generally called “platonism” or “realism,” says that mathematical objects are mind-, space- and time-independent abstract objects. Because they are non-physical, they are also assumed to have no causal interactions with physical beings. They cannot feature in any essential way in any causal explanation of things that happen in the physical world, and nothing that happens in the physical world can change them. So if they exist, but they don’t exist in physical space, nor in our minds, where do they exist? For Plato, there was an alternative realm, where the Forms reside; mathematical objects also reside there. One current response is that it’s the wrong question, a question that doesn’t apply to that type of object, just as “what is the color of 2?” is a wrong question, or “How big are the eyes on that tree?” Members of the category of trees don’t have eyes, members of the category of numbers don’t have colors, and members of the category of abstract objects don’t have places. If there are mathematical objects, then statements about them are true or false (independent of whether we know which is the case). This can be called “realism in truth-value.”So the twin-prime conjecture is either true or false, even though we don’t know yet which it is. Similarly, the continuum hypothesis is independent of the Zermelo-Fraenkel axioms of set theory, but it is either true or false (assuming we have an unambiguous notion of set). One major argument against platonism is called Occam’s razor, also called the “principle of parsimony” — do not multiply entities unnecessarily. So if we can do without mathematical objects, we should. The major philosophical response to this has been the “Quine-Putnam indispensability argument” [14]. It says, we should be committed to the existence of all and only those entities that are indispensable to our best scientific theories. Since mathematical objects are indispensable to our best scientific theories, we should be committed to the existence of mathematical objects. It is in the nature of philosophical discussion that, when there is a deductive argument for some theory, as above, those who disagree with the theory must attack one of the premises. There have been several attacks on this view. These include work by Field [17] trying to show that one can do science without using mathematics; if true, mathematics would not be indispensable. Others have observed that only a very small subset of mathematical objects are actually needed for our scientific theories — in particular, none uses the full set-theoretic hierarchy — and thus, at the least, we do not have to be committed to more than a small collection of mathematical objects. Philosophy of Mathematics: What Has Happened Since Gödel’s Results? 337

The biggest problem for realists is accounting for how humans can have knowledge of mathematics. If mathematical objects are not part of the physical world, and we are physical beings who learn thingsthroughour senses, how can we know anything about mathematical objects? Most work by defenders of realism concentrates on this problem. The best-known mathematical platonist was Kurt Gödel. In 1964 he wrote “But, despite their remoteness from sense experience, we do have something like a perception also of the objects of set theory, as is seen from the fact that the axioms force themselves upon us as being true. I don’t see any reason why we should have less confidence in this kind of perception, i.e., in mathematical intuition, than in sense perception [which also isn’t infallible]” [20]. Philosophers of mathematics since then have rarely tried to follow Gödel’s direction. Rather, a major approach has also been via the Quine-Putnam indispensability argument: we know that mathematical statements are true because we use them in doing science, and the science comes out true. Thus we know that the mathematics must be true. However, this is not entirely satisfactory for a number of reasons. Hence mathematical knowledge remains a major sticking point for realists, and has led to the development of a new philosophical approach, structuralism, which I will discuss in the next section. Another alternative some platonists take is to say that some basic mathematical knowledge comes from experience, and then we make deductions from these basic facts. Resnik [31] is one who has defended this view. Others propose that, since mathematical statements are “necessarily true”, we can develop mathematical knowledge strictly through logic, which is one of the accepted human routes to knowledge. This isn’t quite the same as logicism: we’re not reducing mathematics to logic, but simply using this connection to explain how human beings can know mathematical facts [2]. The semantic question is easy for platonists: mathematical statements are interpreted “at face value” — they mean exactly what they appear to mean. Since there are mathematical objects, statements that appear to be about mathematical objects should be interpreted to be about those objects. “There are four primes between 40 and 55” means just that: there are four objects in the set of prime numbers (41, 43, 47 and 53) that are between 40 and 55. The question of applicabilityis, however, problematic for platonistsfor a similar reason as the epistemological one. Why should objects in some realm independent of the physical world be so useful for science? One possible answer is that the abstract objects we choose to concentrate on are those that are abstracted from real-world problems. Hence their usefulness comes from the fact that we chose to work with objects that would be likely to be useful. However, as the physicist Eugene Wigner observed [36], it has often happened that mathematical objects that appear to have been developed for purely internal reasons have then turned out to be useful in ways not anticipated either by mathematicians or physicists when the concepts were developed. Several philosophers have tried to work on this question, particularly Steiner [35]. Among those who have classified themselves as platonists or realists are Gödel, Quine, Putnam, Resnik, Shapiro [32], and Maddy and Burgess in their earlier writings.

Nominalism and fictionalism For the nominalist, there are no mathematical objects — that is, no abstract, mind- and time-independent objects — or at least, they view the assumption that such objects exist as 338 A Century of Advancing Mathematics similar to adopting a religious view. If there are no such objects, this would seem to imply that statements appearing to be about mathematical objects “don’t refer” — that is, they’re about nothing — and thus cannot possibly be true. This is, however, a major problem, because mathematical knowledge appears to be our most dependable kind of knowledge: if we know anything, we know certain mathematical facts. This is, in fact, the primary argument against nominalism, and thus, by process of elimination, in favor of platonism; Burgess made essentially this argument in [9]. Michael Dummett introduced the distinction between platonism and realism (or anti-realism) in truth value. Some nominalists argue against a belief in abstract objects, but in favor of mathematical statements being true or false (that is, in favor of realism in truth value). Therefore, many nominalists, such as Chihara [11] and Hellman [21], have moved to reinterpreting mathematical statements to find a way that, even though mathematical statements are not literally true, the reinterpreted statements are true, and are such that humans can gain knowledge of them. This is called “paraphrase nominalism,” which claims that we should not read sentences like “3 is prime” at face value, i.e., as being of the form ‘a is F’. Instead such sentences should be paraphrased in one of several ways to reveal their real logical forms. One approach, “if-thenism,” holds that “3 is prime” can be paraphrased by “If there were numbers, then 3 would be prime.” An alternative is to insert modal operators “it is possible that” or “it is necessary that” at the front of such sentences: “It is necessary that 3 is prime” or “It isnecessary that, ifthere is a number 3, it is prime.” We can then gain knowledge of such statements through logic. Field [17] and Yablo are fictionalists, a form of nominalism. For them, there are no mathematical objects, nor any truth or falsity about mathematical statements: they’re just a “façon de parler” that happens to be useful but can be dispensed with if we choose to do so. For fictionalists, mathematical statements are simply “true in our story of mathematics” just as “Sherlock Holmes lived in London” is true in the stories of Conan Doyle and “Sherlock Holmes lived in Scotland” is false in these stories. So how, then, do we gain mathematical knowledge? We study the story of mathematics, and determine whether a given statement is true in that story. There are problems withfictionalism: only Doyle had control over what was happening in his detective stories, but thousands of people get involved in the story of mathematics. One would think that we would have more disagreements about the story, as one group wants the theorem to develop one way, another group a different way. A related view is “conventionalism.” This view holds that sentences like “3 is prime” and “There are infinitely many prime numbers” are analytic. That is, they are true in virtue of linguistic conventions — along the lines of, say, “All bachelors are unmarried.” This takes us back to logicism, which has received renewed attention in recent years, but axioms of mathematics (number theory, or set theory) are not simply analytic — they are not just implicit in the meaning of linguistic conventions. However, maybe they ARE implicit in what we really mean by the integers, or sets — after all, what we try to do with our definitions and axioms is capture the basic concepts. On the question of applicability of mathematics, nominalists run into even more problems than platonists. Why should we be so interested in this story of mathematics (for fictionalists), or in this game of if-then (for paraphrase nominalists)? Why should it be so useful in the world? Science fiction is fun, and occasionally foreshadows actual developments, but scientists do not use it in their papers. Philosophy of Mathematics: What Has Happened Since Gödel’s Results? 339

Structuralism

Structuralismis a new philosophyof mathematics that started in theearly 1980s.It takes off from Benacerraf’s statement, in [4] that“To be3 isno moreor lessthan tobegreater than2 and less than 4. Any object can play the role of 3.” The claim of structuralismis that indeed this is exactly what mathematical objects are: they are places in structures. Structuralism also was developed in response to views being expressed in the mathematical community: Bourbaki’s discussion of “mother structures” and the centrality of structures for mathematics; and many mathematicians’ description of mathematics as the study/science of patterns/structures. For Bourbaki, and many mathematicians studying more modern mathematical concepts (e.g., groups, topological spaces, categories), the important issue isn’t the individual mathematical objects, but how they inter-relate: which groups are isomorphic to which others, for which groups there is a homomorphism from one to the other, etc. Think of the Königsberg bridge problem: the graph involved in this problem can be represented by a certain collection of dots and curves representing the land masses and bridges of Königsberg, or by the city of Königsberg itself, or by a map of the city of Königsberg, etc. Yet from the perspective of the problem and the graph theory that solves it, it doesn’t matter which representation we work with. Does this approach also work for classical analysts and number theorists who are only interested in one or two particular structures where, in a sense, each object has its own characteristics? On reflection, it seems to: when doing number theory, we really don’t care, in any deep sense, what the integers are. If someone wants to believe in them as some kinds of sets, that’s fine. All that is important about them is their relationships: that 1 < 2 < 3 : : :, that each integer has a unique successor, that 2 3 6, etc. D For philosophers, structuralism presents a possible solution to the problems of platonism and nominalism. It answers Benacerraf’s concerns in [4] because now numbers are simply places in the natural number structure, and can be represented by any model of the structure: sets of either type mentioned, or any other sequence ordered like the natural numbers. For platonists, seeing mathematical objects as structures allows our knowledge of mathematics to be explained by our knowledge of more general abstract objects, structures. How this general cognition of structures relates to mathematical proof still needs explanation, however. Structuralistscome in both realist (Resnik [31], Shapiro [33]) and nominalist (Chihara [12]) versions, and they answer this question differently.

Social constructivism

Unlike structuralism, which was started by people in philosophy departments, social constructivism, the other philosophy of mathematics that originated in the last 35 years, was started by mathematicians. Social constructivism has existed in philosophy of science since at least 1979, and appears to have inﬂuenced Philip Davis and Reuben Hersh as early as 1981, intheir book The Mathematical Experience, but onlywaslaid outas a fullphilosophy of mathematics in 1997 (by Hersh [22]) and 1998 (by Ernest [16]). Predecessors of social constructivism include the intuitionists, particularly L. E. J. Brouwer [8], for whom both mathematical objects and mathematical truth are mind- dependent, constructed in each mathematician’s mind. 340 A Century of Advancing Mathematics

For social constructivists, “Mathematical objects are created by humans. Not arbitrarily, but from activity with existing mathematical objects, and from the needs of science and daily life. Once created, mathematical objects can have properties that are difficult for us to discover” [22, p. 16]. Hersh distinguished three types of objects: first, physical objects; second, mental objects — items of individual consciousness — private thoughts before they are communicated to the world and become social. But the third type, where Platonists would say “abstract objects,” he defines as “social entities” or “social-cultural-historic objects.” He gives a lot of examples of these: marriage, divorce, war, jobs, money, prices, sonatas, the U.S. Supreme Court. “All these entities have mental and physical aspects, but none is a mental or physical entity. Every one is a social entity”[22, p. 14]. The Supreme Court isn’t the building in which it is housed, nor the particular judges who are currently its members. It was, however, created at a particular instant in time by the Constitution of the United States, in 1789. Although the court itself is an abstract object (which may come to an end, for example, if the United States disappears or changes its constitution), it has clear interactions with the physical world, making decisions that lead to executions, to reorganization of school systems, etc. So, for social constructivists, mathematical objects are similarly social-cultural-historic objects, created by mathematicians. This in some ways fits very well with how mathematics is actually developed. When groups were initially being developed, the question of whether to include commutativity in the definition was not uniformly agreed upon. Finally the defi- nitionsettled on our current version (withoutcommutativity) and commutative groups were given a separate name, abelian groups. One issue that immediately occurs to people when told that mathematical objects are constructed by mathematicians is, how can it happen that the mathematical community may not be aware of a property that follows from an object it has defined, or of inter- relations among mathematical objects or systems? However, as Hersh observes, knowing enough about some object to uniquely define it does not necessarily mean you know all the deductive consequences of these properties. Thus, although the community develops these objects, they may have unforeseen properties and consequences. Mathematical statements are either true or false, once the mathematical community has developed the objects that the statements describe. Social constructivism gives a response to Benacerraf’s challenge to give an explanation for how mathematicians’ results are systematically true, and how we can come to know them. Mathematical practices are responsible for the basic beliefs being true: the Peano axioms have been accepted as an optimal characterization of the natural numbers. These mathematical practices — for example, being taught from generation to generation — then causally influence why human beings have these beliefs. Mathematicians gain knowledge of more complicated statements through proofs. However, there is a difference between platonists’ proofs and social constructivists’ proofs. For a platonist, if something published in a mathematics paper as a proof contains an error, then it wasn’t a proof, it was a “purported proof.” Real proofs are infallible, and something a student or mathematician writes, claiming it is a proof, either is a proof, or it isn’t. For a social constructivist, a proof is what the mathematical community, at any given moment, says is a proof. Since our notions of rigor, and what constitutes a proof, change over time, this means that something that counts as a proof for one period’s Philosophy of Mathematics: What Has Happened Since Gödel’s Results? 341 mathematicians may not count in later eras. For Hersh “Mathematics can be defined as the subject that consists, primarily, char- acteristically, of conclusive, irresistible reasoning about abstract concepts.” That is, from among all the social-cultural-historical objects, one can distinguish mathematical objects: they “are simply those concepts which are subject to such conclusive reasoning and proof” [23, p. 101]. Social constructivists have good responses to the semantic and applicability questions as well. They agree with platonists that mathematical statements should be interpreted literally. As to the applicability of mathematics, mathematicians study questions suggested by already existing mathematical objects, develop new objects in the process of tryingto figure out how already existing ones work, and also investigate mathematical questions (such as solutions of differential equations) that arise in other sciences. There may even be a hint at an answer to Wigner’s question about the unreasonable effectiveness of mathematics. It not infrequently happens in intellectual history that there are ideas “in the air” and that various fields of the arts respond to these ideas in their own ways, developing responses that are in some ways similar. This may happen with mathematics and physics: the kinds of structures physicists are becoming aware of the need for, are built of concepts that mathematics is, at the same time, developing. There is a range of problems that philosopherssee with social constructivism. The most obvious is that we are finite beings, and in a finite amount of time, can only define a finite number of objects, yet there are uncountably many mathematical objects. Because of this and other problems, philosophers have generally ignored social constructivism, dismissing it with all other viewpoints that see mathematical objects as mental constructs as “psy- chologism.” However, there is one young philosopher, Julian Cole, who has tried to find a variant of social constructivism that will be acceptable to the philosophical community [13].

Philosophy of real mathematics/mathematical practice Another new direction in the philosophyof mathematics is the study of how mathematics is actually done. This is on the boundary between philosophy and sociology of mathematics, but to the extent that it discusses how we come to mathematical knowledge, it is philosophy. Probably the earliest work in this direction was done by Imre Lakatos, a student of Karl Popper. He applied Popper’s work on how scientific theories are either shown to be false or are confirmed, to mathematics with his thesis in 1961, eventually published posthumously in 1976 [25]. Rather than studying mathematics in its deductive published form, he was interested in the process of mathematical discovery. The book is in the form of a classroom dialogue between a teacher and some students, investigating what was presented as a conjecture, Euler’s theorem for polyhedra, V E F 2, via proofs, counterexamples and C D monster-barring (revising definitions or hypotheses to make the theorem true). More recently, David Corfield led a resurgence in this area with [15] in which he argued that too often, philosophers are busy with foundational issues or restrict their discussion to the natural numbers, and ignore how mathematicians actually do research. Among the issues he urges philosophers of mathematics to consider are distinguishing between apparently mathematical entities that receive a large amount of attention and those that 342 A Century of Advancing Mathematics are arbitrarily cooked-up; the unity or connectivity of mathematics; cases of unexpected discovery; how mathematical knowledge grows; what is meant by mathematical progress; what is meant by explanation in mathematics. Among the topics he discusses are automated theorem provers and conjecture formulation, the role of analogy in mathematics, and issues in probabilitytheory. In 2010, the Association for the Philosophy of Mathematical Practice, devoted to further work on these issues, was started. A number of mathematicians, of course, have made contributions in this area. Barry Mazur, in particular, has written several articles about philosophical aspects of mathematical practice, discussing conjecture [27] and the identityof mathematical objects [28] among other topics. In the next two sections, I will discuss some recent philosophical work on topics that cut across particular philosophical schools.

Visualization Recently a number of philosophers and mathematicians have explored aspects of the role of visualization in mathematics. Essentially all mathematicians use diagrams, both in our own research and in teaching mathematics. When we do research, we often use diagrams as we are developing our conjectures, checking out special cases, trying to develop a proof, but when we write papers for publication, we often do not include the diagrams, or include only those most essential to understanding. From the earliest days of mathematical proof, there have been concerns about the use of diagrams in proofs. There are two main reasons: their unreliability and their particularity. There are well-known examples used to support the charge of unreliability, such as the famous diagrammatic “proof” that all triangles are isosceles. The issue of particularity is that most mathematical statements are general truths, while diagrams only show particular cases. So a diagram cannot justify such a truth except by some sort of analogical or broadly inductive extension. When Hilbert attempted to take euclidean geometry and formalize it via axioms, he found that there were several concepts that were taken for granted in euclidean geometry but not ever stated in the assumptions: the concepts of betweenness, of lying on one side of a given line or inside a circle, and the completeness of the real line, for example. These concepts were used on occasion in theorems or proofs, but often they were implicit in a proof, dependent on the diagram. In recent years, several mathematicians and philosophers have reexamined these issues, proposing that, at least under certain circumstances, diagrams can be a reliable method of proof. The ﬁrst was by Barwise (a logician) and Etchemendy (a philosopher) in 1991 [3]. They note that false proofs can be developed algebraically as well as diagrammatically. For example, there are many proofs that 0 1 arrived at by dividing by 0 in a hidden way. D To illustrate legitimate use of diagrammatic proof, they give an example of a well-known proof of the Pythagorean theorem via a diagram that has a square of side a b, with each C corner a right triangle with sides of lengths a and b, and an internal square with side c formed by the hypotenuses of these triangles. One then uses some algebra, together with the fact that the total area of the large square is also the sum of the four triangles plus the area of the internal square, to get the theorem. They claim that it is clearly a “legitimate proof of the Pythagorean theorem.” It involves a combination of “geometric manipulation of a diagram and algebraic manipulation of nondiagrammatic symbols.” The diagrammatic Philosophy of Mathematics: What Has Happened Since Gödel’s Results? 343 elements “play a crucial role in the proof.” Further, the proof (clearly) does not make use of “accidental features” of the diagrams involved [3, p. 12]. In 2007, Giaquinto published a book [19] exploring in much greater detail the use of diagrams in geometric reasoning, and concludes that there are many cases in geometry where diagrammatic reasoning is used in a non-superﬂuous but valid way. He also notes that Nathaniel Miller has proved that a particular formal system of diagrammatic proofs of euclidean geometry, called “FG,” is in fact sound; that is, you cannot get improper proofs in his system. On the other hand, Giaquinto notes that there are serious problems in using diagrammatic proofs in analysis that, unlike geometry, cannot be overcome. As a result, even gaining a level of knowledge short of proof is not possible. For example, could someone have discovered Rolle’s theorem via visualization? Analyzing the range of potential counterexamples, he concludes that, while individual cases might be suggestive, they would pre- vent a conjecture from being knowledge.

Proof Proofs play a range of roles in mathematics. The primary role is verification that a proposed theorem is in fact true. Of probably equal importance, though, is helping mathematicians understand why the result is true. Although proof remains the standard method of verification in mathematics, there have been several challenges to its role starting in the mid 1970s. In order for a proof to verify a theorem, it must either be written in a formal language that allows mechanical checking of each step (which virtually never happens with new mathematical theorems), or must be surveyable by mathematicians so that they understand why each step is valid. The first of these generally conflicts with its second role, since adding the details needed for a formal proof to any but the most trivial theorems makes it both too long and too tedious for mathematicians to gain understanding by reading the proof. The first challenge to the role of proofs, in 1976, was the Appel-Haken proof of the four-color theorem using a computer to verify approximately 2000 specific configurations. Mathematicians can read and understand the reduction to the configurations, and can read the program code to determine whether it appears to be doing what it purports to do. But we cannot check that there has not been an error as the computer compiles and runs the program. The computer algorithm was tested several times, and since then a simpler proof using the same ideas and still relying on computers was published in 1997. Additionallyin 2005, the theorem was proven by Gonthier with general purpose theorem-proving software. So given the range of confirmatory developments, mathematicians are inclined to believe that it has, in fact, been proven. More recently, in 1998 Hales gave a proof of Kepler’s conjecture, a 400-year-old problem, that the most efficient packing of spheres is the hexagonal close packing (the way fruit vendors stack grapefruits). The top-level outline of the proof, eventually published in the Annals of Mathematics, was over 100 pages, without the details, and without the computer code that did the detailed checking. Twelve referees worked for four years on the original full version submitted to the Annals. In the end, they were unable to completely certify the proof, although they reported they were 99 percent certain of it. In the meantime, the classification of finite simple groups was accomplished by a collection of several hundred journal articles published over a fifty-year period. All of these 344 A Century of Advancing Mathematics proofs raise questions in terms of what constitutes an acceptable proof, and the relationship between proofs and mathematicians. To what extent do the above-mentioned proofs give us an understanding of the results, even if they do verify that the results are true? Philosophersof mathematics have normally taken proof to mean formal proof, although they have gradually been trying to come to terms with proof as a methodology in mathematics that is not identical to formal proof. For a recent discussion of this question, see [1]. Meanwhile, the increasing use of computers to explore mathematical objects and relationships has led to a community of mathematicians who are more interested in the results of the computer experiments than in justifying the results with proofs. They have sometimes found robust evidence supporting assertions that have resisted proof. This led to an opinion piece in the Bulletin of the AMS by Jaffe and Quinn [24], “‘Theoretical mathematics’: To- ward a cultural synthesis of mathematics and theoretical physics.” In it they proposed that, just as physicists have split into theoretical and experimental physicists, mathematicians should do so also, with “theoretical” mathematicians being free to use intuitive methods. The response of most mathematicians to their proposal has been quite negative, with a small subcommunity having the opposite response. I think the reason for the negative response is a combination of concern that false results will creep into the literature, together with the fact that most mathematicians are more interested in understanding why a result is true than in simply knowing that it is true.

Foundations revisited Although my doctorate was in mathematical logic, in the early 1980s my interest changed to the philosophy of mathematics and I have not kept up in detail with developments in the ﬁeld. So although the material discussed above I have read in detail, the material I will now discuss I am much less familiar with. I could not, in any case, in the space remaining discuss all the major developments in the foundations of mathematics since G¨odel. What I will try to touch on are those with substantial philosophical relevance.

Set theory: Cohen, independence results, large cardinal axioms In 1938, Gödel proved that, if the usual axioms of set theory are consistent, then one can form a submodel, called the constructiblesets, in which the axiom of choice and the continuum hypothesis are true, thus showing that these axioms are consistent with the rest of the axioms of set theory. Then in 1963, Paul Cohen developed a new method, called forcing, that generated extensions of a given model, and used it to show that the negation of boththe axiom of choice and the continuum hypothesis are also consistent. As a result, both axioms are independent of the other axioms and of each other. At first glance, this seems to be similar to the independence of the parallel postulate in geometry. There are several possible extensions of neutral geometry (euclidean geometry without the parallel postulate), one of which is euclidean geometry, and the others non- euclidean geometries. The question of which is the actual geometry of space then becomes a question for physics, not mathematics or philosophy. The situation for set theory is somewhat different. The question becomes, do we have several different conceptions of set, in one of which these additional axioms are true, oth- Philosophy of Mathematics: What Has Happened Since Gödel’s Results? 345 ers in which one, the other, or both, are false? Or do we actually have one concept of set, and simply have not yet discovered appropriate new axioms that would tell us whether the continuum hypothesis and the axiom of choice are true or false for our actual concept of set? Gödel was in the latter camp, and worked for many years trying to discover appropriate axioms, although he never achieved a result in this direction that he felt was ready to publish. However, most work in set theory since Cohen appears to have taken the first direction, and explored a wide range of possible independent set theories. An assortment of new axioms have been introduced, including the Axiom of Determinacy, the Axiom of De- pendent Choice, and large cardinal axioms (inaccessible cardinals, measurable cardinals). At the International Congress of Mathematicians in 2010, Hugh Woodin [37] introduced a concept, ultimate L, that he suggests may in fact take things in the opposite direction, toward a single concept of set. Using universally Baire sets he introduces a generalization of Gödel’s constructible universe L. Adding as an axiom “V ultimate L” resolves the D continuum hypothesis (positively) and reduces all questions in set theory to large cardinal axioms. Taking set theory in a very different direction is the study of fuzzy set theory, where, rather than an object being either a member or not a member of a given set, it is assigned a degree of membership anywhere on the interval [0,1]. The development of category theory has led some to question whether set theory is an appropriate foundational system for mathematics in any case, since categories and functors often are not proper sets. Topos theory [29] has been proposed as an alternative foundation.

