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Knowledge and Community in Mathematics

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Knowledge and Community in Mathematics 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 e®orts 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", Scienti¯c American 211, no. 9 (1964): 130. ishment of encompassing philosophical perspectives from the centre ¯xe. Of note in this commentary is the contribution of Vladimir Tasi¶c. 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 ¯xed perspectives of acceptable practice in both mathematical activity and its publication be acknowledged as presenting signi¯cant 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 mathe- matician responsible only for discovery, observations, and explanations, not creations. Absolutism also encompasses the logico-formalist schools as well as intuitionism and con- structivism; in short, any perspective which strictly de¯nes 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 de¯ned 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 sys- tems of publication which further re¯ne 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 math- ematical truth supports and is supported by a hierarchical community structure possessed of well-de¯ned 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 in- dictment 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 de¯nition 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 per- spective. 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 math- ematics, but also the nature of the mathematical community. Towards Mathematical Fallibilism This new approach to mathematics|the utilization of advanced computing tech- nology 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 veri¯cation. 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, con¯dence levels can in many cases be made arbitrarily high, it is notable that the concept of a `con¯dence level' has traditionally been a property of statistically-oriented ¯elds. 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, de¯nably 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
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