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A professor of psychology was exploring the creative process and as one of his subjects chose a mathematician who was world famous for his ability to solve problems. They gave him a problem to work on. He wrote something down and immediately scribbled it out. He wrote something else down and scribbled it out. The professor asked him to leave everything on the page. He explained that he was interested in the process, the wrong answers and the right answers. The mathematician sat down. Wrote something. The psychology professor waited in anticipation but the mathematician announced he could not proceed without erasing his mistakes. While the mathematician in this situation is undoubtedly fairly idiosyncratic in how he attacks problems there is a strongly felt separation between the creative process of mathematics and the published or finished product.

A current focal point for experimental mathematics is the journal called Experimental Mathematics. But does it really seek to change the way we do mathematics, or to change the way we write mathematics? We begin by attempting to extract a definition of `experimental' from the Journal's introductory article ([8]) ``About this Journal'' by David Epstein, Silvio Levy and Rafael de la Llave.

The word ``experimental'' is conceived broadly: many mathematical experiments these days are carried out on computers, but others are still the result of pencil-and-paper work, and there are other experimental techniques, like building physical models. ([8] p. 1)

It seems that almost anything can be conceived of as being experimental. Let us try again.

Experiment has always been, and increasingly is, an important method of mathematical discovery. (Gauss declared that his way of arriving at mathematical truths was `through systematic experimentation'.) Yet this tends to be concealed by the tradition of presenting only elegant, well-rounded and rigorous results. ([8] p. 1)

Now we begin to get closer to the truth. Experimentation is still ill defined but is clearly an important part of the mathematical process. It is clearly not new but by implication must be inelegant, lopsided and lax. We, of course, dispute all three of these points and while we do not reply directly to these charges, we hope the reader will be convinced that there need be no compromises made with respect to the quality of the work.

But what is the journal interested in publishing? Their goal seems to be two-fold.

While we value the theorem-proof method of exposition, and while we do not depart from the established view that a result can only become part of mathematical knowledge once it is supported by a logical proof, we consider it anomalous that an important component of the process of mathematical creation is hidden from public discussion. It is to our loss that most of the mathematical community are almost always unaware of how new results have been discovered. ([8] p. 1)

and

The early sharing of insights increases the possibility that they will lead to theorems: an interesting conjecture is often formulated by a researcher who lacks the techniques to formalize a proof, while those who have the techniques at their fingertips have been looking elsewhere.

It appears that through the journal Experimental Mathematics the editors advocate a not undramatic change in writing style. So what does a paper published in that journal look like? A recent example is ``Experimental Evaluation of Euler sums'' by D.H. Bailey, J. Borwein and R. Girgensohn ([2]). The authors describe how their interest in Euler sums was roused by a surprising discovery:

In April 1993, Enrico Au-Yeung, an undergraduate at the University of Waterloo, brought to the attention of one of us the curious fact that

based on a computation to 500,000 terms. This author's reaction was to compute the value of this constant to a higher level of precision in order to dispel this conjecture. Surprisingly, a computation to 30 and later to 100 decimal digits still affirmed it. ([2] p. 17)

This type of serendipitous discovery must go on all the time, but it needs the flash of insight that will place it in a broader context. It is like a gold nugget waiting to be refined --- without a context it would remain a curiosity. The authors now proceeded to provide a context by mounting a full-fledged assault on the problem. They systematically applied an integer relation detection algorithm to large classes of sums of the above type, trying to find evaluations of these sums in terms of zeta functions (see box for details). Some of the experimentally discovered evaluations were then proven rigorously, others remain conjectures. While Au-Yeung's insight may fill us with a sense of amazement, the experimenters' approach appears quite natural and systematic.

The editors of Experimental Mathematics are advocating a change in the way mathematics is written, placing more emphasis on the mathematical process. Imre Lakatos in his influential though controversial book Proofs and Refutations [11] advocated a similar change from what he called the deductivist style of proof to the heuristic style of proof. In the deductivist style, the definitions are carefully tailored to the proofs. The proofs are frequently elegant and short. But it is difficult to see what process led to the discovery of the theorem and its proof. The heuristic style maintains the mathematical rigor but again the emphasis is more on process. One does not merely give the definition but perhaps includes a comment on why this definition was chosen and not another. This is clearly an important shift if the editors wish to meet their second objective, the sharing of insights.



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