Modern logicism: fixing Frege Starting in 1973, a number of philosophers (Dummett [18], Boolos, Wright, Hale) have attempted to fix the paradoxes that arose in Frege’s work. What they do is go back to Hume’s principle, “For any concepts F, G, the number of F’s is identical to the number of G’s if and only if F is equinumerous with G.” Rather than continuefrom there to Basic Law V as Frege did (which led to paradoxes), they use Hume’s principle to define numbers. So 0 can be defined as the number of things for which x x, 1 as the number of the ¤ concept “identical to 0,” etc. Hale and Wright have argued that Hume’s principle is an implicit definition of number, and thus true by stipulation. There are issues with this, since Basic Law V is of the same form as Hume’s principle, but is inconsistent — so when is it acceptable to stipulate such things and when not? For a thorough discussion of this work, see [10]. For more information on recent developments in logicism, intuitionism and formalism, see [26].

Modern intuitionism and computability Although several significant mathematicians, such as Weyl, initially were attracted to Brouwer’s intuitionism, the vast majority of the mathematical community was unwilling to follow its implications and give up most of modern analysis (and modern algebra). Giv- ing up completed infinities and the law of the excluded middle results in a very bizarre real number system (using “free choice sequences” to build non-definable real numbers) and throws out too many theorems we like, such as the intermediate value theorem. 346 A Century of Advancing Mathematics

In the late 1940s A. A. Markov developed an alternative form of constructive mathematics, “which is, essentially, recursive function theory with intuitionistic logic” [7]. “The objects are defined by means of Gödel-numberings, and the procedures are all recursive” [7]. It requires both an understanding of recursive function theory and of analysis, which makes it unattractive for most analysts. Errett Bishop, in the 1960s, led a revival of a variant of intuitionism which he called constructive mathematics. In 1965, he published [6], in which he gave a more attractive definition of real numbers, and developed constructively true theorems that are classically equivalent to the important classical theorems of analysis. This work has some interest even for classical mathematicians, because he was very good at getting new information from the constructions needed for proving these theorems. That is, while it’s nice to know, via a non-constructive proof, that a certain object exists, it’s more useful to actually be able to construct the object. Followers of Bishop have worked on developing modern algebraic concepts constructively. Although this work has not become part of mainstream mathematics, questions about what functions can be given constructively is of considerable interest to computer science. There has been a lot of work, first by logicians and later also by computer scientists developing the assorted hierarchies related to recursive sets and computational complexity: the arithmetic, analytic, and projective hierarchies involving sets generated by statements using quantification over second-order arithmetic; functions computable in polynomial time, non-deterministic polynomial time, exponential time (generating the famous P NP D problem). Gödel’s incompleteness theorems have led to a wide range of extensions, including un- decidability results for various mathematical systems and the unsolvability of the halting problem for Turing machines. The unsolvabilityof Hilbert’s tenth problem (finding a general algorithm to determine whether a given diophantine equation has solutions in integers) through proving that every recursively enumerable set is diophantine involves much work beyond Gödel’s incompleteness theorems but those theorems are part of its ancestry. The Paris-Harrington theorem, in 1977, was the first natural example of a true statement about integers, the strengthened Ramsey theorem about colorings of n-element subsets of an initial segment of natural numbers, that was shown to be unprovable in Peano arithmetic. The question of finding useful explicit true arithmetical statements that are independent of arithmetic or even set theory with large cardinals remained open. Harvey Friedman has developed numerous such combinatorial principles about the integers which are independent of very large cardinal extensions of ZFC.

Formalism/proof theory Gödel’s incompleteness theorems showed that the full Hilbert program, to formalize all of mathematics via a (presumably recursive, or better, finite) set of axioms and prove that this formalization is consistent, is not possible. Soon after the incompleteness theorems, logicians, beginning with Gentzen in the 1930s, started work on a range of modified Hilbert programs, a field that became known as proof theory. The hope was that consistency proofs might give insight into the constructive content of arithmetic. Relative consistency proofs by Gentzen and Schütte involve induction up to assorted ordinals. For these results to have Philosophy of Mathematics: What Has Happened Since Gödel’s Results? 347 foundational significance would require a constructive argument justifying such induction. Kreisel and Feferman suggested a program of looking for reductions of systems weaker than full classical mathematics to theories not too much stronger than finitary mathematics. A lot of work of this sort has been done, but its philosophical implications are debatable. Friedman and Simpson’s program of reverse mathematics approaches these questions from a different perspective: they look at how much of classical mathematics can be reduced to finitary mathematics. That is, “which theorems of classical mathematics are provable in weak subsystems of analysis which are reducible to finitary mathematics?” [38]. Reverse mathematics is an axiomatic version of recursive mathematics. Its importance is in showing how much contemporary mathematics does not depend on working in ZF, but rather in a limited second-order extension of arithmetic. “A typical result is that the Hahn-Banach theorem of functional analysis is provable in a theory known as WKL0 (for “weak König 0 lemma”); WKL0 is conservative over PRA [primitive recursive arithmetic] for 2 sentences (i.e., sentences of the form x yA.x; y/”[38]. 8 9 Q Some philosophers (including Lucas and Penrose [30]) have tried to use Gödel’s incompleteness theorems to demonstrate that the human mind goes beyond any mechanism or formal system. These attempts have largely been rejected by philosophers who have argued that, for the argument to work, people would have to be able to always see whether a formalized theory is consistent.

Non-standard analysis and infinitesimals One direct application of Gödel’s completeness theorem (and the compactness theorem) is that there are non-standard models of the natural numbers and the real numbers — that is, models that have all the first-order properties of these systems, yet also have infinite elements (greater than all standard numbers). The use of infinitesimals had been banished from analysis in the 1900s in the process of making it rigorous,in favor of the now standard ı definition of limit, because of the many apparent contradictions that had arisen in mathematics. Abraham Robinson in the 1960s observed that, by taking a nonstandard model of the real numbers, one can introduce infinitesimals completely rigorously (as reciprocals of infinite numbers). (The same models can also be introduced via ultrafilters.) Although mathematically a straightforwardobservation coming from Gödel’s work, this application’s philosophical implications caused a considerable stir. It led to calculus textbooks written using infinitesimals, and several new theorems in analysis (and, in particular, applications to economics) initially being proved using non-standard analysis, although the proofs were then translated into standard proofs quite routinely. An important example was Robinson and Bernstein’s proof that every polynomially compact linear operator on a Hilbert space has an invariant subspace, which Halmos immediately rewrote using standard techniques. This is a good illustration of the difference between methods of discovery in mathematics, where a particular approach may have given someone a perspective leading to solving a problem, versus methods of justification, which, at least so far, can still be turned into classical techniques.

Acknowledgements I would like to thank Michael Morley and Anil Nerode for a few suggestions of items for the section on Foundations revisited, and Thomas Drucker and Roger Simons for helpful comments on the ﬁrst draft. 348 A Century of Advancing Mathematics

Bibliography [1] J. Azzouni, The relationship of derivations in artificial languages to ordinary rigorous mathematical proof, Philosophia Mathematica (3) 21, 2013, pp. 247–254. [2] M. Balaguer, Mathematical Platonism, pp. 180–204 in B. Gold and R. Simons, eds., Proof and Other Dilemmas: Mathematics and Philosophy, Mathematical Association of America, Washington DC, 2008. [3] J. Barwise and J. Etchemendy,Visual Information and Valid Reasoning in W. Zimmerman and S. Cunningham, eds., Visualizing in Teaching and Learning Mathematics, Mathematical Asso- ciation of America, Washington DC, 1991. [4] P. Benacerraf, What Numbers Could Not Be, reprinted, pp. 272–294, in Benacerraf and Put- nam, Philosophy of Mathematics: Selected Readings, 2nd ed., Cambridge University Press, Cambridge, 1983. [5] ———, Mathematical Truth, reprinted, pp. 403–420, in Benacerraf and Putnam, Philosophy of Mathematics: Selected Readings, 2nd ed., Cambridge University Press, Cambridge, 1983. [6] E. Bishop, Foundations of Constructive Analysis, McGraw Hill, New York 1967. [7] D. Bridges and E. Palmgren, Constructive Mathematics, The Stanford Encyclopedia of Philosophy (Winter 2013 Edition), Edward N. Zalta (ed.), plato.stanford.edu/ archives/win2013/entries/mathematics-constructive/. [8] L. E. J. Brouwer, Intuitionisme en Formalisme, Gronigen, Noordhoof, 1912; translated as “In- tuitionism and Formalism,” pp. 77–89 in Benacerraf and Putnam, Philosophy of Mathematics: Selected Readings, 2nd ed., Cambridge University Press, Cambridge, 1983. [9] J. Burgess, Why I am not a nominalist, Notre Dame Journal of Symbolic Logic 24 (1983), pp. 93–105. [10] ———, Fixing Frege, Princeton University Press, Princeton, 2005. [11] C. Chihara, Constructibility and Mathematical Existence, Oxford University Press, Oxford, 1990. [12] ———, A Structural Account of Mathematics, Oxford University Press, Oxford, 2004. [13] J. Cole, Mathematical Domains: Social Constructs? pp. 109–128 in B. Gold and R. Simons, eds., Proof and Other Dilemmas: Mathematics and Philosophy, Mathematical Association of America, Washington DC, 2008. [14] M. Colyvan, Indispensability Arguments in the Philosophy of Mathematics, The Stan- ford Encyclopedia of Philosophy (Spring 2011 Edition), Edward N. Zalta (ed.), plato.stanford.edu/archives/spr2011/entries/mathphil-indis/. [15] D. Corfield, Towards a Philosophy of Real Mathematics, Cambridge University Press, Cam- bridge, 2003. [16] P. Ernest, Social Constructivism as a Philosophy of Mathematics, State University of New York Press, Albany, 1998. [17] H. Field, Science Without Numbers, Princeton University Press, Princeton, 1980. [18] M. Dummett, Frege: Philosophy of Mathematics, Harvard University Press, Cambridge, MA, 1991. [19] M. Giaquinto, Visual Thinking in Mathematics: An Epistemological Study, Oxford University Press, Oxford, 2007. [20] Kurt Gödel, What is Cantor’s continuum problem? 1964, reprinted, pp. 470–485, in Benacerraf and Putnam, Philosophy of Mathematics: Selected Readings, 2nd ed., Cambridge University Press, Cambridge, 1983. [21] G. Hellman, Mathematics Without Numbers, Oxford University Press, Oxford, 1989. Philosophy of Mathematics: What Has Happened Since Gödel’s Results? 349

[22] R. Hersh, What is Mathematics, Really? Oxford University Press, Oxford, 1997. [23] ———, Mathematical Practice as a Scientific Problem, pp. 96–107 in B. Gold and R. Simons, eds., Proof and Other Dilemmas: Mathematics and Philosophy, Mathematical Association of America, Washington DC, 2008. [24] A. Jaffe and F. Quinn, Theoretical mathematics: towards a cultural synthesis of mathematics and theoretical physics, Bull. Amer. Math. Soc. 29 (1993), pp. 1–13. [25] I. Lakatos, Proofs and Refutations: the Logic of Mathematical Discovery,Cambridge University Press, Cambridge, 1976. [26] S. Lindström et al, eds., Logicism, Intuitionism, and Formalism: What Has Become of Them?, Springer, 2009. [27] B. Mazur, Conjecture, Synthese 111 (1997), pp. 197–210. [28] ———, When is One Thing Equal to Some Other Thing? pp. 221–241 in B. Gold and R. Simons, eds., Proof and Other Dilemmas: Mathematics and Philosophy, Mathematical Associ- ation of America, Washington DC, 2008. [29] C. McLarty, Elementary Categories, Elementary Toposes, Oxford University Press, Oxford, 1996. [30] R. Penrose, The Emperor’sNew Mind: ConcerningComputers,Minds, and the Laws of Physics, Oxford University Press, New York, 1989. [31] M. Resnik, Mathematics as a Science of Patterns, Oxford University Press, Oxford, 1997. [32] S. Shapiro, Mathematics and Reality, Philosophy of Science 50 (1983), pp. 523–548. [33] ———, Philosophy of Mathematics: Structure and Ontology, Oxford University Press, New York, 1997. [34] Stanford Encyclopedia of Philosophy at plato.stanford.edu/. [35] M. Steiner, The Applicability of Mathematics as a Philosophical Problem, Harvard University Press, Cambridge MA, 1998. [36] E. Wigner, The Unreasonable Effectiveness of Mathematics in the Natural Sciences, pp. 222– 237 in his Symmetries and Reflections, Indiana University Press, Bloomington IN, 1967. [37] W. H. Woodin, TheSearchfor Ultimate L, Ziwet Lectureat the University of Michigan, Novem- ber 2010, at www.math.lsa.umich.edu/video/ziwet1.pdf. [38] R. Zach, Hilbert’s Program, The Stanford Encyclopedia of Philosophy (Spring 2009 Edi- tion), Edward N. Zalta (ed.), plato.stanford.edu/archives/spr2009/entries/ hilbert-program/.

Monmouth University, Mathematics Department, 400 Cedar Avenue West, Long Branch, NJ 07764 [email protected]

Twelve Classics People who Love Mathematics Should Know; or, ‘‘What do you mean, you haven’t read E. T. Bell?"

Gerald L. Alexanderson Santa Clara University

I’m old enough to think that everyone in our ﬁeld must be familiar with Men of Mathemat- ics, but I ﬁnd that some of my younger colleagues have never heard of it. I hope to remedy this sad example of a generation gap. The following list of twelve classics in mathematics would be appropriate as recommended reading in a general mathematics course for non- majors, but I would go further: they should be required reading even for undergraduate mathematics majors. We should never neglect our majors.

Men of Mathematics, by Eric Temple Bell (Simon & Schuster, 1937) Yes, the title may offend some people these days, but in 1937 readers probably thought it perfectly natural. However insensitive the title, it has been a thrilling discovery for many mathematics students over a lot of years and it can still be read with pleasure today. Bell, a professor and serious algebraist at Caltech—a winner of the Bôcher Prize for his memoir “Arithmetical paraphrases”—was an author of several general books on mathematics, including the very useful history, The Development of Mathematics (McGraw-Hill, 1940). I realize now, of course, that some of Bell’s stories of mathematicians—ranging from Zeno to Cantor—were occasionally embellished with exaggerated claims, but what great stories they are! When I read them I was not much concerned about the real Galois. I wanted to think of him and of Abel as being great romantic figures. And they happened to be mathematicians. Fans of English poetry have their heroes likeKeats, Shelley and Byron, who died young in exotic places like Rome and Greece. Then there was Elizabeth Barrett Browning, who did not die young—she lived to be 55—but was an eminently romantic figure in her time. By contrast, I had Abel and Galois, and the beautiful Sonya Kovalevskaya. Abel and Galois managed to die at a very youngage, which always adds to the reputationof geniuses. And though Norway and Paris are perhaps no match for Greece and Rome, Paris seemed very romantic to me in my youth. Galois even fought a duel! That is a story right out of grand opera—an art form I very much liked then and still do. Abel’s reputation did not rely totally on words and formulas. The heroic statue of Abel by Gustav Vigeland in the Royal Park in Oslo is as good an over-the-top romantic piece of sculpture as one could imagine, far surpassing the impact of Edward Onslow Ford’s white

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Where does Romanticism reside—triumphant Abel or recumbent Shelley? marble figure of Shelley at University College, Oxford. The difference is that Abel looks triumphant, astride two prone figures representing, we are told, the quintic polynomial equation and an elliptic function (it’s hard to know whether they are good likenesses!), whereas the recumbent Shelley looks dead, as he would, having just washed ashore after drowning at sea. I prefer my heroes to be triumphant and successful. Sonya Kovalevskaya appears in Men of Mathematics only in thesection about her mentor, Karl Weierstrass, but if the book were written today she, Emilie´ du Châtelet, Sophie Germain and the considerably later Emmy Noether, at the very least, would deserve having their own chapters. Bell wrote extensively about mathematics, but he also wrote science fantasy books under the nom de plume, John Taine, so the man was an experienced and unusually gifted writer.It turnsout that his lifewas more complicated than his colleagues, or even hisfamily, might have suspected. During the 1980s Don Albers and I, both having been influenced to go into mathematics by Bell’s book, began exploring the idea of writing something biographical about Bell. This was made feasible by the fact that Bell’s son, Taine, was a pediatrician who was practicing in Watsonville, California. So we were able to get firsthand accounts of Bell and examine family records and memorabilia. Since we were busy doing other things and slow to get to this project, we persuaded Constance Reid, Hilbert’s biographer, to take over. She discovered all kinds of “secrets” about Bell that he had covered up for years. Her book about him appeared as The Search for E. T. Bell/Also Known as John Taine, published by the MAA in 1993. The story of Bell’s lifeis comparable to some of his stories of the lives he wrote about,but with one difference. The details of his lifeare almost certainly entirely true.

Hilbert, by Constance Reid (Springer, 1970) Constance Reid was not a mathematician, but she was the sister of a very well-knownmath- ematician, Julia Robinson, who was married to another mathematician, Raphael Robinson, Twelve Classics People who Love Mathematics Should Know 353 a number theorist at Berkeley. Julia became widely known in the 1970s for her work on Hilbert’s famous tenth problem, eventually solved completely by Yuri Matiyasevich, who provided a full proof built on work by Julia and several others. The solution is viewed as one of the great achievements of twentieth-century mathematics. Julia went on to become the first woman to be president of the American Mathematical Society and the first woman in mathematics to become a member of the National Academy of Sciences (US). This biographyofHilbertcame outwhen I was well alongin my career, but itconvinced me of the stature of this great mathematician: his probity, his wisdom, and the universal respect that he inspired. Hilbert was no rebel like Galois. He lived the conventional life of an academic at Göttingen. A solid member of the bourgeoisie, he seemed to possess an almost universal knowledge of the mathematics of his day, and his advocating for Emmy Noether, for example, confirmed his place in history as a superb human being. In writing about Hilbert, Reid faced skepticism from the German mathematical community: what made an American woman—who wasn’t even a mathematician—think she could write about the great Hilbert? She ended up not only writing a universally admired book, she also performed a service by rescuing boxes of Hilbert’s papers that had been stored in Göttingen without any sorting or attention since Hilbert’s death. Left as they were, they might well have deteriorated over time and been lost. It wasn’t easy, however, finding someone to publish it. Reid told me she first had in mind a modern version of Bell’s book, not providing real biographies but instead, as she put it, describing lives “in the Plutarchian sense,” moving on from where Bell had left off. Earlier she had written a couple of books on mathematics for the general reader, so she went to her publisher, Robert Crowell, and told him she wanted to write about lives of mathematicians. He pointed out that the only thing that would sell worse than a biography of a mathematician would be a book about South America! She persisted, however, and found that when she got to Hilbert she wanted to write a whole book about him. When she finished the manuscript, she took it to Crowell, but by then Richard Courant was insisting that it should be published by Springer. They ended up publishingit. Klaus Peters was then a mathematics editor at Springer and, to his great credit and throughout his distinguished career, he had taken risks on a lot of projects that were just slightly out of the ordinary in their day. This book was one of his best moves. Constance told me that she thought that Crowell would have published it, but when he heard that Springer was bringing it out, he was probably much relieved. My only disappointment in the volume is that it does not include a short recording of Hilbert’s voice, that to Springer’s credit, was included in another book of theirs that appeared in 1971: Hilbert: Gedenkband, herausgegeben von K. Reidemeister. The recording1 is an excerpt from Hilbert’s often quoted 1930 address in Königsberg, “Naturerkennen und Logik” and concludes with the famous admonition: “Wir müssen wissen, wir werden wissen (We must know, we will know).” Fighting words! I cannot pass up a chance to point out that shortly after the book on Hilbert, Reid published (again with Springer) a book, Richard Courant in Göttingen and New York: the Story of an Improbable Mathematician. I was interested in a comment by Don Albers recently: he said that he read the book in one day. That was my experience, except that I read it in

1This recording can be heard at the MAA’s website: www.maa.org/publications/periodicals/ convergence/david-hilberts-radio-address-german-and-english. 354 A Century of Advancing Mathematics one night. I started it and couldn’t put it down till I finished it in the early hours of the morning. Anyone interested in Göttingen and the work of Courant has to be susceptible to the appeal of this extremely well-written book. In the 1930s Courant replicated at New York University the success in Göttingen by bringing over many émigrés from Germany to join the faculty (particularly in applied mathematics) at NYU.

A Mathematician’s Apology, by G. H. Hardy (Cambridge University Press, 1940) This short volume, roughly 100 pages, was written late in Hardy’s life and it is a wistful reflection of a great man looking back on his life as a creative mathematician. It is a vale- dictory, rather autumnal in tone—a summing up of his accomplishments but with regret that he no longer had the power to attack great problems. He had to fall back on hispassion for following cricket. It was not the same. Hardy’s life at Cambridge is the subject of legend. Bringing the genius Ramanujan to England from India, making his astonishing work known, and collaborating with him, would have been enough to establish Hardy’s eminence in the mathematical world, but he independently contributed powerfully to mathematics in other areas. Some of his greatest work was done in collaboration with J. E. Littlewood.There are those who claim that there were three great mathematicians in Britain in the twentieth century: Hardy, Littlewood, and Hardy-Littlewood! Though reported to have been rather reserved, Hardy was also an attractive and colorful figure and his endearing eccentricities were legendary. The literature abounds in tales of his life-long difficult relationship with God, his overall erudition, and his amusing lists, often ranked, of mathematicians, poets, and cricket players. Five years before his death he published a little-known work, Bertrand Russell and Trinity, in which he defended Russell who had been dismissed by the Fellows at TrinityCollege for his polit- ically unpopular views. Hardy failed to get Russell reinstated, but the published arguments provide a fascinating glimpse into the academic life at Trinity at the time. He quotes a letter from A. E. Housman who expressed distress over the treatment of Russell but, with the poet’s world-weary view, added “This ought not to be, but the world is as God made it.” Hardy held appointments to venerable chairs at both Oxford and Cambridge and generally played an important role in the intellectual life in England in the first half of the century. One of his close friends was the economist John Maynard Keynes. Beginning with the reprinting of the Apology in 1967, C. P. Snow, a novelist and a good friend of Hardy’s, added a foreword of personal reminiscences. Snow claims that Hardy modestly ranked himself as number five among mathematicians in the world at that time. Unfortunately I have not found the preceding names on that list. Hardy was a passionate defender of pure mathematics and claimed that he was proudof never having produced any mathematics that would turn out to be “useful.” With the rise of importance of the Hardy-Weinberg Principle in population studies, one can only speculate what he might have to say today about his life’s work.Atle Selberg toldme that in his view, A Mathematician’s Apology goes beyond other books in this genre and is indeed a work of literature.

Mathematical Discovery, by George P´olya (John Wiley & Sons, 1962, 1965) It is tough to choose something from among P´olya’s many books, but for sheer reading pleasure I have picked my personal favorite, Mathematical Discovery. It is, however, a dif- Twelve Classics People who Love Mathematics Should Know 355

ficult call, between this two-volume set of elementary problems clustered to demonstrate problem-solving techniques, and the earlier two-volume set, Mathematics and Plausible Reasoning (Princeton University Press, 1954), much in the same format but with problems at a more sophisticated level. Two other possibilities still remain: the much earlier set, Auf- gaben und Lehrsätze aus der Analysis (Problems and Theorems in Analysis), with Gábor Szeg˝o, of 1925, and the small best-selling book, How to Solve It, published in 1945. How To Solve It continues to be much in demand, even after almost seventy years in print. Fur- ther, it has been translated into something like twenty-five languages, an amazing record for a mathematics book. There is no question that it has charm, but the mathematics included is rather too elementary to be very appealing to me and I prefer the eye-opening topics and problems in the other collections. In Mathematical Discovery we find small gatherings of problems that move in various directions to explore analogues in 3-space of the Pythagorean theorem in the plane, for example. Or we can follow a set of problems that use recursion to find the sum of the kth powers of the first n positive integers, a problem solved by the great Jacob Bernoulli many years earlier, but only with some effort, as he pointed out in his Ars Conjectandi in 1713. There are variations on Pascal’s triangle, including Leibniz’s harmonic triangle, and figurate numbers, and the use of homothetic points in geometry. Of course, if one tires of problems that are so elementary, then one can move on to Mathematics and Plausible Reasoning to explore problems like (1) Euler’s Basel problem, finding the value of .2/ (the sum of the reciprocals of the squares of the positive integers); or (2) Euler’s discovery of the recursive formula for the general partition function for integers, using the pentagonal number theorem, something Euler had failed to prove; or (3) a variation on (2) but instead on the recursion formula for the sum of divisors function; or (4) riffs on Waring’s problem such as counting the number of ways of representing multiples of four as sums of four odd squares; or (5) the discovery of patterns in slicing the plane with lines, 3-space with planes, and so on (Steiner’s problem). All of these books are full of beautiful mathematics, with the added advantage that they show ways of attacking natural but non-trivial, and often truly challenging, problems.

