Knowledge and Community in Mathematics

Jonathan Borwein1 and Terry Stanway2

Mathematical Knowledge - As We Knew It

Each society has its regime of truth, its “general politics” of truth: that is, the types of discourse which it accepts and makes function as true; the mechanisms and instances which enable one to distinguish true and false statements, the means by which each is sanctioned; the techniques and procedures accorded value in the acquisition of truth; the status of those who are charged with saying what counts as truth.3 (Michel Foucault)

Henri Lebesgue once remarked that “a mathematician, in so far as he is a mathematician, need not preoccupy himself with philosophy.” He went on to add that this was “an opinion,

moreover, which has been expressed by many philosophers.”4 The idea that mathematicians can do mathematics without a precise philosophical understanding of what they are doing is, by observation, mercifully true. However, while a neglect of philosophical issues does not impede mathematical discussion, discussion about mathematics tends to quickly become embroiled in philosophy and perforce, encompass the question of the nature of mathematical knowledge. Within this discussion, some attention has been paid to the resonance between the failure of twentieth century eﬀorts to enunciate a comprehensive, absolute foundation for mathematics and the postmodern deconstruction of meaning and its corresponding ban-

1Faculty of Computer Science, Dalhousie University, Nova Scotia, Canada. B3H 1W5 [email protected] 2Centre for Experimental and Constructive Mathematics, Simon Fraser University, Burnaby, British Columbia, Canada. V5A 1S6 [email protected] 3Michel Foucault, “Truth and Power”, Power/Knowledge: Selected Interviews and Other Writings 1972- 1977, edited by Colin Gordon 4Freeman Dyson, “Mathematics in the Physical Sciences”, Scientific American 211, no. 9 (1964): 130. ishment of encompassing philosophical perspectives from the centre fixe. Of note in this commentary is the contribution of Vladimir Tasić. In his book, Math-

ematics and the Roots of Postmodern Thought, he comments on the broad range of ideas about the interrelationship between language, meaning, and society that are commonly con- sidered to fall under the umbrella of postmodernism. Stating that, “attempts to make sense of this elusive concept threaten to outnumber attempts to square the circle”, he focuses his attention on two relatively well-developed aspects of postmodern theory: “poststruc- turalism” and “deconstruction”.5 He argues that the development of these theories, in the works of Derrida and others, resonates with the debates surrounding foundationism which

preoccupied the philosophy of mathematics in the early stages of the last century and may

even have been partly informed by those debates. Our present purpose is not to revisit the connections between the foundationist debates and the advent of postmodern thought, but rather to describe and discuss some of the ways in which epistemological relativism and other postmodern perspectives are manifest in the changing ways in which mathematicians do mathematics and express mathematical knowledge. The analysis is not intended to be a lament; but it does contain an element of warning. It is central to our purpose that the erosion of universally ﬁxed perspectives of acceptable practice in both mathematical activity and its publication be acknowledged as presenting signiﬁcant challenges to the mathematical

community.

Absolutism and Typographic Mathematics

I believe that mathematical reality lies outside us, that our function is to dis- cover or observe it, and that the theorems which we prove, and which we describe grandiloquently as our “creations”, are simply the notes of our observations.6

5Vladimir Tasi´c, Mathematics and the Roots of Postmodern Thought. (Oxford: Oxford University Press, 2001), 5. 6G.H. Hardy, A Mathematician’s Apology. (London: Cambridge University Press, 1967), 21.

2 (G.H. Hardy)

We follow the example of Paul Ernest and others and cast under the banner of absolutism descriptions of mathematical knowledge that exclude any element of uncertainty or subjec- tivity.7 The quote from Hardy is frequently cited as capturing the essence of Mathematical

Platonism, a philosophical perspective that accepts any reasonable methodology and places a minimum amount of responsibility on the shoulders of the mathematician. An undigested Platonism is commonly viewed to be the default perspective of the research mathematician and, in locating mathematical reality outside human thought, ultimately holds the mathematician responsible only for discovery, observations, and explanations, not creations. Absolutism also encompasses the logico-formalist schools as well as intuitionism and constructivism; in short, any perspective which strictly defines what constitutes mathematical knowledge or how mathematical knowledge is created or uncovered. Few would oppose the assertion that an absolutist perspective, predominately in the de facto Platonist sense, has been the dominant epistemology amongst working mathematicians since antiquity. Perhaps not as evident are the strong connections between epistemological perspective, community structure, and the technologies which support both mathematical activity and mathematical discourse. The media culture of typographic mathematics is defined by centres of publication and a system of community elites which determines what, and by extension who, is published. The abiding ethic calls upon mathematicians to respect academic credentialism and the systems of publication which further refine community hierarchies. Community protocols exalt the published, peer-reviewed article as the highest form of mathematical discourse.

