II'i=~.~--w_,~[,[,~ David E. Rowe, Editor 1
t takes a good deal of historical imag- a system of ideas ultimately grounded Euclidean ~ ination to picture the kinds of de- in undefined terms and arbitrary ax- bates that accompanied the slow ioms. A modern-day Platonist would ve- process which ultimately led to the ac- hemently object to this characterization, Geometry and ceptance of non-Euclidean geometry a which puts too much emphasis on little more than a century ago. The dif- purely arbitrary constructions rather ficulty stems mainly from our tendency than conceiving of geometrical figures Physical to think of geometry as a branch of pure as idealized instantiations of perfect mathematics rather than as a science forms. Space with deep empirical roots, the oldest Those who might like to see what natural science so to speak. For many such a debate looked like around 1900 DAVID E. ROWE of us, there is a natural tendency to need only read the correspondence be- think of geometry in idealized, Platonic tween the philosopher Gottlob Frege terras. So to gain a sense of how late- and the mathematician David Hilbert nineteenth-century authorities debated [Gabriel 1980]. Their dispute began over the true geometry of physical when Frege wrote Hilbert after reading space, it may help to remember the the opening pages of Hilbert's Foun- etymological roots of the word geome- dations of Geometry [Hilbert 1899], the try: geo plus metria literally meant to work that did so much to make the measure the earth. In fact, Herodotus modern axiomatic approach fashion- reported that this was originally an able. Although one of the founding fa- Egyptian science; each spring the Egyp- thers of modern logic, Frege simply tians were forced to remeasure the land could not accept Hilbert's contention after the Nile River flooded its banks, that the fundamental concepts of geom- altering property lines. Among those etry had no intrinsic meaning when engaged in this land survey were the seen from a purely logical point of view. legendary Egyptian rope-stretchers, the For Frege, points, lines, and planes harpedonaptai, who were occasionally were not simply empty words. They depicted in artwork relating to Egyp- were in some deep sense real; geome- tian ceremonials. try was the science that studied the We are apt to smile when reading properties of real figures composed of Herodotus's remarks, dismissing these them. Hilbert, to be sure, was by no as just another example of the Greek means adw)cating a modern formalist tendency to think of ancient Egypt as approach to geometry that broke with the fount of all wisdom. Herodotus was the classical tradition. In essence, his ax- famous for repeating such lore, and ioms for Euclidean geometry were here he was apparently confusing merely a refinement of those presented geometry with the science of geodesy, in Euclid's Elements. In 1905 he em- and the latter has little to do with the phasized that "the aim of every science former; at least not anymore. We do not is... to set up a network of concepts customarily think of circles, triangles, or based on axioms to which we are led the five Platonic solids as real figures: naturally by intuition and experiencg' they are far too perfect, the products of [Corry 2004, 124]. Hilbert thus recog- the mind's eye. Of course, there is still nized the empirical roots of geometri- plenty of room for disagreement. A for- cal knowledge, but he also emphasized Send submissions to David E. Rowe, malist will stress that geometrical figures that the question as to how and why Fachbereich 17--Mathematik, are mere conventions or, at best, im- Euclidean geometry conformed to our Johannes Gutenberg University, ages we attach to fictive objects that spatial perceptions lay outside the reahn D55099 Mainz, Germany. have no purpose other than to illustrate of mathematical and logical investiga-
2006 Springer Science+Business Media, Inc., Volume 28, Number 2, 2006 51 tions. For these, all that mattered at bot- Herodotus wrote about the Egyptian before the time of Pythagoras. It tells us tom was proving the consistency of a roots of the science of geometry. something fundamental about planar certain set of axioms. This reductionist measurements in right triangles: the viewpoint was sheer anathema in the Gauss, Measurement, and square on the hypotenuse equals the eyes of Gottlob Frege. the Pythagorean Theorem sum of the squares on the two other Most mathematicians, to the extent Gauss, after all, was not only a mathe- sides. Some have conjectured that the that they grasped what was at stake, matician and astronomer, he was also Egyptian harpedonaptai, whom Dem- sided with Hilbert in this debate. Over a professional surveyor who at least oc- ocritus once praised, used the converse the course of the next three decades the casionally waded through the marshy of the Pythagorean Theorem to lay out status of the continuum nevertheless hinterlands of Hanover taking sightings right angles at the corners of temples played a major role in the larger foun- in order to construct a net of triangles and pyramids. The claimants suggest dations debates between formalists and that would span this largely uncharted that these professional surveyors used intuitionists. Those rather esoteric dis- region [BOhler 1981, 95-103]. This work a rope tied with 12 knots at equal dis- cussions, however, left traditional real- helped inspire a profound contribution tance from each other; by pulling the ist assumptions about the nature of geo- to pure geometry: Gauss's study of the rope taut, they could form a 3-4-5 right metrical knowledge behind. For despite intrinsic geometry of surfaces, which triangle. It's a nice idea, but nothing their strong differences, the proponents helped launch a theory of measurement more. Archaeologists can still measure of formalism and intuitionism were both in geometry that opened the way to the angles of Egyptian buildings, of guided by their respective visions