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The Experimental Mathematician: the Pleasure of Discovery and the Role of Proof International Journal of Computers for Mathematical Learning (2005) 10: 75–108 DOI 10.1007/s10758-005-5216-x Ó Springer 2005 JONATHAN M. BORWEIN THE EXPERIMENTAL MATHEMATICIAN: THE PLEASURE OF DISCOVERY AND THE ROLE OF PROOF ‘...where almost one quarter hour was spent, each beholding the other with admiration before one word was spoken: at last Mr. Briggs began ‘‘My Lord, I have undertaken this long journey purposely to see your person, and to know by what wit or ingenuity you first came to think of this most excellent help unto Astronomy, viz. the Logarithms: but my Lord, being by you found out, I wonder nobody else found it out before, when now being known it appears so easy.’’ ’1 ABSTRACT. The emergence of powerful mathematical computing environments, the growing availability of correspondingly powerful (multi-processor) computers and the pervasive presence of the internet allow for mathematicians, students and teachers, to proceed heuristically and ‘quasi-inductively’. We may increasingly use symbolic and nu- meric computation, visualization tools, simulation and data mining. The unique features of our discipline make this both more problematic and more challenging. For example, there is still no truly satisfactory way of displaying mathematical notation on the web; and we care more about the reliability of our literature than does any other science. The traditional role of proof in mathematics is arguably under siege – for reasons both good and bad. AMS Classifications: 00A30, 00A35, 97C50 KEY WORDS: aesthetics, constructivism, experimental mathematics, humanist philosophy, insight, integer relations, proof 1. EXPERIMENTAL MATH: AN INTRODUCTION ‘‘There is a story told of the mathematician Claude Chevalley (1909–1984), who, as a true Bourbaki, was extremely opposed to the use of images in geometric reasoning. He is said to have been giving a very abstract and algebraic lecture when he got stuck. After a moment of pondering, he turned to the blackboard, and, trying to hide what he was doing, drew a little diagram, looked at it for a moment, then quickly erased it, and turned back to the audience and proceeded with the lecture... 76 JONATHAN M. BORWEIN ...The computer offers those less expert, and less stubborn than Chevalley, access to the kinds of images that could only be imagined in the heads of the most gifted mathematicians, ...’’ (Nathalie Sinclair2) For my coauthors and I, Experimental Mathematics (Borwein and Bailey, 2003) connotes the use of the computer for some or all of: 1. Gaining insight and intuition. 2. Discovering new patterns and relationships. 3. Graphing to expose math principles. 4. Testing and especially falsifying conjectures. 5. Exploring a possible result to see if it merits formal proof. 6. Suggesting approaches for formal proof. 7. Computing replacing lengthy hand derivations. 8. Confirming analytically derived results. This process is studied very nicely by Nathalie Sinclair in the context of pre-service teacher training.3 Limned by examples, I shall also raise questions such as: What constitutes secure mathematical knowledge? When is com- putation convincing? Are humans less fallible? What tools are available? What methodologies? What about the ‘law of the small numbers’? How is mathematics actually done? How should it be? Who cares for certainty? What is the role of proof? And I shall offer some personal conclusions from more than twenty years of intensive exploitation of the computer as an adjunct to mathematical discovery. 1.1. The Centre for Experimental Math About 12 years ago I was offered the signal opportunity to found the Centre for Experimental and Constructive Mathematics (CECM) at Simon Fraser University. On its web-site (www.cecm.sfu.ca) I wrote ‘‘At CECM we are interested in developing methods for exploiting mathematical computation as a tool in the development of mathematical intuition, in hypotheses building, in the generation of symbolically assisted proofs, and in the construction of a flexible computer environment in which researchers and research students can undertake such research. That is, in doing ‘Experimental Mathematics.’’’ The decision to build CECM was based on: (i) more than a dec- ade’s personal experience, largely since the advent of the personal computer, of the value of computing as an adjunct to mathematical THE EXPERIMENTAL MATHEMATICIAN 77 insight and correctness; (ii) on a growing conviction that the future of mathematics would rely much more on collaboration and intelligent computation; (iii) that such developments needed to be enshrined in, and were equally valuable for, mathematical education; and (iv) that experimental mathematics is fun. A decade or more later, my colleagues and I are even more convinced of the value of our venture – and the ‘mathematical universe is unfolding’ much as we anticipated. Our efforts and philosophy are described in some detail in the recent books (Borwein and Bailey, 2003; Borwein et al., 2004) and in the survey articles (Borwein et al., 