A Beautiful Mind (33 page)

Newman described Nash as a “very poetic, different kind of thinker.”
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In this instance, Nash used differential calculus, not geometric pictures or algebraic manipulations, methods that were classical outgrowths of nineteenth-century calculus. The technique is now referred to as the Nash-Moser theorem, although there is no dispute that Nash was its originator.
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Jürgen Moser was to show how Nash’s technique could be modified and applied to celestial mechanics — the movement of planets — especially for establishing the stability of periodic orbits.
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Nash solved the problem in two steps. He discovered that one could embed a Riemannian manifold in a three-dimensional space if one ignored smoothness.
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One had, so to speak, to crumple it up. It was a remarkable result, a strange and interesting result, but a mathematical curiosity, or so it seemed.
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Mathematicians were interested in embedding without wrinkles, embedding in which the smoothness of the manifold could be preserved.

In his autobiographical essay, Nash wrote:

So as it happened, as soon as I heard in conversation at MIT about the question of embeddability being open I began to study it. The first break led to a curious result about the embeddability being realizable in surprisingly low-dimensional ambient spaces provided that one would accept that the embedding would have only limited smoothness. And later, with “heavy analysis,” the problem was solved in terms of embedding with a more proper degree of smoothness.
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Nash presented his initial, “curious” result at a seminar in Princeton, most likely in the spring of 1953, at around the same time that Ambrose wrote his scathing letter to Halmos. Emil Artin was in the audience. He made no secret of his doubts.

“Well, that’s all well and good, but what about the embedding theorem?” said Artin. “You’ll never get it.”

“I’ll get it next week,” Nash shot back.
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One night, possibly en route to this very talk, Nash was hurtling down the Merritt Parkway.
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Poldy Flatto was riding with him as far as the Bronx. Flatto, like all the other graduate students, knew that Nash was working on the embedding problem. Most likely to get Nash’s goat and have the pleasure of watching his reaction, he mentioned that Jacob Schwartz, a brilliant young mathematician at Yale whom Nash knew slightly, was also working on the problem.

Nash became quite agitated. He gripped the steering wheel and almost shouted at Flatto, asking whether he had meant to say that Schwartz had solved the problem. “I didn’t say that,” Flatto corrected. “I said I heard he was working on it.”

“Working on it?” Nash replied, his whole body now the picture of relaxation. “Well, then there’s nothing to worry about. He doesn’t have the insights I have.”

Schwartz was indeed working on the same problem. Later, after Nash had produced his solution, Schwartz wrote a book on the subject of implicit-function theorems. He recalled in 1996:

I got half the idea independently, but I couldn’t get the other half. It’s easy to see an approximate statement to the effect that not every surface can be exactly embedded, but that you can come arbitrarily close. I got that idea and I was able to produce the proof of the easy half in a day. But then I realized that there was a technical problem. I worked on it for a month and couldn’t see any way to make headway. I ran into an absolute stone wall. I didn’t know what to do. Nash worked on that problem for two years with a sort of ferocious, fantastic tenacity until he broke through it.
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Week after week, Nash would turn up in Levinson’s office, much as he had in Spencer’s at Princeton. He would describe to Levinson what he had done and Levinson would show him why it didn’t work. Isadore Singer, a fellow Moore instructor, recalled:

He’d show the solutions to Levinson. The first few times he was dead wrong. But he didn’t give up. As he saw the problem get harder and harder, he applied himself more, and more and more. He was motivated just to show everybody how good he was, sure, but on the other hand he didn’t give up even when the problem turned out to be much harder than expected. He put more and more of himself into it.
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There is no way of knowing what enables one man to crack a big problem while another man, also brilliant, fails. Some geniuses have been sprinters who have solved problems quickly. Nash was a long-distance runner. If Nash defied von Neumann in his approach to the theory of games, he now took on the received wisdom of nearly a century. He went into a classical domain where everybody believed that they understood what was possible and not possible. “It took enormous courage to attack these problems,” said Paul Cohen, a mathematician at Stanford University and a Fields medalist.”
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His tolerance for solitude, great confidence in his own intuition, indifference to criticism — all detectable at a young age but now prominent and impermeable features of his personality — served him well. He was a hard worker by habit. He worked mostly at night in his MIT office — from ten in the evening until 3:00
A.M.
— and on weekends as well, with, as one observer said, “no references but his own mind” and his “supreme self-confidence.” Schwartz called it “the ability to continue punching the wall until the stone breaks.”

The most eloquent description of Nash’s single-minded attack on the problem comes from Moser:

The difficulty [that Levinson had pointed out], to anyone in his right mind, would have stopped them cold and caused them to abandon the problem. But Nash was different. If he had a hunch, conventional criticisms didn’t stop him. He had no background knowledge. It was totally uncanny. Nobody could understand how somebody like that could do it. He was the only person I ever saw with that kind of power, just brute mental power.
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The editors of the
Annals of Mathematics
hardly knew what to make of Nash’s manuscript when it landed on their desks at the end of October 1954. It hardly had the look of a mathematics paper. It was as thick as a book, printed by hand rather than typed, and chaotic. It made use of concepts and terminology more familiar to engineers than to mathematicians. So they sent it to a mathematician at Brown University, Herbert Federer, an Austrian-born refugee from Nazism and a pioneer in surface area theory, who, although only thirty-four, already had a reputation for high standards, superb taste, and an unusual willingness to tackle difficult manuscripts.
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Mathematics is often described, quite rightly, as the most solitary of endeavors. But when a serious mathematician announces that he has found the solution to an important problem, at least one other serious mathematician, and sometimes several, as a matter of longstanding tradition that goes back hundreds of years, will set aside his own work for weeks and months at a time, as one former collaborator of Federer’s put it, “to make a go of it and straighten everything out.”
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Nash’s manuscript presented Federer with a sensationally complicated puzzle and he attacked the task with relish.