of probing the geometry of space itself. course, but our access to the mathe- pure mathematics, independent of its Only about a half century earlier, two matical knowledge that lay behind the relevance to other disciplines, like as- leading French mathematicians, Clairaut architectural splendours of ancient tronomy and physics. and Maupertuis, had studied the shape Egypt is highly limited. Papyri can eas- This suggests that a fundamental shift of the earth's surface, showing that it ily disintegrate with time, and only two took place around 1900 regarding the formed an oblate spheroid. As one have been found that provide much in- status of geometrical knowledge. This moved northward, they discovered, the sight into the mathematical methods of reorientation was certainly profound, curvature of the earth flattened, just as the time: the Rhind and Moscow papyri, but it seems to have been rather quickly Newtonian theory predicted. Mauper- which were presumably used as train- forgotten in the wake of other, even tuis's celebrated expedition to Lapland ing manuals for Egyptian scribes. Nei- more dramatic developments. Soon af- brought him fame and the nickname of ther contains anything close to the terward Einstein's general theory of rel- "the earth flattener." It also provided the Pythagorean Theorem. ativity would lead to a flurry of new dis- French Academy with stunning proof Historians of Mesopotamian mathe- cussions about the interplay between that Descartes's theory of gravity could matics have been luckier; they have had space, time, and matter. Leading math- not be right, thereby overcoming the plenty of source material available ever ematicians like Hilbert and Hermann last major bastion of resistance against since it became possible to decipher the Weyl became strong proponents of Ein- Newtonianism in France. Thus precise clay cuneiform tablets archaeologists stein's ideas, even as they sharply dis- measurements of the earth's curvature began turning up a little more than a agreed about epistemological issues re- had already exerted a deep impact on century ago. Some of the Babylonian lating to the mathematical continuum, a modern science. mathematical texts reveal not just a concept of central importance for the In the 1820s Gauss took the mea- passing familiarity with the Pythagorean geometer. surement of the earth as his point of Theorem but even a masterful use of it Looking backward from the 1920s, it departure for an abstract theory of sur- for numerical computations, like ap- would seem that the opposing views of faces, asking whether and how a sci- proximating the value of ~ or calcu- formalists and intuitionists actually re- entist could determine the curvature of lating Pythagorean triples. Herodotus flect distinctly modern attitudes about an arbitrary surface through measure- claimed that "the Greeks learnt the the nature of geometrical knowledge ments made only along the surface it- rroAog, the gnomon, and the twelve that would have been scarcely think- self, without knowing anything at all parts of the day from the Babylonians" able prior to 1900. Up until then, geom- about the way in which the surface [Heath 1956, vol. 1, 370]. etry was always conceived as somehow might be embedded in space. To talk Still, justly or not, we tend to credit wedded to a physical world that dis- about curvature as an intrinsic property the Greeks with being the first to give played discernible geometrical features. of a surface requires a careful recon- a proof of this ancient theorem. But Take the developments that led to the sideration of concepts like the measure since nearly all information about early birth of modern science in the seven- of distances between points and angles Greek mathematical texts is lost, we can teenth century: anyone who studies the between curves. So let's briefly review only speculate about the context of dis- works of Copernicus, Kepler, Galileo, the historical role of measurement in covery; we know nothing about the or Newton cannot help but notice the geometry. original proof itself. What we can ob- deep impact of geometry on their con- Consider the Pythagorean Theorem, serve is that right triangles play a cen- ceptions of the natural world. But the a ubiquitous result familiar to many cul- tral role in Euclid's Elements [Heath same can be said of Gauss, whose ca- tures and already found by Babylonian 1956, vol. 1]. Moreover, the Pythagorean reer ought to make us rethink what mathematicians more than a millennium Theorem and its converse appear as 1.47
52 THE MATHEMATICALINTELLIGENCER and surfaces, and they used calculus to study their differential properties. By mid-century they had even begun to leap by analogy into higher dimensions, and above all they liked to use com- plex numbers in connection with a mathematical realm of four dimensions. But to the extent they identified them- selves as geometers, mathematicians drew their inspiration from phenomena they could somehow visualize or imag- ine in ordinary Euclidean space. In the German tradition one spoke of anschauliche Geometrie, a term that does not really translate well into Eng- lish. The popular textbook with this title by Hilbert and Cohn-Vossen, published in the United States as Mathematics and the Imagination, serves as a reminder that Hilbert by no means held to a nar- row formalist view, despite the influ- ence of his work on axiomatization. In the preface, he wrote: In mathematics, as in any scientific research, we find two tendencies present. On the one hand, the ten- dency toward abstraction seeks to crystallize the logical relations in- herent in the maze of material that is being studied, and to correlate the material in a systematic and orderly manner. On the other hand, the ten- dency toward intuitive understand- ing fosters a more immediate grasp of the objects one studies, a live rap- port with them, so to speak, which stresses the concrete meaning of their relations. [Hilbert and