The centralized nature of publication and distribution both sustains and is sustained by the community’s hierarchies of knowledge management. Publishing houses, the peer review process, editorial boards, and the subscription-based distribution system each require a mea-

7Paul Ernest, Social Constructivism As a Philosophy of Mathematics, (Albany: State University of New York Press, 1998), 13.

3 sure of central control. The centralized protocols of typographic discourse resonate strongly with absolutist notions of mathematical knowledge. The emphasis on an encompassing mathematical truth supports and is supported by a hierarchical community structure possessed of well-deﬁned methods of knowledge validation and publication. These norms support a system of community elites to which ascension is granted through a successful history with community publication media, most importantly the refereed article.

The interrelationships between community practice, structure, and epistemology are deep-rooted. Rigid epistemologies require centralized protocols of knowledge validation and these protocols are only sustainable in media environments which embrace centralized modes of publication and distribution. As an aside, we emphasize that this is not meant as an indictment of publishers as bestowers of possibly unmerited authority—though the present disjunct between digitally “published” eprints which are read and typographically published reprints which are cited is quite striking. Rather, it is a description of a time-honoured and robust deﬁnition of merit in a typographical publishing environment. In the latter part of the twentieth century, a critique of absolutist notions of mathematical knowledge emerged in the form of the experimental mathematics methodology and the social constructivist perspective. In the next section, we consider how evolving notions of mathematical knowledge and new media are combining to change not only the way mathematicians do and publish mathematics, but also the nature of the mathematical community.

Towards Mathematical Fallibilism

This new approach to mathematics—the utilization of advanced computing technology in mathematical research—is often called experimental mathematics. The computer provides the mathematician with a laboratory in which he or she can

4 perform experiments: analyzing examples, testing out new ideas, or searching for patterns.8 (David Bailey and Jonathan Borwein)

The experimental methodology embraces digital computation as a means of discovery and veriﬁcation. Described in detail in two recently published volumes, Mathematics by Experiment: Plausible Reasoning in the 21st Century and Experimentation in Mathematics: Computational Paths to Discovery, the methodology as outlined by the authors (joined by Roland Girgensohn in the later work), accepts, as part of the experimental process, stan- dards of certainty in mathematical knowledge which are more akin to the empirical sciences than they are to mathematics. As an experimental tool, the computer can provide strong,

but typically not conclusive, evidence regarding the validity of an assertion. While with

appropriate validity checking, confidence levels can in many cases be made arbitrarily high, it is notable that the concept of a ‘confidence level’ has traditionally been a property of statistically-oriented fields. It is important to note that the authors are not calling for a new standard of certainty in mathematical knowledge but rather the appropriate use of a methodology which may produce, as a product of its methods, definably uncertain transi- tional knowledge.

What the authors do advocate is closer attention to and acceptance of degrees of certainty in mathematical knowledge. This recommendation is made on the basis of argued assertions

such as:

1. Almost certain mathematical knowledge is valid if treated appropriately;

2. In some cases ‘almost certain’ is as good as it gets;

3. In some cases an almost certain computationally derived assertion is at least as strong as a complex formal assertion.

8J.M. Borwein and D.H. Bailey, Mathematics by Experiment: Plausible Reasoning in the 21st Century, A.K. Peters Ltd, 2003. ISBN: 1-56881-211-6, 2-3.

5 The ﬁrst assertion is addressed by the methodology itself and in Mathematics by Ex- periment, the authors discuss in detail and by way of example the appropriate treatment of

‘almost certain’ knowledge. The second is a recognition of the limitations imposed by G¨odel’s Incompleteness Theorem, not to mention human frailty. The third is more challenging for it addresses the idea that ‘certainty’ is in part a function of the community’s knowledge validation protocols. By way of example, the authors write:

. . . perhaps only 200 people alive can, given enough time, digest all of Andrew Wiles’ extraordinarily sophisticated proof of Fermat’s Last Theorem. If there is even a one percent chance that each has overlooked the same subtle error (and they may be psychologically predisposed so to do, given the numerous earlier results that Wiles’ result relies on), then we must conclude that computational results are in many cases actually more secure than the proof of Fermat’s Last Theorem.9

Three Mathematical Examples

Our ﬁrst and pithiest example answers a question set by Donald Knuth10 who asked for a closed form evaluation of the expression below.

Example 1: Evaluate

½ ¾ X∞ kk 1 − √ = −0.084069508727655996461 .... k! ek k=1 2 π k

It is currently easy to compute 20 or 200 digits of this sum. Using the ‘smart lookup’

9Borwein and Bailey, p. 10. 10Posed as MAA Problem 10832, November 2002. Solution details are given on pages 15-17 of Borwein, Bailey and Girgensohn.

