Mathematical constants II by Steven R. Finch, pp. 769, £125 (hard), ISBN 978-1-10847-059-9, Cambridge University Press (2018)

Abstract

Of course, as the authors quite rightly point out (p. 9), the great advantage of computers is that they allow the consideration of far more examples than are possible to handle by hand; hence, easier leading to recognizing over-all patterns. They also wisely caution on p. 9 that ‘Experimentation should be a complement to—and never a substitute for—careful theoretical considerations.’ The book is replete with many famous historical examples like the prime number theorem, first conjectured by Gauss, Mersenne and Fermat primes, the Collatz conjecture, the ABC conjecture, etc. The book also contains many interesting investigations and conjectures that were entirely new to me, such as the LEGO pyramids problem. The book assumes the use of Maple, and a short introductory chapter to basic programming in Maple is provided. However, Maple is not essential as many of the investigations are easily adapted to other similar computational software such as Mathematica or Sage. Other chapters in the book focus on Iteration and Recursion, Visualization, Symbolic Inversion, Pseudorandomness, Time, Memory and Precision, as well as applications of Linear Algebra and Graph Theory. It is good to see the authors regularly warning readers about the dangers of overgeneralization from computational evidence, and provide many illustrative examples throughout, including some nice ones that I’ve not seen before. In a 2004 paper of mine on “The role and function of quasi-empirical methods in mathematics”, I distinguish, discuss and illustrate the following five important functions of experimentation, namely, conjecturing, verification, global refutation, heuristic refutation, and understanding. I was thereafter very happy to find that the authors more or less adequately address and give detailed examples of each of these functions. I was also delighted to see the authors also allude to the so-called “explanatory” function of proof when writing on p. 151 that “... when two sets of data coincide for a long time it is advisable to assert that they are in fact the same, and attempt to understand why.” Perhaps the only small shortcoming of the book for myself is that the book does not include any dynamic geometry software examples for 2D or 3D. Other than that, I very highly recommend the book not only as a great, first time introductory course to experimental mathematics for mathematics students, but also compulsory reading for mathematics educators from high school to college and university level. I’d like to applaud the authors for their interesting, innovative, well-structured book that is sure to serve for some time as a standard against which other future books of a similar kind will be measured. 10.1017/mag.2021.97 MICHAEL DE VILLIERS Research Unit for Mathematics Education (RUMEUS), University of Stellenbosch, © The Mathematical Association 2021 Stellenbosch 7600, South Africa