The collaboration between author and referee took months. A large correspondence, many telephone conversations, and numerous drafts ensued. Nash did not submit the revised version of the paper until nearly the end of the following summer. His acknowledgment to Federer was, by Nash’s standards, effusive: “I am profoundly indebted to H. Federer, to whom may be traced most of the improvement over the first chaotic formulation of this work.”
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Armand Borel, who was a visiting professor at Chicago when Nash gave a lecture on his embedding theorem, remembers the audience’s shocked reaction. “Nobody believed his proof at first,” he recalled in 1995. “People were very skeptical. It looked like a [beguiling] idea. But when there’s no technique, you are skeptical. You dream about a vision. Usually you’re missing something. People did not challenge him publicly, but they talked privately.”
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(Characteristically, Nash’s report to his parents merely said “talks went well.”)
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Gian-Carlo Rota, professor of mathematics and philosophy at MIT, confirmed Borel’s account. “One of the great experts on the subject told me that if one of his graduate students had proposed such an outlandish idea he’d throw him out of his office.”
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The result was so unexpected, and Nash’s methods so novel, that even the experts had tremendous difficulty understanding what he had done. Nash used to leave drafts lying around the MIT common room.
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A former MIT graduate student recalls a long and confused discussion between Ambrose, Singer, and Masatake Kuranishi (a mathematician at Columbia University who later applied Nash’s result) in which each one tried to explain Nash’s result to the other, without much success.
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Jack Schwartz recalled:

Nash’s solution was not just novel, but very mysterious, a mysterious set of weird inequalities that all came together. In my explication of it I sort of looked at what happened and could generalize and give an abstract form and realize it was applicable to situations other than the specific one he treated. But I didn’t quite get to the bottom of it either.
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Later, Heinz Hopf, professor of mathematics in Zurich and a past president of the International Mathematical Union, “a great man with a small build, friendly, radiating a warm glow, who knew everything about differential geometry,” gave a talk on Nash’s embedding theorem in New York.
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Usually Hopf’s lectures were models of crystalline clarity. Moser, who was in the audience, recalled: “So we thought, ‘NOW we’ll understand what Nash did.’ He was naturally skeptical. He would have been an important validator of Nash’s work. But as the lecture went on, my God, Hopf was befuddled himself. He couldn’t convey a complete picture. He was completely overwhelmed.”
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Several years later, Jürgen Moser tried to get Nash to explain how he had overcome the difficulties that Levinson had originally pointed out. “I did not learn so much from him. When he talked, he was vague, hand waving, ‘You have to control this. You have to watch out for that.’ You couldn’t follow him. But his written paper was complete and correct.”
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Federer not only edited Nash’s paper to make it more accessible, but also was the first to convince the mathematical community that Nash’s theorem was indeed correct.

Martin’s surprise proposal, in the early part of 1953, to offer Nash a permanent faculty position set off a storm of controversy among the eighteen-member mathematics faculty.
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Levinson and Wiener were among Nash’s strongest supporters. But others, like Warren Ambrose and George Whitehead, the distinguished topolo-gist, were opposed. Moore Instructorships weren’t meant to lead to tenure-track positions. More to the point, Nash had made plenty of enemies and few friends in
his first year and a half. His disdainful manner toward his colleagues and his poor record as a teacher rubbed many the wrong way.

Mostly, however, Nash’s opponents were of the opinion that he hadn’t proved he could produce. Whitehead recalled, “He talked big. Some of us were not sure he could live up to his claims.”
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Ambrose, not surprisingly, felt similarly. Even Nash’s champions could not have been completely certain. Flatto remembered one occasion on which Nash came to Levinson’s office to ask Levinson whether he’d read a draft of his embedding paper. Levinson said, “To tell you the truth I don’t have enough background in this area to pass judgment.”
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When Nash finally succeeded, Ambrose did what a fine mathematician and sterling human being would do. His applause was as loud as or louder than anyone else’s. The bantering became friendlier and, among other things, Ambrose took to telling his musical friends that Nash’s whistling was the purest, most beautiful tone he had ever heard.
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21
Singularity

Nash was leading all these separate lives. Completely separate lives.

— A
RTHUR
M
ATTUCK
, 1997

A
LL THROUGH HIS CHILDHOOD
, adolescence, and brilliant student career, Nash had seemed largely to live inside his own head, immune to the emotional forces that bind people together. His overriding interest was in patterns, not people, and his greatest need was making sense of the chaos within and without by employing, to the largest possible extent, the resources of his own powerful, fearless, fertile mind. His apparent lack of ordinary human needs was, if anything, a matter of pride and satisfaction to him, confirming his own uniqueness. He thought of himself as a rationalist, a free thinker, a sort of Spock of the starship
Enterprise.
But now, as he entered early adulthood, this unfettered persona was shown to be partly a fiction or at least partly superseded. In those first years at MIT, he discovered that he had some of the same wishes as others. The cerebral, playful, calculating, and episodic connections that had once sufficed no longer served. In five short years, between the ages of twenty-four and twenty-nine, Nash became emotionally involved with at least three other men. He acquired and then abandoned a secret mistress who bore his child. And he courted — or rather was courted by — a woman who became his wife.