Like Nash, though with far less cause for optimism, I came to Princeton in my youth hoping to make an important contribution to mathematics. I've known dozens of the mathematicians and economists whose portraits Ms. Nasar paints with uncanny accuracy. Reading her book, I hear voices from my past, and they all sound familiar.

It's not just the people she's got right; it's also the places, the milieu, the social structure. No place on earth feels quite like the math common room at Princeton, where great minds and ambitious young scholars meet for tea, making both a glorious history and an exhilarating future seem palpably present. Ms. Nasar's book shows exactly how that room feels. As for the movie--well, as I said, it's a lovely fiction. In more ways than one.

Many critics have noted that the movie bowdlerizes Nash's life, omitting mention of his bisexuality and divorce, among other things. But those omissions at least serve a dramatic purpose. Far more unsettling is that the movie rewrites not just Nash's life but his thought.

Take the bizarre and ludicrous scene--apparently invented--where Nash, out drinking with his fellow graduate students, achieves the flash of insight that he's been desperately seeking for months.

If he and his friends all hit on the same woman, Nash reasons, they'll devastate one another's chances while letting other, slightly less desirable, women get away. "Adam Smith needs revision!" he declares triumphantly. To his baffled classmates, he explains: "Adam Smith said the best result comes from everyone in the group doing what's best for himself, right? Adam Smith was wrong!" The message: Sometimes it's better to cooperate!

Well, duh. Does anyone believe that the benefits of cooperation could have eluded the astute Adam Smith? Isn't this the same Adam Smith who famously remarked that "people of the same trade seldom meet together, even for merriment or diversion, but the conversation ends in a conspiracy against the public, or in some contrivance to raise prices"? Surely Smith would have appreciated the value of a contrivance among horny graduate students to stay out of one another's way.

That's not obvious, but it's true. It was proved, as a matter of pure mathematics, by economists like Gerard Debreu, Kenneth Arrow and Lionel McKenzie, beginning around the time Nash was supposedly having that fateful drink. Surely that would have been good enough for Nash, who through all his breakdowns never lost his respect for mathematical reasoning.

In the real world, as opposed to the movie, Nash complemented Smith without supplanting him. The Invisible Hand tells you that good things happen when people compete in free markets. Nash laid the game-theoretic mathematical foundations for figuring out what happens when people compete in other ways. It turns out that without markets, many different things can happen, not all of them good. That means we should have more respect for markets, not less.

What the fictionalized Nash should have said--and the real Nash would have said--is that mating competitions can turn out badly because mates are not allocated through a competitive price system. But thank goodness other goods--from hamburgers to locomotives--are allocated by prices, yielding the desirable outcomes that Adam Smith promised us all along.

The key tool in this analysis is Nash's notion of equilibrium: a situation where everyone does the best he can subject to what everyone else is doing--that is, "everyone in the group doing what's best for himself." In the presence of competitive price systems, a Nash equilibrium is usually a desirable outcome. "Adam Smith needs revision" indeed!

It's said that Einstein, on being asked whether he kept a notebook to write down all his ideas, replied: "I've only had three ideas in my life!" Nash had approximately the same number, and they were very good ideas, though not as good as Einstein's. Besides his Nobel Prize-worthy work in game theory, Nash solved some impressively difficult problems in pure mathematics--showing, for example, that certain abstractly defined geometric objects are in a sense more concrete than anybody had a right to expect.

But solving problems is not the same thing as laying new foundations, unifying vast areas of mathematics and uncovering deep analogies between disparate fields. In Nash's time, more than the usual number of mathematical giants walked the earth--men like Andre Weil, Alexandre Grothendieck, Jean-Pierre Serre and Nash's old friend John Milnor. Despite all the hype associated with the movie--and despite the brilliance of his work--Nash was never quite in their league.

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