6 facility in the Inverse Symbolic Calculator 11 rapidly returns:

2 ζ (1/2) 0.0840695087276559964 ≈ + √ . 3 2 π

We thus have a prediction which Maple 9.5 on a laptop conﬁrms to 100 places in under 6 seconds and to 500 in 40 seconds. Arguably we are done. ¤

The second example, originates with a multiple integral which arises in Gaussian and spherical models of ferromagnetism and in the theory of random walks. This leads to an impressive closed form evaluation due to G. N. Watson: Example 2:

√ Z π Z π Z π µ ¶ µ ¶ c 1 ( 3 − 1) 2 1 2 11 W3 = dx dy dz = Γ Γ . −π −π −π 3 − cos (x) − cos (y) − cos (z) 96 24 24

The most self-contained derivation of this very subtle Green’s function result is recent and is due to Joyce and Zucker.12 Computational conﬁrmation to very high precision is however easy. c Further experimental analysis involved writing W3 as a product of only Γ−values. This form of the answer is then susceptible to integer relation techniques. To high precision, an Integer Relation algorithm returns:

0= -1.* log[w3] + -1.* log[gamma[1/24]] + 4.*log[gamma[3/24]]

+ -8.*log[gamma[5/24]] + 1.* log[gamma[7/24]] + 14.*log[gamma[9/24]]

+ -6.*log[gamma[11/24]] + -9.*log[gamma[13/24]]

+ 18.*log[gamma[15/24]] + -2.*log[gamma[17/24]]-7.*log[gamma[19/24]]

11At www.cecm.sfu.ca/projects/ISC/ISCmain.html 12See pages 117–121 of J.M. Borwein, D.H. Bailey, and R. Girgensohn Experimentation in Mathematics: Computational Paths to Discovery, A.K. Peters Ltd, 2003. ISBN: 1-56881-136-5.

7 Proving this discovery is achieved by comparing the outcome with Watson’s result and √ establishing the implicit Γ−representation of ( 3 − 1)2/96. c Similar searches suggest there is no similar four dimensional closed form for W4. Fortu- R c ∞ 4 nately, a one variable integral representation is at hand in W4 := 0 exp(−4t)I0 (t) dt, where

I0 is the Bessel integral of the ﬁrst kind. The high cost of four dimensional numeric integra- tion is thus avoided. A numerical search for identities then involves the careful computation of exp(−t) I0(t) carefully, using

X∞ t2n exp(−t) I (t) = exp(−t) 0 22n(n!)2 n=0

for t up to roughly 1.2 · d, where d is the number of signiﬁcant digits needed, and

Q 1 XN n (2k − 1)2 exp(−t) I (t) = √ k=1 0 (8t)nn! 2πt n=0

for larger t, where the limit N of the second summation is chosen to be the ﬁrst index n such that the summand is less than 10−d. (This is an asymptotic expansion so taking more terms than N may increase, not decrease the error.) c Bailey and Borwein found that W4 is not expressible as a product of powers of Γ(k/120) (for 0 < k < 120) with coeﬃcients of less than 12 digits. This result does not, of course, rule out the possibility of a larger relation, but it does cast experimental doubt that such a relation exists—more than enough to stop looking. ¤ The third example emphasizes the growing role of visual discovery.

Example 3: Recent continued fraction work by Borwein and Crandall illustrates the

methodology’s embracing of computer-aided visualization as a means of discovery. They

8 Figure 1: The starting point depends on the choice of unit vectors, a and b.

investigated the dynamical system deﬁned by: t0 := t1 := 1 and

µ ¶ 1 1 t ←- t + ω 1 − t , n n n−1 n−1 n n−2

2 2 where ωn = a , b are distinct unit vectors, for n even, odd respectively—that occur in the original continued fraction. Treated as a black box all that can be veriﬁed numerically is that tn → 0 slowly. Pictorially one learns more, as illustrated by Figure 1. √ Figure 2 illustrates the ﬁne structure that appears when the system is scaled by n and odd and even iterates are coloured distinctly.

With a lot of work everything seen in these pictures is now explained. Indeed from these four cases one is compelled to conjecture that the attractor is ﬁnite of cardinality N exactly when the input, a or b, is an Nth root of unity; otherwise it is a circle. Which conjecture one then repeatedly may test. ¤

The idea that what is accepted as mathematical knowledge is, to some degree, dependent upon a community’s methods of knowledge acceptance is an idea that is central to the social constructivist school of mathematical philosophy.

