A Beautiful Mind
Page 23
Thanks to MIT’s support and the compromises they struck, Levinson and the others kept their jobs. But the whole dispiriting affair, which had been preceded by months of harassment and threats, left deep scars on everyone involved. Martin, in particular, was shattered and deeply depressed, and was unable, nearly forty-five years later, to talk about it.13 Levinson’s younger daughter, a student in junior high school, suffered a breakdown and was diagnosed with manic depression. Levinson and his wife blamed it partly on her being harassed by the FBI.14 And those on the periphery, ostensibly unaffected, learned a lesson, namely that the world they so very much took for granted was dangerously fragile and vulnerable to forces beyond its control.
Nash took no part in the heated discussions among some of the graduate students over the morality of the mathematicians’ decision to cooperate with the government.15 Any discussion of morality raised for him the specter of hypocrisy. But the angry, frightening, turbulent time would supply him with some of the prosecutory demons that came to haunt him later.16
20
Geometry
There are two kinds of mathematical contributions: work that’s important to the history of mathematics and work that’s simply a triumph of the human spirit.
—PAUL J. COHEN, 1996
IN THE SPRING OF 1953, Paul Halmos, a mathematician at the University of Chicago, received the following letter from his old friend Warren Ambrose, a colleague of Nash’s:
There’s no significant news from here, as always. Martin is appointing John Nash to an Assistant Professorship (not the Nash at Illinois, the one out of Princeton by Steenrod) and I’m pretty annoyed at that. Nash is a childish bright guy who wants to be “basically original,” which I suppose is fine for those who have some basic originality in them. He also makes a damned fool of himself in various ways contrary to this philosophy. He recently heard of the unsolved problem about imbedding a Riemannian manifold isometrically in Euclidean space, felt that this was his sort of thing, provided the problem were sufficiently worthwhile to justify his efforts; so he proceeded to write to everyone in the math society to check on that, was told that it probably was, and proceeded to announce that he had solved it, modulo details, and told Mackey he would like to talk about it at the Harvard colloquium. Meanwhile he went to Levinson to inquire about a differential equation that intervened and Levinson says it is a system of partial differential equations and if he could only [get] to the essentially simpler analog of a single ordinary differential equation it would be a damned good paper — and Nash had only the vaguest notions about the whole thing. So it is generally conceded he is getting nowhere and making an even bigger ass of himself than he has been previously supposed by those with less insight than myself. But we’ve got him and saved ourselves the possibility of having gotten a real mathematician. He’s a bright guy but conceited as Hell, childish as Wiener, hasty as X, obstreperous as Y, for arbitrary X and Y.1
• • •
Ambrose had every reason to be both skeptical and annoyed.
Ambrose was a moody, intense, somewhat frustrated mathematician in his late thirties, full, as his letter indicates, of black humor.2 He was a radical and nonconformist. He married three times. He gave a lecture on “Why I am an atheist.” He once tried to defend some left-wing demonstrators against police in Argentina — and got himself beaten up and jailed for his efforts. He was also a jazz fanatic, a personal friend of Charlie Parker, and a fine trumpet player.3 Handsome, solidly built, with a boxer’s broken nose — the consequence of an accident in an elevator! — he was one of the most popular members of the department. He and Nash clashed from the start.
Ambrose’s manner was calculated to give an impression of stupidity: “I’m a simple man, I can’t understand this.” Robert Aumann recalled: “Ambrose came to class one day with one shoelace tied and the other untied. ‘Did you know your right shoelace is untied?’ we asked. ’Oh, my God,’ he said, ’I tied the left one and thought the other must be tied by considerations of symmetry.’ ”4
The older faculty in the department mostly ignored Nash’s putdowns and jibes. Ambrose did not. Soon a tit-for-tat rivalry was under way. Ambrose was famous, among other things, for detail. His blackboard notes were so dense that rather than attempt the impossible task of copying them, one of his assistants used to photograph them.5 Nash, who disliked laborious, step-by-step expositions, found much to mock. When Ambrose wrote what Nash considered an ugly argument on the blackboard during a seminar, Nash would mutter, “Hack, Hack,” from the back of the room.6
Nash made Ambrose the target of several pranks. “Seminar on the REAL mathematics!” read a sign that Nash posted one day. “The seminar will meet weekly Thursdays at 2 P.M. in the Common Room.” Thursday at 2:00 P.M. was the hour that Ambrose taught his graduate course in analysis.7 On another occasion, after Ambrose delivered a lecture at the Harvard mathematics colloquium, Nash arranged to have a large bouquet of red roses delivered to the podium as if Ambrose were a ballerina taking her bows.8
Ambrose needled back. He wrote “Fuck Myself” on the “To Do” list that Nash kept hanging over his desk on a clipboard.9 It was he who nicknamed Nash “Gnash” for constantly making belittling remarks about other mathematicians.10 And, during a discussion in the common room, after one of Nash’s diatribes about hacks and drones, Ambrose said disgustedly, “If you’re so good, why don’t you solve the embedding problem for manifolds?” — a notoriously difficult problem that had been around since it was posed by Riemann.11
So Nash did.