Cohn- Vossen 1965, preface]. It would be exaggerated to call Hilbert-Cohn-Vossen a "picture-book" approach to geometry, but without the visuals their text would certainly lose its effectiveness. Throughout much of Western history, the discipline of geom- Figure I. Returning from the Lapland Expedition, Maupertuis became famous as the etry was closely linked with logic. But "Earth Flattener" who confirmed Newton's theory. many experts appreciated that the source of geometrical knowledge owed nothing to logical rigor. Hilbert and and 1.48, the two culminating proposi- without realizing that this is a distinctly Cohn-Vossen make this abundantly tions in Book I of the Elements. There- modern way of thinking. Classically, clear: they mainly just describe and ex- after Euclid makes use of it in several geometry was always about figures in plain what the reader is supposed to of the most important propositions of space, whereas space itself was never see. This kind of seeing, though, re- Books II and III. the object of study. One did not go quires imagination, and to be imagined about thinking of different kinds of a figure or configuration must have Euclidean Traditions and spaces, or even space in the plural. some relation to objects in real space, anschauliche Geometrie True, nineteenth-century mathemati- which takes us back to Euclid again. Today we talk about various models for cians differentiated between the metri- Euclidean geometry, in its original all kinds of geometries and spaces, cal and projective properties of curves garb, was traditionally regarded not as
9 2006 Springer Science+ Business Media, Inc., Volume 28, Number 2, 2006 one model among many for geometry, postulate as the crux of what made Eu- with another alternative, since we can but rather as the model for a rigorous clid's presentation of geometry Euclid- still read ancient works on spherics, or scientific system based on deductive ar- ean. what came to be known as spherical gument. As late as the 1880s, Charles Some of Euclid's modern rivals pre- geometry. The shortest distance be- Dodgson, better known today as Lewis ferred Playfair's more elegant formula- tween two points on the surface of a Carroll, was a valiant spokesman for this tion: sphere forms the arc of a great circle. conservative approach to teaching Playfair's Axiom.. Given a line I and Taking such geodesics as the counter- geometry. The tightly constructed argu- a point P not on it, there exists no more parts to straight lines in the plane, it is ments in the first two books of Euclid's than one line through P parallel to /. easy to see that the sum of the angles Elements left an indelible impression on in a spherical triangle exceeds 180 ~ (this the logician Carroll. angle sum varies: the smaller the trian- In Euclid and his Modern Rivals [Car- "The description of right gle, the smaller the sum of its angles) roll 1885], one of the oddest dramatic lines and circles, upon But if the Greeks already knew several works ever written (and surely never which geometry is founded, fundamental results of spherical geom- performed), Carroll brought Euclid's belongs to mechanics. etry, why did it take so long for math- ghost back to life to face his challengers. Geometry does not teach ematicians to accept the validity of non- This play without a plot quickly turns us to draw these lines, but Euclidean geometries? Why did they not into a bizarre mathematical dialogue in requires them to be realize that the geometry on the surface which Euclid defends his text and drawn." [Newton 1 726] of a sphere already provided a coun- leaves it to the judges in Hades to de- terexample to Euclid's theory of paral- cide whether any of the thirteen other lels? The answer to this puzzle deserves modernized treatments of plane geom- But Lewis Carroll clearly disagreed serious scrutiny as it not only tells us a etry deserved to take its place. The with them [Carroll 1885, 40-47, 77-84]. good deal about the intellectual context showdown that ensues has all parties His arguments seem to me both sound of geometrical investigations in antiq- citing the Elements, chapter and verse, and convincing. Playfair's version of the uity but also how the discipline of scurrying to discover which textbook Parallel Postulate is in the spirit of an geometry was perceived for many cen- most effectively honed the minds of existence theorem in modern mathe- turies afterward. young Englishmen. Just to be sure that matics; it lacks the constructive charac- If we widen the scope of our inquiry you have the right picture here: the au- ter that makes Euclid's rendition so use- into the ancient sciences, it becomes thors whose books come under discus- ful. Euclid's Fifth Postulate thus enables clear that spherics was studied for many sion are all respectable English gentle- the geometer to know in advance not centuries as a foundation for the higher men, among the leading mathematicians merely that two lines, when extended, disciplines of astronomy and astrology. of their day. So these rival texts obvi- will intersect, but also where the point For example, we learn from Aristotle ously did not breathe a word about the of intersection will occur, namely on the that the early fourth-century B.C. geome- new-fangled non-Euclidean geometries side where the angle sum is smaller ter Eudoxus of Cnidus developed a the- that had since found their way across than two right angles. Euclid makes cru- ory of homocentric spheres to study the channel from the Continent. Such cial use of this property, for example, heavenly bodies, including the retro- monstrosities clearly had no place in in Prop. 1.44 while executing a para- grade motion of planets like Mars. the college curriculum. Euclid's rivals bolic application of area for a given tri- About a century later a contemporary merely sought to upgrade the very same angle. Carroll clearly understood the of Euclid, Autolycus of Pitane, wrote On body of knowledge one found in the