A Beautiful Mind Part 6

One point which should particularly be noticed is the linearity hypothesis in Nash's theorem. This is a direct application of the von Neumann-Morgenstern theory of numerical utility; the claim that it is possible to measure the relative desirability of different possible outcomes by a real-valued function which is linear with respect to probabilities... . My own belief is that this is quite reasonable as a normative theory, but that it may not be realistic as a descriptive theory.

Evidently, Nash's theory was not a finished answer to the problem of understanding compet.i.tive situations. In fact, it should be emphasized that no simple mathematical theory can provide a complete answer, since the psychology of the players and the mechanism of their interaction may be crucial to a more precise understanding.20

Nevertheless, decades later, economists, differing with Milnor, came to regard this "failure" of an experiment as a very worthwhile one. Casual as the experiment was in one sense, it became a model for a new method of economic research, one that had never before been tried in the two hundred years since Adam Smith dreamed up the Invisible Hand. The feeling was that even if the experiments weren't sophisticated enough to show how people's brains work, watching the way people played games could draw researchers' attention to elements of interaction - such as signaling or implicit threats - that couldn't be derived axiomatically.21 By the time the experiment was run the relations.h.i.+p between Nash and Milnor had become strained, and Milnor had moved out of the Georgina Avenue apartment.

Milnor says now that Nash made a s.e.xual overture toward him. "I was very naive and very h.o.m.ophobic," said Milnor. "It wasn't the kind of thing people talked about then."22 But what Nash felt toward Milnor may have been something close to love. A dozen years later, in a letter to Milnor, Nash wrote: "Concerning love, I know a conjugation: amo, amas, amat, amamus, amatis, amant. Perhaps amas is also the imperative, love! Perhaps one must be very masculine to use the imperative." But what Nash felt toward Milnor may have been something close to love. A dozen years later, in a letter to Milnor, Nash wrote: "Concerning love, I know a conjugation: amo, amas, amat, amamus, amatis, amant. Perhaps amas is also the imperative, love! Perhaps one must be very masculine to use the imperative."23

CHAPTER 19

Reds Spring 1953 Spring 1953

Now, the thing I think would interest the committee very greatly, if you could possibly explain to them ... Doctor... how you can account for what would seem to be an abnormally large percentage of communists at MIT?

-ROBERT L. K L. KUNZIG, Counsel HUAC, April 22, 1953

THE C COLD W WAR promised to be the sugar daddy of the MIT mathematics department, but McCarthy ism - which blamed the setbacks in that war on sinister conspiracies and domestic subversion - threatened to devour it. promised to be the sugar daddy of the MIT mathematics department, but McCarthy ism - which blamed the setbacks in that war on sinister conspiracies and domestic subversion - threatened to devour it.

While Nash and his graduate student friends were shooting each other down and playing games in the mathematics common room, FBI investigators were fanning out around Cambridge, rifling through trash cans, placing individuals under surveillance, and questioning neighbors, colleagues, students, and even children.1 Their targets, as Nash and everyone else at MIT would learn in early 1953, included the chairman and the deputy chairman of the MIT mathematics department, as well as a tenured full professor of mathematics, Dirk Struik - all three one-time members, indeed, leading members, of the Cambridge cell of the Communist Party. All three were subpoenaed by the House Un-American Activities Committee. Their targets, as Nash and everyone else at MIT would learn in early 1953, included the chairman and the deputy chairman of the MIT mathematics department, as well as a tenured full professor of mathematics, Dirk Struik - all three one-time members, indeed, leading members, of the Cambridge cell of the Communist Party. All three were subpoenaed by the House Un-American Activities Committee.2 It was a state of siege and everyone in the mathematics department felt the threat. It was a state of siege and everyone in the mathematics department felt the threat.

At the time, Nash was no doubt far more preoccupied with the draft - not to mention growing complications of his personal life - than with the possible repercussions for himself of the persecution of his benefactors. Nevertheless, the whole episode was a warning that the world he and other mathematicians inhabited was an extremely fragile one. A congressional committee could destroy your career, just as your draft board could send you halfway around the world.

