Mathematics by Experiment, 2nd Edition: Plausible Reasoning in the 21st Century

High end mathematical theory has veered away from actually doing arithmetic because most of the problems being addressed required a tremendous amount of calculation and that calculation was difficult in the days of only pen and paper. About twenty years ago the advent of big/fast computers (by the standards of those days) began to allow the ready solution of these problems without requiring large numbers of people doing the computing. Borwein and Bailey have been pioneers in the exploration of the types of mathematical problems that would lend themselves to solution using digital computational means. This book describes this new approach to mathematics, commonly called 'experimental mathematics.' Obviously in computer related mathematics it began with a lot of emphasis on prime numbers, on calculating the value of Pi to ever greater precision. It has since moved on to many other classes of problems, and the work of the principle researchers in the field is summarized here.

If one peruses the mathematical literature for the last one hundred years one will notice that in most cases no diagrams or pictures appear. The level of rigor in all cases is impressive though, but unfortunately this makes the understanding of the results much more difficult. There seems to be an inverse relationship between rigor and understanding in mathematics, at least for those who are new to the subject at hand. In order to gain this understanding, the drawing of pictures and diagrams is useful, along with a certain amount of experimentation with the concepts at hand. Intuition, how mysterious, and however ill defined, plays a role in both the understanding of mathematical results and in their discovery. Many mathematicians do not want to acknowledge this, as a visitation to a typical conference will readily verify. The attitude has been expressed that mathematics has "always been abstract" and therefore that pictures or diagrams violate its spirit. Even a well-known geometry center whose goal was to use sophisticated computer graphics to visualize complex mathematical objects lost its funding, to the consternation of a few but with glee to most. Thus the way to discovery of mathematics, i.e. the heavy use of intuition, the disorganized shuffling of concepts, and the experimental doodling, has been masked by the final product of this process: a superb example of logical rigor and organization called modern mathematics. The authors of this book however think otherwise, and they give the best apology for the role of experimental mathematics than anyone else in the literature. The book is packed with highly interesting examples and challenging exercises, all of which are ample proof of the need for doing experimentation in mathematics. In addition to these considerations, the book is just plain fun to read, and even though time constraints may prohibit the working out of every exercise, the book could be used profitably in a graduate course in mathematics or even possibly in an undergraduate course at the senior level. Hopefully this approach to scholarship in mathematics will take hold in this century, and mathematicians will not only write down their final results with all their splendid rigor, but also how they got there. This would serve to educate younger generations of mathematicians in just how discovery in mathematics is done and increase their efficacy in the same. The book will also assist those who are trying to build machines capable of discovering novel results in mathematics. Machine proofs of difficult theorems and conjectures are now a reality, and in the twenty-first century we will no doubt see many more of these. This book therefore contains a lot of hints about how to proceed in mathematics. Its acceptance will depend on how well it does its job in the creation of new mathematical results and in the teaching of them. Results in mathematics that seem plausible serve to make conjectures and motivate the construction of

See my book review that appeared in American Scientist http://www.math.rutgers.edu/~zeilberg/mamarim/mamarimhtml/mathexp.html

The collaborative work of Jonathan Borwein and David Bailey, Mathematics By Experiment: Plausible Reasoning In The 21st Century provides a complex and informative text for advanced mathematics students which offs an historical context and rationale behind experimental mathematics, as well as how modern technology enables the analysis of new examples and the discovery of patterns in a previously unimaginable "laboratory" of raw processing power. A thoroughly detailed work, Mathematics By Experiment offers a veritable wealth of meticulously presented examples which are most especially recommended for graduate-level mathematics studies.

"Mathematics by Experiment" is a ground-breaking book about a new way of doing math that generated so much excitement it was reviewed in "Scientific American" six months before it got into print. The authors are long-time collaborators David Bailey, chief technologist in the Computational Research Department of Lawrence Berkeley National Laboratory, and Jonathan Borwein, professor of science at Simon Fraser University in Vancouver, B.C. They write that applied mathematicians and many scientists and engineers were quick to embrace computer technology, while pure mathematicians -- whose field gave rise to computers in the first place, through the work of beautiful minds like Alan Turing's -- were slower to see the possibilities. Two decades ago, when Bailey and Borwein started collaborating, "there appeared to be a widespread view in the field that 'real mathematicians don't compute.'" Their book is testament to a paradigm shift in the making. Hardware has "skyrocketed in power and plummeted in cost," and powerful mathematical software has come on the market. Just as important, "a new generation of mathematicians is eagerly becoming skilled at using these tools" -- people comfortable with the notion that "the computer provides the mathematician with a 'laboratory' in which he or she can perform experiments: analyzing examples, testing out new ideas, or searching for patterns." In this virtual laboratory Bailey and Borwein, with other colleagues, were among the first to discover a number of remarkable new algorithms, among them an extraordinary, simple formula for finding any hexadecimal or binary digit of pi without knowing any of the preceding digits. Further research led to proof that a wide class of fundamental constants are mathematically "normal" -- probably including pi, alhough that remains to be proved. Their section on "proof versus truth" is an example of the gems even a mathematical tyro can find among these equations. Bailey and Borwein don't claim computers can supply rigorous proofs. Rather, the computer is a way to discover truths -- and avenues for approaching formal proofs. But often, the authors add, "computations constitute very strong evidence..., at least as compelling as some of the more complex formal proofs in the literature." Drawing on their own work and that of others, Bailey and Borwein not only explain experimental mathematics in a lively, surprisingly accessible fashion but give many engaging examples of the "new paradigm" in action.