distinction between theorems and prob- the Moving Sphere, one of the oldest ex- Elements. Needless to say, the author of lems in geometry. He even has Euclid's tant mathematical texts. Another related Alice in Wonderland gave Euclid's ghost address the proposal by the work by the same author, On Risings ghost full satisfaction, routing the thir- British Committee for the Improvement and Settings', describes the paths of the teen rivals with scholarly acumen and of Geometrical Teaching that called for sun and stars throughout the year. A witty jibes. Nor should we be surprised presenting theorems and problems sep- more poetic account of the geometry of that the meatiest arguments on both arately. the heavens can be found near the be- sides were reserved for Euclid's con- ginning of Plato's Timaeus, where he troversial fifth postulate concerning par- What about describes how the cosmos was con- allel lines: Spherical Geometry? structed by the Demiurge using two That if a straight line falling on two Over time, mathematicians came to re- great circles. These works stood at the straight lines makes the interior an- alize that the Pythagorean Theorem is beginning of a long tradition, typified gles on the same side less than two mathematically equivalent to the paral- by the work of the thirteenth-century right angles, the straight lines, if pro- lel postulate, and both are equivalent to Augustinian cleric John of Holywood, duced indefinitely, will meet on that the proposition that the sum of the an- better known as Johannes de Sacro- side on which the angles are less gles in a triangle equals 180 ~ But why bosco, whose Sphaera was widely stud- than two right angles. did the Greeks feel compelled to limit ied by European astronomers up until Historically, mathematicians had geometry to what we now see as one the time of Christoph Clavius, an older long focused attention on this parallel special case? Clearly they were familiar contemporary of Galileo. By studying
THE MATHEMATICAL INTELLIGENCER Sacrobosco, one could learn the rudi- who lived shortly after Aristotle, around the works of Archimedes were taken up ments of spherical geometry, enough 300 B.C., they had become firmly es- by the mathematicians of the Renais- for an ambitious reader to proceed on tablished categories. Astronomy and sance. Only with the Copernican Revo- to Ptolemy's Almagest. Yet there is physics had virtually nothing to do with lution did this bifurcated worldview something fundamentally wrong with each other. Whereas spherics belonged come to an end. Galileo, the mechani- this formulation. For by combining to the former, geometry had close ties cian, sought to refute both Ptolemy and spherics with geometry to describe a tra- to ancient mechanics, which was not a Aristotle, but it took the even more dition that extended from Autolycus to branch of natural science at all, but daring ideas of Kepler and Newton to Sacrobosco, I have invoked a hybrid con- rather was synonymous with ancient finish the job. In the course of doing cept--spherical geometry--that makes technology. Mechanicians constructed so, the prospect of an infinite space no sense within that historical setting. machines just as geometers constructed emerged for the first time. In the ancient world, the motions of figures and diagrams. Among the heavenly bodies were thought to be Greeks, Archimedes was a virtuoso in On the Shoulders perfectly circular, unlike the natural mo- both disciplines. Indeed, his work suc- of Ancient Giants tions of terrestrial objects, which rise or cessfully bridged the gap between me- Newton effectively created the disci- fall as they seek their natural place in chanics and geometry, as he relied pline of celestial mechanics in his Prin- the world. Physical objects, as we know heavily on the law of the lever to cipia [Newton 1687]. In the forty years them here on earth, move about in the "weigh" geometrical figures before remaining to him, he had ample time space surrounding us. The traditional proving theorems about their areas and to comment on his legendary success, purpose of plane and solid geometry volumes. Geometry's close links to tech- and his assessments have been repeated was to study the properties of simple, nology, machines, and the science of countless times. It is tempting to dis- idealized figures in this terrestrial realm. mechanics became even stronger after count his famous pronouncement that Whether or not the roots of this science were Egyptian, it surely drew on cen- turies of human experience with phys- ical objects in space. To fill up parts of space, builders used bricks of a uniform size and shape, rectangular solids, not round ones or solids with curved sur- faces. Straightness and flatness were the primary spatial qualities one imagined in everyday life. A solid, such as a sphere, was obviously in some basic sense a more complex object than a solid figure bounded by plane figures. Both are treated in the last three books of Euclid's Elemen& but it is clear that the latter figures, particularly the five Platonic solids, were regarded as fun- damental. Some of these regular solids occur in nature in crystalline forms, and Plato identified four of them as the shapes of the four primary elements: earth, water, air, and fire. For the Greeks, the sphere had a deep cosmic significance: the planets and fixed stars were conceived as carried about on gi- ant invisible spheres. The natural rota- tional motion of spheres could be sim- ulated here on earth, of course, but the truly natural motions of terrestrial ob- jects were rectilinear: straight up and down. Of course the earth itself was not seen as fiat, but it was mainly on a cos- mic scale that the sphere came force- fully into play in Greek science. These distinctions were described in detail by Aristotle, who drew heavily on Figure 2. In Kepler's Mysterium Cosmograpbicum, the five Platonic solids structured earlier authors. By the period of Euclid, the Universe.
