God Created the Integers: Mathematical Breakthroughs that Changed History

God Created the Integers: The Mathematical Breakthroughs that Changed History, Edited with commentary by Stephen Hawking, 2005, xiii + 1160 pp., $29.95,  ISBN 0-7624-1922-9, Running Press Book Publishers, 120 South 22nd St., Philadelphia, PA 19103-4399, 215-567-5080 or www.runningpress.com

This book is evidently conceived as a companion to the author's previous work, On the Shoulders of Giants (2003). Both share the same format. Selections from the works of great mathematicians (in the volume under review) or physicists (in the previous volume) are preceded by brief biographies written by Hawking. While only five physicists appear in Giants (Copernicus, Galileo, Kepler, Newton, and Einstein), each quoted extensively, the current volume includes  chapters on seventeen mathematicians: Euclid, Archimedes, Diophantus, Descartes, Newton, Laplace, Fourier, Gauss, Cauchy, Boole, Riemann, Weierstrass, Dedekind, Cantor, Lebesgue, Gödel, and Turing, and  is slightly shorter in length so the excerpts are shorter.  Only Newton appears in both.

The selections are offered in chronological order so that reading the whole book spans two and a half millennia of mathematics history. The majority of the text discusses events subsequent to the sixteenth century. One could contest the particular choice of mathematicians included but 2500 years is a long time and some choices had to be made. The complete volume presents a reasonable survey for the period of history covered.

Disappointingly, it is difficult to recognize the book’s intended audience. Certainly, it is not written for the mathematics historian. The biographical sketches contain nothing that is not well known to even the most casual reader in the history of mathematics. Indeed, where the mathematician under consideration appears in both, there is almost nothing of a biographical nature that cannot also be found in E. T. Bell’s Men of Mathematics (Bell’s known errors notwithstanding). Few of the excerpts are complete and most are available from other sources. The dust cover blurb states that three of them are translated into English for the first time in this volume. While this is a positive addition to the literature, I have been unable to ascertain precisely which three.

God Created the Integers is not for the curious layman or the average mathematics student. The overwhelming bulk of the text is devoted to excerpts from the published works of the mathematicians under consideration. Probably few laymen will benefit from reading Gödel’s “On Formally Undecidable Propositions of Mathematics”,  Gauss' “Disquisitiones Arithmeticae”, or even  Newton's “Principia” unless the entries are extensively footnoted, which they are not. 

On the other hand, a student taking a course in Number Theory, for example, might just enjoy reading Gauss' exposition of the topic, which suggests that this text might be useful as a resource for a course on the history of mathematics. This is what I was hoping for when I agreed to do this review and it is possible, given the right circumstances.

If I were teaching a two semester course in the history of mathematics I would seriously consider this text as the primary resource for the second semester. The mathematicians and excerpts beginning with Descartes and ending with Gödel and Turing provide a good overview of the progress and sweep of Western mathematics from the sixteenth century forward. Supplementary materials would be required, of course, but this book provides a workable central core. Alas, the course I teach confines the entire history of mathematics to a single semester so I, personally, cannot use it.

Eugene Boman, Associate Professor of Mathematics, The Pennsylvania State University, Middletown,PA.

See also the MAA Review by Fernando Q. Gouvêa.