God Created the Integers: The Mathematical Breakthroughs that Changed History

Edited by:
  • Stephen Hawking
Running Press: 2005. 1,160 pp. $29.95. 0762419229 | ISBN: 0-762-41922-9

The subtitle — the mathematical breakthroughs that changed history — is not mere advertising. It is a yardstick, in fact various yardsticks, for assessing the book as a whole. Stephen Hawking brings us generous selections of Euclid's Elements, the works of Archimedes and Diophantus, René Descartes' Geometry entire, Pierre-Simon Laplace's A Philosophical Essay on Probability, and material from Joseph Fourier, Carl Friedrich Gauss, Augustin-Louis Cauchy, Bernhard Riemann, Georg Cantor and Henri-Léon Lebesgue. There are also the first 11 chapters of George Boole's An Investigation into the Laws of Thought, Richard Dedekind's Essays on the Theory of Numbers, smaller extracts from Newton's Principia, and papers by Karl Weierstrass, Kurt Gödel and Alan Turing. All these have some claim to have changed history, except perhaps the work by Diophantus, remarkable though it is. The same claim can be made of quite a few other works too, but as this book has nearly 1,200 pages it would be churlish to complain.

The items are well chosen. They are an interesting mix of the well known and the unexpected, and cover a range of topics in mathematics from geometry to mathematical analysis, probability and the modern foundations of mathematics. They range over an extensive period and, because many are presented either whole or at least extensively, they can be read with pleasure.

Every anthology faces the issue of what to do with the best-known pieces. Include them and some complain that they are too well known; omit them and others lament their loss. If this book is the only collection of original works of mathematics in translation that a reader owns, there is a lot to be said for it. But as an addition to the small but useful number of collections in English it is more annoying, because the opportunity was not taken to translate more works. All but four of the items here are already available in English. The new ones are the work by Cauchy (some of which already exists in English elsewhere), some of the papers by Riemann (one of which is already in English), the passage by Weierstrass, and the extracts from the work of Lebesgue.

It is one thing to use an old translation if there is no significant improvement to be had in making a new one, but it is quite another to perpetuate an inadequate work. Here the Diophantus is annoying because it is the version by Thomas Little Heath. He turned the work into a densely written but by now old-fashioned school problem book, and the commentary, given in extensive footnotes, moves the reader even further from Diophantus' habit of mind. It is true that Diophantus' example is so strange that it reminds us how little we know about the ancient Greeks, but we do know how it was received in the Arab world and in the modern West, and it can be said to be only one of several works that helped bring about the ‘modern algebra’ of the seventeenth century. It would have been better if Hawking had included instead al-Khwarizmi's work on al-jabr, from which algebra takes its name (see ‘A culture of knowledge’).

As for Euclid and Archimedes, there are 240 pages of Heath's editions with their erudite but dated commentaries. These may well help the reader with the mathematics, but historical scholarship has moved on. Unfortunately, the new comments are not much better. Each item is introduced with remarks that vary from the personal and insightful to the tired and incorrect. Among the latter is the supposed Greek crisis of the incommensurables, which was once presumed to have derailed the pythagoreans and which most historians these days think had little effect. The generally more accurate account of the life and work of Archimedes fails to mention the fact that the only manuscript of The Method — the most interesting of his works, in which he explained how he came to his discoveries — has recently re-entered the public domain after disappearing for most of the twentieth century.

Similar comments could be made about the more modern entries. These too are generally accurate but the origin of the information is not stated, so readers have no chance to catch up with contemporary scholarship in the history of mathematics. Nor can they find out how to sustain the flame of interest this book surely hopes to kindle, which is a pity, because Hawking's comments have an infectious enthusiasm for their subject and the book contains some great works.