God Created the Integers: The Mathematical Breakthroughs that Changed History

Historians of mathematics are always looking for sourcebooks, especially for use in class. Finding translated versions of many classical texts is quite difficult. When they exist, they turn out to have been published twenty years ago in very small print runs and are long out of print. So a book that brings together a selection of historically significant works is very useful. Several such sourcebooks already exist, so Stephen Hawking's God Created the Integers enters into a crowded field. The main question for a reviewer, then, is whether Hawking's collection offers something new.

First of all, there is Hawking's brief introduction and the commentaries on each of the authors included. These strike me as rather perfunctory at the beginning, then better when the material comes closer to modern science. They tend to be mostly biographical. Hawking is not a historian; inevitably, there are annoying mistakes in these introductions. Many of these amount to the repetition of "folklore," but some are more serious.

Then we come to the selections themselves, and things get more idiosyncratic. The book opens with selections from Euclid. They are all from Heath's translation of the Elements, and they include Heath's copious notes (in very very small print). The beginning of book I is included: definitions, postulates, and common notions. Then (without much warning to the reader) we are suddenly at Proposition 47, the Pythagorean theorem. This all fits on three pages. There follow 15 pages of notes by Heath. This is very weird: one could almost fit all of book I in those 15 pages. Wouldn't that serve the reader better?

And so it goes. For Euclid, we then get all of book V (Eudoxus' theory of proportions), all of book VII, and little bits of books IX and X. Next comes Archimedes (in Heath's paraphrased translation), with both parts of "The Sphere and the Cylinder", "Measurement of a Circle", "The Sand Reckoner", and the "Method". To finish off Greek mathematics, we get selections from Heath's book on Diophantus, which of course does not even claim to be a translation.

After this obligatory nod to the ancient Greeks, we take a flying leap to early modern times, and get selections from Descartes, Newton, Laplace, Fourier, Gauss, Cauchy. (No, nothing from Euler!) At this point, we're well into modern mathematics, and the book concludes with selections from Boole, Riemann, Weierstrass, Dedekind, Cantor, Lebesgue, Gödel, and Turing. Most of these (and some of the earlier material too) will be inaccessible to most non-mathematician readers.

As with the previous collection of this type edited by Hawking, On the Shoulders of Giants, one gets the feeling that the selections were determined more by the availability of translations in the public domain than by any other considerations. According to the preface, four of the selections were translated specifically for this volume. (I think they are: a selection from Cauchy's course on differential calculus, two selections from Riemann, a selection from Weierstrass.) Almost all the others are reproductions of material from various Dover publications, even when, as in the case of Newton, parts of Archimedes, and Laplace, there are better translations available.

In the end, I'm not too impressed. A selection of mathematical texts of historical importance (as in "The Mathematical Breakthroughs that Changed History") that excludes all of Arabic mathematics, all of Medieval and Renaissance mathematics, and that doesn't include Stevin, Fermat, the Bernoullis, Euler, Lagrange, Abel, Galois, Hilbert, Poincaré... it all seems weirdly lopsided. There is nothing on statistics, and very little that is recent. In fact, the only texts included here that were published after 1920 are by Gödel, and Turing, as if 20th century mathematics were dominated by logic.

I don't think that the publishers of the other sourcebooks need to worry about the competition from this one.

Fernando Q. Gouvêa is Professor of Mathematics at Colby College and the co-author, with William P. Berlinghoff, of Math through the Ages. He somehow finds time to also be the editor of MAA Reviews.