Proofs and Refutations, by Imre Lakatos (Cambridge University Press, 1976) George Pólya introduced me to this book when it came out and I have read it and read it again over many years. It is a brilliantSocratic dialogue on the evolutionof Euler’s formula for polyhedra. It shows something that mathematicians often face—the task of trying to prove something they do not quite understand when they formulate a conjecture. As Pólya put it, you have to guess a mathematical theorem before you can prove it. In the case of Euler’s formula for polyhedra the question is first of all, “What is a polyhedron?” Then, what if itis notconvex? Or, what if ithas a hole in it? What if the faces have holes in them? What if we look at analogues in higher dimensions? Though the original statement might seem quite straightforward, the problem evolves as one becomes more and more aware of pitfallsalong the way. The end result is a much more refined statement of the problem. The writing is a tour de force, though the task of keeping track of who is saying what, and how their views can alter the problem, can be a challenge. The “students” in the dialogue are given Greek letters as names and there are practically enough of them to use up the whole Greek alphabet. But here we come to what I find the most enjoyable aspect of 356 A Century of Advancing Mathematics the book: the footnotes are wonderful. One often hears complaints about books and papers that have an excessive number of footnotes. They make a text look scary. If anything, my reaction to the many, many footnotes in Proofs and Refutations is that the footnotes are more interesting than the text. The narrative in the footnotes on the history of the problem is so compelling that I tend to open the book just to read the footnotes. Now, how often does that happen?

A Survey of Geometry, by Howard Eves (Allyn and Bacon, 1963, 1965) In my first years after graduate school I was asked to teach a special sequence of courses in the history and philosophyof mathematics for a newly established HonorsDivision at Santa Clara University. It was a “great books,” tutorial-based,honorsprogram of a form then fash- ionable and inspired by the great University of Chicago president, Robert Hutchins. This movement continues today at St. John’s Annapolis and Santa Fe, among a few others. I was given a task I was ill-equippedto handle, but the group of leaders and tutorsin the program were willing to turn it over to me because they felt obliged to do something about science but did not want to touch it themselves. They preferred to talk about philosophy, literature, languages, economics, and politics instead. This program was an almost independent unit within Santa Clara, a comprehensive university. It didn’t last all that long and was replaced by another type of honors program that I was tapped to organize and direct a few years later. It was less expensive! But in that earlier program the school poured a lot of money into recruiting students from all over the country. I had a class of extraordinarily curious and capable students, many of whom I’m still in touch with today. Members of that group have gone on to academic positions, medicine, politics, and the law. Unfortunately no one became a mathematician, though one became a biologist at a major research university. It was a close-knit group then and they still have reunions to which I am invited. Initially they did not show much interest in science or mathematics, but they were really smart. One young woman who sat in the front row was very nice but she scared me. In my long career of teaching, there are few students I can say that about. She made me think of Pólya’s story about having John von Neumann in class at the ETH in Zurich. One day Pólya tossed out a plausible conjecture but mentioned that no one had been able to prove it. Minutes later von Neumann’s hand went up and he proceeded to go to the board and write out a proof. After that Pólya said he was afraid of von Neumann. Knowing that story, with this young woman in the class I did not toss out any unsolved problems. Now to the reason this digression might be relevant: since I was given so much freedom in doing what I wanted in these courses and Howard Eves’s two-volume set, A Survey of Geometry, was issued about then, I decided to show them some projective geometry. Eves’s books are just full of what now are assumed to be hopelessly old-fashioned topics in a part of mathematics often ignored. One is the wondrous Pascal-Brianchon theorem (in the case of a circle, sometimes referred to as Pascal’s mystic hexagram theorem). Brianchon’s version is this: If ABCDEF is a hexagon circumscribed about a circle, then AD, BE, CF are concurrent, that is, the lines all pass through the same point. The Pascal form of the theorem says that if we consider the points L, M , and N of intersection of the three pairs of opposite sides AB and DE, BC and EF , CD and FA of a (not necessarily convex) hexagon ABCDEF inscribed in a circle, then the three points L, M , and N lie on a single line. Twelve Classics People who Love Mathematics Should Know 357

C A

L M N

D F E The Pascal-Brianchon theorem—L, M,and N mystically collinear.

The geometrically sophisticated will recognize that these are dual theorems, and further that they apply more generally to ellipses and even other conics. After showing the class what all of this means I went on to try to convince them that these are unexpected and very beautiful results. Pascal, who was a philosopher as well as a mathematician, was very proud of his achievement, which he discovered when he was only 16. Then I went on to read to them from Eves: “There are ... 60 possible ways of forming a hexagon from 6 points on a circle, and, by Pascal’s theorem, to each hexagon corresponds a Pascal line. These 60 Pascal lines pass three by three through 20 points, called Steiner points, which in turn lie four by four on 15 lines, called Plücker lines. The Pascal lines also concur three by three in another set of points, called Kirkman points, of which there are 60. Corresponding to each Steiner point, there are three Kirkman points such that all four lie upon a line, called a Cayley line. There are 20 of these Cayley lines, and they pass four by four through 15 points, called Salmon points.” I then moved off in a quite different direction. These special honors students had a large common room/lounge on the second floor of the library for their exclusive use. A week or so after the class described above, I began to get questionsfrom faculty. “What on earth are those students of yours doingin the common room?” I had no idea. Eventually I was curious enough to investigate. They had covered the floor with butcher paper, constructed an ellipse in the middle, and were proceeding to draw Pascal lines, Steiner points, Plücker lines, and on and on. Of course, they could never complete the task because after a while I’m quite sure some of the points and lines would have been lying somewhere between here and Berkeley, or maybe even in Nevada! So it remained an unfinished project, but to me it demonstrated that mathematics has an appeal even to those who might claim that they don’t really care much about it. You just have to pique their curiosity with something surprising. This is only one example of the intriguing facts in Eves’s books. Between 1968 and 1988 he assembled a set of six volumes of anecdotes and stories of mathematics and mathematicians with variations on the title In Mathematical Circles, broken up in the first two volumes into “quadrants” of 90 anecdotes per quadrant. Those were followed by Mathe- matical Circles Revisited, Mathematical Circles Squared, Mathematical Circles Adieu, and Return to Mathematical Circles. They were all published by Prindle, Weber, and Schmidt of Boston, and the publisher spared no expense in producing them. They are in handsome uniform bindings, printed on high rag content paper with deckle edges, and in slipcases! 358 A Century of Advancing Mathematics

They are a joy to see on the bookshelf and to hold. In 2002 they were reissued in an attractive but more conventional format in three volumes, published by the MAA. Both sets are, alas, in sharp contrast to the two-volume set, A Survey of Geometry, where the publisher put out the second volume in a noticeably different trim size from that of the ﬁrst!

What Is Mathematics?, by Richard Courant and Herbert Robbins (Oxford University Press, 1941) Though the historical aspects of the classical Greek problems of straightedge and compass constructions are not as detailed here as they are in other descriptions, the mathematical arguments for showing nonconstructibility are clear and elementary so that readers need not know much algebra to understand them. The techniques in this book do not suffice to show the insolvability of the quintic equation or the impossibility of squaring the circle— those require more powerful methods—but one can easily use what is here to show the impossibilityof trisecting the angle, doubling the size of a cube, and constructing a regular heptagon. And there is sufficient explanation of the stories behind the problems to get the attention of students. I regularly teach a sophomore-level one-quarter course in abstract algebra and cover a fair amount of group theory, but I always come up short trying to include enough material about fields to take care of the case of the quintic. Last year, though, I had some fun with the class. I told them the story of the citizens of Delos consulting the oracle at Delphi about ending a plague that was sweeping through Greece. One version of the story is that the oracle told them that they could end the plague by doubling the size of the cubical altar to Apollo. They doubled the edge and, of course, did not succeed in doubling the volume. We know that what was needed was the construction by straightedge and compass of the cube root of two.Since the work of Pierre-Laurent Wantzel in the early nineteenth century, we know that this is impossible. But I told my students that the ancient Greeks failed to construct this number and the plague continued. I then added that this is why we see so few ancient Greeks today. The class seemed to find that plausible. That’s where it stood until I asked on the final examination why we see so few ancient Greeks around these days—a question worth five points for any answer they might give. There were various predictable answers: not even ancient Greeks live for more than 2000 years; the Greeks wasted their time philosophizing, not enough time living; and so on. But my favorite answer was: even with all of the technical advances in modern times, we have not been able to find a way to construct, with straightedge and compass, ancient Greeks. Though Courant and Robbins include some discussion of later developments concerning constructions, I think the more satisfying accounts are in Eves. He writes in some detail about the work in Lorenzo Mascheroni’s Geometria del Compasso (1797) wherein Mascheroni proved that insofar as the given and required objects are points, any straightedge and compass construction can be done with the compass alone. (Of course, without a straightedge one cannot actually draw a line, but one can still determine the line with the appropriate two points. So the straightedge is in a sense superfluous.) It was not until 1928 that a book was found in a used book store in Copenhagen, written by Georg Mohr, that demonstrated that this result had been discovered by Mohr in 1672. The book is Eu- clides Danicus, published in Copenhagen and in Amsterdam that year. It is not surprising that copies of this book are extremely scarce, but a few years back a copy turned up in an auction in San Francisco. Fortunately, few people knew of its extreme rarity and the story Twelve Classics People who Love Mathematics Should Know 359 surrounding it, so the price was right. The theorem is now known as the Mohr-Mascheroni theorem. But that’s not the end of the story. Between 1822 and 1833 Victor Poncelet and Jacob Steiner proved that although a straightedge alone does not suffice to do the whole range of straightedge and compass constructions, one need have only a straightedge and some circle drawn by a compass, even if only once. Now, to me that’s surprising. I am often disappointed that seemingly so few mathematicians, even among those professing to know some geometry, appear to know this startling result. I realize, it’s not like knowing lots of topology or algebraic geometry or heavy-duty analysis, but I think anyone claiming to know some of the culture of mathematics should have a passing acquaintance with this kind of unexpected non-intuitive result. Just assume, though, that we are looking for something representing topology. In Courant and Robbins there is a nice explanation of the Brouwer fixed point theorem that concerns the following question: if the liquid in a glass is set in motion by stirring it in such a manner that particles on the surface remain on the surface but move around on it to other positions, then at any given instant the position of the particles on the surface defines a continuous transformation of the original distribution of the particles. Any such transformation leaves at least one point fixed, that is, there is at least one point that does not move but stays in its original position. Again, unexpected? Yes, at least to me. In Courant and Robbins there are pictures, no longer well-known, of a right circular cone with the Dandelin spheres that make geometrically obvious the locus definitions of the conic sections—the so-called Dandelin-Quetelet theorem. And there are sections on (1) an easily understoodproofthat there are only five regular polyhedrain 3-space; (2) a picture to convince us that the Jordan curve theorem (a simple closed curve in the plane divides the plane into two domains, the inside and the outside) is not as “obvious” as one might think; (3) a now out-of-date discussion, alas, of the four-color problem (any geographical map in the plane can be “colored” with at most four colors); and (4) a proof by Steiner of the isoperimetric theorem in the plane (the circle contains the largest area among all closed curves with prescribed length), with further discussion of the Plateau problem, the soap film problem of minimizing surface area for surfaces connecting given curves in space. The book is filled with beautiful and often familiar mathematical statements, but even professional mathematicians might not know just how to set about proving them. Of course, the date the book appeared, 1941, suggests a rather traditional view of what constitutes mathematics, obviously missing mention of recent advances. Shortly after publication G. H. Hardy wrote what could only be called a rave review of Courant and Robbinsin Nature, specifically citing many of the sections mentioned above. This volume remains a great classic today. In January 1992, at the Joint Mathemat- ics Meetings in Baltimore, Maryland, the MAA gave a special recognition award to Her- bert Robbins to celebrate the 50th anniversary of the appearance of What Is Mathematics? Courant had died in 1972.

The Enjoyment of Mathematics, by Hans Rademacher and Otto Toeplitz (Princeton University Press, 1957) This book for a more or less general audience was published by Princeton in 1957, but it was a translation from German, originally printed in 1935. The ﬁrst appearance had the rather unwieldy title: Von Zahlen und Figuren: Proben Mathematischen Denkens f¨ur 360 A Century of Advancing Mathematics

Liebhafer der Mathematik.The translationwas done by Herbert Zuckerman, a Universityof Washington number theorist, who also added two new sections. So taking into account the taste of the authors and translator, we are not surprised that the emphasis on number theory is stronger than that in other volumes described here. It does repeat, however, the proof that there are only ﬁve regular polyhedra in 3-space, discussed in Courant and Robbins, in the context of Euler’s formula for polyhedra, on which the proof depends. There is also some good material on curves of constant width. The discussion of Waring’s problem is interesting and probably not widely known. A gem, however, is the not completely trivial proof that thirty is the largest positive integer that has the property that all of the numbers less that thirty and relatively prime to thirty are indeed prime numbers. That little exercise has enlivened number theory classes for me for years. And recently I tossed out the problem at lunch with some of my colleagues and it killed the conversation completely. A story about Hadamard came to mind. We all know that where possible, traditionally, International Mathematical Congresses take place in cities that border on the sea or a nice lake or some other body of water so there can be a cruise on the sea or an excursion nearby. At theCongress in Bolognain 1928(where Benito Mussolini was president of the Congress!) arrangements had to be made for a long train ride to the seashore at Ravenna so that members of the Congress could spend a day by the Adriatic. On the train ride back Hadamard was tired and wanted to take a nap, but everyone was chattering away about mathematics. So he tossed out a challenging mathematical problem. Suddenly it went quiet and Hadamard was allowed to get some sleep.

Mathematical Recreations and Essays, by W. W. Rouse Ball (Macmillan, 1892) I am at Santa Clara University, one of the two oldest institutionsof higher learning in Cal- ifornia (the other is the University of the Pacific), both founded in 1851, two years after the Gold Rush. The populationof California at that time largely consisted of Native Amer- icans, some miners and ranchers. So when it came time to send their sons off to get an education (and they sent only sons off to school in those days), many families sent them to Santa Clara where these future ranchers learned Latin and Greek! Because we are on the site of one of the 21 California missions, our library has holdings that go back to the 1770s. When I joined the faculty here I was surprised and pleased to find some unexpected material: complete runs of The American Mathematical Monthly, the AMS Bulletin, and the American Journal of Mathematics.So I could browse through issues of the American Jour- nal all the way back to when it was founded in 1878 and edited by J. J. Sylvester. I found the library to be surprisingly good, but one book seemed to be shelved rather awkwardly. Rouse Ball’s famous popular mathematics book, due to its title, was shelved among books on football and soccer. I promptly took care of that by asking that it be given a 510 number (yes, we were on the Dewey Decimal System at that time, though it was subsequently given an appropriate QA number). Though there are a couple of later editions in the open stacks in the library, that early copy is now in storage in a bin in our vast space-age Automatic Retrieval System, available only by a computer request whereby one of the cranes in the system (I call them robots) goes and locates the bin containing the book and brings it to the circulation desk. (The cranes are suitably named: Ichabod, Hart, and Stephen.) Libraries have changed. Perhaps at this point a cautionary word is in order: though there are still a couple of copies of Ball’s book in the open stacks, there is the possibility that an older Twelve Classics People who Love Mathematics Should Know 361 book like this, if not checked out occasionally, can be discarded or banished to an automatic retrieval system and never discovered by a curious student browsing in the stacks. That would be a shame, because thisbook is still a joy to read, especially in the revision by H.S.M. Coxeter of 1974. Ball was a fellow at Trinity College, Cambridge, and wrote various other books, notably a history of mathematics, but it is this collection that is most admired. It is reported that it went through over eighty editions and reprintings. He includes chapters on a variety of mathematical topics: arithmetic curiosities and geometric paradoxes, for example. One such is a nice description of Besicovitch’s solution to the Kakeya problem: that the area swept out by a straight line segment of unit length in the plane, which is reversed in direction by a continuous motion during which it takes every possible orientation, can be made as small as possible. That is not obvious to me. Beyond that there are polyhedra, chess- board problems, magic squares, map colorings, Euler’s Königsberg bridge problem, mazes and space-filling curves, and much more. But a chapter I would like to mention here is one that is so odd that I have not seen anything quite like it elsewhere. It is about calculating prodigies over the centuries. These are people with extraordinary talents for doing mental calculations or committing to memory vast amounts of information. One case is the Bidder family, father and brothers. One brother had memorized the Bible and could respond with chapter and verse to a quoted set of words from the text. He was the google.com of his day. Another brother worked as an actuary. Midcareer the books of the firm were destroyed in a fire and this man had such an incredible memory for numbers that he was able to reconstruct the records entirely from memory. It took him six months to do it. Alas, shortly thereafter he died of “brain fever.” There’s a lesson to be learned here. If a problem is too hard, it is perhaps prudent to move on to something else. It’s just not worth risking a case of brain fever (whatever that is!).

Some Films I know, they’re not books, but in the past couple of decades there have been some extraordinary documentaries made for television that I believe should be viewed by anyone interested in mathematics—and by some who think they are not. In the NOVA series on PBS a video, in recent years hard to find but now available on YouTube, called Letters from an Indian Clerk, tells the famous story of the life of Srinivasa Ramanujan. It is visually stunning with many views of India and Cambridge and it contains an interview with Ramanujan’s wife, Janaki Ammal, who was still alive when the film was made. She had married Ramanujan at the age of nine in a marriage arranged by his mother. The couple lived together for a rather short time before he left for England and they were apart until he returned to India and worked on a manuscript known later as the “lost notebooks” in the months remaining before his untimely death in 1920 at the age of 33. In the film his wife is interviewed and this tiny, frail woman, then in her 90s, says, “All I can tell you is that day and night he worked on sums. He didn’t do anything else. He wasn’t interested in anything else. Just sums. He wouldn’t stop work even to eat. We had to make rice balls for him and place them in the palm of his hand. Isn’t that extraordinary?” Through history there must have been many spouses of mathematicians who would sympathize with this woman as they wondered about how their husbands or wives spent their time, “doing their sums.” The film may have inspired Robert Kanigel to write his brilliantbook, The Man Who Knew Infinity/A Life of the Genius Ramanujan (Scribner’s, 1991). 362 A Century of Advancing Mathematics

The Proof is the story of Andrew Wiles’ finishing off Fermat’s Last Theorem. The day following the appearance of this film on television I was approached at our Faculty Club by colleagues from departments like English and History, telling me with great enthusiasm how much they had enjoyed seeing the film. Wiles came across as unpretentious and a really nice guy. My friends were moved by his description of his life having been up to that time a long quest for the solution to that problem. Mathematicians watching the film also had the pleasure of hearing that relatively recent mathematics—the Taniyama-Shimura conjecture, in this case—had been used to crack such a venerable classical problem. George Csicsery’s ZALA Films produced N is a Number, a visually sumptuous film about the legendary Paul Erd˝os, opening as it does with an academic procession at Cam- bridge where Erd˝os was receiving an honorary degree. No one can match the English in organizing a procession and this one, surrounded by ancient academic buildings, is certainly a treat for the eye. Beyond that there are enlightening conversations with people who knew Erd˝os, a genius who was widely known in the mathematical and the non-mathematical worlds. I have often been asked, on an airplane, say, what I do for a living. When I answer that I am a mathematics professor, either I am assumed to be a whiz at doing my taxes (standard misinformation about mathematics) or, if the questioner is more sophisticated, I am asked, “What’s your Erd˝os number?” (The Erd˝os number measures how far someone is removed from Erd˝os through a chain of coauthorships of papers. The concept has become widely known, enough so that one of my freshmen engineering students wrote about me on “Rate My Professor” that “[He] has an Erd˝os number of 2. How awesome is that!!!??”) In the vast amount of information out there about Erd˝os (there are two full-length biographies), one fact is sometimes lost. Even people who knew him personally sometimes did not get past the eccentricities to see how incredibly kind he was. He was also curious about everything, not just mathematics. The first time he visited my own campus, I drove over to Stanford to pick him up and when he got out of the car here he looked around and asked, “What was the temperature in this valley during the ice age?” Since I was then and am now quite old, I should have answered, “I don’t remember.” But that response only occurred to me later with esprit d’escalier. Csicsery has produced other films about mathematicians, among them, Julia Robinson and Hilbert’s Tenth Problem, and I Want to Be a Mathematician/A Conversation with Paul Halmos. Both are rewarding.

Mathematics in Western Culture, by Morris Kline (Oxford University Press, 1953) Though in recent years there has been a surfeit of books published purporting to connect mathematics with the arts, one is wise to be a bit skeptical. Those who claim to ﬁnd Fi- bonacci numbers in the work of Bach may be sincere, but we have to wonder whether some of these observed phenomena just might be an example of Richard Guy’s Strong Law of Small Numbers. The numbers 1, 2, 3, 5, and 8 may occur more often than huge numbers but not because they happen to be Fibonacci numbers. And golden means in ancient architecture may explain something, but it may just be that rectangles with sides in something close to that ratio look rather natural. Kline bases his observations on much more evidence. For example, we see Pascal’s mystic hexagram theorem again, but this time in the context of perspective, a concept certainly having relevance in painting and architecture. We also ﬁnd the idea of projections in map-making and astronomy, Newton’s musings on philoso- Twelve Classics People who Love Mathematics Should Know 363 phy, and the study of musical scales. The emphasis is on culture, however, not so much on mathematics. Some years later Kline published a large volume called Mathematical Thought from Ancient to Modern Times (Oxford University Press, 1972) on the history of mathematics, but quite unlike the traditional history books of Cajori or Struik or Boyer. Those run through the centuries chronicling the various discoveries. Kline’s book, as the title suggests, concentrates on movements, new ways of looking at questions, even though it is largely chronological. It explores—often at some length—areas that he was interested in. I consult it often.

Excavation and Other Verse, by Katharine O’Brien (Anthoensen Press, 1967) This may seem a curious addition to a list of classics among general books about mathematics. Still, it contains some gems of light verse, often with mathematical connections. O’Brien was a professor of mathematics at the University of New England in Portland, Maine, having received her doctorate in mathematics at Brown, a student of the eminent J. D. Tamarkin. I include this to demonstrate some kind of connection between mathematics and the art of poetry, but mainly because one of the poems sums up pretty well how those of us who are passionate about books feel about our collections. It is called Aftermath and here it is: It worries me some: whatever’ll become of my papers and books when I go? They’d bring nothing-at-all at a second hand stall— It’s a pity to leave them below. Now Hardy’s a treasure and Banach a pleasure and the Knopps a delight for the mind. There’s P´olya und Szeg˝o—well, I go where they go— couldn’t bear it to leave them behind. My Titchmarsh I’ll need, also Hausdorff—agreed! And Courant und Hilbert—oh yes! And reprints of theses of various species and copies of Bull. A.M.S. If I should be early in reaching the pearly, or tardily answer the call, could gracious St. Peter do anything sweeter than let in my bookcase and all?