@ 2006 Springer Science+Business Media, inc., Volume 28, Number 2, 2006 55 of the Principia. Not that he was merely paying homage to the ancients or try- ing to accommodate modern readers, as some historians of physics have sug- gested. Since most physicists learn celestial mechanics based on the ana- lytic methods first developed by Euler, D'Alembert, and others, they have a hard time squaring what they find in the Principia with their image of Newton as co-founder of the calculus. Thus some have occasionally argued that Newton must have originally derived the results in his Principia by using the calculus. He then supposedly chose to couch the whole thing in the language of tradi- tional geomeu 7 in order not to over- whelm readers with mathematical ter- minology and techniques that were new and unfamiliar. There seems not to be a scintilla of evidence to support this claim, nor do l know of any leading Newton scholar who thinks that New- ton just dressed up the Principia in geometry to make it easier to swallow (if he had done so, we would have to conclude that he failed pretty miserably, since his contemporaries found it a very tough read, too). What we do find in Newton's published and unpublished writings are numerous explicit state- ments and arguments expressing why he preferred to use geometry as the nat- ural mathematical language for treating problems in mechanics. And by geom- etry, he meant traditional synthetic geometry in the tradition of Euclid and the Greeks. Newton believed that space had an absolute reality that endowed it with physical properties and geometrical he "stood on the shoulders of giants"- cept that Newton's system of mechanics structure. He attributed inertial effects on most occasions Newton was not would help quantify. This is the tradition like the centrifugal forces that accom- prone to such modesty--but perhaps of terrestrial mechanics represented by pany rotations to the effort required to we can make sense of this by asking Archimedes and later by Leonardo da move against the grain of space, so to just whose shoulders he might have had Vinci and Galileo. None of these Re- speak. His first taw, describing the na- in mind. naissance figures ever dreamed, of ture of force-free motion, tells us what We might start by reading the pref- course, that the heavens themselves it means to move with the grain of ab- ace to the Principia: "The description might be understood as a giant ma- solute space, namely in a straight line of right lines and circles, upon which chine. That was Newton's grand vision, with uniform speed. The principle of geomeFy is founded, be)ongs to me- made famous by 1he image of a Deity relativity, in its original form, is then a chanics. Geometry does not teach us to that designed the worm like an intricate simple consequence of Newton's first draw these lines, but requires them to clock. two laws. Because the laws of me- be drawn" [Newton 1726]. Newton went Newton was ahead of his time in so chanics all deal with forces acting on on to mention briefly the conception of many respects that it is easy to overlook bodies, and because these forces are mechanics set forth by Pappus of how steeped he was in the traditions of directly linked to accelerations by New- Alexandria around 300 A.D. In his Col- the past. Unlike Descartes, he revered ~on's second law, no physical experi- lection Pappus described five types of Euclid and the ancients, and this surely ment can distinguish between two in- machines designed to save work, a con- accounts for the arcane geometric style ertial frames of reference. Both move
56 THE MATHEMATICAL INTELLIGENGER with uniform velocity with respect to more to the conviction that the necessity the astronomer Wilhelm Bessel, Gauss absolute space and hence with respect of our geometry cannot be proved .... wrote: "We must admit with humility to each other. So no accelerations arise, Geometry should be ranked not with that, while number is purely a product unlike Newton's famous example of the arithmetic, which is purely aprioristic, but of our minds, space has a reality out- rotating water bucket. with mechanics." [Dunnington 2004, 180] side our minds, so that we cannot com- Leibniz and others objected vocifer- Ten years later, the "Prince of Math- pletely prescribe its properties a priori." ously to Newton's quasi-theological ematicians" published his pioneering Eight years later, Bessel successfully doctrines regarding absolute space and work on the intrinsic geometry of sur- measured stellar parallax under the as- time. But toward the end of the eigh- faces in which he introduced the no- sumption that light travelled along Eu- teenth century Immanuel Kant gave tion we today call Gaussian curvature clidean geodesics. But what if it did not? them a central place in his epistemol- [Gauss 1828]. In one sense, this notion On the surface of a sphere, the sum ogy. Kant's Critique of Pure Reason