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Books: Some oblique angles on life

Do you have to be mad to be a mathematical genius? No, argues Ian Stewart - just enchanted by the magic of numbers; The Man Who Loved Only Numbers by Paul Hoffman Fourth Estate, pounds 12.99, 256pp; A Beautiful Mind by Sylvia Nasar Faber & Faber, pounds 17.99, 448pp
How old is the epsilon? Is it a boss or a slave?" If you hadn't encountered Paul Erdos before, it was an odd and rather disquieting question, but his friends and acquaintances, accustomed to his private language, had no difficulty in translating. He was asking about a child: was it a girl or a boy?

Erdos, the most prolific mathematician of modern times, loved children. His term for them was a mathematician's joke: in analysis, epsilon denotes an arbitrarily small quantity. The boss/slave terminology was a kind of private joke in which Erdos poked gentle fun at a world he understood but whose values were seldom aligned with his own. He referred to God as "SF', "supreme fascist", but reserved his greatest reverence for "The Book", an imaginary volume in which God had written down the best proofs of the best theorems.

John Nash was another brilliant mathematician and a rough contemporary of Erdos. At the age of 66, he was awarded the Nobel prize - not a bad trick when there is no Nobel for mathematics. But for much of his career, Nash suffered from paranoid schizophrenia.

So is mathematical ability akin to madness? Theodore Kaczynski, the Unabomber, is often cited as some kind of proof, mostly by people who feel uncomfortable at their own inability to handle mathematics and want reassurance that anyone who can do better is crazy. I think the answer is "no". After all, the Unabomber had to have had some profession, be it dentist or wall-of- death motorcyclist. But now biographies of both these remarkable mathematicians are available, and they provide some interesting first-hand evidence.

Erdos first: Paul Hoffman - former editor of Discover magazine, and publisher of Encyclopaedia Britannica - has written a wonderful, playful, insightful life of this century's most unusual mathematician. Erdos published 1,475 papers, "many of them monumental and all of them substantial". He engaged in more collaborations than any other mathematician ever, 485 to be exact. If you are a mathematician, your "Erdos number" is the length of the smallest chain of joint papers that links you to Erdos.

In March 1913, while Erdos's mother was in hospital giving birth to him, his two young sisters both died of septic scarlet fever. Young Paul was kept at home a lot, on the assumption that this would protect him against contagious diseases. Self-absorbed and self-sufficient, he was a strange, though sometimes charming, child. Possibly his unusual childhood was the cause of his later unworldliness: he never owned a house or rented an apartment.

Instead, he travelled the globe, staying with colleagues and friends. Most of the money he earned he gave away to deserving causes. The friends remember him with a mixture of exasperation and affection, but it is the affection that shines through.

Erdos's most famous feat was a proof by elementary methods of the "prime number theorem", which gives a good approximation to the number of primes less than a given value. His greatest love, however, was Ramsey theory, which stems from a curious discovery made by Frank Ramsey, an atheist whose brother Michael became Archbishop of Canterbury.

How many people must there be at a party in order that either three of them all know each other or three of them are mutual strangers? Six. If we ask the same question for foursomes, there must be at least 18 people at the party. For fivesomes, the best anyone knows is that the answer lies between 43 and 49; for sixsomes, between 102 and 165. Questions like this have useful applications outside party chit-chat, but it is striking how difficult they are. Erdos loved problems like that.

Nash is the subject of Sylvia Nasar's A Beautiful Mind, less playful than Hoffman's book but more meticulous and equally gripping. Like Erdos, Nash was a strange child, but it is the differences in their lives that are most revealing.

Nash published little, but what he did publish was absolutely spectacular. He was extraordinarily competitive, confidently anticipating the award of the Fields Medal, the mathematician's equivalent of a Nobel, desperately hurt when he was passed over. He missed by a hair's breadth, not that he could have known that. The committee may well have had it in mind to consider him again four years later, but by then Nash was suffering from delusions about messages from extraterrestrials.

While it is not against the rules to award a subject's most prestigious medal to a schizophrenic, it is a rare occurrence. All the more amazing, then, that in 1994 Nash (by then fully recovered) received the Nobel prize in economics for his work on game theory. Here Sylvia Nasar has done a fascinating job of reconstructing the Nobel deliberations: the infighting, the academic politics. By the slenderest of margins, we discover, justice was done.

There have been mad mathematicians, then, but the question is to what extent madness, or at least eccentricity, is essential for mathematical creativity. And here we must avoid being selective. Hardly any of the other mathematicians in these books are crazy. John von Neumann, inventor of game theory, was an accomplished political operator; Lloyd Shapley, who worked at the RAND corporation, was a war hero.

Usually creative people, in arts and business as well as in science and mathematics, need not be crazy - but they do need to be driven. And if the human mind is driven too hard, it may bend, even snap. That may be what happened to Nash, though there is evidence that his condition was in part genetic.

Erdos, on the other hand, was simply too sane for his own good. He had no interest in money, and proved by example that he had no need for it. He spent 19 hours a day working on his beloved mathematics - and that was all he had ever wanted to do.