The social constructivist thesis is that mathematics is a social construction, a

9 Figure 2: The attractors for various |a| = |b| = 1.

cultural product, fallible like any other branch of knowledge.13 (Paul Ernest)

Associated most notably with the writing of Paul Ernest, an English mathematician and

Professor in the Philosophy of Mathematics Education, social constructivism seeks to deﬁne mathematical knowledge and epistemology through the social structure and interactions of the mathematical community and society as a whole. In Social Constructivism As a

Philosophy of Mathematics, Ernest carefully traces the intellectual pedigree for his thesis, a pedigree that encompasses the writings of Wittgenstein, Lakatos, Davis, and Hersh among others.14

For our purpose, it is useful to note that the philosophical aspects of the experimental methodology combined with the social constructivist perspective provide a pragmatic alternative to Platonism; an alternative which furthermore avoids the Platonist pitfalls. The apparent paradox in suggesting that the dominant community view of mathematics— Platonism—is at odds with a social constructivist accounting is at least partially countered by the observation that we and our critics have inhabited quite distinct communities. The impact of one on the other was well described by Dewey a century ago:

Old ideas give way slowly; for they are more than abstract logical forms and categories. They are habits, predispositions, deeply engrained attitudes of aversion

13 14Ernest, p. 39ﬀ

10 and preference. ··· Old questions are solved by disappearing, evaporating, while new questions corresponding to the changed attitude of endeavor and preference

take their place. Doubtless the greatest dissolvent in contemporary thought of old questions, the greatest precipitant of new methods, new intentions, new problems, is the one eﬀected by the scientiﬁc revolution that found its climax in the “Origin of Species.” 15 (John Dewey) new mathematics, new media, and new community protocols

With a proclivity towards centralized modes of knowledge validation, absolutist epistemologies are supported by well-defined community structures and publication protocols. In contrast, both the experimental methodology and social constructivist perspective resonate with a more fluid community structure in which communities, along with their implicit and ex- plicit hierarchies, form and dissolve in response to the establishment of common purposes. The experimental methodology, with its embracing of computational methods, de-emphasizes individual accomplishment by encouraging collaboration not only between mathematicians but between mathematicians and researchers from various branches of computer science. Conceiving of mathematical knowledge as a function of the social structure and interactions of mathematical communities, the social constructivist perspective is inherently ac- cepting of a realignment of community authority away from easily identified elites and in the direction of those who can most effectively harness the potential for collaboration and publication afforded by new media. The capacity for mass publication no longer resides ex- clusively in the hands of publishing houses; any workstation equipped with a LATEX compiler and the appropriate interpreters is all that is needed. The changes that are occurring in the ways we do mathematics, the ways we publish mathematical research, and the nature of the mathematical community leave little opportunity for resistance or nostalgia. From a purely

15Quoted from The Inﬂuence of Darwin on Philosophy, 1910.

11 pragmatic perspective, the community has little choice but to accept a broader deﬁnition of valid mathematical knowledge and valid mathematical publication. In fact, in the transition between publishing protocols based upon mechanical type-setting to protocols supported by digital media, we are already witnessing the beginnings of a realignment of elites and hierarchies and a corresponding re-evaluation of the mathematical skill-set. Before considering more carefully the changes that are occurring in mathematics, we turn our attention to some arguably and, perhaps hopefully, immutable aspects of mathematical knowledge.

Some Societal Aspects of Mathematical Knowledge

The question of the ultimate foundations and the ultimate meaning of mathematics remains open: we do not know in what direction it will ﬁnd its ﬁnal solution

or even whether a ﬁnal objective answer can be expected at all. ’Mathematiz- ing’ may well be a creative activity of man, like language or music, of primary originality, whose Historical decisions defy complete objective rationalisation.16 (Hermann Weyl)

Membership in a community implies mutual identiﬁcation with other members which is manifest in an assumption of some level of shared language, knowledge, attitudes, and practices. Deeply woven into the sensibilities of mathematical research communities, and to varying degrees, the sensibilities of society as a whole, are some assumptions about the role of mathematical knowledge in society and what constitutes essential mathematical knowledge. These assumptions are part of the mythology of mathematical communities and the larger society and it is reasonable to assume that they will not be readily surrendered in the face of evolving ideas about the epistemology of mathematics or changes in the methods of practicing and publishing mathematics.

16Cited in: Obituary: David Hilbert 1862 - 1943, RSBIOS, 4, 1944, pp. 547 - 553

12 mathematics as fundamental knowledge

Mathematics is the tool specially suited for dealing with abstract concepts of any kind and there is no limit to its power in this ﬁeld.17 (Paul Dirac)

In the epistemological universe, mathematics is conceived as a large mass about which orbits many other bodies of knowledge and whose gravity exerts influence throughout. The medieval recognition of the centrality of mathematics was reflected in the quadrivium which ascribed to the sciences of number: arithmetic, geometry, astronomy, and music, four out of the seven designated liberal arts. Today, mathematics is viewed by many as an impenetrable, but essential, subject that is at the foundation of much of the knowledge that informs our understanding of the scientific universe and human affairs. We are somehow reassured by the idea of a Federal Reserve Chairman who purportedly solves differential equations in his spare time.