Afterword Alas, there were many other contenders for this list and I’ll mention just a couple of them brieﬂy: (1) Ralph Boas’s A Primer of Real Functions (MAA (Carus), 1960), beautifully written and full of surprises; and (2) Clifton Fadiman’s Fantasia Mathematica (Simon and Schuster, 1952), a collection of excerpts about mathematics from a wide variety of largely literary sources—Mark Twain, Aldous Huxley, A. E. Housman, James Branch Ca- bell, Arthur Schnitzler, Robert Graves, Arthur Koestler, H. G. Wells, and George P´olya 364 A Century of Advancing Mathematics

(who is quoted as having described a principle “as being so perfectly general that no particular application of it is possible” and that “geometry is the art of correct reasoning on incorrect figures.” This book is paired with another Fadiman collection published by Simon and Schuster in 1962, The Mathematical Magpie.In the latter one finds Winston Churchill’s description of his encounter with mathematics: “I had a feeling once about Mathematics— that I saw it all. Depth beyond Depth was revealed to me—the Byss and the Abyss. I saw— as one might see the transit of Venus or even the Lord Mayor’s Show—a quantity passing through infinity and changing its sign from plus to minus. I saw exactly why it happened and why the tergiversation was inevitable—but it was after dinner and I let it go.” Since reading that for the first time I have been trying to find an occasion to use “tergiversation” in casual conversation, but without success. Curiously, if you check Fadiman’s publications on Wikipedia, these two volumes appear in a list of children’s books! Worth noting is another series of small books written in something like free verse by Lillian R. Lieber (illustrated by Hugh Gray Lieber). One is The Education of T. C. Mits (W. W. Norton, 1942). Others in the series are on group theory, logic, relativity, and noneu- clidean geometry, often conveying serious ideas in charming format and language. T. C. Mits, incidentally, stands for “the celebrated man in the street.” A reader of my list of favorites would have every reason to be disappointed that I have not included great monographs or textbooks on mainstream mathematics and have instead concentrated on general books that could, for the most part, be read by members of the uninitiatedpublic. One could easily write down a similar list of books that one encountered during one’s formative years in mathematics that opened up new vistas into unfamiliar areas of the subject. Again, my choices would be rather classical. Among such volumes, though, I would pass up some of O’Brien’s choices—Banach and Hausdorff, for example. Otherwise, there is obvious overlap between her list and mine: (1) Knopp’s extraordinarily beautiful survey: Theory and Application of Infinite Series (Hafner, 1947); (2) Courant’s two-volume Differential and Integral Calculus (Interscience, 1957), just full of good problems and intuitive insights; (3) Pólya and Szeg˝o’s Aufgaben und Lehrsätze aus der Analysis (Springer, 1925), the problem books to end all problem books; (4) Halmos’s Finite Dimen- sional Vector Spaces (Van Nostrand, 1958), based on notes by von Neumann and written with typical Halmos clarity; (5) Hardy and Wright’s An Introductionto the Theory of Num- bers (Oxford University Press, 1938), a great text with fascinating endnotes; (6) Schilling’s Theory of Valuations (American Mathematical Society, 1950), opening with an illuminat- ing introduction to p-adics; (7) G. H. Hardy’s A Course of Pure Mathematics (Cambridge University Press, 1908), the introduction to analysis for generations of students; and, of course, (8) O’Brien’s choice, Titchmarsh’s The Theory of Functions (Oxford University Press, 1932). These are books I keep going back to. But a detailed discussion of those can wait for another time. Meanwhile readers may be prompted to think of their own lists of classics they would like to take with them when (or if) they go. Department of Mathematics and Computer Science, Santa Clara University, 500 El Camino Real, Santa Clara, CA 95053 [email protected] The Dramatic Life of Mathematics A Centennial History of the Intersection of Mathematics and Theater in a Prologue, Three Acts, and an Epilogue

Stephen D. Abbott Middlebury College

Prologue God’s truth, Septimus, if there is an equation for a curve like a bell, there must be an equation for one like a bluebell. —Thomasina, from Arcadia Tom Stoppard’s Arcadia opened at the National Theatre in London on April 13, 1993. Two months later, on June 23 at the Isaac Newton Institute in nearby Cambridge, Andrew Wiles went public with his initial proof of Fermat’s Last Theorem. This was almost certainly the first time a Cambridge mathematics conference had a direct bearing on events at the National, but indeed, Wiles’s surprise announcement sent Stoppard scrambling to update the program. In the first scene of Stoppard’s new play, thirteen-year-old Thomasina Coverly has been assigned the task of proving FLT but, having overheard some house gossip earlier in that day, she stubbornlyinterruptsher studies to ask for a definition of “carnal embrace.” Septimus, her tutor, tries to steer her back to Fermat but to no avail. The battle of wills concludes with the following exchange: Thomasina: If you do not teach me the true meaning of things, who will? Septimus: Ah. Yes, I am ashamed. Carnal embrace is sexual congress, which is the insertion of the male genital organ into the female genital organ for purposes of procreation and pleasure. Fermat’s last theorem, by contrast, asserts that when x, y, and z are whole numbers raised to the power of n, the sum of the first two can never equal the third when n is greater than 2. Thomasina: Eurghhh! Septimus: Nevertheless, that is the theorem. Arcadia was groundbreaking on a number of levels, starting with its engagement of mathematics. Fermat is just the tip of the iceberg in this regard. A key piece of inspiration for Stoppard was James Gleick’s 1987 book Chaos, and the playwright wasn’t just interested in the pretty pictures. In Stoppard’s play, young Thomasina grows bored with Septimus’s classical algebra and Euclidean geometry and sets off to invent a new kind of mathematics that can better capture the jaggedness and unpredictability of nature. But

365 366 A Century of Advancing Mathematics the year is 1809. With computers nowhere in sight, Thomasina puts pen to paper and fills her maths primer with rough sketches of what her intuition senses are revolutionary ideas. “This margin being too mean for my purpose,” she writes to the twenty-first century scholars who will find their way to her notebooks, “the reader must look elsewhere for the New Geometry of Irregular Forms discovered by Thomasina Coverly.” As charming as it is clever, the real miracle of Arcadia is not so much that it contains a great deal of mathematics, but that it makes the content of the math integral to the play’s moving emotional arc. The early reviews were effusive. “A perfect marriage of ideas and high comedy,” wrote The Times. The Daily Telegraph called it “a masterpiece.” In 1993, Stoppard had been authoring successful plays for thirty years, and even he had a sense that somethingspecial had occurred withthisscript. “I feel foronce that I stumbled ontoa really good narrative idea. Arcadia has got a classical kind of story and, whether we are writing about science or French maids, this whole thing is about storytelling first and foremost” [4, p. 441]. Tom Stoppard put mathematics onto the pages of the arts section of every paper, and two months later Andrew Wiles moved it to the front page, above the fold. This one-two punch of popularity marked the beginning of a seismic change in the way mathematics in- tersected with popular culture, especially through the arts. In the two decades since Arcadia premiered, mathematics has become regular source material for scriptwriters—of successful plays certainly but also of Academy Award-winning films and highly rated television dramas. To say that Stoppard singlehandedly brought about the genre of the math play is an overstatement, but looking at bibliographies of plays about mathematics there is an unmistakable demarcation separating the era before Arcadia from the one that came after. There is good fun to be had in surveying the wildly creative ways that mathematics has been incorporated onto the stage in the last twenty years—and we shall tend to that momentarily—but the more interesting assignment is to scour the eighty years that came before Arcadia for signs of mathematical life in the theater. At first glance there is not much to speak of. Bertolt Brecht’s Life of Galileo (1945) is an important drama about the politics of science but does not mention mathematics. The same assessment can be made about The Physicists, by Friedrich Durrenmatt (1962), which features characters named Newton and Einstein, and even one named Möbius. But a closer look reveals that, in fact, there are precursors to Stoppard’s explicit engagement of mathematics. To find them we have to wander off the beaten path, far beyond the main stages of Europe; our first stop is the small town of Zakopane, Poland.

Act I: Non-euclidean theater

Papa! I don’t want to hear any of that. What have all of you done to inﬁnity? —Irene Brainiowicz from Tumor Brainiowicz

One hundred years ago, Stanislaw Ignacy Witkiewicz was a 29-year-old officer in the Rus- sian army. This was an odd place to find a non-conformist, Polish dramatist and reports are that Witkiewicz was highly ineffective in this capacity. This particular experience did, however, provide Witkiewicz with a ringside seat for the Communist revolution, a transforma- tive event for the young artist. Before this, his creative energies were largely unconcerned The Dramatic Life of Mathematics 367 with the real world, but the Russian Revolution of 1917 served as a bridge between his metaphysical musings and the social and political realities of his time. It would influence every aspect of his artistic vision for the remainder of his life. After the war, Witkeiwicz returned to a newly independent Poland and embarked on a staggeringly productive decade. Publishingand painting under the name Witkacy (a confla- tion of his middle and last names) he authored roughly fifty plays in addition to a series of novels and philosophical investigations. One of Witkacy’s earliest plays, and the first to be produced, was Tumor Brainiowicz: A Drama in Three Acts with a Prologue. The play gets its title from the lead character, described as “a famous mathematician of humble origins.” Whatever stereotype this might conjure up is inevitably wrong. Tumor Brainiowicz is built like a wild ox and, in his frequent gnashing and bellowing tantrums, has the disposition of one as well. As his name suggests, there is also something dangerous about him. He has produced a gaggle of children by three marriages, and we get the sense early on that he is similarly prolific in terms of his mathematical output. “I have lived alone, immersed in my calculations,” Tumor tells his wife during one of his frequent outbursts. “I begat a new world. Cantor, Georg Cantor, is a mere infant compared to my definitions of infinity, and Frege and Russell are the paltry decanting of the Greek void into the void of our own times with their definition of number, compared to what I thought up this morning.” Naming his new class of transfinite numbers after himself, Brainiowicz’s “tumors” are so potentially sinister that the Mathematical Central and General Office (M. C. G. O.) sends out special agents to arrest and hogtie Brainiowicz like the beast he is. The mathematics serves as a kind of catalyst for the chaos. The paradoxical nature of infinity is a good match for the overall turbulent tone of the play, which is melodramatic, comic, and borders on the nonsensical. Brainiowicz’s genius puts him at odds with the authorities, his unexplained desire to be a poet is at odds with his mathematics, and his slovenly roots put him at odds with Iza, his stepdaughter, who is also the object of Tumor’s uncontrollable passions. Brainiowicz: (fists clenched, to Iza) Oh! If I could first calculate your differential, analyze each infinitesimal particle of your cursed, russet-colored blood, each element of your hot, parched whiteness, and then take it, mash it, integrate it and finally comprehend wherein lies the infernal strength of your unattainability that burns and consumes me down to the very last tissue of my lowborn, slobbish flesh. Iza’s response is the taunting retort, “My infinity is no symbol.” And how was the first known math play received by critics? “It seems that we are watching and hearing the ravings of a syphilitic in the last stages of creeping paralysis,” ranted one reviewer who went on to declare that “Witkiewicz’s play is a total uncertainty from which nothing can ever arise. For it does not have a place in any line of development ... It should be put in alcohol and studied by psychopaths” [1, p. 15]. Amid this kind of criticism, the Cracow Mathematical Society had to act quickly to present the playwright with a bouquet of lilies—which reportedly occurred during the intermission on one of the two nights that the play actually ran [1, p. 42]. It’s conceivable that Witkacy took some comfort in the reviews of his first play. His nods to Cantor, Frege, and Russell show he was interested in the revolutions happening in 368 A Century of Advancing Mathematics mathematics, and this is because he was attempting a similar feat in the worldof theater. At the close of the nineteenthcentury, Henrik Ibsen had opened the door to realistic drama that the likes of George Bernard Shaw and Anton Chekhov exploited to great effect. This was a case of art imitating life. These great playwrights probed the social and psychological realities of their day and created detailed and lifelike portraits of what they found. “Realism,” as it came to be called, was so highly effective that it still characterizes mainstream theater a century later. Witkacy had no interest in writing plays in this style. His fascination with mathematics, especially modern geometry, begins to make sense when viewed in the larger context of the playwright’s efforts to depart from the dominant trends in drama at the time. In the same way that Euclid’s postulates were chosen to capture space as we perceive it, modern realism in the theater was bent on holding a mirror up to human predicaments, shunning exaggerated melodrama in favor of showing events as they really are. If euclidean geometry was the geometry of empirical space, the geometry that provided the framework for the Newtonian mechanics that made sense out of everything we see, then Witkacy would naturally be drawn to the new non-euclidean alternatives that were available to him. Looking over the long list of his plays, two in particular come with mathematically provocative subtitles: Gyubal Wahazar: A non-Euclidean Drama in Four Acts and The Wa- ter Hen: A Spherical Tragedy in Three Acts. While “hyperbolic” is a perfectly apt description for Gyubal Wahazar on a number of levels, the spherical tragedy of The Water Hen is arguably Witkeiwicz’s most satisfying attempt at implementing his evolving philosophy of “Pure Form” on the stage.

The Water Hen A tiger devoured him in Janjapara Jungle ... I assure you he died beautifully. His belly was torn to pieces and he suffered terribly. But up to the last moment he was reading Russell and Whitehead’s Principia Mathematica. You know—all those symbols. —Lady Alice of Nevermore in The Water Hen Despite what the name might suggest, The Water Hen is actually a person—a 26-year- old woman, “pretty, but not at all seductive,” whom we meet in the opening scene. She is standing in an open field at sunset waiting impatiently for Edgar to finish loading his double-barreled shotgun. Water Hen: (gently reproachful) Could you be a littlequicker about it? Edgar (finishing loading) All right—I’m ready, I’m ready. (Shoulders the gun and aims at her—a pause) I can’t. Damn it. (He lowers the gun.) In a manner that suggests that they are old friends, the two begin haggling over the meaning of existence with Edgar exhibiting his latent indecisiveness. “How stupid he is!” the Water Hen taunts him. “Greatness is always irrevocable . . . Whatever one can do several times is by its very nature trivial. You want to be great, and yet you don’t want to do anything that can’t be undone.” The Water Hen makes a compelling argument for her own execution and, sure enough, Edgar eventually concedes the point and fires both barrels of his gun in her direction. “One miss,” she remarks with no inflection in her voice, “the other straight through the heart.” The Dramatic Life of Mathematics 369

So what in the world is going on here? The answer depends on what world we are in. As the play progresses we get more and more clues as to what Witkacy intended by subtitling his play “a spherical tragedy.” We meet a young boy who has no conception of what the word “never” means. Then we notice that the standard temporal hierarchy between generations seems out of kilter. In Act II we encounter another character named Edgar and a third who is described as looking exactly like Edgar. Throughout this strange trip, the original Edgar from the ﬁrst scene falls deeper and deeper into existential despair. It’s hard to summarize in a tidy way, but essentially the playwright is exploring the implications for a non-euclidean geometry of human psychology—a mental existence where all lines intersect and the universe of possibilities is without boundary but yet ﬁnite and cyclical. What would “never” mean in such a place? Would anything be “irrevocable?” Amid all the overlapping names and themes, the most jarring turn of events occurs in the second act when the previously deceased Water Hen enters in the middle of a dinner party tocheck on theprogress of her old friend.“Will you beangry if I ask yousomething?” she says to Edgar who is miserable in his new marriage and being pressured by his father to become an artist. “How’s the greatness problem coming?”

Pure form and pure mathematics In An Introduction to the Theory of Pure Form in the Theater, penned by Witkacy in 1920, the playwright notes that, “the breaking of certain inessential bad habits touching the real- life aspect of works of art opens new horizons for formal possibilities.” Even with all the explicit allusions to Russell and Cantor, it is really at this conceptual level that Witkacy’s plays achieve their deepest affinity with mathematics. The shattering of the myth of a single, correct geometry for space set the mathematical imagination free, and Witkacy wanted the same thing for himself and his fellow artists. “The idea is to make it possible to deform either life or the world of fantasy with complete freedom,” he explains, “so as to create a whole whose meaning would be defined only by its purely scenic internal construction, and not by the demands of consistent psychology and action according to assumptions from real life.” The analogy with geometry is very rich because Witkacy and the pioneers of non- euclidean geometry were all fighting against the same demon of deep-seated prejudice toward realism. As much as one tries to make “point” and “line” undefined terms, their age-old euclidean interpretations are a very hard habit to break. In the same way, the sight of a human actor on Witkacy’s stage inevitably brings with it a host of unwarranted hypotheses about the “humanness” or “reality” of the action. If mimicking our perceptions of the world is no longer the task of theater then, like the mathematician, the playwright is free to adopt whatever set of hypotheses suit his purpose. There are some restrictions, however. A fundamental feature of geometry, euclidean or otherwise, is the internal consistency required of the axioms. They don’t have to jibe with our empirical observations, but they are required to have their own internal logic that must be free of contradictions. Here again, Witkacy’s conception of Pure Form follows the mathematics very closely: We can imagine such a play as having complete freedom with respect to absolutely everything from the point of view of real life, and yet being extraordinarily closely knit and highly wrought in the way the action is tied together. The task would be 370 A Century of Advancing Mathematics

to ﬁll several hours on the stage with a performance possessing its own internal, formal logic, independent of anything in “real life” [2, p. 295].

Real-life tragedies In a spherical tragedy there is nowhere to go except where you have already been, and in Act III an exasperated Edgar re-dons his clothes from Act I, reloads his shotgun, and kills the Water Hen a second time. In keeping with the self-conscious jokes about art and drama that have been ubiquitous throughout, Edgar’s father proposes one final time that his son pursue the arts. Edgar responds by pulling a revolver from his pocket and shooting himself in the head. The play concludes with the incongruous sounds of Edgar’s father playing a game of bridge, interspersed by exploding shells from a revolution starting in the streets. The darkness of this final image is made worse when we remember the real-life insanity that is to be visited upon Poland and the rest of Europe in the years ahead. Witkacy remained active and productive as an artist, eventually turning his attention to fiction and philosophy in the 1930s, but all the while growing more despondent at the political realities closing in around him. When the Nazis invaded Poland in September of 1939, Witkacy fled on foot with other refugees but his health was not good and his heart was not in it. Aware that the Russians were invading as well, Witkacy committed suicide in a country field. He died in the company of his traveling companion and lover at the time, to whom he reportedly said, “I will not live as less than myself” [1, p. 20].

Act II: Existential quantifiers Moment upon moment, pattering down, like the millet grains of ... that old Greek, and alllife longyou waitfor that to mountup to a life. —Hamm, from Endgame

Witkacy’s mathematically inspired experiments to produce a purely formal brand of theater were doomed to reside at the outer fringes of popular entertainment. Perhaps this seems inevitable, given their eclectic nature, but in the ensuing decades another dramatist attempting a very similar program managed to find an enormous audience. Avant-garde theater, which was born in the imagination of playwrights like Witkacy, reached a pinnacle when the Irish novelist Samuel Beckett realized that the stage could be a more effective medium in which to create his universal world of nothingness. As well known as Beckett is, what is not widely appreciated is the role mathematics played in his long journey as a writer. There are several reasons for this, the first being that, like Witkacy, Beckett’s primary interest was with mathematical form rather than content. A second reason is that his mathematical influence is much more apparent in his novels than in his more widely known stage plays. Still,the mathematics is there, and once we know to be looking for it, it has a way of jumping off the stage. Here for example is Clov’s opening speech in Endgame: Clov (fixed gaze, tonelessly): Finished, it’s finished, nearly finished, it must be nearly finished. (Pause.) Grain upon grain, one by one, and one day, suddenly, there’s a heap, a little heap, the impossible heap. (Pause.) I can’t be punished anymore. (Pause.) I’ll go now to my kitchen, ten feet by ten feet by ten feet, and wait for him to whistle me. The Dramatic Life of Mathematics 371

(Pause.) Nice dimensions, nice proportions,I’ll lean on the table, and look at the wall, and wait for him to whistle me.

The “grain upon grain” that leads eventually to the “the impossible heap” is an allusion to Zeno’s lesser-known paradox of the millet seed. If a single millet seed makes no sound as it falls, Zeno argued, how then does a heap of seeds make an audible thump? Generaliz- ing, how can a collection of nothings—the word “zero” is uttered upwards of ten times in Endgame—accumulate into something nontrivial? Clov also mentions his perfectly cubic kitchen, “ten feet by ten feet by ten feet ... Nice dimensions, nice proportions.” The kitchen, which we never see in Endgame, is Clov’s space. Hamm’s domain—the single interior set of the play—is a metaphorical circle where the blind and lame Hamm spends the entire play in an armchair with metal casters located center stage. At one point Hamm impulsively requests Clov to roll him around the room, but before long he must be returned to his proper location:

Hamm: Back to my place! (Clov pushes chair back to center.) Is that my place? Clov: Yes, that’s your place. Hamm: Am I right in the center? Clov: I’ll measure it. Hamm: More or less! More or less! Clov: (moving chair slightly): There!

Endgame by no means deserves the moniker “math play,” but it shares a great deal with Witkacy’s philosophy of Pure Form. The internal consistency of Beckett’s world is sharply at odds with our perceived sense of our own, and once again we have to jettisonour various base assumptions to make sense of the world of the play in front of us. Not coincidentally, these assumptions are exactly the things we use to give our lives a sense of stability and purpose. Hamm’s parents live in two large upstage garbage bins, the “earth” is out one window while the “sea” is out the other. By the end of the play, Beckett has untethered every essential concept we have from its traditional mooring—fathers and sons, love and death, past and future, happiness and misery. It is utterly bleak and fundamentally hopeless, but to leave it at that is to miss the subtle texture Beckett adds that somehow justiﬁes the stoicism his characters routinely exhibit. “You’re on earth,” Hamm says more than once in Endgame, “and there is no cure for that.”

Beyond language Some of the strongest evidence that Beckett lookedto mathematics as a means to pursue his brand of theatrical formalization comes from seeing where his journey eventually took him. One argument for why Beckett began writing in French in the 1950s is that the language was more sterilized in his mind. Since French was not his native tongue, the words did not come laden with connotations the way English words did, and it was easier for him to adopt a formalized point of view. Eventually, Beckett moved beyond the use of text altogether, creating a number of mimed pieces that often relied on mathematical structure as the primary mechanism for providing a sense of form. The pinnacle of this progression 372 A Century of Advancing Mathematics

A B

C D Figure 1. A stage map for Quad towards a purely mathematical vocabulary is Quad, which Beckett wrote for television in 1981. The script for Quad looks like something lifted from an elementary discrete math book. It starts with a labeled graph (see Figure 1) which is meant to be a map of the stage with the length of a side set at six paces. Beckett then outlines four isomorphic walking courses throughthe graph that four players are meant to traverse, in a staggered overlapping fashion. Each player wears a hooded gown of a different color and is accompanied by an identifying percussion instrument (e.g., drum, gong, triangle, woodblock)so there are multiplesensory ways to experience the choreographed combinatorics. As it turned out, the non-planar nature of Beckett’s original graph caused trouble when multiple players all arrived at E at the same time. This prompted a revised graph (see Figure 2) that enabled the walkers to avoid collisions.

A B

3 4 2 4 1 3 2 E 4 1 3 1 2

C D Figure 2. Revised stage map for Quad

Watching a performance of Quad can elicit a range of responses (try it, via YouTube) but as is usually the case with Beckett, to ask what it means is to miss the point. Quad is abstract theater—a Beckettian version of Witkacy’s Pure Form—and is more aptly appreciated in the way one might experience a highly structured musical canon.

