The sad story of mad genius John Nash.

People with high IQs tend to be nearsighted. This is not because they read a lot or stare at computer screens too much. That common-sense hypothesis has been discredited by research. Rather, it is a matter of genetics. The same genes that tend to elevate IQ also tend to affect the shape of the eyeball in a way that leads to myopia. This relationship--known in genetics as "pleiotropy"--seems to be completely accidental, a quirk of evolution.

Could there be a similar pleiotropy between madness and mathematics? Reading this absolutely fascinating biography by Sylvia Nasar, an economics writer for the New York Times, I began to wonder. Its subject, John Nash, is a mathematical genius who went crazy at the age of 30 and then, after several decades of flamboyant lunacy, was awarded the Nobel Prize in economics for something he had discovered as a graduate student. (He is now about to turn 70.) Nash is among the latest in a long and distinguished line of mathematicians--stretching back to that morbid paranoiac, Isaac Newton--who have been certifiably insane during parts of their lives.

Just in the last 100 years or so, most of the heroic figures in the foundations of mathematics have landed in mental asylums or have died by their own hand. The greatest of them, Kurt Gödel, starved himself to death in the belief that his colleagues were putting poison in his food. Of the two pioneers of game theory--the field in which Nash garnered his Nobel--one, Ernst Zermelo, was hospitalized for psychosis. The other, John Von Neumann, may not have been clinically insane, but he did serve as the real-life model for the title character in Stanley Kubrick's Dr. Strangelove.

So maybe there is an accidental, pleiotropic connection between madness and mathematics. Or maybe it isn't so accidental. Mathematicians are, after all, people who fancy that they commune with perfect Platonic objects--abstract spaces, infinite numbers, zeta functions--that are invisible to normal humans. They spend their days piecing together complicated, scrupulously logical tales about these hallucinatory entities, which they believe are vastly more important than anything in the actual world. Is this not a kind of a folie à n (where n equals the number of pure mathematicians worldwide)?

ABeautiful Mind reveals quite a lot about the psychic continuum leading from mathematical genius to madness. It is also a very peculiar redemption story: how three decades of raging schizophrenia, capped by an unexpected Nobel Prize, can transmute a cruel shit into a frail but decent human being.

As a boy growing up in the hills of West Virginia, Nash enjoyed torturing animals and building homemade bombs with two other unpopular youngsters, one of whom was accidentally killed by a blast. (Given Nash's childhood keenness for explosives and his later penchant for sending odd packages to prominent strangers through the mail, it's a wonder the FBI never got on to him as a Unabomber suspect.) He made his way to Carnegie Tech, where he was a classmate of Andy Warhol's, and thence to Princeton--the world capital of mathematics at the time--at the age of 20.

In sheer appearance, this cold and aloof Southerner stood out from his fellow math prodigies. A "beautiful dark-haired young man," "handsome as a god," he was 6 feet 1 inch tall, with broad shoulders, a heavily muscled chest (which he liked to show off with see-through Dacron shirts), a tapered waist, and "rather limp and beautiful hands" accentuated by long fingernails. Within two years of entering Princeton, Nash had framed and proved the most important proposition in the theory of games.

Mathematically, this was no big deal. Game theory was a somewhat fashionable pursuit for mathematicians in those postwar days, when it looked as if it might do for military science and economics what Newton's calculus had done for physics. But they were bored with it by the early 1950s. Economists, after a few decades of hesitation, picked it up in the '80s and made it a cornerstone of their discipline.

Agame is just a conflict situation with a bunch of participants, or "players." The players could be poker pals, oligopolists competing to corner a market, or nuclear powers trying to dominate each other. Each player has several strategy options to choose from. What Nash showed was that in every such game there is what has become known as a "Nash equilibrium": a set of strategies, one for each player, such that no player can improve his situation by switching to a different strategy. His proof was elegant but slight. A game is guaranteed to have a Nash equilibrium, it turns out, for the same reason that in a cup of coffee that is being stirred, at least one coffee molecule must remain absolutely still. Both are direct consequences of a "fixed-point theorem" in the branch of mathematics known as topology. This theorem says that for any continuous rearrangement of a domain of things, there will necessarily exist at least one thing in that domain that will remain unchanged--the "fixed point." Nash found a way of applying this to the domain of all game strategies so that the guaranteed fixed point was the equilibrium for the game--clever, but the earlier topological theorem did all the work. Still, for an economics theorem, that counts as profound. Economists have been known to win Nobel Prizes for rediscovering theorems in elementary calculus.

Nash's breakthrough in game theory got him recruited by the Rand Corp., which was then a secretive military think tank in Santa Monica (its name is an acronym for "research and development"). However, the achievement did not greatly impress his fellow mathematicians. To do that, Nash, on a wager, disposed of a deep problem that had baffled the profession since the 19^th century: He showed that any Riemannian manifold possessing a special kind of "smoothness" can be embedded in Euclidean space. Manifolds, one must understand, are fairly wild and exotic beasts in mathematics. A famous example is the Klein bottle, a kind of higher-dimensional Moebius strip whose inside is somehow the same as its outside. Euclidean space, by contrast, is orderly and bourgeois. To demonstrate that "impossible" manifolds could be coaxed into living in Euclidean space is counterintuitive and pretty exciting. Nash did this by constructing a bizarre set of inequalities that left his fellow mathematicians thoroughly befuddled.