Books by Hellman, Hal

Hellman, Hal. Great Feuds in Mathematics. Hoboken, NJ: John Wiley & Sons, 2006. ISBN 0-471-64877-9.

Since antiquity, many philosophers have looked upon mathematics as one thing, perhaps the only thing, that we can know for sure, “the last fortress of certitude” (p. 200). Certainly then, mathematicians must be dispassionate explorers of this frontier of knowledge, and mathematical research a grand collaborative endeavour, building upon the work of the past and weaving the various threads of inquiry into a seamless intellectual fabric. Well, not exactly….

Mathematicians are human, and mathematical research is a human activity like any other, so regardless of the austere crystalline perfection of the final product, the process of getting there can be as messy, contentious, and consequently entertaining as any other enterprise undertaken by talking apes. This book chronicles ten of the most significant and savage disputes in the history of mathematics. The bones of contention vary from the tried-and-true question of priority (Tartaglia vs. Cardano on the solution to cubic polynomials, Newton vs. Leibniz on the origin of the differential and integral calculus), the relation of mathematics to the physical sciences (Sylvester vs. Huxley), the legitimacy of the infinite in mathematics (Kronecker vs. Cantor, Borel vs. Zermelo), the proper foundation for mathematics (Poincaré vs. Russell, Hilbert vs. Brouwer), and even sibling rivalry (Jakob vs. Johann Bernoulli). A final chapter recounts the incessantly disputed question of whether mathematicians discover structures that are “out there” (as John D. Barrow puts it, “Pi in the Sky”) or invent what is ultimately as much a human construct as music or literature.

The focus is primarily on people and events, less so on the mathematical questions behind the conflict; if you're unfamiliar with the issues involved, you may want to look them up in other references. The stories presented here are an excellent antidote to the retrospective view of many accounts which present mathematical history as a steady march forward, with each generation building upon the work of the previous. The reality is much more messy, with the directions of inquiry chosen for reasons of ego and national pride as often as inherent merit, and the paths not taken often as interesting as those which were. Even if you believe (as I do) that mathematics is “out there”, the human struggle to discover and figure out how it all fits together is interesting and ultimately inspiring, and this book provides a glimpse into that ongoing quest.

December 2007