Waiting for Gödel They give birth astride of a grave, the light gleams an instant, then it’s night once more. —Pozzo, from Godot It’s impossible, or at least impolite, to have a discussion about Samuel Beckett and not at least mention Waiting for Godot. Written earlier than Endgame, Godot contains no mathe- The Dramatic Life of Mathematics 373 matical imagery and none of the experiments with randomness or combinatorics that crop up in Beckett’s novels and later plays. There are, however, two points to be made about this seminal play that are relevant to our survey of the intersections between mathematics and theater. The first is that there is a surprising, if unintentional, kinship between Waiting for Godot and twentieth-century logic in the way that each exploits the power and paradoxes of self-referential structures. In mathematics, self-reference is well-known to have both constructive and destructive consequences. It is at the heart of Russell’s paradox, which in a very direct way led to Russell and Whitehead’s historic effort to inoculate the foundations of mathematics from the dangers of self-reference via formal systems. Kurt Gödel then found a way to create self-referential constructions inside these formal systems, which paved the way for his celebrated incompleteness theorems. Self-reference is also a powerful tool in art, and its allure is very apparent in Godot. In fact, there is a way in which the success of the play completely depends upon it. Vladimir and Estragon spend the entirety of two full acts with essentially nothing to do. This is their tragic predicament, but there is a comic self-referential level where their dialogue takes on extra layers of meaning for the members of the audience. Vladimir: Charming evening we’re having. Estragon: Unforgettable. Vladimir: And it’s not over. Estragon: Apparently not. Vladimir: It’s only beginning. Estragon: It’s awful. Vladimir: Worst than the pantomime. Estragon: The circus. Vladimir: The music-hall. Estragon: The circus. Vladimiris so bored at one pointthat he goes to relieve himself midwaythroughthefirst act! How, then, does a play that constantly draws attention to its own monotony become so profoundly moving? The answer, in part, is that a sense of empathy is built up between the people in the audience passing the time watching the characters on stage doing their best to pass the time. We start to viscerally feel that what is happening to Vladimir and Estragon is happening to us. Amusing at first, this deep identification between audience and actor has a chilling effect as these mildly self-deprecating moments are gradually replaced in the course of the evening by darker revelations. When Godot fails to arrive once more at the end of Act II, the sense of desperation is palpable. The second reason to mention Godot in our journey to put Arcadia in some kind of context is that Samuel Beckett was a profoundly important source of inspiration for Tom Stoppard. Reflecting on why he chose playwriting over, say, novels, Stoppard specifically pointed to Godot as altering his sense of what a play could be. “Godot was a shocking event,” Stoppard said, “because it completely redefined the minima of a valid theatrical transaction. Up untilthen, to have a play at all you had to have x’, you couldn’thave a tenth of x’ and still have a play” [4, p. 73]. 374 A Century of Advancing Mathematics

Act III: Mathematical theater in the modern age It costs little to watch and little more if you happen to get caught up in the action. —The Player, from Rosencrantz and Guildenstern Are Dead First performed in 1966, Rosencrantz and Guildenstern Are Dead (R&G) was the play that made the 28-year-old Tom Stoppard a household name in English theater. Just as in Godot, the curtain rises in R&G to reveal two men alone in a place of no distinguishing character who, like Vladimir and Estragon before them, will spend the rest of the play in a failed quest to make sense of just why they are there. Unlike Beckett’s play, however, everyone in the theater knows full well what awaits Stoppard’s duo. Even if the audience is a bit foggy on its knowledge of Hamlet, the play’s title reminds us that the two minor characters from Shakespeare’s play are destined for a tragic end. At rise, the two Elizabethans are passing time flipping coins. “There is an art to the building up of suspense,” Guildenstern says to officially start the play. The barren interior and the meta-theatrical opening line are unmistakable nods to Beckett—and so is the embrace of mathematics. To set the mood, Stoppard has the clever idea to suspend the law of large numbers. For seven or eight flips the coins come up heads, and the large sack of change in Rosencrantz’s possession suggests that this has been going on for a long time. Rosencrantz, the metaphorical tail of the pair, is oblivious to what is happening, although he has manners enough to be embarrassed at winning so much money from his friend. The heady Guildenstern, however, is unnerved but doing his best to stay calm. Guil: This must be indicative of something, besides the redistribution of wealth. List of possible explanations. One: I am willing it. Inside where nothing shows, I am the essence of a man spinning double-headed coins, and betting against himself in private atonement for an unremembered past. (He spins a coin at Ros.) Ros: Heads. Guil: Two: Time has stopped dead, and the single experience of one coin being spun has been repeated ninety times . . . (He flips a coin and tosses it to Ros.) On the whole doubtful.Three: divine intervention ... Four: a spectacular vindication of the principle that each individual coin spun individually (he spins one) is as likely to come down heads as tails and therefore should cause no surprise each individual time it does. (It does. He tosses it to Ros.) In fact what is going on is that the fates of Rosencrantz and Guildenstern are under the deterministic control of the script of Hamlet. The complementary head and tail of a single coin is an apt metaphor for the relationship between Stoppard’s play and Shakespeare’s play. Onstage for us, Rosencrantz and Guildenstern are unwittingly waiting to play their part in Shakespeare’s famous tragedy. Sure enough, when one of their coins does finally come up tails, Claudius and Gertrude enter, and we are thrust into the middle of a production of Hamlet, at least for the few brief moments in which Rosencrantz and Guildenstern are required. For poor Rosencrantz and Guildenstern the effect is bewildering—for the rest of us it is comic amazement. R&G also harkens back to Godot in the way that it exploits self-reference.“I don’t like to think about these things,“ Stoppard said, “but if forced to I suppose there is some theme of the commentator making points about the material which he is part of ... The device of The Dramatic Life of Mathematics 375 having a voice outside the play, though belonging to a character in the play” [3, p. 121]. In fact, Stoppard is able to push this theatrical device much farther than Beckett, thanks in part to Shakespeare who endowed Hamlet with a troupe of traveling players. In Shakespeare’s script, Hamlet has the players re-enact the murder of Hamlet’s father by Claudius. By having Rosencrantz and Guildenstern stumble on a dress rehearsal of this production,Stoppard is able to extend the troupe’s performance into more or less a complete synopsis of Ham- let. Bertrand Russell exposed the logical paradoxes that can arise when mathematical sets are allowed to contain themselves as elements, and Stoppard finds an interesting theatrical analogue in R&G. Riveted by a performance of the very play they are in, Rosencrantz and Guildenstern are just short of grasping the fact that they are watching a rendition of their own executions. Stoppard’s play is much more a comedy than Godot, but it does acquire some of the same chilling undercurrents as Rosencrantz and Guildenstern are drawn helplessly toward their inevitabledemise. They have the illusionof free will but are aware that stronger forces are at work. Throughout three full acts they are never fully granted the enlightenment they seek, and in the end they disappear into a tableau that is, in essence, the last scene of Ham- let. The English Ambassador from Shakespeare’s script arrives to announce the deaths of Rosencrantz and Guildenstern, but it is the utter futility of Guildenstern’s parting words— “we’ll know better next time”—that hangs heaviest over the audience at the final curtain.

Arcadia See? In an ocean of ashes, islands of order. Patterns making themselves out of nothing. —Valentine, from Arcadia The critical acclaim of Rosencrantz and Guildenstern Are Dead marked the beginning of a long and distinguished career for Tom Stoppard that to date has included four Tony Awards, an Academy Award for Best Screenplay and a knighthood. It is ironic that one of England’s greatest intellectual playwrights was not born English (he was born Tomas Straussler in Czechoslovakia), nor did he attend university. The fact that Stoppard was primarily self-educated is arguably why, throughout his career, he has felt comfortable access- ing ideas from across the intellectual spectrum, including the mathematical and scientific end. Zeno’s paradoxes and Bertrand Russell find their way into Jumpers (1972) and a bit of geometry graces Squaring the Circle (1984). The 1986 play Hapgood contains a long explanation of Euler’s Königsberg bridge problem as well as a healthy dose of quantum mechanics. The math and science in Hapgood was an order of magnitude more in-depth than anything Stoppard had attempted before, and critics thanked him by characterizing his new play as an impenetrable lecture. “It would need a seeing-eye dog with A-level physics to guide most of us through what was going on,” was a typical response [4, p. 378]. It is all the more remarkable then that five years later Stoppard returned to mathematics and science as source material for Arcadia, which many view as Stoppard’s best work. The early nineteenth-century world of Thomasina and Septimus is in fact only half of the play. The other half takes place in the same country house, but in the present, where three academics are engaged in researching the antics of the nineteenth-century cast. Not coincidentally, the earlier period sits at a transitional point in history between the Enlightenment and the Romantic Era, and one way to view the entire play is as an exploration of the ten- 376 A Century of Advancing Mathematics sion between romantic and classical ideas, broadly defined. The play is full of allusions to art and architecture, but mathematics and science are frequently center stage. In Stoppard’s telling, fractal geometry represents a romantic counterpart to classical euclidean geometry. Newton’s laws are classical physics while the second law of thermodynamics is given a romantic identity by describing it with double entendres such as “the action of bodies in heat.” Even Septimus’s incongruous juxtaposition of carnal embrace with Fermat’s Last Theorem can be interpreted as another manifestation of this theme. By focusing on the chasm between science and the humanities, Stoppard actually found a way to bring the sides closer together. The blowhard Byron scholar is given his moments to pontificate. “A great poet is always timely,” he howls, “a great philosopher is an urgent need. There’s no rush for Isaac Newton. We were quite happy with Aristotle’s cosmos. Personally, I preferred it. Fifty-five crystal spheres geared to God’s crankshaft is my idea of a satisfying universe.” The bluster makes for good theater, but the rebuttal is where we hear the playwright’s real convictions. In arguably the play’s most important speech, Hannah, the modern-day historian tells Valentine the mathematician that “comparing what we are looking for misses the point. It’s wanting to know that makes us matter.”

Copenhagen, Proof, and beyond In the years following Arcadia, mathematics and science began cropping up regularly in mainstream stage plays and musicals. Copenhagen (1998) by Michael Frayn explores the wartime visit of Werner Heisenberg to his friend and mentor Niels Bohr. Dense with the details of nuclear fission and quantum uncertainty, Copenhagen was widely embraced by audiences and critics, and eventually won the 2000 Tony Award for Best Play when it came to New York. At about the same time, David Auburn’s play Proof was playing to sold-out houses at the Manhattan Theatre Club. Three of the four characters in Proof are mathematicians, and although not much explicit mathematics is discussed, the aesthetic nature of mathematics is thoughtfully portrayed and central to the larger message of the play. Could a family drama full of passion for mathematics have succeeded in a pre-Arcadia world? This is impossible to answer, but the fact is that no one thought to use mathematics in a play meant for popular consumption—untilsuddenly everyone was doing it. As it was, Auburn’s play won a Pulitzer Prize and was made into a major motion picture. Alongside these high profile examples of mathematical theater has come a steady stream of lesser-known contributions to the genre. The celebrated collaboration between G.H. Hardy and Srinivasa Ramanujan is the subject matter for Partition (2003), written by Ira Hauptman, and in a less direct way it served as inspiration for The Five Hysterical Girls Theorem (2000) written by Rinne Groff. Whereas Partition is a fanciful account of a real historical friendship, The Five Hysterical Girls Theorem is a purely fictitious farce about an international mathematics conference that features a protagonist loosely based on Paul Erd˝os. Playwright Lauren Gunderson has made a career writing plays that engage mathematics and science. Notable among her contributions is Leap (2004), a whimsical account of Newton’s annus mirabilis, and Emilie (2009), which tells the story of the eighteenth- century female scientist Emilie´ Du Châtelet who translated Newton’s Principia into French while serving as mistress, muse, and intellectual rival to Voltaire. The Dramatic Life of Mathematics 377

As these examples suggest, biography and historical fiction are the dominant forms for much new mathematical theater. Seventeenth Night (2004) by Apostolos Doxiadis tells the story of the final days of Kurt Gödel’s life and Georg Cantor appears alongside his philosophical nemesis Leopold Kronecker in a scene in the experimental play Infinities (2002) written by John Barrow. The story of Andrew Wiles’s proof of Fermat’s Last Theorem is comically portrayed in the 2000 musical Fermat’s Last Tango. The drama, and tragedy, of Alan Turing’s life is the subject of at least four plays. The most well-known of these is Breaking the Code (1986) by Hugh Whitmore but a more ambitious attempt to theatrically engage Turing’s mathematical ideas can be found in Lovesong of the Electric Bear (2003) by the late British playwright Snoo Wilson.

Epilogue

A mathematician, like a painter or a poet, is a maker of patterns. —G. H. Hardy quoted in A Disappearing Number

In 2007, the acclaimed London-based theater company Complicite premiered a new play called A Disappearing Number. Devised by the company, the primary source of inspiration was G. H. Hardy’s essay A Mathematician’s Apology. The touching and tragic story of Hardy’s collaboration with Srinivasa Ramanujan is a major thread of the piece, but it is intertwined with a fictionalized modern-day love story that enhances the emotional range of the play. One of the modern characters, a mathematician named Ruth, appears in the play’s opening scene, which is worth recounting. Audiences arriving for a production of A Disappearing Number are greeted with a curious sight. There is no curtain, the lights are up, and the stage resembles a typical university lecture hall. Eventually Ruth arrives, nervously thanks her audience for coming, and sets off on a lecture about infinite series. For the vast majority of the audience, this is the first mathematics lesson they have attended in decades, and the response is an awkward sense of amusement. “Let’s consider these sets of numbers,” Ruth says as she begins scribbling lists of integers on the whiteboard. The sequence of prime numbers elicits some suppressed giggles; the use of standard sigma notationfor infinitesummations earns a hearty guffaw. As the difficulty of the mathematics increases, so does the intensity of the audience’s uneasy laughter. When Ruth introduces the Riemann zeta function the house comes down. Just at the moment Ruth plunges into the impenetrable details of the mathematics, she is joined onstage by a tall, distinguished gentleman. He watches her for a bit (she does not acknowledge him), and then he addresses the audience. “You are probably wondering if this is the whole show,” he says to everyone’s great re- lief, in an accent that suggests he is fromIndia. “My name is AnindaRao, and thisisRuth,” he explains—but then he stops. His posture changes, his accent disappears, and he begins a confession. “Actually, that’s a lie. I am an actor playing Aninda, she’s an actress playing Ruth ... This phone, for ... ‘Hello mum?’ ... no mum, no ring tone! This door doesn’t lead anywhere! I can push these walls right off!” Exposing the artifice of theater, Aninda— or whoever he is—physically dismantles the lecture hall piece by piece into the theatrical ether. But there is an ironic twist. Theater is an illusion, he is telling us, and everything 378 A Century of Advancing Mathematics we see up on the stage tonight is fake—everything, that is, except the mathematics. “[The mathematics] is real,” he says. “In fact, we could say that this is the only real thing here.” With this gesture, Aninda is paying a compliment to mathematics at the expense of theater. But on further reflection, it is clear that A Disappearing Number is really an exercise in mutual admiration. Mathematics is celebrated to be sure, but the play also uses mathematics for its own purposes. Before the evening is over, mathematics is employed to explore themes of love, grief, creativity and permanence, all of which transcend disciplinary boundaries as well as cultural ones. This is the power of art—to expose similarities between disparate objects, to find patterns amid apparent chaos, to derive abstract general truths from special cases. That all of these characteristics describe the business of mathematics just as well is hardly a coincidence. “Mathematics must be justified as art, if it can be justified at all,” is how Hardy put it in his Apology, and A Disappearing Number reveals, in both form and content, precisely what he meant. Hardy staked his life’s worth on the belief that his theorems had an intrinsic value—not because they were useful in any way whatsoever, but because they possessed an undefin- able beauty. In a world that prioritizes utility and profit, Hardy cast his lot with the poets and playwrights. That mathematical ideas have been so thoroughly embraced by vision- ary theater practitioners for over a century not only vindicates Hardy, but it suggests that Aninda—and the rest of us—should not be too quick to set mathematical reality above the realities explored by Hardy’s fellow artists in the theater. [Portions of this essay were adapted from previous articles by the author.]

Bibliography [1] D. Gerould. Stanislaw Ignacy Witkiewicz, Seven Plays. Martin E. Segal Theatre Center Publica- tions, New York, NY, 2004. [2] C.S. Durer, D. Gerould. The Madman and The Nun, and Other Plays by Stanislaw Ignacy Witkiewicz. University of Washington Press, Seattle, WA, 1968. [3] M. Gussow. Conversations with Stoppard.Grove Press, New York, NY, 1995. [4] I. Nadal. Tom Stoppard: A Life. Palgrave Macmillan, New York, NY, 2002.

Department of Mathematics, Middlebury College, 303 College St., Middlebury, VT 05753 [email protected] 2007: The Year of Euler

William Dunham Muhlenberg College

Many years ago, on my first visit to the offices of the Mathematical Association of America, I entered what seemed to be a generic lobby, with a receptionist at the main desk and portraits of past dignitarieslining the walls. It looked more or less likeany other organizational headquarters in the Dupont Circle area of Washington, DC. But then I noticed an old book resting upon a pedestal in the lobby’s center. This volume, dense with Latin words and mathematical symbols, was Leonhard Euler’s great text, Introductio in analysin infinitorum. Its presence confirmed that this was a mathematics organization, even as its position of honor proclaimed that Euler was a mathematician of special significance. I begin with thisrecollection because it foreshadowed a uniqueepisode from the MAA’s first century: the “Year of Euler.” Leonhard Euler was born in Switzerland on April 15, 1707, so his three-hundredth birthday rolled around in 2007. As this milestone approached, Executive Director Tina Straley, Director of Publications Don Albers, and other MAA leaders prepared to mark the tercentenary with a major celebration. To put it simply, the MAA decided to throw Euler a party. From its inception, the MAA had recognized mathematical birthdays, like Gauss’s two- hundredth in 1977 and Newton’s three-hundredth in 1942. Before the Euler tercentenary, however, the MAA had never mounted a year-long campaign focused on a single individual from the past. This would blaze new ground, and the results promised to be interesting. Of course, Euler had been no stranger to the MAA. Both the quantity and quality of his output have always been the stuff of legend. His publications range so widely across the mathematical landscape that there is, in Euler’s work, something for everyone. But I believe there is another reason for his popularity among MAA members. Euler began his career, in the early eighteenth century, knowing only euclidean geometry, elementary algebra, trigonometry, and calculus (the last of which was barely a generation old). In short, his starting point was essentially that of today’s incoming college students, and thus his reasoning is easily accessible to our mathematics majors. Yet, from these modest beginnings, he pushed forward to such sophisticated topics as the calculus of variations, the gamma integral, the phi-function, and the theory of partitions. All of this makes Leonhard Euler, in my view, the perfect mathematician for an MAA audience. Interestingly, The American Mathematical Monthly was writing about Euler before there was an MAA. In the Monthly’s issue of December, 1897, editor B. F. Finkel provided

379 380 A Century of Advancing Mathematics his readers with a short biography of Euler, whom he described as “... one of the greatest and most proliﬁc mathematicians that the world has produced.” The characterization was, and remains, apt. Subsequent MAA publications featured scores of articles about Euler’s work. Of particular note was the Mathematics Magazine of November, 1983, which marked the two- hundredth anniversary of Euler’s death. Edited by Doris Schattschneider, the issue was devoted entirely to Euler and boasted contributionsfrom such luminaries as Paul Erd˝os, Jerry Alexanderson, Underwood Dudley, and Morris Kline. It became something of a collector’s item for the math history crowd. Down through the years, many papers that received special recognition from the MAA carried Euler’s name in their titles. These included winners of the Chauvenet Prize in 1963, the Lester R. Ford Award in 1970 and 1975, the Allendoerfer Award in 2002, and the George P´olya Award in 1992 and 1995. That’s not bad for a mathematician whose research was centuries old. And, in 1999, I published a book in the MAA’s Dolciani series titled Euler: The Master of Us All. In it, I considered a selection of his most brilliant theorems, drawn from number theory, calculus, algebra, complex variables, geometry, and discrete mathematics. The book’s reception indicated that interest in Euler was as strong as ever. All of this suggests a longstanding love affair between the MAA and Leonhard Euler. But the 2007 celebration wouldratchet up the intensity to include programs aimed at mathematicians, others aimed at the general public, and an ambitious series of books to be called “The Euler Tercentenary Collection.” For starters, any big initiative needs a poster, and the MAA created one, appropriately titled “The Year of Euler.” One challenge for the poster’s designers was to incorporate a handful of formulas that would represent Euler at his best. The four that were chosen, and that can be seen running clockwise from the top left, were: ei cos i sin (Euler’s identity) D C 1 1 1 (Euler’s product-sum formula) k D 1 1=p k 1 p XD Y V E F 2 (Euler’s polyhedron formula) C D 2 1 1 (the Basel problem) k2 D 6 k 1 XD Here indeed is a collection of greatest hits. Let me say a word about each. Euler’s identity is shown below as it appeared in his Introductio of 1748. Admittedly this looks a bit old-fashioned. Back then, Euler wrote the imaginary constant as “p 1” and dutifully put a period after “cos” and “sin” because these were abbreviations of the Latin cosinus and sinus. Original notation notwithstanding,this equation surely belongs on 2007: The Year of Euler 381

the poster. With its fusion of the real and complex, of the exponential and trigonometric, Euler’s identity is often cited as the most beautiful formula in all of mathematics. Next we see the product-sum formula, equating the harmonic series with a certain infinite product. Euler derived this peculiar result in a 1737 paper. Because both the sum and the product diverge, the modern reader might be puzzled as to the meaning, and skeptical as to the utility, of such an expression. But Euler, who was neither puzzled nor skeptical, used it to prove the divergence of p 1=p, where the sum is taken over all the primes. In P 382 A Century of Advancing Mathematics describing this Eulerian triumph, AndréWeil wrote, “One may well regard these investigations as marking the birth of analytic number theory” [2]. The polyhedron formula, which Euler discovered in 1750, is a first step on the road from geometry to topology. In fact, Dave Richeson based an entire book — Euler’s Gem: The Polyhedron Formula and the Birth of Topology — on this result and its consequences. His book, fittingly, won the MAA’s Euler Prize in 2008. 1 Finally, we have k1 1 k2 , the series of reciprocals of the squares. In 1689 Jacob Bernoulli, living in Basel,D used the comparison test to show that this series converged. P He then challenged the mathematical community to find its exact value—the so-called “Basel Problem.” It remained an open question until Euler came along. When, in 1734, he demonstrated that the series (most improbably) summed to 2=6, the young Euler made an international splash. Ed Sandifer described this as “... the problem that first makes him famous and launches him into the scientific elite of his times.” [1] So let there be no doubt: the poster’s designers chose well. The poster advertised other features of the tercentenary. A short course, organized by Ed Sandifer and Rob Bradley, was held at the Joint Meetings in New Orleans in January of 2007. In June, I ran a week-long PREP Workshop on Euler at the MAA Carriage House in Washington, DC. Immediately thereafter, a group of 26 mathematicians, under the direction of Victor and Phyllis Katz, headed off to Europe on a kind of pilgrimage. Since 2003, the MAA had been sponsoring trips abroad to mathematically significant places. This one — officially called the “MAA Euler Study Tour” — carried the participants from Russia to Switzerland to Germany. Over a period of two weeks, they visited streets bearing Euler’s name, places where he had lived and worked, and a pair of institutions associated with his career. In Berlin,they saw hisstately home in the city where he had resided for a quarter of a century, and, in a St. Petersburg archive, they had a chance to inspect Euler’s handwritten notebooks, some of which have yet to be published. One participant described this archival visit as the “experience of a lifetime.”

Euler’s home in Berlin 2007: The Year of Euler 383

First-person accounts of the study tour, with photos, can be found in the December, 2007 issue of MAA FOCUS. There was more. At the 2007 MathFest in San Jose, amid the Euler talks and book exhibits, we had a visit from the Swiss Consul General, who came down from San Francisco for the occasion. This was one of many times the Swiss honored their favorite son during the Year of Euler. For me personally, the most remarkable of these came when I was invited to speak at the Swiss Embassy in Washington, DC. It was a once-in-a-lifetime experience. The Embassy had sent out invitations to colleges and universities, to federal agencies with scientific missions, and to local members of the Swiss community. My audience thus ranged from college faculty to NASA engineers to Smithsonian fellows to the general public. I tried to pitch my remarks in such a way that everyone, whether technically trained or not, grasped something of Euler’s impact on the world of mathematics. Then, vin d’honneur and Swiss chocolates capped off a memorable evening. But the most tangible product of the MAA’s celebration was a series of five books released during 2007. These had a common format and matching design so as to be a unified set. Publishing them within a tight deadline posed a challenge of the first order, one that consumed many hundredsof hours of MAA time. Special commendation is due to Don Albers for the fundamental concept, to Jerry Alexanderson for his editorial expertise, and to Elaine Pedreira and Beverly Ruedi for their skills in shepherding the project through to a successful conclusion. The volumes, numbered sequentially, were: 1. The Early Mathematics of Leonhard Euler by C. Edward Sandifer Here, in chronological order, are summaries of the first few dozen papers that Euler wrote. It begins with his earliest article from 1725, a short result on isochronal curves 384 A Century of Advancing Mathematics published when he was just eighteen years old, and continues to a 1741 essay on the utility of higher mathematics. Along the way, Sandifer provides a “rating system” — from no stars to three stars — to indicate the significance of the various results. The book gives a fascinating perspective on Euler’s year-by-year development as a mathematician. 2. The Genius of Euler: Reflections on His Life and Work, edited by William Dunham This is a selection of articles about Euler that had previously appeared in print — most of them in MAA publications— from the late nineteenth to the early twenty-first centuries. Its contributing authors run from W. W. Rouse Ball to George Pólya, from Florian Cajori to Carl Boyer. Collectively, they are a Who’s Who of mathematical expositors. 3. How Euler Did It, by C. Edward Sandifer Since 2003, Sandifer had been writing columns with this title for the MAA website. Among other topics, he described Euler’s research on amicable numbers, on the knight’s tour in chess, and on the arc length of an ellipse. In this volume, Sandifer collected his columns to produce a most engaging survey of Euler’s work. 4. Euler and Modern Science, edited by N. N. Bogolyubov, G. K Mikhailov, and A. P. Yushkevich, translated by Robert Burns Because St. Petersburg was Euler’s home from 1727 to 1741 and again from 1766 to the end of his life, it is fitting that a Russian work should appear among the volumes. This book focuses on Euler’s contributions outside of pure mathematics, contributions that influenced so much of eighteenth-century science. Not surprisingly, it proved to be the most challenging of the five books, as the MAA scrambled to acquire the rights, produce the translation, and move to publication within the demanding time constraints. But they did it. 5. Euler at 300: An Appreciation, edited by Robert Bradley, Lawrence D’Antonio, and C. Edward Sandifer The final book contains twenty-one papers about Euler that had been presented at mathematics meetings over the previous years and were being published here for the first time. For instance, we find an article by Christopher Baltus on the “Euler-Bernoulli proof” of the fundamental theorem of algebra, one by Stacy Langton on Euler and the quadrature of lunes, and one by Ed Sandifer on Euler as naval scientist (title: Euler Rows the Boat). Taken together, these five volumes give an extensive overview of Euler’s mathematics, from the elementary to the advanced, from the pure to the applied, and from the justly famous to the less well known. In their 1700 pages, the books provide a vast and valuable resource for anyone interested in the work of this extraordinary mathematician. And there was one last flourish. As 2007 wound down, the MAA devoted much of the November issue of Math Horizons toEuler ... and even put himonthecover inappropriate seasonal garb. After books and lectures and workshops, after solid scholarship and a bit of levity, the Year of Euler finally came to an end. It is hard to imagine how an MAA member would have remained untouched by the festivities. Although the tercentenary was over, the MAA’s interest in Euler was not. In 2011, the MAA announced that its website would host the Euler Archive, an enormous databank containing facsimiles of virtually everything Euler had published. This was the brainchild of two Dartmouth graduate students, Dominic Klyve and Lee Stemkoski, who discussed its 2007: The Year of Euler 385

creation in an MAA FOCUS article from January, 2007. Because of their efforts, anyone interested in, say, Euler’s number theory can go to the MAA site, ﬁnd the Euler Archive, and search his papers on the subject. These will appear as originally published, sometimes accompanied by a translation and/or commentary. The initial efforts of Klyve and Stemkoski, and the subsequent support of the Mathematical Association of America, have made Euler’s work readily available to scholars the world over. Of course, the Year of Euler occupied just 1% of the MAA’s ﬁrst century and therefore stands as something of a sidebar to our long and impressive history. It nonetheless warrants this brief account, for it was a unique organizational tribute to one who so profoundly shaped mathematics as we know it. And it can serve as a model, somewhere down the road, should the MAA choose to throw another birthday bash.

