The Computer as Crucible: An Introduction to Experimental Mathematics

Contemporary experimental mathematics — broadly, the use of computers in mathematics for its own sake — is too new to have a very precise definition. In the last five years several books on the subject have appeared, but this is the first one to attempt to introduce the subject to a broader mathematical audience. The authors are mathematicians of different flavors: one (Borwein) has spent much of his career working in and around experimental mathematics and the other (Devlin), originally a logician, now works in the general area of mathematical cognition. They share a common interest in experimentation in mathematics.

What kind of experimentation? At this stage in its development, the best one can do is to make a list. In this book, these are some of the things they consider:

• Symbolic computation using a computer algebra system like Mathematica;
• Integer-relation algorithms like PSLQ;
• High precision integer and floating point arithmetic;
• High precision evaluation of integrals and summation of infinite series; and
• Identification of functions based on characteristics of their graphs.

What makes experimental mathematics different? In the authors’ words, “...the experimental process is regarded not as a precursor to a proof... Rather, experimentation is regarded as a significant part of mathematics in its own right, to be published, to be considered by others, and (of particular importance) to contribute to our overall mathematical knowledge.” (Authors’ emphasis.)

They seem to have a lot of fun doing it. There are only snippets in the book, appetizers to draw in the reader. For example, investigations of the values of the Riemann zeta function at integers 2 and greater, a certain integral involving products of the sinc function, and the mysteries of the Meijer G-function are there to draw you in. It’s also worth emphasizing that this is not mere fiddling with the computer. Much of the stuff here is likely to draw the unwary back into some old-fashioned classical analysis.

This is not a textbook, though it could be used as a supplementary text or for a reading course. It has no exercises but does include “Explorations” with follow-up examples and suggestions of things for the reader to try as well as a corresponding “Answers and Reflections” chapter at the end.

Finally, why the “computer as crucible”? I suspect this is meant to suggest an alchemical aspect to the enterprise, a forging of new rare materials from old ones.

Bill Satzer (wjsatzer@mmm.com) is a senior intellectual property scientist at 3M Company, having previously been a lab manager at 3M for composites and electromagnetic materials. His training is in dynamical systems and particularly celestial mechanics; his current interests are broadly in applied mathematics and the teaching of mathematics.