1996; Borwein and Carless, 1999; Bailey and Borwein, 2000; Borwein and Borwein, 2001). More technical accounts of some of our tools and successes are detailed in (Borwein and Bradley (1997) and Borwein and Lisone˘ k (2000). About 10 years ago the term ‘experimental mathematics’ was often treated as an oxymoron. Now there is a highly visible and high quality journal of the same name. About 15 years ago, most self-respecting research pure mathematicians would not admit to using computers as an adjunct to research. Now they will talk about the topic whether or not they have any expertise. The centrality of infor- mation technology to our era and the growing need for concrete im- plementable answers suggests why we had attached the word ‘Constructive’ to CECM – and it motivated my recent move to Dal- housie to establish a new Distributed Research Institute and Virtual Environment, D-DRIVE (www.cs.dal.ca/ddrive). While some things have happened much more slowly than we guessed (e.g., good character recognition for mathematics, any sub- stantial impact on classroom parole) others have happened much more rapidly (e.g., the explosion of the world wide web4, the quality of graphics and animations, the speed and power of computers). Crudely, the tools with broad societal or economic value arrive rapidly, those interesting primarily in our niche do not. Research mathematicians for the most part neither think deeply about nor are terribly concerned with either pedagogy or the phi- losophy of mathematics. Nonetheless, aesthetic and philosophical notions have always permeated (pure and applied) mathematics. And the top researchers have always been driven by an aesthetic impera- tive: ‘‘We all believe that mathematics is an art. The author of a book, the lecturer in a classroom tries to convey the structural beauty of mathematics to his readers, to his listeners. In this attempt, he must always fail. Mathematics is logical to be sure, 78 JONATHAN M. BORWEIN each conclusion is drawn from previously derived statements. Yet the whole of it, the real piece of art, is not linear; worse than that, its perception should be instantaneous. We have all experienced on some rare occasions the feeling of ela- tion in realizing that we have enabled our listeners to see at a moment’s glance the whole architecture and all its ramifications.’’ (Emil Artin, 1898–1962)5 Elsewhere, I have similarly argued for aesthetics before utility (Borwein, 2004, in press). The opportunities to tie research and teaching to aesthetics are almost boundless – at all levels of the curriculum.6 This is in part due to the increasing power and sophis- tication of visualization, geometry, algebra and other mathematical software. That said, in my online lectures and resources,7 and in many of the references one will find numerous examples of the utility of experimental mathematics. In this article, my primary concern is to explore the relationship between proof (deduction) and experiment (induction). I borrow quite shamelessly from my earlier writings. There is a disconcerting pressure at all levels of the curriculum to derogate the role of proof. This is in part motivated by the aridity of some traditional teaching (e.g., of Euclid), by the alternatives now being offered by good software, by the difficulty of teaching and learning the tools of the traditional trade, and perhaps by laziness. My own attitude is perhaps best summed up by a cartoon in a book on learning to program in APL (a very high level language). The blurb above reads Remember 10 minutes of computation is worth 10 hours of thought. The blurb below reads Remember 10 minutes of thought is worth 10 hours of computation. Just as the unlived life is not much worth examining, proof and rigour should be in the service of things worth proving. And equally foolish, but pervasive, is encour- aging students to ‘discover’ fatuous generalizations of uninteresting facts. As an antidote, In Section 2, I start by discussing and illus- trating a few of George Polya’s views. Before doing so, I review the structure of this article. Section 2 discusses some of George Polya’s view on heuristic mathematics, while Section 3 visits opinions of various eminent mathematicians. Section 4 discusses my own view and their genesis. Section 5 contains a set of mathematical examples amplifying the prior discussion. Sections 6 and 7 provide two fuller examples of computer discovery, and in Section 8 I return to more philosophical matters – in particular, a discussion of proof versus truth and the nature of secure mathematical knowledge. THE EXPERIMENTAL MATHEMATICIAN 79 2. POLYA ON PICTURE-WRITING ‘‘[I]ntuition comes to us much earlier and with much less outside influence than formal arguments which we cannot really understand unless we have reached a relatively high level of logical experience and sophistication.’’ (Geroge Polya)8 Polya, in his engaging eponymous 1956 American Mathematical Monthly article on picture writing, provided three provoking exam- ples of converting pictorial representations of problems into gener- ating function solutions: 1.
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