Bibliography [1] C. Edward Sandifer, The Early Mathematics of Leonhard Euler, Mathematical Association of America, 2007, p. 157. [2] Andr´eWeil, Number Theory: An Approach Through History, Birkh¨auser, 1984 , p. 267.

Department of Mathematics, Muhlenberg College, Allentown, PA 18104 [email protected]

The Putnam Competition Origin, Lore, Structure

Leonard F. Klosinski Santa Clara University

George Csicsery, the noted documentary filmmaker, recently released a film about the 2006 American team that participated in the International Mathematical Olympiad, called “Hard Problems: the Road to the World’s Toughest Math Contest.” Well, maybe it is the toughest mathematical contest for the students participating. But there’s another mathematical competition for more experienced students of mathematics, one with an even longer history and with problems that stump even the professors. The William Lowell Putnam Mathemat- ical Competition, long administered by the Mathematical Association of America (MAA), just celebrated its seventy-fifth anniversary. Its list of winners, like that of the International Mathematical Olympiad, includes some of the most important American mathematicians of the twentieth century.

History of the competition In an essay on the growth of American mathematics published by the MAA in the Amer- ican bicentennial year,1976 [3, pp. 50–52], Garrett Birkhoff of Harvard, son of one of the very ﬁrst American mathematicians to have a truly international reputation in mathematics, George David Birkhoff, wrote of the beginnings of the Putnam Competition. It was established in the 1930s by Elizabeth Lowell Putnam, the widow of William Lowell Putnam and sister of the then president of Harvard, Abbott Lawrence Lowell. They were all members of a distinguished intellectual and social aristocracy in Boston in the nineteenth and twentieth centuries. Lowell’s brother, Percival, for whom the Lowell Observatory in Flagstaff, Ari- zona, is named, set theagendathatled to thediscovery ofPluto,thoughhedidnotlivetosee the actual discovery. He was an eminent scientist in his day. Abbott Lawrence Lowell was president of Harvard for a near record 24 years (1909–1933) and profoundly inﬂuenced the direction of that University. At Harvard as a student he had majored in mathematics graduating in 1877 summa cum laude. As president he was instrumental in reforming the mathematics curriculum. A lawyer by profession he reminded generations of students that “there was no better preparation for the law than the study of mathematics,” good advice still today. Along with William Lowell Putnam, Abbott chaired a “Visiting Committee” that moved to require every undergraduate to take at least one course in mathematics or philosophy. He viewed calculus, along with the phonetic alphabet and the Hindu-Arabic

387 388 A Century of Advancing Mathematics decimal notation, as being among the greatest inventions of all time. Until 1911 first-year calculus was still viewed as acceptable as part of graduate study of mathematics. William Lowell Putnam had also been a mathematics major at Harvard, graduating in 1882 magna cum laude. It was his widow—citing her husband’s commitment to making “excellence in scholarship as honorable among undergraduates as athletic prowess”—who worked closely with the elder Birkhoff in setting up the earliest Putnam Mathematical Competition, reflecting her husband’s belief in the effectiveness of team competitions, in- stilling in them “pride in the achievements of their team as a whole and the standing of the institution which it represented.” The details of the history of the founding of the Putnam Competition can be found in Garrett Birkhoff’s article “The William Putnam Mathematical Competition:Early History” in [4, pp. 603–606]. Mrs. Putnam, carrying out her husband’s vision—laid out in an article he wrote for the Harvard Graduates’ Magazine in 1921—set up a trust of $125,000 in 1927 to fund a competition and the first was held in 1928, between Harvard and Yale; but the subject was English. An invitation to repeat this competition was issued, but the schools declined and efforts to set up something on economics with Cambridge Univer- sity failed. After various attempts to establish a mathematical competition—motivated in part by the example of the Hungarians with their famous Eötvös Competition first given in 1894 and often credited for the flowering of mathematics in Hungary in the early part of the twentieth century—the first Putnam Competition in the form we know it today was given in 1938 after Mrs. Putnam’s death. The trust is administered by a board of trustees consisting largely of family members. Garrett Birkhoff in [4, p. 604] wrote of the connection of others in the family to mathematics: “George Putnam, a trustee of the Putnam Fund, had also majored in mathematics and was active in his turn on the same Visiting Commit- tee, while the present form of the Putnam was being planned. Roger Lowell Putnam also majored in mathematics at Harvard, graduating magna cum laude in 1915, while William Lowell Putnam’s grandson, McGeorge Bundy (later Dean of the Faculty at Harvard and Na- tional Security Advisor to two presidents), and his brother Harvey majored in mathematics at Yale.” From the beginning the Competition attracted some of the best and brightest students in colleges and universities in the United States and Canada. In the first twenty-five years of the contest we find among the Putnam Fellows (those placing among the top five each year) names that are, at least in mathematical circles, household names: in the first year there were Irving Kaplansky (then at Toronto) and George Mackey (then at Rice but soon to be at Harvard). And later there were those who became Fields Medalists: John Milnor, David Mumford, Daniel Quillen; a solver of a Hilbert problem, Andrew Gleason; a famous physicist, Richard Feynman; and many, many extraordinarily productive mathematicians such as Richard Arens, Felix Browder, Eugenio Calabi, Max Rosenlicht, Harold Widom, Eugene Rodemich, Alfred Hales, Robin Hartshorne, Marty Isaacs, Mel Hochster, Edward Bender, Elwyn Berlekamp, Roger Howe, Dennis Hejhal, Don Zagier, Neal Koblitz, Jeff Lagarias, David Vogan, Noam Elkies, Bjorn Poonen, Ravi Vakil, Kiran Kedlaya, Ioana Dumitriu, Lenhard Ng, and Melanie Wood, among many others. Of course, there are outstanding mathematicians who do mathematics in varying ways. Some “win” the Putnam and have amazing powers for solving difficult elementary problems under time pressure and others who put together profound ideas that form great mathematical structures. But The Putnam Competition: Origin, Lore, Structure 389 over its many years, the Putnam Competition has fulfilled one of the early stated goals— identifying mathematical talent. Anecdotal information about Putnam winners has been written about by two former Directors of the Competition,L. E. Bush [4, pp. 609–622]and L. M. Kelly [4, pp. 634–38], the latter being a response to rather detailed comments on the qualityof the problems by the eminent British mathematician L. J. Mordell at St. John’s College, Cambridge. The concerns raised are similar to those discussed over many years by Cambridge mathematicians critical of the methods for choosing Senior Wranglers. This brings to mind a comment by Garrett Birkhoff that in the nineteenth century the selection of the Senior Wrangler was intended to identify those “most likely to succeed” in either mathematics or the law, and it is paralleled by Putnam’s concern for locating talent for going into either mathematics or the law. It is perhaps not surprising since preparing a good argument for a successful brief is similar to writing up a good tight proof. Other sources of information on the Com- petition, with a detailed study of team members and high ranking individuals, who went on to receive Fields Medals, the Abel Prize, and various other research awards or become members of the U.S. National Academy of Sciences are [5], [6], [7], and [8]. In the early 1990s the Putnam family expressed concern that few women were listed among Putnam winners, or even among participants, so a special prize in honor of the founder, Elizabeth Lowell Putnam, was proposed, to be given from time to time to a woman with significant achievement in the Competition. Some people involved in the discussion were concerned about singling out women for a special prize so it was agreed that participation would be optional; any woman taking the contest and wishing to be considered for the prize had to indicate an interest in being considered. The objective was to encourage women to take the contest. Since 1992 there have been nine winners of that Prize, with several—Ioana Dumitriu of NYU, Melanie Wood of Duke, Ana Caraiani of Princeton, and Alison B. Miller of Harvard—who were winners in more than one year.

Answers to some often-asked questions The ranking of Putnam teams is somewhat unusual. The three team members are identified in advance by the institutions participating in the Competition, but in spite of the motivating examples of teams from athletics, the Putnam team performance is not the result of a joint effort on the part of the members. The team members participate as individuals, with no collaboration and in the end, the team ranks are based on the sum of the ranks of the individual student participants. This was put in place because there was concern that if the team performance were based on the sum of points earned by the three members, there was the risk of Harvard’s winning every year. (The view of Harvard’s power was perhaps influenced by the fact that those setting the rules were overwhelmingly products of Harvard.) So this method was devised to avoid that. It has not been entirely successful since in the first 73 years of the competition, Harvard placed first in 29 of those years (with an additional ten years in second place and thirteen in third place). Runners-up for the first place slot were Caltech (ten years) and MIT (six years). For top positions, Princeton did respectably by achieving second place eleven times. In a further attempt to limit the number of times Harvard would “win,” there was a rule that following one contest, the next set of problems would be made up by faculty from the institution with the winning team. (The first team 390 A Century of Advancing Mathematics to place first was from the University of Toronto.) Then, of course, it was necessary to follow that with a rule that institutionscould not “win” two years in a row since the winning team the second year would be taking a contest made up by faculty at its own institution. This procedure was abandoned after five years but that then coincided with a period during World War II when the Competition was suspended, thus making it a bit complicated to state just how old the Putnam Competition actually is. After the War, the first contest was constructed by George Pólya, the renowned expert on problems and originally from Hungary, Tibor Radó, another Hungarian, and Irving Kaplansky, a Putnam Fellow from the first Competition of 1938. Subsequent committees to construct the problems each year have been appointed for three-year terms by the MAA. The grading of the Putnam Competitionis done a few weeks after the administrationof the test, with graders assembled in some location where the weather is not too bad in late December. The identity of the graders is not divulged publicly,according to the rules set out in the Putnam-MAA document governing the administration of the Competition, though there are occasionally articles written by former graders who recount their experiences in grading. Just as members of the Committee making up questions for the Competition are brought to the task from various parts of the United States and Canada, so too are graders flown in for the general grading sessions. To assure as much accuracy and uniformity of the grading as possible, the papers of the top 200 participants and the members of the top 35 teams are regraded. It is a time-consuming process and explains why the results are not usually known until March. On the initial grading, the graders work in teams and normally assign grades only after the team agrees on the number of points to assign for each particular attempt at a solution. The regrading also tends to eliminate any difference between the point assignment at the beginningand at the end of a grading session. Needless to say, as the papers are read the graders often come up with correct alternate solutions,and similarly sometimes discover a weakness in arguments read earlier. Incoming members of the Questions Committee sometimes assume that to make up a new contest all one needs to do is go back in the problems literature and find some good problems. It’s not as easy as that, however, because the community of problem solvers already knows those problems. The Committee each year has to produce a set of new problems, not recycled problems. And connoisseurs of problems eagerly await a new collection of attractive and appealing problems. These often arise in the research being done by members of the Committee. And, as one can see in reading the solutions and commentary of some of the summary collections of Putnam Competition problems, some of the challenges can even lead to extensions and generalizations of the problems on the Contest. See [1], [4], and [9]. They can actually prompt research projects. A persistent question concerns how partial credit is assigned by the grading teams. In fact, little partial credit is given, and the ten points for a problem are assigned for perfect solutions, with the loss of a point or two resulting from a failure to make some relevant observation,but for full credit the solutionhas tobe complete. On the otherhand littleif any credit is given for attempts, however promising they may appear. There are rarely, if ever, scores in the 3–7 range. Total scores are often disappointingto participants who are used to getting generous partial credit in the examinations they take in their classes. Putnam scores often disappointfaculty supervisors who expect higher scores for their students. Ultimately, the median score on the test can be at or near zero. The Putnam Competition: Origin, Lore, Structure 391

The object of the Competition, as set out by the founders, is to identify the very best and most promising students in mathematics in the country. At the same time, I am aware of the demoralizing effect the test can have on undergraduates. For that reason the members of the Questions Committee are asked to include problems of varying degrees of difﬁculty. Informally the opening two problems in the morning and again in the afternoon are meant to be approachable by large numbers of participants. Of course, as any experienced teacher of mathematics knows, it is not easy to design problems that are predictably easy, or challenging, for their students. So all too often I ﬁnd that at the end of the grading, we have a low median score. In the last ten years a median score of zero has occurred only three times. Even the hardest problem I can recall in recent years—B-6 in 2011—still received one perfect score of ten. Participants include some really good problem solvers. But as Garrett Birkhoff pointed out, the test is such that a participant has reason to be proud if he or she gets a score of, say, ten or twenty— inother contexts that might be embarrassing—on the Putnam Competition it is indeed a real achievement.

A growing competition Directors of the Putnam Competition—there have been five: L. E. Bush, L. M. Kelly, J. H. McKay, A. P. Hillman, and myself—with the probable exception of the early directors, have been former participants as students. When we look back on the questions that baffled us when we took the Contest and compare those questions with those that appear on the Contest today, it seems that there has been a steady increase in difficulty. Nevertheless the number of contestants has increased dramatically over the years, albeit not monotonically. In 1959, when I participated there were 632 others from 135 colleges and universities; by 2011 those figures had increased to 4440 student participants from 572 colleges and universities. We do not have records on the number of graders involved in 1959, but by 2013 I had to increase the number to forty, an all-time high. When I was a student the team prizes ranged from $100 to $500; now they range from $10,000 to $25,000. And Putnam Fellows then received $75 each, with an additional five students receiving $35 each. This past year the Putnam Fellows received $2500 each, with an additional twenty students receiving awards of $250 to $1000. It may even be keeping up with inflation! In the world of problem competitions, business is just fine.

Bibliography [1] Alexanderson, Gerald L., Leonard F. Klosinski, and Loren C. Larson, The William Lowell Put- nam Mathematical Competition: 1965–1984, Washington, DC, Mathematical Association of America, 1985. [2] ———, How Putnam Fellows view the competition. FOCUS, December2004, 14–15. [3] Birkhoff, Garrett, Some leaders in American mathematics. In Tarwater, Dalton, ed., The Bicen- tennial Tribute to American Mathematics 1776–1976, Washington, DC, Mathematical Associa- tion of America, 1977, pp. 50–52 . [4] ———, The William Lowell Putnam Mathematical Competition: Early History. In Gleason, Andrew M., R. E. Greenwood,L. M. Kelly, The William Lowell Putnam Mathematical Competi- tion/Problems and Solutions: 1938–1964,Washington, DC, Mathematical Association of Amer- ica, 1980. 392 A Century of Advancing Mathematics

[5] Gallian, Joseph A., Fifty years of Putnam trivia, Amer. Math. Monthly 96 (1989), 711–713. [6] ———, The ﬁrst sixty-six years of the Putnam Competition, Amer. Math. Monthly 111 (2004), 691–699. [7] ———, Putnam trivia for the 90s, Amer. Math. Monthly 107 (2000), 733–735. [8] ———, History of the Putnam Competition, http://www.d.umn.edu/˜jgallian/ [9] Kedlaya, Kiran S., Bjorn Poonen, and Ravi Vakil, The William Lowell Putnam Mathematical Competition 1985–2000/Problems, Solutions and Commentary. Washington, DC, Mathematical Association of America, 2002.

Department of Mathematics and Computer Science, Santa Clara University, 500 El Camino Real, Santa Clara, CA 95053-0290 [email protected] Getting Involved with the MAA A Path Less Traveled

Ezra ‘‘Bud" Brown Virginia Tech

Many of us in the MAA are given student memberships as undergraduates, or graduate students, or high-school students. Here is a member with quite a different story. You had a talent and a taste for mathematics, went through school and somehow or other ended up with a PhD in mathematics with a dissertation in number theory and a tenure-track job as a college professor at a university that was “on the way up.” This was in the day when research mathematicians talked about teaching behind closed doors — if they talked about teaching at all. Although you always enjoyed teaching, research was your big thing, so you subscribed to the Jour- nal of Number Theory, published like mad, joined a scholarly organization, and attended professional meetings focusing on research in general or in number theory. Of course, your university subscribed to quite a number of research journals, and to the reviewing journals Mathematical Reviews and Zentralblatt. Your department also took an interesting magazine devoted to expository mathematics called The American Mathematical Monthly. Some- where along the way, you were tenured and promoted. You received your professional organization’s ofﬁcial magazine, and went to meetings — and things began to change. Meetings devoted to your specialty were, and still are, highly enjoyable. However, you noticed that the business meetings of your professional society were becoming less mathematical and more political, more contentious, and a whole lot less enjoyable. Also, for whatever reason, the articles appearing in that ofﬁcial magazine were hard to read and were not particularly well written. It was as if the writers had all heard your major professor’s advice (“You should writeyour mathematics as if you wanted someone to read it”) and had all decided to do the opposite. Those big meetings weren’t fun anymore. What to do?

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Just about this time, your department recruited a new full professor, a prominent young researcher named David Roselle who had been one of your teachers in graduate school, had written a letter of recommendation for your current job, and who just happened to be involved with the MAA at the national level. The two of you chatted about the situation, and he suggested that you might want to join the MAA (“You meet the nicest people there”), one of the benefits being your own copy of the Monthly. So you change professional organizations and become an MAA member. Of course, one of the first things you discovered is that the Monthly was and still is a wonderful source of problems, especially useful for the weekly undergraduate problem session that had been running in your department for several years. You immediately get involved in that problem session. And for the next 24 years, your involvement in the MAA consists of reading the Monthly and answering a call by one of the terrific Monthly editors to review problem submissions and edit solution sets. Oh yes, and you attend a grand total of one section meeting. At that meeting, you meet the legendary Herta Freitag, attend an invited address by the prominent number theorist Carl Pomerance, and are introduced to a young graduate student named Art Benjamin. One section meeting in 24 years. Pitiful. But again, things began to change. You are promoted to full professor, and about that time, two things happen. The first is that you realize that in number theory, you are never going to be Gauss. The second is that you teach an undergraduate course in number theory that does not go well. In fact, it is a complete disaster. Your student evaluations contain many creative descriptions about how awful a teacher you were — and they were right. So you take the evaluations home and burn them in the fireplace. But you can’t burn the words. So you resolve to do something about it. You receive and heed plenty of sound advice from several outstanding teachers on your campus. You get your ego out of the classroom. You put your students and the subject matter at the center of each class. You put into practice many of the ideas that are presented at MAA workshops— although you did not know it at the time. You put into practice many of the ideas that are presented to Project NExT Fellows — although at that time, Project NExT was years in the future. You get involved on your campus with a calculus-readiness program called the Emerging Scholars Program (ESP) that is modeled after Uri Treisman’s collaborative workshops at Berkeley and at the University of Texas. You mentor graduate students and young faculty colleagues in teaching. You take on the enjoyable task of helping colleagues prepare dossiers for possible teaching awards. And then inJanuary of 1999,your department head tellsyou to go tothe springmeeting of the MD/DC/VA Section, which is your MAA section, because (as he puts it) the people there are interested in our ESP student-success program and want to give us some money. At the meeting, you discover that your department head had brought you there under false pretenses, and that you were really there to receive your section’s teaching award. You are rendered totally speechless. And then you think to yourself that if these folks have done this for you, the very least that you can do is go to their meetings and get involved in their section, which is of course your section. So you do, and the only section meeting you miss from that day down to this is one that takes place the day after you have cataract surgery. Getting Involved with the MAA: A Path Less Traveled 395

Around this time, you go to a Joint Math Meeting in Baltimore to give a talk about getting students involved in undergraduate research. You see a long-timefriend and number theory colleague named Underwood Dudley, who tells you that he has been named editor of The College Mathematics Journal and invites you to submit some expository articles. So you write up two pieces for the CMJ and they are published. Because of those two articles, you end up at MathFest in 2000 (at UCLA) and 2001 (at Madison), where you meet Frank Farris, editor of Mathematics Magazine. Frank suggests that you might want to write an expository article for Math Mag, and so you do. You go to MathFest in 2003, where you meet and are able to personally thank one of the referees of that article, the great number theorist and combinatorialist Richard Guy. You give a talk at a meeting of the Northeastern Section in 2004 and have a conversation with one of the other speakers — none other than Art Benjamin. The result of this chat is that the two of you organize an Invited Paper Session called Gems of Number Theory for MathFest 2005 in Albuquerque. It turns out to be a roaring success. The MAA asks the two of you to put together a collection of articles on number theory that would be accessible to students who have had, or who are taking, a ﬁrst course in number theory. The two of you do just that, and the MAA publishes that collection under the title Biscuits of Number Theory. Apparently, lots of people like it. Meanwhile, back in the MD/DC/VASection, you join the section’s teaching committee, then you chair it, then you are named the section’s Program Chair, and then you are elected section Governor. During this time, the section begins holding its own version of Math Jeopardy at the spring section meetings, and the organizers ask you to be Alex Trebek. So, 24 years as a mostly-inactive MAA member have been followed by thirteen years of being quite involved with the MAA at both the sectional and the national level. And the question one might ask is: Why? There are many reasons why, but the main reason why is quite simple: as Dave Roselle said almost forty years ago, “You meet the nicest people there!” Department of Mathematics, Virginia Tech, MS 0123 568 McBryde Hall, Blacksburg, VA 24061 [email protected]

Henry L. Alder

Donald J. Albers and Gerald L. Alexanderson MAA Santa Clara University

Through the 1940s and into the late 1950sthe administrativeworkof theMAA was handled at offices on the campus of the University of Buffalo. From 1943 to 1947 W. B. Carver of Cornell was the MAA Secretary-Treasurer, succeeded in 1948 by Harry M. Gehman who served as Secretary-Treasurer until 1959 and then as Treasurer from 1960 to 1967. He also served as Executive Director from 1960 to 1968. So, to some extent, the MAA was a one- man operation during that time [4]. The job of Secretary-Treasurer was split in 1960 when Henry L. Alder of the University of California, Davis, became Secretary and remained in that office until 1974, a term length matched but not sur- passed more recently by Martha Siegel. The move to space in a large commercial office building on Connecticut Avenue facing Dupont Circle in Washington coincided with Gehman’s being replaced as Executive Director by Al Willcox who came to the MAA from the faculty of Amherst College in 1968. One of us (Alexanderson) became a member of the Board of Governors a few years later, shortly after the move into the current Vaughn Building, the first building in the three-buildingMAA headquarters complex just off Dupont Circle. Then the other of us, Albers, became Second Vice President in 1983; one year later Alexanderson became First Vice President. Earlier we had gotten to know Harry Gehman because when he retired he moved to Los Altos, California, to be close to his son (and perhaps because even Northern California is warmer than Buffalo) and became a member of the Northern California Section. But in spite of these connections with the administrationof the MAA, to us the MAA always meant Henry Alder in those days. While he was Secretary and for many years after, includinghis term as President, the Editorial Director was Raoul Hailpern (assisted by his wife Fanny) and they continued to work from the Buffalo campus. At that stage Buffalo remained the home of the Monthly. In 1998 the two of us got together with Henry to reminisce about earlier times in the