The high value that society places on an understanding of basic mathematics is reﬂected in UNESCO’s speciﬁcation of numeracy, along with literacy and essential life skills, as a fundamental educational objective. This place of privilege bestows upon the mathematical research community some unique responsibilities. Among them, the articulation of mathematical ideas to research, business, and public policy communities whose prime objective is not the furthering of mathematical knowledge. As well, as concerns are raised in many jurisdictions about poor performance in mathematics at the grade school level, research communities are asked to participate in the general discussion about mathematical education. the mathematical canon

I will be glad if I have succeeded in impressing the idea that it is not only pleasant to read at times the works of the old mathematical authors, but this may occasion-

17Dirac writing in the preface to The principles of Quantum Mechanics Oxford, 1930.

13 ally be of use for the actual advancement of science.18 (Constantin Carath´eodory)

The mathematical community is the custodian of an extensive collection of core knowledge to a larger degree than any other basic discipline with the arguable exception of the combined fields of rhetoric and literature. Preserved largely by the high degree of harmo- nization of grade school and undergraduate university curricula, this mathematical canon is at once a touchstone of shared experience of community members and an imposing barrier to anyone who might seek to participate in the discourse of the community without hav- ing some understanding of the various relationships between the topics of core knowledge. While the exact definition of the canon is far from precise, to varying degrees of mastery it certainly includes Euclidean geometry, differential equations, elementary algebra, number theory, combinatorics, and probability. It is worth noting parenthetically that while mathematical notation can act as a barrier to mathematical discourse, its universality helps promote the universality of the canon.

At the level of individual works and speciﬁc problems, mathematicians display a high degree of respect for historical antecedent. Mathematics has advanced largely through the careful aggregation of a mathematical literature whose reliability has been established by a slow but thorough process of formal and informal scrutiny. Unlike the other sciences, mathematical works and problems need not be recent to be pertinent. Tom Hales’ recent computer assisted solution of Kepler’s problem makes this point and many others. Kepler’s conjecture: the densest way to stack spheres is in a pyramid is perhaps the oldest problem in discrete geometry. It is also the most interesting recent example of computer-assisted proof. The publication of Hales’ result in the Annals of Mathematics, with an “only 99% checked” disclaimer, has triggered varied reactions.19

18Speaking to an MAA meeting in 1936. 19See “In Math, Computers Don’t Lie. Or Do They?”, The New York Times April 6, 2004.

14 the mathematical aesthetic

The mathematician’s patterns, like the painter’s or the poet’s must be beautiful; the ideas, like the colors or the words must ﬁt together in a harmonious way.

Beauty is the ﬁrst test: there is no permanent place in this world for ugly mathematics.20 (G. H. Hardy)

Another distinguishing preoccupation of the mathematical community is the notion of a mathematical aesthetic. It is commonly held that good mathematics reflects this aesthetic and that a developed sense of the mathematical aesthetic is an attribute of a good mathematician. The following exemplifies the “infinity in the palm of your hand” encapsulation of complexity which is one aspect of the aesthetic sense in mathematics.

Z 1 1 1 1 1 1 + 2 + 3 + 4 + ... = x dx 2 3 4 0 x

Discovered in 1697 by Johannes Bernoulli, this formula has been dubbed the Sophomore’s Dream in recognition of the surprising similarities it reveals between a series and its integral equivalent. Its proof is not too simple and not too hard, and the formula oﬀers the mix of surprise and simplicity that seems central to the mathematical aesthetic. By contrast several of the recent very long proofs are neither simple nor beautiful.

To see a World in a Grain of Sand; and a Heaven in a Wild Flower; Hold Inﬁnity in the palm of your hand; And Eternity in an hour. (William Blake)

Freedom and Discipline

In this section, we make some observations about the tension between conformity and diver- sity which is present in the protocols of both typographically and digitally oriented commu- 20G.H. Hardy, A Mathematician’s Apology. (London: Cambridge University Press, 1967), 21.

15 nities.