was was a refinement of the classical notion of the angles in a geodesic triangle ex- widely regarded as a tour de force that of curvature introduced by Leonhard ceeds 180 ~, but Gauss recognized there tamed the excesses of Continental ra- Euler in the eighteenth century. In was a second distinct possibility. In a tionalism and metaphysics while over- Euler's theory there are two principal letter from November 1824 he wrote: coming the scepticism of Humean em- curvatures associated with each point of There is no doubt that it can be rig- piricism. Newtonian space and time a surface. These are obtained by tak- orously established that the sum of provided Kant with the keys that led to ing the surface normal at each point as the angles of a rectilinear triangle a new synthesis. He argued that our the axis for a pencil of planes. Euler cannot exceed 180 ~. But it is other- knowledge of space and time had an proved that by rotating these planes wise with the statement that the sum utterly different character than all other of the angles cannot be less than forms of knowing: it was neither ana- 180~ this is the real Gordian knot, lytic nor a posterior--meaning that we "I am coming more and the rocks which cause the wreck of cannot know the properties of space more to the conviction all .... I have been occupied with and time by means of deductive rea- that the necessity of our the problem over thirty years and I soning nor by appealing to sense ex- geometry cannot be doubt if anyone has given it more perience. Nevertheless, we can formu- proved .... "[C. F. Gauss] serious attention, though I have late true synthetic a priori propositions never published anything concern- about them because they provide the ing it [Gauss 1900, 187]. foundations for all other forms of about this axis there will be two par- Gauss clearly ruled out spherical knowledge. Thus, according to Kant, ticular ones that cut the surface in geometry, presumably because its geo- space and time are the necessary curves with a maximum and minimum desics have finite length, contradicting preconditions for knowing; they supply plane curvature at the given point. Euclid's second postulate that a straight the transcendent categories that give Moreover, these two special planes will line can always be extended. But mankind the ability to know. (This was always be perpendicular to one an- though he dismissed the possibility of heady stuff, of course, but Kant's views other. Thus they determine two prin- triangles with an angle sum exceeding were enormously influential throughout cipal directions with plane curvatures 180 ~ he went on to note how the nineteenth century, an era when K1, K2. These, however, are not intrin- The assumption that the sum of the professional philosophers were still sic invariants of the surface since they three angles of a triangle is less than widely read.) depend on knowing how it sits in the 180 ~ leads to a special geometry, surrounding space. Remarkably, how- quite different from ours [i.e. Eu- Gauss and the Intrinsic ever, the product K1 K2 = K turns out clidean geometry], which is ab- Geometry of Surfaces to be an intrinsic invariant, as Gauss solutely consistent and which I have In different ways Euclid, Newton, and was able to prove in his Theorema developed to my entire satisfaction, Kant were still massive authorities dur- Egregium: so that I can solve every problem in ing the nineteenth century, and each re- If two surfaces are isometric, then it with the exception of the deter- inforced the established view that space they have the same Gaussian curva- mination of a constant, which can- carried a geometrical structure that was ture at corresponding points. not be fixed apriori. The larger one Euclidean. That position seemed invul- assumes this constant, the closer one nerable throughout most of the century, There seemed to be no reason to re- approaches Euclidean geometry and in part because no other mathematical gard these purely mathematical ideas as an infinitely large value makes the alternative seemed conceivable. It was a threat to Euclidean geometry so long two coincide. If non-Euclidean not until the 1860s that mathematicians as space itself was taken to be fiat. geometry were the true one, and began to take the possibility of a non- Gauss, however, thought differently that constant in some relation to Euclidean geometry seriously, this de- about this matter. Publicly he said noth- such magnitudes as are in the do- spite the fact that Carl Friedrich Gauss ing, but privately he made several allu- main of our measurements here on had entertained this idea throughout sions to the possibility that the parallel earth or in the heavens, then it could much of his career. In 1817 Gauss wrote postulate might actually fail to hold in be found out a posteriori. [Dun- to a colleague: "I am coming more and physical space. In a letter from 1830 to nington 2004, 181-182].
