Edited, with commentary, by Stephen Hawking, Running Press. Hardback ISBN 0762419229, £19.99 ($29.95).
“God created the integers, all the rest is the work of Man,” are the words of 18th-century mathematician Leopold Kronecker, whose thought-provoking statement makes a fitting title to this collection. Following on the path of his previous collection Standing on the Shoulders of Giants, Hawking has brought together representative works of the most influential mathematicians of the past 2500 years, from Euclid to Alan Turing. (Incidentally, Kronecker did not make the cut to be included, but his best friend Karl Weierstrass did.)
The collection outlines the life of each mathematician before reproducing a selection of original work. The sections are self-contained so the book can be read over time or out of sequence. Reading it in one go has its advantages, however: the beautifully terse ancient Greek text contrasts well with the flowery style of George Boole, for instance. Also, there are recurrent themes, such as the problem of the continuity of a function, or David Hilbert’s challenges, which are tackled by several mathematicians in this volume.
Each section starts with an introduction where Hawking’s delightful pen takes us through the mathematician’s biography and then patiently through the most important points of his works. Hawking’s introductions are informative, understandable and in places amusing – for instance, the hilarious story of when Kurt Goedel is taken by his friend, Albert Einstein, for his US nationality hearings.
There is a wealth of information in the original works reproduced and I mention here a few points that I found interesting.
Euclid’s The Elements concerns geometry, and number theory – geometry is the means of visualizing important proofs, such as the infinitude of prime numbers. Archimedes, although better known for his engineering skills, was one of the best mathematicians of antiquity. One of the works reproduced is The Sand Reckoner, Archimedes’ ambitious attempt to measure the mass of the visible universe using the measured size of the Earth and the Sun to arrive at his answer: 1063 grains of sand. What is interesting is his treatment of errors – although he knows the size of the Earth, he conservatively assumes a figure 10 times as big.
The collection covers another Greek, Diophantus, although he is perhaps best known from a footnote that the mathematician Pierre de Fermat (not covered) wrote in the margin next to one of his theorems, which became known as Fermat’s last theorem. It puzzled mathematicians for many years until finally proved in 1992.
Isaac Newton has probably the shortest space allocated in the book, but quantity is not proportional to importance. Newton was not only a great physicist, but also a brilliant mathematician: he invented calculus. In another link between mathematics and physics, Joseph Fourier derived his trigonometric series while trying to solve a physics problem on heat transfer.
Carl Friedrich Gauss, a mathematical prodigy, is considered by many as the greatest mathematician of all time. Less well known is the fact that he worked for a period as a surveyor and that he achieved international fame when he calculated the position of asteroid Ceres in 1801.
Bernhard Riemann generalized geometry in a way that proved essential to Albert Einstein more than 60 years later. Ironically, Riemann was worrying about deviations from the Euclidian model for the infinitesimally small.
While Hilbert does not feature in this volume, his statement of the three most important challenges of mathematics inspired mathematicians who do appear. Goedel would prove the incompleteness and inconsistency of mathematics, whereas Turing would disprove the decidability of mathematics some years later.
When reading about the lives of the great men featured a few worrying recurrent themes appear, some in line with caricatures of mathematicians. Mental problems and an unfulfilled private life occur with alarming frequency. Less expected, perhaps, is the struggle for professional recognition.
Unfortunately, the book is not perfect and I do have a few gripes. The Greek text is deeply flawed and quite unacceptable for such a publication; the typesetting is appalling and proofreading evidently was non-existent. Running Press would do well to rectify this on future editions. Throughout the book, the hallmarks of a rushed job are everywhere. There are far too many typing errors, which is especially bad when formulae are concerned; it is not always clear if a footnote was from the original author, or inserted by the editor; and finally, some figures (the ones appearing in Henri Lebesgue’s section) were of surprisingly low quality.
Even so, this is an impressive collection of works that are part of our intellectual heritage. It is an important addition to every library, but bedside (or poolside) reading it is not.