The only avenue towards wisdom is by freedom in the presence of knowledge. But the only avenue towards knowledge is by discipline in the acquirement of ordered fact.21 (Alfred North Whitehead)

Included in the introduction to his essay The Rhythmic Claims of Freedom and Discipline,

Whitehead’s comments about the importance of the give and take between freedom and discipline in education can be extended to more general knowledge management domains. In the discourse of mathematical research, tendencies towards freedom and discipline, decentralization and centralization, the organic and the ordered, coexist in both typographic and digital environments. While it may be true that typographic norms are characterized by centralized nodes of publication and authority and the community order that they impose, an examination of the mathematical landscape in the mid-twentieth century reveals strong tendencies towards decentralization occurring independently of the influence of digital media. Mutually reinforcing trends, including an increase in the number of PhD’s, an increase in the number of journals and published articles, and the application of advanced mathematical methods to fields outside the domain of the traditional mathematical sciences combined to challenge the tendency to maintain centralized community structures. The result was, and continues to be, a replication of a centralized community structure in increasingly specialized domains of interest. In mathematics more than in any other field of research, the knowledge explosion has led to increased specialization, with new fields giving birth to new journals and the organizational structures which support them. While the structures and protocols which describe the digital mathematical community are still taking shape, it would be inaccurate to suggest that the tendency of digital media to promote freedom and decentralized norms of knowledge sharing is unmatched by tendencies

21Alfred North Whitehead, The Aims of Education, (New York: The Free Press, 1957), 30.

16 to impose control and order. If the natively centralized norms of typographic mathematics manifest decentralization as knowledge fragmentation, we are presently observing tendencies

emerging from digital mathematics communities to find order and control in the knowledge atomization that results from the codification of mathematical knowledge at the level of micro-ontologies. The W3C’s MathML initiative and the European Union’s OpenMath project are complementary efforts to construct a comprehensive, fine-grained codification of mathematical knowledge that binds semantics to notation and the context in which the notation is used.22 The tongue-in-cheek indictment of typographic subject specialization as producing experts who learn more and more about less and less until achieving complete knowledge of nothing-at-all becomes, under the digital norms, the increasingly detailed description of increasingly restricted concepts until one arrives at a complete description of nothing-at-all. Ontologies become micro-ontologies and risk becoming “non-tologies”. If typographic modes of knowledge validation and publication are collapsing under the weight of subject specialization, the digital ideal of a comprehensive meta-mathematical descriptive and semantic framework which embraces all mathematics may also prove to be overreaching.

Some Implications

Communication of mathematical research and scholarship is undergoing profound change as new technology creates new ways to disseminate and access the literature. More than technology is changing, however, the culture and practices of

those who create, disseminate, and archive the mathematical literature are changing as well. For the sake of present and future mathematicians, we should shape those changes to make them suit the needs of the discipline.23 (International

22For background on these projects, see: www.w3.org/Math/ and www.openmath.org respectively. 23The IMU’s Committee on Electronic Information and Communication (CEIC) re- ports to the IMU on matters concerning the digital publication of mathematics. See:

17 Math Union Committee on Electronic Information and Communication)

. . . to suggest that the normal processes of scholarship work well on the whole and in the long run is in no way contradictory to the view that the processes of selection and sifting which are essential to the scholarly process are ﬁlled with

error and sometimes prejudice.24 (Kenneth Arrow)

Our present idea of a mathematical research community is built on the foundation of the protocols and hierarchies which define the practices of typographic mathematics. At this point, how the combined effects of digital media will affect the nature of the community remains an open question however some trends are emerging:

1. Changing modes of collaboration: With the facilitation of collaboration afforded by digital networks, individual authorship is increasingly ceding place to joint authorship. It is possible that forms of community authorship, such as are common in the Open Source programming community, may find a place in mathematical research. Michael Kohlhase and Romeo Anghelache have proposed a version-based content management system for mathematical communities which would permit multiple users to make joint contributions to a common research effort.25 The system facilitates collabo-

ration by attaching version control to electronic document management. Such systems, should they be adopted, challenge not only the notion of authorship but also the idea of what constitutes a valid form of publication.

www.ceic.math.ca/Publications/Recommendations/3 best practices.shtml 24E. Roy Weintraub and Ted Gayer, “Equilibrium Proofmaking”, Journal of the History of Economic Thought, 23 (Dec. 2001), 421-442. This provides a remarkably detailed analysis of the genesis and publication of the Arrow-Debreu theorem. 25Michael Kohlhase and Romeo Anghelache, “Towards Collaborative Content Management and Version Control for Structured Mathematical Knowledge”, Lecture Notes in Computer Science no. 2594: Mathemat- ical Knowledge Management: proceedings of The Second International Conference, Andrea Asperti, Bruno Buchberger, and James C. Davenport editors, (Berlin: Springer-Verlag, 2003) 45.

