Mathematics by Experiment: Plausible Reasoning in the 21st Century

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Summary: A Mathematical Paradigm Shift
Review: "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.

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Summary: A thoroughly detailed work
Review: 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.