The Experimental Mathematician: the Pleasure of Discovery and the Role of Proof

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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 numeric 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. . .

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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. 2. 3. 4. 5. 6. 7. 8.

Gaining insight and intuition. Discovering new patterns and relationships. Graphing to expose math principles. Testing and especially falsifying conjectures. Exploring a possible result to see if it merits formal proof. Suggesting approaches for formal proof. Computing replacing lengthy hand derivations. 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 computation convincing? Are humans less fallible? What tools are available? What me

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