2006 Springer Science+Business Media, Inc., Volume 28, Number 2, 2006 57 Soon after Gauss's death in 1855, his ther he nor any of his contemporaries given in advance. It only gives nom- friend Sartorius von Waltershausen seem to have regarded this as any more inal definitions for them, while the wrote that he had actually attempted to than a curiosity. Not until 1866 when essential means of determining them test the hypothesis that space might be Eugenio Beltrami obtained a surface of appear in the form of axioms. The curved by measuring the angles in a constant negative curvature by revolv- relationship of these presumptions is large triangle that he used in his sur- ing a tractrix curve about its axis, did it left in the dark; one sees neither vey of Hanover. The triangle had ver- become possible to visualize such a whether nor how far their connec- tices located on the mountain peaks of non-Euclidean geometry. tion is necessary or a priori even Hohenhagen, Inselberg, and Brocken, This lack of Anschaulichkeit had possible. From Euclid to Legendre, which served as a reference system for something to do with the slow reception to name the most renowned of the system of smaller triangles. Several of the work of Lobachevsky and Bolyai, modern writers on geometry, this writers have argued that this famous but the delay was also due to a lack of darkness has been lifted neither by story is just a myth, but even if true, intellectual courage on the part of the the mathematicians nor the philoso- Gauss apparently concluded that the leading mathematical minds of Europe. phers who have labored upon it deviation of the sum of the angle mea- Surely the history of non-Euclidean [Riemann 1854, 133]. surements from 180 ~ was smaller than geometry would have unfolded quite dif- Riemann's approach abandoned re- the margin of error. So this test would ferently had Gauss made public his liance on a theory of parallels, turn- have merely confirmed that the geom- views on the theory of parallels. ing instead to a theory of distance etry of space is either fiat or else its In December 1853, Bernhard Rie- based on a generalization of the curvature was too small to be detected. mann submitted his post-doctoral the- Pythagorean Theorem. On a human sis to the G6ttingen philosophical fac- scale, he noted that the metric prop- Slow Acceptance of Non- ulty along with three proposed topics erties of space accorded well with Eu- Euclidean Geometry for the final lecture required of all new clidean geometry. However, "the em- If one is tempted to speak of revolu- members. The elderly Gauss presided pirical concepts on which the metric tions in the history of mathematics (a on this occasion and requested that Rie- determinations of space are based-- debatable point), then one might well mann speak about the third topic on the concepts of a rigid body and a regard the advent of non-Euclidean the list: "On the Hypotheses that Lie at light ray--lose their validity in the in- geometry in the nineteenth century as the Foundations of Geometry." Rie- finitely small; it is therefore quite a striking example [Bonola, Rosenfeld]. mann was undoubtedly surprised by likely that the metric relations of space Mathematicians tinkered for centuries that decision and none too pleased in the infinitely small do not agree trying to find a completely elementary about it. He complained to his brother with the assumptions of geometry, proof that Euclid's fifth postulate was that this was the only topic he had not and in fact one would have to accept true. Most, including the eighteenth- properly prepared at the time he sub- this as soon as the phenomena can century Jesuit Giovanni Girolamo Sac- mitted his thesis. The lecture took place thereby be explained in a simpler cheri, were convinced that it was not a the following June, and according to way" [Riemann 1854, 149]. postulate at all, but rather a theorem. Richard Dedekind it made a deep im- As for geometry in the large, Riemann By the 1820s and 1830s, the Russian pression on Gauss, as it "surpassed all emphasized that our intuition makes it mathematician Nicholas Lobachevsky his expectations. In the greatest aston- hard to conceive of physical space as and a young Hungarian named Janos ishment, on the way back from the fac- bounded, whereas a space of infinite ex- Bolyai showed that one could develop ulty meeting he spoke to Wilhelm tent poses real difficulties for cosmology. an exotic system of geometry in which Weber about the depth of the ideas pre- This suggested the possibility that our the fifth postulate was false and, instead sented by Riemann, expressing the cosmos might have the structure of a 3- of having only one line in the plane that greatest appreciation and an excite- dimensional manifold of constant posi- passes through a given point without ment rare for him" [Riemann 1892, 517] tive curvature. Riemann said all this and meeting a given line, there will be in- This was the famous lecture in which much more in 1854, but he took these finitely many. Riemann explained how the notion of thoughts with him to the grave. Neither It would take over three decades be- Gaussian curvature could be extended Gauss nor anyone else urged him to fore the publications of Lobachevsky beyond surfaces to manifolds with an publish his manuscript or pursue their and Bolyai gained belated recognition; arbitrary number of dimensions. In par- consequences further. not before the 1860s did mathemati- ticular, this meant that one could study Nearly unapproachable during his cians begin to take the new theory se- the intrinsic geometry of three-dimen- lifetime, Gauss passed from the scene riously. A major obstacle for this new sional spaces. Riemann began with this without so much as once publicly ad- non-Euclidean geometry was the lack assessment of how little progress had dressing the dogma that the geometry of of a "real-world model" in Euclidean 3- been made in clarifying the foundations space had to be Euclidean. After Rie- space comparable to the sphere. The of geometrical research: mann's death in 1866, Dedekind was ap- eighteenth-century Alsatian mathemati- It is well known that geometry pre- pointed editor of his Collected Works, cian J. L. Lambert had found an analytic supposes not only the concept of and it was he who stumbled upon the model by studying the properties of a space but also the first fundamental manuscript on the foundations of geom- sphere with imaginary radius, but nei- notions for constructions in space as etry among his deceased friend's papers.