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MAA and we asked ourselves how we happened to get involved with the Association. We had always assumed that Henry acted as an advocate for each of us, in his capacity as Sec- retary, in getting us appointed to national committees and eventually nominated for office. Henry denied it. When we asked him how he got involved in the MAA, he didn’t know that either, though he admitted that in 1960, then MAA President Carl Allendoerfer had approached him and asked him whether he would be interested in being national Secre- tary. (He knew Allendoerfer from their both having been on the Board of Governors of the Pacific Journal of Mathematics.) All three of us had gotten our start in the Northern California Section. As section chair Henry had served in 1956, Alexanderson in 1971 and Albers in 1974; Alexanderson as section secretary-treasurer in 1967–69 and 1973 [1]. In the 1998 interview, Henry recalled that in the late 1950s “What particularly struck me was the involvement from mathematicians from all institutions of higher education—universities, colleges and community colleges. I was very impressed by their involvement as officers. If you look at officers at that time, they came from UC Berkeley, Stanford, UC Davis, San Jose State, San Francisco State, etc.” Henry attended meetings of the Executive Committee of the Northern California Sec- tion for more than twenty years, providing important historical information for those of us with less than perfect memories. Henry was born in Germany in 1922 and educated in Switzerland, spending his early academic life at the Eidgenösische Technische Hochschule in Zurich, followed by a PhD at the University of California, Berkeley, writing his dissertation under the supervision of the number theorist, D. H. Lehmer, remembered today for the Lucas-Lehmer test for Mersenne number primality. Alder’s research was in the theory of partitions. He wrote the well- regarded Partition Identities—from Euler to the Present, for The American Mathematical Monthly in 1969 [2]. It won the Lester R. Ford Award for Expository Writing from the MAA in 1970 [3]. When Henry was MAA Secretary, and later when he was President, he knew just about everything there was to know about the organization. When Ken Ross was chosen to succeed Henry as Secretary, just in time for Henry to run for President, Henry took Ken aside and told him that he would have a great deal of power as Secretary “and pointed out that Lenin was Secretary of the Communist Party, never President.” He was right, of course: a Secretary stays on, often many years in the position; a President serves a single short term. If knowledge is power, then the long tenure of the Secretary serves one well in shaping an institution. Of course, Allendoerfer had told him that the task of a Secretary is to tend to “the care and feeding of Presidents.” Henry quoted that regularly. Not only did Henry provide care and feeding of presidents, he also provided invaluable assistance to Executive Directors Al Willcox and Marcia Sward. His phenomenal memory, knowledge of overall MAA structure and committee functions, and efficient and timely responses to questions were legendary. He carried a briefcase filled with important MAA correspondence and minutes to meetings and often at committee meetings he would extract a document from his briefcase which would shed light on matters under discussion. Henry generated an enormous correspondence and would frequently call Al or Marcia in the MAA’s Washington office at 8:00 A.M. Eastern Time. That meant, of course, that he was up well before 5:00 A.M. Pacific Time taking care of MAA business. Did we mention that Henry Alder was dedicated to his job as Secretary? Henry L. Alder 399

Henry was an extraordinary teacher, often cited by Davis graduates as enormously influential in their development as teachers and mathematicians. In 1976 he won the UC Davis Award for Distinguished Teaching. He wrote out a series of instructions for beginning teaching assistants, advice that should be obvious but perhaps comes as a surprise to many: “Don’t keep your back to the class all of the time;” “Always say out loud what you are writing on the board, don’t just stand there and write;” and had he been writing in more recent times it may have included something like: “Don’t assume that it is sufficient to keep your eyes on the monitor of your laptop through the whole class period, reading what is shown by Power Point on the screen.” It was all good common sense. With E. B. Roessler, a colleague at Davis, he wrote a textbook on probabilityand statistics that lived on through many editions. Henry also understood very well what makes an MAA section work. He formulated rules that have since become well-known “Alder Rules.” His influence remains in the Northern California Section: (1) “No one should be asked to give a major address at a section meeting unless someone on the planning committee has actually heard the person speak; hearsay about a speaker’s purported skill as a lecturer is not sufficient.” Failure to follow this rule is often evident at both section and national meetings. (2) “Rotate meeting sites through institutions of different types to make sure one is reaching the various constituencies: major research institutions, second-tier public universities, liberal arts colleges, two-year colleges, government-supported or industrial research facilities.” (3) “Do the same when choosing candidates for the section chair to make sure every constituency has a chance over a five- to seven-year period to have someone from that group as a major officer.” (4) Further, any group organizing a meeting should keep in mind that “an MAA talk should be given for the benefit of the audience, not the benefit of the speaker.” (Of course some talks are for the benefit of the speaker if that someone is interviewing for a job and needs to explain his or her research. But that is not what an MAA talk is all about.) (5) One other factor in a section’s success that Henry cited was the long-standing custom of past officers’ attending section Executive Committee and Program Committee meetings, providing long-term continuity. Though Henry held some of the most important offices in the national organization, he offered lots and lots of his time to helping his section remain healthy. We recall that when he probably had less than a year to live—a victim of cancer— he made the long trek from Davis to the South Bay Area to attend a section meeting. And he seemed to be having an awfully good time. Since 2004 the Association each year at the national meetings has given usually three, occasionally two, Henry L. Alder Awards for Distinguished Teaching by a Beginning Col- lege or University Mathematics Faculty Member, so Henry’s name lives on at meetings of the MAA. One of the early winners has recently been elected President of the Association.

Bibliography [1] D. J. Albers, Henry and Jerry, in From Galileo (1939) to Santa Clara (2001): An update on the history of the Northern California Section/Mathematical Association of America, Edited by Leonard F. Klosinski, MAA–Northern California Section, Santa Clara, CA, 2001. [2] H. L. Alder, Partition Identities—From Euler to the Present, Amer. Math. Monthly 76 (1969) 733–746. 400 A Century of Advancing Mathematics

[3] G. L. Alexanderson, Remembering Henry L. Alder (1922–2002), FOCUS 23, No. 1 (January 2004) 4–6. [4] K. O. May, (ed.), The Mathematical Association of America: The First Fifty Years, MAA, 1972.

Gerald Alexanderson Department of Mathematics and Computer Science, Santa Clara University, 500 El Camino Real, Santa Clara, CA 95053 [email protected] Donald Albers 925 Arbor Rd., Menlo Park, CA 94025 [email protected] Lida K. Barrett

Kenneth A. Ross University of Oregon

As a project for the 2015 centennial of the MAA, members of the history subcommittee of the MAA centennial committee have been interviewing prominent members of the mathematical community. The excerpts here are based on an interview that took place August 11, 2006, in Knoxville, Tennessee.

When did you get interested in mathematics? What were the circumstances?

In the ﬁfth grade. I was bored in school and acted up, so I had to stay after school and do long-division problems. I got really skilled at arithmetic in the process of doing problems as rapidly as possible. Later in junior high I was encouraged by Miss Emma French to try for themath team. I was selected for the team and we trained hard, learninglots of shortcuts for calculations and properties of mathematics.

Lida Barrett in the 1940s

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Where did you go to school? I was born and grew up in Houston; I went to public schools, including San Jacinto High School. I was barely sixteen when I graduated; note that there were eleven grades in school at that time and I skipped one. I went to Rice Institute (now Rice University), where I graduated at eighteen. At Rice I started out as a biology major but switched to math as a sophomore. I was encouraged to be a math major by the department chair, Hubert Evelyn Bray, who hired me to grade papers. I had a lousy calculus teacher, a graduate student who eventually became an outstanding mathematician, and I really learned calculus while taking advanced courses and grading papers. After Rice, I worked as a mathematician for Schlumberger Well Services. In order to get into their research division, I needed a graduate degree or experience working in the field, which at that time they wouldnot let women do. In the summer of 1947, I was recruited to teach at the Texas State College for Women (TSCW). Bray recommended me when they were searching for someone that summer. The department head was Harlan C. Miller, a Moore PhD She encouraged me to go to [the University of] Texas for summer school in the summer of 1948. She had taught a course just for me on the number system using the Moore method, and she encouraged me to take a course with Moore that summer. I was offered a graduate assistantship, so I did not return to TSCW but stayed at Texas. I took courses from R. L. Moore every year. I received a master’s in 1949. I married John Barrett in 1950, whom I met in graduate school at Texas. John obtained his PhD in 1951.John went to theUniversity of Delaware, so Mooreand J. R. Klineworked it out so I could study with Kline, who was Moore’s first PhD and department head at the University of Pennsylvania. I had to take comprehensive exams at Penn; there weren’t any at Texas, where they used grades and class participation instead of comprehensives. I only took five courses at Penn, so I spent a summer cramming for the comprehensives. The day after I turned in my thesis, Kline had a heart attack. His young colleague Dick Anderson took over and did the hard work of directing my work, editing and getting the thesis writtenup. I got my PhD from Penn in 1954, and my official thesis adviser was J. R. Kline. But Dick Anderson deserves a lot of credit. In 1954–1955, I attended seminars at Penn, and John continued teaching at Delaware. In 1955–1956,John was a visitor at Yale on NSF funding and worked with Einar Hille, and I taught at the University of Connecticut at Waterbury. The most important mathematical support in my life came from my husband. After he completed his PhD, he insisted that I complete mine. During the three years I commuted from Newark, Delaware, to the Univer- sity of Pennsylvania in Philadelphia, he did a major portion of the housework, had dinner ready when he picked me up at the train, and was totally supportive of my graduate study.

When did you join the MAA? Who influenced you in this decision? I joined the MAA in 1955 while my husband, John, was visiting Yale and I was teaching at the University of Connecticut branch at Waterbury. Ed Begle told us that mathematicians ought to be MAA members.

How did you get actively involved in the MAA? We went to meetings, and in the 1960s we both served as MAA visiting lecturers. Lida K. Barrett 403

What accomplishments in the MAA are you especially proud of?

I had two agendas: improving the committee structure and broadening participation. I was involved in modifying the MAA’s process for choosing committee members and chairs so that we now have more diversity and broader participation. We instituted terms for committee members and limited the number of committees any individual could serve on. I worked hard to ensure that the broader participation included more minorities. Ray Johnson, at the University of Maryland, was particularly helpful in identifying African- American mathematicians. I found Bob Megginson, a Native American, who was then on the faculty of Eastern Illinois. I urged my successor, Debbie Haimo, toward the council concept that the MAA subsequently adopted.

Let me explain why I was a different sort of MAA president. When I was an associate provost at Northern Illinois University, I was sent to a five-week seminar at the Harvard Institute of Educational Management. This experience sharpened my awareness and concerns of how organizations worked. This included committee structure and finances. The staffing of the finance division and of publicationsat MAA headquarters was modified. I’m pleased to take some credit for the changes that made it possible to bring Don Albers to the MAA. Another accomplishment of mine is that I helped defeat a proposal for the MAA to move its headquarters to Alexandria, Virginia. That was a key decision because some people, including Al Willcox and [Ken Ross], favored the move. One of my interests was the relationship between research in undergraduate education and teaching practice. I was interested in the MAA publications presenting some of this material. At one point an editor of the Monthly did not include anything on education. There was some discussion about a possible new journal in this area, but [it was] generally

Barrett in 2012. 404 A Century of Advancing Mathematics agreed that there were not sufﬁcient articles to justify a journal. However, it was agreed that editors would be encouraged to consider quality education articles for inclusion in the MAA journals.

During your career, what personalities have stood out in the mathematics community? When my husband and I started attending math meetings in the 1950s, they were much smaller. In fact, they used to list the names of those who attended meetings in the reports in the AMS Bulletin. It was wonderful to have direct contact with the leaders in the community. I remember meeting and talking with the presidents of MAA and AMS. The Moore PhDs who were presidents of MAA (Ed Moise, Gail Young, R. H. Bing, G. T. Whyburn, Dick Anderson) were all leaders whom I knew on a personal basis. Saunders Mac Lane was well spoken, sometimes outspoken, with his ideas and leadership in math activities. Leon Cohen’s work at NSF [National Science Foundation] contributed to what was happening in research. Baley Price worked hard to make CBMS [Conference Board of Mathematical Sciences] effective, and he led a move to have it more closely integrate the working of the math organizations. Shirley Hill, as the ﬁrst head of the Mathematical Sciences Educa- tion Board at the National Academy of Sciences, was a force in bringing education to the attention of the whole mathematics community. The secretaries of the MAA have all played important roles. Presidents come and go. Secretaries provide continuity: Dave Roselle, Ken Ross, Jerry Alexanderson. Don Albers has done so much for MAA publications. Certainly all the presidents of the MAA during my years as a member have had an impact.

2063 Lincoln St., Eugene, OR 97403 [email protected] Ralph P. Boas

Daniel Zelinsky Northwestern University

Ralph Philip Boas was a prolific analyst and a powerful contributor to the mathematics community. He wrote nearly two hundred papers on real and complex variables and served as president of the MAA, vice president of the AMS, and editor of several journals. He was connected with Mathematical Reviews from its founding by Neugebauer in 1940 and served as its executive editor from 1945 to 1950. He was editor of The American Mathematical Monthly from 1977 to 1980 and was the recipient of the Association’s Gung and Hu Distinguished Service to Mathematics Award in 1981. He was my close friend and most valued colleague for his last 42 years. Our daughters grew up together and are still good friends. His wisdom, energy and wit were models for me and were instrumentalin shaping the mathematics department at Northwestern University where I lived from 1949 to 1993. Ralph was born in 1912 and died in 1992, survived by his wife Mary, a physicist and author of a long-lived text Mathematical Methods in the Phys- ical Sciences, and by a daughter and two sons, one of whom, Harold Boas, is a distinguishedprofessor of mathematics at Texas A&M University. The Northwestern mathematics department had been a good one in the 1920s but by 1948 the faculty members doing serious research were reduced to Ernst Hellinger, who retired in 1949, and W. T. Reid, an old timer from the calculus of variations group at the University of Chicago. The rest of the teaching staff consisted of one fresh PhD, one specialist in quality control, a few educators, a statistician who was more interested in printing and puns, and a bunch of graduate students. Boas arrived in 1950 and in a few years rebuilt a very good research group. He also cooperated with Reid—to continue the project begun by Hellinger—to build a fine mathematics library. This project workedwell and the library is now called the R. P.Boas Library. Rebuilding the research faculty culminated with the appointment of Jean Dieudonné whose stay at Northwestern, 1953–1959, was the longest of his career until then. He left to join the faculty at Institut des Hautes Etudes´ Scientifiques outside Paris. Dieudonné

405 406 A Century of Advancing Mathematics had helped found this institute in 1958 partly to provide a position for his former student Alexander Grothendieck who, in spite of his enormous importance, could not get a position at any major French university because he was stateless, not a French citizen. At this time Dieudonnéwas drafting and typing Grothendieck’s influential Eléments´ de Géométrie Algébrique which revolutionized the approach to algebraic geometry and many similar subjects. Dieudonnéhad also done this for much of Bourbaki’s Eléments´ de Mathématique in the 1930s. So we at Northwestern were immersed in Bourbaki and Grothendieck. The French mathematicians, including Dieudonné, who founded Bourbaki zealously insisted that he was not a fiction. Boas had an interesting correspondence because of this. He wrote the article on mathematics in an annual volume of the Encyclopedia Bri- tannica, where he reported the appearance of another volume in Bourbaki’s Eléments´ de Mathématique.He remarked that Bourbaki was the pseudonym of a group of French mathematicians. Soon he received a letter from Bourbaki (we always assumed it was from André Weil) from somewhere in the Himalayas saying, “You miserable worm! How dare you imply that I do not exist? I shall spread the word that Ralph Boas does not exist but is just a pseudonym for a group of editors of Math Reviews.” Boas was interested in pseudonyms and wordplay; one of his early papers appeared under the pseudonym H. Pétard, after Shakespeare’s line in Hamlet, “hoist on his own petard” (a petard was a military explosive device) [2]. It sparked a host of imitators, several reprinted in [1]. He toyed with other pseudonyms like Pondiczery or Lemontré (lemon tree), or Zitronenbaum, the first of which he used in more than one paper [3, 4]. His interest in pseudonyms was part of his fondness for arcane properties of infinite series and unusual or unexpected twists of language, like “white as the may” where “the may” is another name for the hawthorn; or “the moon is green cheese” where the color is not green but white like unripe (unfermented, fresh) cheese. He enjoyed a putative journal Trivia Mathematica which was concocted by Aurel Wint- ner and Norbert Wiener, but I believed he had a hand in it.It consisted of titles of imaginary articles like the one that always reminds me of Ralph’s spirit: Method of steepest descent on weakly bounding bi-cycles.

Bibliography [1] G. L. Alexanderson and D. H. Mugler, Lion-Hunting and Other Mathematical Pursuits, MAA, Washington, DC, 1995. [2] H. P´etard, A Contribution to the Theory of Mathematical Big Game Hunting, Amer. Math. Monthly 45 (1938) 446–447. [3] E. S. Pondiczery, Power Problems in Abstract Spaces, Duke Math. J. 11 (1944) 835–837. [4] ———, A Function-Theoretical Paradox, Amer. Math. Monthly 45 (1938) 307.

8100 Connecticut Avenue, Apartment 1009, Chevy Chase, MD 20815-2817 [email protected] Leonard Gillman — Reminiscences

Martha J. Siegel Towson University

My acquaintance with Len Gillman began when we arrived, separately but simultaneously, at the University of Rochester in September 1960. He came to Rochester from Purdue to be department chair. I came to begin graduate school. He immediately made the graduate students feel welcome as part of the professional life of the department. As I had an undergraduate degree in mathematics education and some classroom experience as a student teacher, I quickly became the “teaching” expert among the graduate students. Most of them were men; all had no experience in front of a classroom and almost all of us were TAs. I truly believe that they came for advice in their classrooms because Len sent them to me as part of their assignment. Len orchestrated most of what went on in the department. It appeared that many things were spontaneous, but in retrospect, I am sure he planned our lives carefully. The Common Room was the hub of the place. Coffee and mathematical conversation were on at all hours. And somehow Len knew everyone’s business. Assistantships paid very little, but tuitionwas covered and most of us had real places to sleep at night. Several of the foreign students may have slept in the Common Room for a while, but they somehow found affordable quarters by “divine” intervention. Len was a hard-drivinginstructor,who expected perfection from students.But he helped them reach that goal by marking up papers with careful and exacting comments on one’s mathematical correctness and one’s exposition. As a ﬁrst-year graduate student at Rochester, I was subjected to these demands in a Set Theory and the Real Numbers course. I had many a paper returned with corrections in one color for a mathematical error and corrections in another color to indicate that my exposition needed improvement. We were limited in the length of a proof, with the advice to make it elegant as well as transparent.

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Social life extended beyond the university. Len was very determined that graduate students have a balanced life. He insisted that I invite a particular young man to a department party at hishome early in my first year. The guy had been coming around to the department to drive me home now and then. I tried to demur, claiming that I did not want to encourage said youngman. But Len likedhim and thought I should too... how had he even managed to talk to him? Okay, Len was right ... I married the guy eventually! Len maintained that students who were happy in their social relationships would do better in their studies so he took a keen interest in the social life of the department and of particular students. He insisted that graduate students attend colloquia and that they be invited to parties with the faculty afterwards. This provided me and my fellow students with a very broad view of mathematics and mathematicians. He also emphasized music and encouraged many students who loved music to pursue it even while doing mathematics. Eastman School is part of the University of Rochester and I have been told that some students came to graduate school in mathematics at Rochester because of the music available there. Quartets in the Commons Room were not uncommon. It also instilled in many of us the strong sense of mathematical community — locally, nationally, and even internationally. When Sierpinski came to deliver a colloquium (in French), the graduate students were required to attend and to come to Len and Reba’s house afterwards for a reception. We were told that we would have to go through a reception line and (with Len standing next to Sierpinski) we would have to speak French to the guest of honor. I knew some Spanish, but no French, but like everyone else, I practiced some gracious phrases for this ordeal. I was instructed to bring my boyfriend (the guy Lenny liked) to the party too. Keep in mind that this was the early 1960s. The boyfriend was an active member of the Congress of Racial Equality (CORE) and wore the black and white lapel button of the organization.He kind of slid behind me as we went through the reception line to greet Sierpinski, and let me do the talking in French to the guest of honor, but Sierpin- ski hardly gave me a glance. Instead, he immediately said something (in English) to my boyfriend complimenting him on working for CORE. And then Sierpinski slipped him a few bills as a donation. Of course, Len noticed that immediately and remembered it forever. All graduate students were members of the AMS, but Len made sure we were aware of and shared interest in the activities and the journals of the MAA. Although he wasn’t the person who first appointed me to anything at the MAA, he encouraged me at every step once I was a member. Doris Schattschneider and I both served as Editors of Mathematics Magazine. Doris was a student in that Set Theory class with Len, and we both agree that any success we had as editors was at least partly attributable to that demanding course in 1960–61. Len was one of several people at Rochester who wouldn’tlet me quit before getting my degree, even after I got married, had two children and my father died. I had a wonderful advisor, Joop Kemperman, whom Gillman had lured from Purdue. I worked in stochastic processes and I taught a lot. Len would find money for me so I wouldn’t have to teach in the summer. He wouldn’t let me leave without my degree. One day I went in to see him to tell himI was not coming back. I complained that my research was goingpoorly,I was very frustrated, and I could not see myself ever finishing the thesis. He maneuvered me out the door with parting words, “Get a larger trash can.” For his faith in me I have been extremely grateful. Leonard Gillman — Reminiscences 409

Sometimes his faith was misplaced. Len and Reba’s daughter was in high school, maybe in college, and she started to hang out with the grad students in the department. She clearly was interested in one of the guys. Len was worried about her going to our parties, so he assigned me and my husband to be her secret chaperones. We protested, to no avail. But we didn’t do a very good job. I just couldn’t warm to that role. Len engineered a lot of close relationships at Rochester. Since faculty and students were so friendly to one another, young faculty members like Ken Ross and Wistar Comfort palled around with some of us. The married students and the young married facultyhad lots of fun together. I shared Ken’s office (at Ken’s invitation) when I was assigned to do the big lecture of a beginning statistics course at Rochester. There were 80–100 students in the class, from freshmen to first-year medical students and the problem sessions were taught by department TAs. The responsibility was awesome, as the only course in statistics I had had was Mathematical Statistics following a semester of probability. Len assigned the best graduate students to me, and I taught them statistics at the same time I taught myself and the 80–100 students who paid for the course! It was a great resume builder. The Lenny oftheMAA was theLenny I knew privately. Hewas astickler fordetails and pretty stubborn, but he thought out his positions carefully and mulled them over and over. Ralph Raimi reminded me of a time when Len was workingon the new math (SMSG) with Ed Begle. Apparently there was a discussion about using the word “numeral” instead of “number” for the elementary school material they were writing.Begle and Gillman thought that young elementary school students should not be burdened with the distinction,but they were overruled by others in the group. Begle wanted more in the way of argument against that decision. Later he was discussing the issue with Gillman. Len mulled this over for a while. When he saw Begle again, he asked him to write down his telephone number. Then he asked Begle to write down his social security number. Pretty quickly, Begle saw the point and they both enjoyed laughing at the folly of the others. Len Gillman left Rochester in 1969 to become chair at Texas. He lured some of Rochester’s best faculty to join him in Austin. Len served as Treasurer of the MAA from 1973 to 1986. He was not an easy person to work with. He was a perfectionist and even if you wrote him a short note, you would have to watch your grammar, your spelling, and your syntax. He could use very few words to say a lot about someone. Nevertheless, he dealt with investments and budgets of the Association with extreme diligence. It was during his term as Treasurer that the MAA was looking at relocating headquarters from Washington to the Virginia suburbs. He was vehemently opposed. The buildings at 1529 and 1527 Eighteenth Street, NW are in some ways his legacy. Len served as MAA President in 1987–88.He tried hard to appoint women to important committees when he had the chance and was a dedicated member of the Committee on Minority Participation from its inception in 1989. When he was Treasurer, Len approached Ken Ross about being MAA Secretary. In fact, it was at a Denver meeting in January 1983. Ken Ross had been serving as Associate Secre- tary of the AMS and had had little interaction with the MAA. After some persuasion, Ken agreed to stand for the job, was elected, and his term as Secretary began in January 1984. Around 1988, Ken and his wife, Ruth, concluded that the Secretary job was too darn big. So Ken got the Executive and Finance Committees (predating the Executive Committee) to 410 A Century of Advancing Mathematics agree and they split the responsibilities by creating the Associate Secretary position. Ken warmed quickly to the idea of being the first Associate Secretary, because he saw that as his lifetime specialty, having served in that role in the AMS. Inviting Ken Ross to consider becoming Secretary was a great favor to the MAA and to Ken. Ken claims that it changed his life, because he found the MAA to be wonderful. And Len’s ability to convince Ken to be the Secretary was an important gift to the MAA. Having Ken as the first Associate Secretary insured that the job was well done from the get-go. Of course, eventually, Ken himself was elected MAA President. In those days, most people knew that Len had an advanced degree from Juilliard in piano. Len got the degree in piano before the PhD in mathematics. He loved music and introduced wonderful concerts to national meetings. His piano recitals at national meetings drew a large and appreciative audience. He frequently invited other mathematicians with very high levels of musical accomplishment to join him in trios and quartets. At the one- hundredth birthday of the AMS, Lenny led the crowd of nearly 2000 mathematicians in singing“Happy Birthday.” He and Reba were very active in the Gilbert and SullivanSociety in Austin, frequently performing in its productions. Even into his nineties he continued to play the piano and stay involved with Gilbert and Sullivan productions. Len died in 2009. I had known Len since September of 1960. In many ways he is my mathematical father and mentor. He and his wife Reba have shared many of our family celebrations and mathematical triumphs.My children threw a surprise birthdayparty for me in 1995. They invitedfriends listed in my address book— no matter where they lived. Reba and Len came from Austin for the fête. I was stunned and honored. When Len died, I was able to deliver a eulogy at his memorial service. He left a legacy that included dedication to family, friends, and profession. He opened his home, his office, and his heart to the people he loved. And because people knew he loved them, they tried their best to live up to his expectations. It has been an honorto have had thechance to knowLen Gillman.His sense of humor— always with the best bon mot for the situation—his talent as a mathematician, teacher, writer, musician, and caring mentor set a high standard that few of us can attain. I miss him. Mathematics Department, Towson University, 8000 York Road, Towson, MD 21252 [email protected] Paul Halmos: No Apologies

John Ewing Math for America

It is a melancholy experience for a professional mathematician to ﬁnd himself writing about mathematics. The function of a mathematician is to do something, to prove new theorems, to add to mathematics, and not to talk about what he or other mathematicians have done. Statesmen despise publicists, painters despise art- critics, and physiologists, physicists, or mathematicians have usually similar feelings: there is no scorn more profound,or on the whole more justiﬁable, than that of the men who make for the men who explain. Exposition, criticism, appreciation, is work for second-rate minds. — G. H. Hardy, A Mathematician’s Apology (1940)

Hardy’s famous lament was written near the end of his life when he felt his research career was over. It is a sentiment often felt by aging mathematicians, even when they don’t admit it. But it was not a sentiment shared by Paul Halmos. His abiding interest in writing and speaking about mathematics—in communicating its elegance, beauty, mystery, and fascination—was a hallmark of his life. Paul loved mathematics; he loved being a mathematician; and he was unapologetic about his passion.