18 2. The ascendancy of grey literature: Under typographic norms, mathematical research has traditionally been conducted with reference to journals and through the

informal consultation with colleagues. Digital media, with its non-discriminating capacity for facilitating instantaneous publication, has placed a wide range of sources at the disposal of the research mathematician. Ranging from Computer Algebra System routines to Home Pages and conference programmes, these sources all provide information that may support mathematical research. In particular, it is possible that a published paper may not be the most appropriate form of publication to emerge from a multi-user content management such as proposed by Kohlhase and Anghelache. It may be that the contributors deem it more appropriate to let the result of their eﬀorts

stand with its organic development exposed through a history of its versions.

3. Changing modes of knowledge authentication: The refereeing process, already under overload induced stress, depends upon a highly controlled publication process. In the distributed publication environment aﬀorded by digital media, new methods of knowledge authentication will necessarily emerge. By necessity, the idea of authentication based on the ethics of referees will be replaced by authentication based on various types of valuation parameters. Services that track citations are currently being used for this purpose by the Web document servers CiteSeer and citebase among. others26

Certainly the ability to compute informedly with formulae in a preprint can dramati-

cally reduce the reader’s or referee’s concern about whether the result is reliable. More than we typically admit or teach our students, mathematicians work without proof if secure in the correctness of their thought processes.

4. Shifts in epistemology: The increasing acceptance of the experimental methodology and social constructivist perspective is leading to a broader deﬁnition of valid knowl-

26citeseer.ist.psu.edu and citebase.eprints.org respectively

19 Figure 3: What you draw is what you see. Roots of polynomials with coeﬃcients 1 or -1 up to degree 18. The colouration is determined by a normalized sensitivity of the coeﬃcients of the polynomials to slight variations around the values of the zeros with red indicating low sensitivity and violet indicating high sensitivity. The bands visible in the lower right picture are unexplained, but believed to be real—not an artifact.

edge and valid forms of knowledge representation. The rapidly expanding capacity

of computers to facilitate visualization and perform symbolic computations is plac- ing increased emphasis on visual arguments and interactive interfaces, thereby making practicable the call by Philip Davis and others a quarter century ago to admit visual proofs more fully into our canon.

The price of metaphor is eternal vigilance (Arturo Rosenblueth & Norbert Wiener)

For example, experimentation with various ways of representing stability of computation led to the four images in Figure 3. They rely on perturbing some quantity and recomputing the image, then coloring to reﬂect the change. Some features are ubiqui- tous while some like the bands only show up in certain settings. Nonetheless, they are thought not to be an artifact of roundoﬀ or other error but to be a real yet unexplained phenomenon.

5. Re-evaluation of valued skills and knowledge: Complementing a reassessment of assumptions about mathematical knowledge, there will be a corresponding reassess-

20 ment of core mathematical knowledge and methods. Mathematical creativity may evolve to depend less upon the type of virtuosity which characterized twentieth cen-

tury mathematicians and more upon an ability to use a variety of approaches and draw together and synthesize materials from a range of sources. This is as much a transfer of attitudes as a transfer of skill sets; the experimental method presupposes an experimental mind-set.

6. Increased community dynamism: Relative to computer and network mediated research, the static social entities which intermesh with the typographic research environment extend the timeline for research and publication and support stability in inter-personal relationships. Collaborations, when they arise, are often career-long, if

not life-long, in their duration. The highly productive friendship between G.H. Hardy and John Littlewood provides a perhaps extreme example. While long-term collaborations are not excluded, the form of collaboration supported by digital media tends to admit a much more ﬂuid community dynamic. Collaborations and coalitions will form as needed and dissolve just as quickly. The four authors of The Siam 100-digit Chal- lenge: A Study In High-accuracy Numerical Computing 27 never met while solving Nick Trefethen’s 2002 ten challenge problems which form the basis for their lovely book.

At the extreme end of the scale, distributed computing can facilitate virtually anony- mous collaboration. In 2000, Colin Percival used the Bailey-Borwein-Plouﬀe algorithm

and connected 1734 machines from 56 countries to determine the quadrillionth bits of π. Accessing an equivalent of more than 250 cpu years, this calculation along with Toy Story Two and other recent movies, ranks as one of the largest computations ever.