58 THE MATHEMATICALINTELLIGENCER Its publication in 1867 sparked immedi- G. Waldo Dunnington, 2004. Carl Frledrich Isaac Newton, 1726. Philosophiae Naturalis ate interest not only in Germany, but in Gauss, Titan of Science, New York: Haffner, Prlncipia Mathematica, 3rd ed., Alexandre Italy and Great Britain as well. Still, it 1955; repr. Mathematical Association of Koyre, I. Bernard Cohen, and Anne Whitman, would take several more years before America, 2004. eds. Cambridge, Mass.: Harvard University non-Euclidean geometry found wide- G. Gabriel, et al. eds. 1980. Gottlob Frege- Press, 1972. spread acceptance among mathemati- Philosophical and Mathematical Correspon- Moritz Pasch, 1882. Vorlesungen dber neueren cians, many of whom remained con- dence. Chicago: University of Chicago Press. Geometrie. Leipzig: Teubner. vinced that Euclid still reigned supreme. C. F. Gauss, 1828. Disquisitiones generales circa Bernhard Riemann, 1854. "0ber die Hypothe- superficies curvas. GOttingen: Dieterich. sen, welche der Geometrie zugrunde liegen," FIIEI~IEllENCE$ --. 1900. Gauss Werke Bd. VIII, GOttingen: Abhandlungen der Kgl. Gesellschaft der Roberto Bonola, 1955. Non-Euclidean Geom- KOnigliche Gesellschaft der Wissenschaften. Wissenschaften zu Gdttingen. 13 (1867): etry: A Critical and Historical Study of its De- T. L. Heath, ed., 1956. The Thirteen Books of 133-152. velopment. New York: Dover. Euclid's Elements. New York: Dover. 3 vols. --. 1892. Bernard Riemann's Gesammelte Walter Kaufmann-BQhler, 1981. Gauss: A Bio- David Hilbert, 1899. "Die Grundlagen der Mathematische Werke und Wissenschaftlicher graphical Study, New York: Springer-Verlag. Geometrie," in Festschrift zur Feier der Ent- Nachlass, hrsg. unter Mitwirkung yon R. Lewis Carroll, 1885. Euclid and his Modem Ri- hdllung des Gauss- Weber Denkmals. Lepzig: Dedekind und H. Weber, 2. Aufl. Leipzig: vals. London: Macmillan, repr. New York: B.G. Teubner. Pp. 3-92. Teubner. Dover, 1973. D. Hilbert and S. Cohn-Vossen, 1965. Geom- B. A. Rosenfeld, 1988. A H/story of Non- Leo Corry, 2004. David Hi/bert and the Axiom- etry and the Imagination. New York, Chelsea. Euclidean Geometry: Evolution of the Con- atization of Physics (1898-1918), Dordrecht: Isaac Newton, 1687. PhilosophiaeNaturalis Prin- cept of a Geometric Space. Trans. Abe Kluwer, 2004. cipia Mathematica. London: Joseph Streater. Shenitzer. New York: Springer-Verlag.
ROCKY MOUNTAIN MATHEMATICA Join us this summer for a Mathematica workshop in the scenic and cool mountain environment of Frisco, Colorado, 9100 feet above sea level. From July 9-14, 2006, Ed Packel (Lake Forest College) and Start Wagon (Macalester College), both accredited Wolfram Research instructors with much experience, teach an introductory course (for those with little or no Mathematica background) and an intermediate course on the use of Mathematica. Anyone who uses mathematics in teaching, research, or industry can benefit from these courses. The introductory course will cover topics that include getting comfortable with the front end, 2- and 3-dimensional graphics, symbolic and numerical computing, and the basics of Mathematica programming. Pedagogical issues relating to calculus and other undergraduate courses are discussed. The intermediate course covers programming, graphics, symbolics, the front end, and the solution of numerical problems related to integration, equation-solving, differential equations, and linear algebra. Comments from participant Paul Grant, Univ. of California at Davis: The instructors had boundless enthusiasm for and an encyclopedic knowledge of Mathematiea. I was able to complete an animation project that I will be showing to my students. I highly recommend these courses to anyone wanting to learn Mathematica from the ground up or enhance their knowledge of the program. For more info, contact packel @lakeforest.edu or [email protected], or visit rmm.lfc.edu.
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