As G. H. Hardy gloomily noted in *A Mathematician’s Apology*, “most people can do nothing at all well”. Of those few who can do one thing well, many keep doing it (while they can), plowing and replowing their furrow, unable or unwilling to excel at anything new. John von Neumann was different. He could do *everything* well. Anything he touched turned to gold.

Von Neumann was born in Hungary in 1903, came to the United States in 1930, where he was one of the six original members of the mathematical section of the Institute for Advanced Study, and died distressingly young in 1957. His mathematical talent was recognized early, and though he earned a degree in chemical engineering and no doubt would have been a great chemical engineer, he devoted his professional life to mathematics. He began in set theory, trying to perfect its axioms and giving the standard definition of ordinal number. He could have spent his mathematical life amid the foundations, but he moved on to Hilbert space, introducing what are now called von Neumann algebras and proving an ergodic theorem.

He continued to move outward, to physics. He showed that Hilbert space could be used to give a mathematical description of quantum mechanics. After physics came economics, where he proved the minimax theorem of game theory. The second world war drew his attention to applied mathematics where, among other things, his work on shock waves helped determine the altitude at which atomic bombs should be detonated to maximize destruction. The war also necessitated much computation, so von Neumann devised the architecture that all digital computers have. Computers led him to look at automata and their relation to thinking and consciousness, but his death intervened before he could make what would probably have been fundamental advances. What a mathematical life!

Biographies of such a towering figure have been scarce. Norman MacRae’s *John von Neumann* (1999) is well written, as could be expected from a former editor of *The Economist*, and has many virtues, but suffers from the author’s self-indulgence and his lack of mathematical training. *Prisoner’s Dilemma* by William Poundstone (1992) concentrates on game theory and also, as the authors of the book under review point out, repeats too many von Neumann stories, many of which are exaggerated, irrelevant, or both. William Aspray’s *John von Neumann and the Origins of Modern Computing* (1990) is more a contribution to the history of computers than a biography.

The jacket copy for *The World as a Mathematical Game* says that the book “provides the first comprehensive scientific and intellectual biography of John von Neumann”. It covers the ground well and thoroughly in 175 pages of text, moving chronologically through his astonishing career. Its title may overstate a little von Neumann’s belief in the power of mathematics over the world, but it was a belief that he had. As S. J. Heims, quoted by the authors, said, “He seemed to regard the empirical world, probably even life and mind, as comprehensible only in terms of abstract formal structure”.

The translation from the Italian, by Ian McGilvray, reads smoothly. I suppose that it reflects the authors’ style, which runs, unfortunately in my view, to overlong sequences of 40-word sentences.

The authors have written their book for a general audience. There are hardly any equations and no mathematics is explained, except for two pages on matrix games (which, in the absence of any similar material, seems out of place). I was sometimes frustrated, thinking, as the authors talked about von Neumann’s mathematical achievements, “Yes, but what did he *do*?” But the authors were not writing for mathematicians.

It’s not clear how much a truly general reader will get out of the book. Sentences like “According to the “bourbakists”, the whole of mathematics could be reduced to the interweaving of three fundamental axiomatic structures: *algebraic* structures, *order* structures and *topological* structures” seem to demand quite a bit of mathematical background for comprehension. The authors may have set themselves an impossible task. I suppose that the market will decide how successful they have been.

In any event, it is good to have this outline of von Neumann’s work. His definitive biography remains to be written.

Woody Dudley retired from teaching mathematics in 2004. He graduated from college in the year that von Neumann died; though he could conceivably have met him, their paths never crossed.