27Folkmar Bornemann, Dirk Laurie, Stan Wagon, J¨orgWaldvogel, SIAM 2004

21 The computation was based on the computer-discovered identity

µ ¶ ½ ¾ X∞ 1 k 4 2 1 1 π = − − − , 16 8k + 1 8k + 4 8k + 5 8k + 6 k=0

which allows binary digits to be computed independently.28

A Temporary Epilogue

The plural of ’anecdote’ is not ’evidence’.29 (Alan L. Leshner)

These trends are presently combining to shape a new community ethic. Under the dictates of typographic norms, ethical behaviour in mathematical research involves adhering to well- established protocols of research and publication. While the balance of personal freedom against community order which deﬁnes the ethic of digitally oriented mathematical research communities may never be as ﬁrm or as enforceable by community protocols, some principles are emerging. The CEIC’s statement of best current practices for mathematicians provides

a snapshot of the developing consensus on this question. Stating that “those who write, disseminate, and store mathematical literature should act in ways that serve the interests of mathematics, ﬁrst and foremost”, the recommendations advocate that mathematicians take full advantage of digital media by publishing structured documents which are appropriately linked and marked-up with meta-data.30 Researchers are also advised to maintain personal homepages with links to their articles and to submit their work to pre-print and archive servers. Acknowledging the complexity of the issue, the ﬁnal CEIC recommendation concerns the question of copyright and makes no attempt to recommend a set course of action but

28See Borwein and Bailey, Chapter 3. 29The publisher of Science speaking at the Canadian Federal Science and Technology Forum, Oct 2, 2002. 30CEIC Recommendations. See: http://www.ceic.math.ca

22 rather simply advises mathematicians to be aware of copyright law and custom and consider carefully the options. Extending back to Britain’s first copyright law, The Statute of Anne, enacted in 1710, the idea of copyright is historically bound to typographic publication and the protocols of typographic society. Digital copyright law is an emerging field and it is presently unclear how copyright, and the economic models of knowledge distribution that depend upon it, will adapt to the emerging digital publishing environment. The relatively liberal epistemology offered by the experimental method and the social constructivist perspective and the potential for distributed research and publication afforded by digital media will reshape the protocols and hierarchies of mathematical research communities. Along with long-held beliefs about what constitutes mathematical knowledge and how it is validated and published, at stake are our personal assumptions about the nature of mathematical communities and mathematical knowledge.31

While the norms of typographic mathematics are not without faults and weaknesses, we are familiar with them to the point that they instill in us a form of faith; a faith that if we play along, on balance we will be granted fair access to opportunity. As the centralized protocols of typographic mathematics give way to the weakly deﬁned protocols of digital mathematics, it may seem that we are ceding a system that provided a way to agree upon mathematical truth for an environment undermined by relativism that will mix veriﬁably true statements with statements that guarantee only the probability of truth and an environment which furthermore is bereft of reliable systems for assessing the validity of publications. The coincidental weakening of community authority structures as typographic elites are rendered increasingly irrelevant by digital publishing protocols may make it seem as though the social imperatives that bind the mathematical community have been weakened. Any sense of loss is the mathematician’s version of postmodern malaise and we hope and predict

31As one of our referees has noted: “The law is clearly 25 years behind info-technology.“ He continues “What is at stake here is not only intellectual property but the whole system of priorities, fees, royalties, accolades, recognition of accomplishments, jobs.”

23 that, as the community incorporates these changes, the malaise will be short-lived. That incorporation is taking place, there can be no doubt. In higher education, we now assume

that our students can access and share information via the Web and we require that they learn how to reliably use vast mathematical software packages whose internal algorithms are not necessarily accessible to them even in principle. One reason that, in the mathematical case, the “unbearable lightness” may prove to be bearable after all is that while fundamental assumptions about mathematical knowledge may be reinterpreted, they will survive. In particular, the idea of mathematical knowledge as being central to the advancement of science and human aﬀairs, the idea of a mathematical

canon and its components, and the idea of a mathematical aesthetic will each ﬁnd expression

in the context of the emerging epistemology and protocols of research and publication. In closing, we note that to the extent that there may be an opportunity to shape the epistemology, protocols, and fundamental assumptions that guide the mathematical research communities of the future, that opportunity is most eﬀectively seized upon during these initial stages of digital mathematical research and publishing.

···

Whether we scientists are inspired, bored, or infuriated by philosophy, all our theorizing and experimentation depends on particular philosophical background assumptions. This hidden inﬂuence is an acute embarrassment to many researchers, and it is therefore not often acknowledged. Such fundamental notions as reality, space, time, and causality–notions found at the core of the scientiﬁc enterprise–all rely on particular metaphysical assumptions about the

world.32 (Christof Koch)

32In “Thinking About the Conscious Mind,” a review of John R. Searle’s Mind. A Brief Introduction, Oxford University Press, 2004.

24 The assumptions that we have sought to address in this article are those that deﬁne how mathematical reality is investigated, created, and shared by mathematicians working within the social context of the mathematical community and its many sub-communities. We have maintained that those assumptions are strongly guided by technology and epistemology and that furthermore, technological and epistemological change are revealing the assumptions to be more fragile than until recently, we might have reasonably assumed.

November 22, 2004