Experimentation in Mathematics. Computational Path to Discovery
Despite the slightly different titles, the books form two volumes of the same publication. Mathematical experiments performed on computers are more and more important factors of further development in the mathematics. The text is carefully written and the plentiful short remarks on related topics are pleasant to read. The material is mostly accessible without knowledge of advanced parts of modern mathematics. Certain familiarity with computer algebra programs like Maple or Mathematica is helpful for a reader willing to try these things in practice. The first volume of the work contains a gentle introduction to experimental mathematics in its historical context and to its methodology, using a series of well-chosen examples. The first chapter shows it as a rapidly developing field of mathematics, the second contains further numerous illustrative examples. These quite often provoke a reader to work or play with the examples on his own PC. The third chapter describes the progress in computing π, where both authors made significant contributions (BBP formula for computing π). A chapter on normality of numbers deals with another fascinating problem in which experimental mathematics promises at least some path to future discoveries. In contrast to the fact that the book introduces to the reader a highly up-to-date part of mathematics, the next chapter offers another look at classical themes like the fundamental theorem of algebra, the Gamma function or Stirling's formula. After an exposition on basic tools, the book is closed by a chapter with the title ‘Making sense of experimental mathematics’.
For those wanting to know more on the progression from numerical experiments to hypotheses and finally to deep mathematical theorems proven within the frame of “classical mathematics”, the authors prepared the second volume of the work together with R. Girgensohn. In fact, both volumes can be read independently. The second volume starts with a chapter on sequences, series, products and integrals, which is followed by the chapter on Fourier series and integrals. These chapters form one third of the book and will be of interest to any open-minded person with a deeper interest in mathematics, equipped with a basic knowledge of undergraduate mathematics. The following chapters on zeta functions and multi-zeta functions, partition, powers, primes and polynomials are more specialised. The final two chapters invite a reader to explore deeper methods and tools of experimental mathematics. Each chapter in both volumes is closed by commentaries and additional examples. The amount of collected material is tremendous. It is impossible to describe the content of the whole work in detail in just a few lines. These are very nice and provoking books showing that experiments both were and are an important part of the development of mathematics. New computer-based tools are broadly used mainly “outside” mathematics and the book shows “… how today, the use of advanced computing technology provides mathematicians with an amazing, previously unimaginable ‘laboratory’, in which examples can be analysed, new ideas tested, and patterns discovered.” I strongly recommend visiting e.g. web-site on URL http://www.expmath.info, where some parts of the first volume are also available. Also the very good typesetting and nice graphical appearance of the whole work are worth mentioning. It can be recommended not only to libraries but to all members of the mathematical community.