Two years later at the University of Chicago, Nash began a lecture describing his first really big theorem by saying, “I did this because of a bet.”12: Nash’s opening statement spoke volumes about who he was. He was a mathematician who viewed mathematics not as a grand scheme, but as a collection of challenging problems. In the taxonomy of mathematicians, there are problem solvers and theoreticians, and, by temperament, Nash belonged to the first group. He was not a game theorist, analyst, algebraist, geometer, topologist, or mathematical physicist. But he zeroed in on areas in these fields where essentially nobody had achieved anything. The thing was to find an interesting question that he could say something about.
Before taking on Ambrose’s challenge, Nash wanted to be certain that solving the problem would cover him with glory. He not only quizzed various experts on the problem’s importance, but, according to Felix Browder, another Moore Instructor, claimed to have proved the result long before he actually had.13 When a mathematician at Harvard confronted Nash, recalled Browder, “Nash explained that he wanted to find out whether it was worth working on.”14
“The discussion of manifolds was everywhere,” said Joseph Kohn in 1995, gesturing to the air around him. “The precise question that Ambrose asked Nash in the common room one day was the following: Is it possible to embed any Riemannian manifold in a Euclidean space?”15
It’s a “deep philosophical question” concerning the foundations of geometry that virtually every mathematician — from Riemann and Hilbert to Elie-Joseph Cartan and Hermann Weyl — working in the field of differential geometry for the past century had asked himself.16 The question, first posed explicitly by Ludwig Schlafli in the 1870s, had evolved naturally from a progression of other questions that had been posed and partly answered beginning in the mid-nineteenth century.17 First mathematicians studied ordinary curves, then surfaces, and finally, thanks to Riemann, a sickly German genius and one of the great figures of nineteenth-century mathematics, geometric objects in higher dimensions. Riemann discovered examples of manifolds inside Euclidean spaces. But in the early 1950s interest shifted to manifolds partly because of the large role that distorted space and time relationships had in Einstein’s theory of relativity.
Nash’s own description of the embedding problem in his 1995 Nobel autobiography hints at the reason he wished to make sure that solving the problem would be worth the effort: “This problem, although classical, was not much talked about as an outstand
ing problem. It was not like, for example, the four-color conjecture.”18
Embedding involves portraying a geometric object as — or, a bit more precisely, making it a subset of — some space in some dimension. Take the surface of a balloon. You can’t put it on a blackboard, which is a two-dimensional space. But you can make it a subset of spaces of three or more dimensions. Now take a slightly more complicated object, say a Klein bottle. A Klein bottle looks like a tin can whose lid and bottom have been removed and whose top has been stretched around and reconnected through the side to the bottom. If you think about it, it’s obvious that if you try that in three-dimensional space, the thing intersects itself. That’s bad from a mathematical point of view because the neighborhood in the immediate vicinity of the intersection looks weird and irregular, and attempts to calculate various attributes like distance or rates of change in that part of the object tend to blow up. But put the same Klein bottle into a space of four dimensions and the thing no longer intersects itself. Like a ball embedded in three-space, a Klein bottle in four-space becomes a perfectly well-behaved manifold.
Nash’s theorem stated that any kind of surface that embodied a special notion of smoothness can actually be embedded in Euclidean space. He showed that you could fold the manifold like a silk handkerchief, without distorting it. Nobody would have expected Nash’s theorem to be true. In fact, everyone would have expected it to be false. “It showed incredible originality,” said Mikhail Gromov, the geometer whose book Partial Differential Relations builds on Nash’s work. He went on:
Many of us have the power to develop existing ideas. We follow paths prepared by others. But most of us could never produce anything comparable to what Nash produced. It’s like lightning striking. Psychologically the barrier he broke is absolutely fantastic. He has completely changed the perspective on partial differential equations. There has been some tendency in recent decades to move from harmony to chaos. Nash says chaos is just around the corner.19
John Conway, the Princeton mathematician who discovered surreal numbers and invented the game of Life, called Nash’s result “one of the most important pieces of mathematical analysis in this century.”20
It was also, one must add, a deliberate jab at then-fashionable approaches to Riemannian manifolds, just as Nash’s approach to the theory of games was a direct challenge to von Neumann’s. Ambrose, for example, was himself involved in a highly abstract and conceptual description of such manifolds at the time. As Jürgen Moser, a young German mathematician who came to know Nash well in the mid-1950s, put it, “Nash didn’t like that style of mathematics at all. He was out to show that this, to his mind, exotic approach was completely unnecessary since any such manifold was simply a submanifold of a high dimensional Euclidean space.”21
Nash’s more important achievement may have been the powerful technique he invented to obtain his result. In order to prove his theorem, Nash had to confront a seemingly insurmountable obstacle, solving a certain set of partial differential equations that were impossible to solve with existing methods.
That obstacle cropped up in many mathematical and physical problems. It was the difficulty that Levinson, according to Ambrose’s letter, pointed out to Nash, and it is a difficulty that crops up in many, many problems — in particular, nonlinear problems.22 Typically, in solving an equation, the thing that is given is some function, and one finds estimates of derivatives of a solution in terms of derivatives of the given function. Nash’s solution was remarkable in that the a priori estimates lost derivatives. Nobody knew how to deal with such equations. Nash invented a novel iterative method — a procedure for making a series of educated guesses — for finding roots of equations, and combined it with a technique for smoothing to counteract the loss of derivatives.23
Newman described Nash as a “very poetic, different kind of thinker.”24 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.25 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.26
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.27 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.28 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.29
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.30
One night, possibly en route to this very talk, Nash was hurtling down the Merritt Parkway.31 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.32
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.33
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.”34 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.35