Paul Halmos, 1981

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This passion was evident in his written exposition, which was proliﬁc. More than half his nearly 200 published works were expository, ranging from short papers on a topic (The Foundation of Probability [1]) to longer historical surveys (American Mathematics from 1940 to the Day before Yesterday [2]). Most of these were published in The American Mathematical Monthly—a publication that was always near to his heart (and for which he eventually became editor-in-chief). Many of his expository papers were book reviews. He believed a review should be a chatty essay about the book’s mathematics rather than a dull report on its table of contents. Book publishers and book authors usually disagreed. Char- acteristically, Paul put his readers ﬁrst—a review was not an advertisement for the book, he said, but rather an excuse to expound mathematics. When he became book review editor for the Bulletin of the AMS, he converted his personal views into policy, and it changed the way people thought about book reviews ever since.

Paul Halmos, 1959

Paul’s passion was also evident in his talks, which were legend for their clarity. He was a brilliant speaker—relaxed, polished, and precise. From his opening remarks to his punctuated ﬁnale, his talks were planned and executed with fanatical attention to detail. But the planning and structure were never intrusive or even apparent, and this was no accident. Paul claimed that he spent 50 hours preparing for a talk. He wrote down all the words, although never read them aloud. He rehearsed, edited, timed, and perfected, and he did these things many times. It showed. He was always disdainful of people who gave ”spontaneous” talks for colloquia or seminars. And his passion for being a mathematician (and an academic) showed in every aspect of his life, from the mundane to the exalted. When he was at Indiana, he commanded a lunch crowd every day by banging on ofﬁce doors with his cane. The lunch conversation invariably was mathematical, from clever problems to professional gossip, and included newcomers and old-timers alike. Together with daily tea, which he attended faithfully, this was a routine that bound the department together. He attended almost every colloquium, often photographing the speaker ahead of time with child-like exuberance. He went to the Paul Halmos: No Apologies 413

Virginia Halmos, 1950

Joint Meetings every year, not because it was a place to foster his research, but because it was a professional responsibility,as well as something he enjoyed. He knew (really knew!) a huge number of mathematicians, their families, and often their students. And he went to the International Congress for similar reasons, not to do research but to be a mathematician. He believed in the American Mathematical Society and the Mathematical Association of America. He was a member of and actively participated in both. Paul and Ginger were inveterate dinner party hosts, bringing groups of mathematicians together with musicians, scientists, and intellectuals of every kind—the more eclectic, the better. Their personal and professional lives were fully and deliberately intertwined. For Paul, being a mathematician meant living a life that was dedicated to academics, not exclusively but predominantly. (He was devoted to his own pets as well as the pets of others; he was obsessive about his daily walks.) For Ginger, being a spouse of a mathematician meant immersing herself in the community, and she always interacted with mathematicians effortlessly and intelligently. Paul had strong beliefs and opinions, and he was unafraid to share those opinions with others. Sometimes these took the form of wise essays that were aimed at young mathematicians early in their career (How to write mathematics [3], How to talk mathematics [4]). Sometimes they merely reﬂected his personal beliefs—he was an avid promoter of the Moore method, and like Hardy, he classiﬁed some mathematicians as ”second-rate minds,” although for Paul these were administrators rather than expositors. Sometimes his opinions were more strident or controversial (Applied mathematics is bad mathematics [5], The calculus turmoil [7]). I met Paul later in his life, when he was nearing sixty. We became good friends, despite the difference in our ages. When he was book review editor of the Bulletin,I was his (unof- 414 A Century of Advancing Mathematics

ficial) assistant editor; when he was editor-in-chief of the Monthly,I was his (official) book review editor. He introduced me to Walter Kaufmann-Buhler, the accomplished Springer editor, and Paul encouraged me to become editor of the Intelligencer. He persuaded Walter to add me to the editorial boards of several Springer series, where I joined him and Fred Gehring—one of the most professionally rewarding roles of my career. I learned a little mathematics from Paul, but not a lot. I learned everything about being a mathematician. Eventually, I spent many hours reading drafts of his automathography, I Want to Be a Mathematician [6]. It helped me to understand what made Paul the way he was. Paul’s life was not a simple one. In one sense, he was throwback to an earlier time, when roles were clearly defined and one’s career defined who you were. He liked that idea—he wanted to be defined as a mathematician. On the other hand, he was remarkably adventurous, in his teaching, his editorial work, and his choice of residence. He was radical about the Moore method, which he believed in deeply. He was an incredibly creative editor, shaping journals and book series alike. He moved from one university to another many times—Chicago, Michigan, Hawaii, Indiana, Santa Barbara, Indiana (again), and Santa Clara—and often traveled to out-of-the-way places to give talks. And he devoted himself to every aspect of mathematical life—research, teaching, writing, editorial, and service (to mathematics). There have been many superb mathematicians in the past century whose research changed mathematics. It would be hard to find a mathematician, however, with as broad a mathematical repertoire as Paul Halmos—research, writing, speaking, editing, and professional wisdom. And for that, Paul was rightfully unapologetic.

Bibliography [1] P. Halmos, The Foundations of probability, Amer. Math. Monthly 51 (1944) 493–510. [2] ———, W. P. Ziemer, W. H. Wheeler, S.H. Moolgavkar, J. H. Ewing, W. H. Gustafson American mathematics from 1940 to the day before yesterday, Amer. Math. Monthly 83 (1976) 67–76. [3] ———, How to write mathematics, L’Enseign. Math. 16 (1970) 123–152. [4] ———, How to talk mathematics, Notices Amer. Math. Soc. 21 (1974) 155–158. [5] ———, Applied mathematics is bad mathematics, Mathematics Tomorrow, Springer-Verlag, New York, 1981, pp. 9–20. [6] ———, I Want to Be a Mathematician, Springer-Verlag, New York, 1985. Reprinted by MAA in 1988. [7] ———, The calculus turmoil, FOCUS, October 1990.

Math for America, 915 Broadway,New York, NY 10010 [email protected] Ivan Niven

Kenneth A. Ross University of Oregon

The early years

Ivan Morton Niven was born in Vancouver, B. C. on October 25, 1915, and lived there until he was 21. His working-class parents had emigrated from Scotland. Ivan grew up “in a school system that was tightly structured and disciplined by today’s standards. (There was little concern about whether we felt good about ourselves.) The purpose was to stuff our empty heads as full of knowledge as possible” [8]. Ivan earned his Bachelor’s and Master’s degrees (1934 and 1936) at the University of British Columbia, and he was awarded his PhD in 1938 at the University of Chicago. He worked with the algebraist and number theorist Leonard Eugene Dickson. His thesis was titled A Waring Problem. Following his work with Dickson, in 1943 he settled some difﬁcult cases of Waring’s problem, one of the greatest problems of number theory and one that put Niven’s name in an illustriouslist of fellowcontributorslike Hardy and Littlewood, Vinogradov, S. S. Pillai, as well as Hilbert himself.

Betty and Ivan Niven, 1989

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Ivan held a postdoctoral research fellowship at the University of Pennsylvania in 1938– 1939. This was a valuable year because he was inﬂuenced there by the eminent Hans Rademacher, and this is where he met Herbert Zuckerman, who became a life-long friend and co-author. Ivan was then on the faculty of the University of Illinois for three years and Purdue University for ﬁve years. At Purdue, Ivan met Paul Erd˝os with whom he collaborated several times. Ivan married Betty Mitchell from Chicago in September 1939. Their son Scott, also a mathematician, was born in February 1942. Betty was active in local community issues and served on local and state planning boards and other committees. She was appointed chair of the Oregon State Housing Council by four different state governors; Betty Niven Drive in Eugene is named in her honor.

The University of Oregon

In 1947, Ivan returned to the Pacific Northwest and joined the faculty of the University of Oregon. Ivan was a key figure in the mathematics department and, in particular, he played a major role in developing a PhD program in mathematics. In fact, he was the advisor for the first three Oregon PhDs in mathematics: Luther Cheo (1950), John Maxfield (1951) and Margaret Maxfield (1951). In all, Ivan had sixteen PhD students. Though he chose never to be department head or a dean at Oregon, Ivan was a key figure across campus and was highly respected for his calm demeanor and sensible ideas. During 1959, when the state of Oregon celebrated its one-hundredth anniversary, three members of the University faculty were chosen to deliver Centennial Lectures. Ivan’s lecture was titled, “Mathematics: A House Built on Sand?” [5]. The evening lecture may have been ambitious for a general audience; the sections in the booklet based on the lecture were titled: crisis in geometry, the use of mathematical models, science and art, a digression on pedagogy, technical examples, unsolved problems, the crisis in set theory, intuitionism, and the final crisis.

Ivan, the communicator

Ivan Niven was the complete mathematician who was noted for outstanding teaching, popular books, a lifelong active research program, and generous service to the general mathematics community. Jerry Alexanderson was an undergraduate at the University of Oregon and took a course in applied mathematics from Ivan. This was well outside of Niven’s primary areas of interest, but the course was so beautifully executed that it helped persuade Alexanderson to major in mathematics. Ivan was an outstanding lecturer, prized for his clarity and enthusiasm and his sense of humor. He was in demand nationwide into the 1980s. In 1951, he gave an invited address to the American Mathematical Society. In 1960 he gave the Mathematical Association of America’s prestigious series of Hedrick Lectures, which Ken Ross found particularly inspiring. During 1962–1966, Ivan was a Travelling Lecturer of the Mathematical Association of America. In August 1990, he presented one of the lectures at the MAA’s Seventy-ﬁfth Anniversary Celebration, which was held at the Ohio State University where the MAA had been founded in 1915, when Ivan was 66 days old. Ivan spoke on “Problems for all Seasons.” Ivan Niven 417

Ivan, the mathematician

Ivan published over sixty papers, some with well-known co-authors such as Samuel Eilen- berg, Paul Erd˝os (six times), Nathan J. Fine, R. D. James, and H. S. Zuckerman (seven times). His areas of expertise were number theory, especially the areas of diophantine approximation and questions of irrationality and transcendence of numbers, and combinatorics. Ivan viewed his most signiﬁcant paper to be, Uniform distribution of sequences of integers [6], which started an entire theory. Three expository articles are of particular interest: his famous 1947 paper containing a simple proof that is irrational — less than one page in the AMS Bulletin [4]; his “relatively simple” three-page proof that is transcendental [3]; and his Monthly article [7] on formal power series, for which he received the Lester R. Ford Award. Ivan wrote seven books, including the Carus Monograph Irrational Numbers, published by the MAA; volumes 1 and 15 of New Mathematical Library series, Numbers: Rational and Irrational and Mathematics of Choice: Or, How to Count Without Counting; the MAA Dolciani Series publication on Maxima and Minima Without Calculus; Diophantine Ap- proximations; and the classic text An Introduction to the Theory of Numbers, co-authored with Herbert S. Zuckerman. A ﬁfth edition, co-authored with Hugh L. Montgomery, was published in 1991. Ivan’s lean and lively 172-page Calculus: An Introductory Approach, was published in 1961. Most of these books are still in print and collectively have been published in eleven different languages. Three mathematical concepts are named after Niven: Niven’s constant, Niven numbers, and Niven’s theorem. However, Niven’s theorem is Corollary 3.12 in Niven’s 1956 classic, Irrational Numbers, where Niven indicates that it was proved by J. M. H. Olmsted in 1945. The asteroid 12513-Niven, discovered by Paul G. Comba in 1998, also bears his name.

Ivan, the public servant

Throughout his career, Ivan was active in the wider mathematical community, especially within the Mathematical Association of America. He was elected First Vice President for the years 1974–1975, and he served as President in 1983–1984. For many years, he was a valuable member of the Executive and Finance Committees, which served as the “Board of Trustees” for the Board of Governors. He was a moderating influence, fiscally conservative, and cautious about new ventures. Ivan served on at least thirty MAA committees. He was a valuable member of all of them, but his greatest contributions were on committees concerned with publications of books, especially the New Mathematical Library series on which he worked with Anneli Lax for nearly thirty years. He also served on the Carus Mathematical Monographs board and the MAA Studies in Mathematics board. In 1989 he was presented the MAA’s highest award for achievement, the Award for Distinguished Service to Mathematics (now called the Yueh-Gin Gung and Dr. Charles Y. Hu Distinguished Service to Mathematics Award). See [9]. Ivan was also active in the MAA at the local level. He served as Governor of the Pacific Northwest Section from 1955 to 1958 and again from 1979 to 1982. He served on nominating committees and other ad hoc sectional committees. Ivan gave invited addresses at 418 A Century of Advancing Mathematics four different meetings of the section, beginning with the second meeting of the section in 1948 and with talks in 1975, 1978 and 1981. In addition, Ivan gave lectures at meetings of at least 24 of the MAA’s other 28 sections.

Ivan’s other lives Ivan especially loved Gilbert and Sullivan, as well as classical music, and he was a sup- porter of the Eugene Symphony, the Oregon Mozart Players, and the Chamber Music So- ciety. He wrote ditties; and he wrote Ivan’s Commandments: Thou shalt make an unceasing effort to see the world as it truly is, not as a product of your desires, not as a work of your imagination, not as a matrix of your special interests, but as an external reality that is no respecter of persons. Thou shalt not deliberately misstate or misrepresent another’s position by exaggeration, by quotation out of context or by confusing a statement and its converse. Neither shalt thou attempt to destroy another’s position by harping on some error or minor defect that in no way affects his principal contention. Thou shalt not claim to know more than thou knowest. Thou shalt judge the merits of a proposal in terms of its own worth, irrespective of the proponents thereof. Thou shalt not exalt trivial matters, nor claim as primary what is at best secondary. For who but a foolish person will resign from his church, his political party or his club because of one or two speeches or occurrences not to his liking. Thou shalt have the grace to concede a point without going into a huff, without claiming that wasn’t what you said, or meant to say, and without saying, “Didn’t you know I was only kidding?” Ivan was also an avid reader and for a long period, before ill health prevented it, he went salmon fishing regularly on charter boats out of Winchester Bay, Oregon. Swimming was his sport, and he swam daily as long as his health allowed. Ivan and Betty were pillars of the Unitarian Church in Eugene; Ivan served two terms as financial secretary and a term as corresponding secretary. He was also instrumental in the church’s purchase of land in 1959 on which a new church was built. Throughout his career, Ivan was the ultimate gentleman and consummate diplomat. In every situation he was thoughtful and wise, and he always gave more than he received. He made significant and lasting contributions to mathematics, and he performed his many tasks in an exemplary way earning him the highest respect of the entire mathematical community. Ivan passed away in Eugene, Oregon, on May 9, 1999, after a series of illnesses.

Bibliography [1] D. J. Albers, G. L. Alexanderson, A Conversation with Ivan Niven, Coll. Math. Jour. 22 (1991), 371–402. [2] G. L. Alexanderson, K. Ross, Ivan Niven Dies at 83, FOCUS, August/September 1999. [3] I. Niven, The transcendenceof , Amer. Math. Monthly, 46 (1939), 469–471. Ivan Niven 419

[4] ———, A simple proof that is irrational, Bull. Amer. Math. Soc., 53 (1947), 509. [5] ———, Mathematics: A House Built on Sand?, published by the University of Oregon, 1959. [6] ———, Uniform distribution of sequences of integers, Trans. Amer. Math. Soc. 98 (1961), 52– 61. [7] ———, Formal power series, Amer. Math. Monthly, 76 (1969), 87–889. [8] ———, Personal Reﬂection, Unitarian Universalist Church in Eugene, October 1987. [9] K. Ross, Award for Distinguished Service to Ivan Niven, Amer. Math. Monthly 96 (1989), 3–4.

2063 Lincoln St., Eugene, OR 97405 [email protected]

George P´olya and the MAA

Gerald L. Alexanderson Santa Clara University

In an earlier time mathematicians in academe were expected to be members of both the American Mathematical Society and the Mathematical Association of America, even when they may have thought of themselves mainly as researchers or, on the other hand, as teachers, depending in part on the kind of institutionthey were afﬁliated with. For George P´olya (1887–1985)there was no doubt about his stature as a research mathematician, but his early interest in problem solving made him a natural participant in the MAA. So he retained an active membership in both. His principal MAA involvement was his participation over many years in the Northern California Section, where he served as Chair in 1947 and Section Governor in 1958–60.

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During much of that time he was still actively continuing his work as a research mathematician. To be sure, his greatest achievements in mathematics came somewhat earlier: the Pólya enumeration theorem in combinatorics, motivated by problems in chemistry; his classification of the seventeen different plane symmetry groups in his 1924 article in the Zeitschrift für Kristallographie, which led to his correspondence with M. C. Escher [6, p. 23]; his massive body of work on entire functions and his 2z-theorem that paved the way for the Gelfond-Schneider solution to Hilbert’s Seventh Problem from the Paris Congress; and his work on hearing the shape of a drum. This last occurred well before Marc Kac’s question in the Monthly [3], Can you hear the shape of a drum?, a problem settled in the negative in 1992 by Carolyn Gordon and some colleagues in a paper in the AMS Bulletin [4]—Kac presented to Pólya a copy of his original paper on the subject with the inscription, “To another drummer, Marc Kac” [1, p. 110]. Pólya’s research spread over a wide range of branches of mathematics. His Inequali- ties, coauthored with Hardy and Littlewood in 1930, remains in print. In probability he is known for coming up with the phrases “central limit theorem” and “random walk,” albeit in German, as well as his development of the idea of the Pólya Urn Scheme. The Pólya Conjecture of 1919, on properties of primes, would have implied the Riemann Hypothesis, had it been true, but unfortunately for number theory, however plausible, counterexamples were shown to exist in 1958 [1, p. 77]. But we need not go on about his many discoveries. The list of theorems, inequalities, conjectures, operators, distributions, and so on takes up seven pages in [1, pp. 237–243]. He was on the faculty of Stanford,a major research university,of course, but throughout his research career he maintained an interest in exposition, teaching, and problem solving, which made him a natural participant in much of the work of the MAA. His interest in problems was evident in his classic work, Aufgaben und Lehrsätze aus der Analysis, coauthored with Gábor Szeg˝oin 1925. (A. O. Gelfond is reported to have been the only Soviet mathematician who solved all of the problems in that vast collection [7, p. 193].) Along these lines, his How To Solve It of 1945 remains in print (with translationsinto nearly thirty languages, and still commonly available in airport shops, often the only mathematics book to be found). During his long life Pólya published over 300 papers and books. He probably retains the record for talks given at meetings of the Northern California Section, fifteen in all, between 1943 and 1979. Between 1946 and 1952 he spoke at every single meeting, a real asset for the various chairs of the Program Committee! His topics ranged from “Estimating Electrostatic Capacity from Geometric Data” to “The Volume of the Sphere and Archimedes’s Discovery of the Integral Calculus” to “On Picture-Writing” to his much-quoted classic, “Some Mathematicians I Have Known” [4,p.53].Of course he was a popular feature of such meetings because he had a gregarious, ebullient personality and much enjoyed visitingwith former students, teachers, and readers of his How To Solve It, Mathematical Discovery, and Mathematics and Plausible Reasoning. But he also knew many in the research community at Stanford and Berkeley and it was very rewarding to be able to listen in on conversations and reminiscences between Pólya and Berkeley colleagues like D. H. and Emma Lehmer, Julia and Raphael Robinson, R. S. Lehman, and many other mathematical luminaries of the 1960s and 1970s. For the MAA he served, not surprisingly, on the committee that makes up problems for the Putnam Competition. Between 1946 and 1962 he produced the problems for the George Polya and the MAA 423

Stanford (later Stanford-Sylvania) Competitive Examination in Mathematics, a precursor of the multitude of contests for high school students now sponsored by the MAA. In later years he taught in many NSF-supported institutes for secondary school teachers of mathematics, thus inspiring a generation of teachers during a lively period of curriculum reform in America. The MAA recognizes Plya’s many contributions through its Pólya Lecturers, who periodically visit MAA Sections to speak at meetings, and the Pólya Awards for outstanding articles in the College Mathematics Journal. He was also the second recipientof theMAA’s prestigious Distinguished Service Award in 1963, following the first such award the previous year to Mina Rees. And when we visit the MAA headquarters in Washington, we find that one of the two principal buildings in the complex is named the Pólya Building,fit recognition for his long influence on the Association.

Bibliography [1] Gerald L. Alexanderson, The Random Walks of George P´olya, Washington, DC, Mathematical Association of America, 2000. [2] Carolyn Gordon, David L. Webb and Scott Wolpert, One cannot hear the shape of a drum, Bull. Amer. Math. Soc. 27 (1992), 134–138. [3] Mark Kac,Can one hear the shapeof a drum? Amer. Math. Monthly 73 (1966) , no. 4, 1–23. [4] Leonard F. Klosinski, From Galileo (1939) to Santa Clara (2001), An Update on the History of the Northern California Section of the Mathematical Association of America, Santa Clara, CA, 2001. [5] George P´olya and Jeremy Kilpatrick, The Stanford Mathematics Problem Book, New York, Teacher’s College Press, 1974. [6] Doris Schattschneider, Visions of Symmetry/Notebooks,Periodic Drawings, and Related Work of M. C. Escher, San Francisco, W. H. Freeman, 1990. [7] Benjamin H. Yandell, The HonorsClass: Hilbert’s Problems and Their Solvers,Natick, MA, AK Peters, 2002.

Department of Mathematics and Computer Science, Santa Clara University, 500 El Camino Real, Santa Clara, CA 95053 [email protected]