The whole thing had begun as a farce.3 McCarthy's original list of communists, announced in February 1950, was studded with academics, including the father of Nash's friend Lloyd Shapley, Harvard astronomy professor Harlow Shapley, whom McCarthy incorrectly identified to reporters as "Howard s.h.i.+pley, astrologer." But as the red hunt gathered momentum, the entire scientific community McCarthy's original list of communists, announced in February 1950, was studded with academics, including the father of Nash's friend Lloyd Shapley, Harvard astronomy professor Harlow Shapley, whom McCarthy incorrectly identified to reporters as "Howard s.h.i.+pley, astrologer." But as the red hunt gathered momentum, the entire scientific community felt vulnerable. Princeton's Solomon Lefschetz would be identified as a possible communist sympathizer by an investigative body. felt vulnerable. Princeton's Solomon Lefschetz would be identified as a possible communist sympathizer by an investigative body.4 Within a year, Robert Oppenheimer, head of the Manhattan Project, one of the most revered scientists in America and the director of the Inst.i.tute for Advanced Study, would be humiliated by the McCarthyites. Within a year, Robert Oppenheimer, head of the Manhattan Project, one of the most revered scientists in America and the director of the Inst.i.tute for Advanced Study, would be humiliated by the McCarthyites.

When the subpoenas were issued, n.o.body knew how MIT would handle the matter. Other universities had responded with immediate firings and suspensions.5 "McCarthyism was a big threat to these schools," Zipporah Levinson, Norman Levinson's widow, recalled. "During the war the government had started pouring money into them. The threat was that the research money would dry up. It was a bread-and-b.u.t.ter issue." "McCarthyism was a big threat to these schools," Zipporah Levinson, Norman Levinson's widow, recalled. "During the war the government had started pouring money into them. The threat was that the research money would dry up. It was a bread-and-b.u.t.ter issue."6 Martin and Levinson were certain that they were about to lose their jobs and wind up blacklisted for good, like so many others. Levinson talked about becoming a plumber and specializing in the repair of furnaces. The investigators had their eye on the three Browder boys - sons of former Communist Party head Earl Browder, who had all studied or were studying mathematics at MIT and were scholars.h.i.+p recipients, as well. Martin and Levinson were certain that they were about to lose their jobs and wind up blacklisted for good, like so many others. Levinson talked about becoming a plumber and specializing in the repair of furnaces. The investigators had their eye on the three Browder boys - sons of former Communist Party head Earl Browder, who had all studied or were studying mathematics at MIT and were scholars.h.i.+p recipients, as well.7 "MIT was turned topsy-turvy," Mrs. Levinson recalled. "The faculty debated and debated how to prove that MIT was patriotic. There was strong pressure to name names."8 As it turned out, Karl Compton, the president of the university and an outspoken liberal who was a supporter of the Chinese revolution and a critic of Chiang Kai-shek, may have felt that he himself would soon be subpoenaed. He hired a white-shoe Boston law firm, Choate, Hall & Steward, to defend Martin, Levinson, and the others for a minimal fee. As it turned out, Karl Compton, the president of the university and an outspoken liberal who was a supporter of the Chinese revolution and a critic of Chiang Kai-shek, may have felt that he himself would soon be subpoenaed. He hired a white-shoe Boston law firm, Choate, Hall & Steward, to defend Martin, Levinson, and the others for a minimal fee.9 By April, when Martin and Levinson were forced to testify, By April, when Martin and Levinson were forced to testify, The Tech The Tech was running daily stories and anti-McCarthy sentiment was running high on campus. was running daily stories and anti-McCarthy sentiment was running high on campus.10 There is no evidence that the FBI ever questioned Nash or any other students or faculty in the department, or asked for depositions, in an effort to establish a link between Levinson's and Martin's Communist Party members.h.i.+p and cla.s.sified defense research - a link that probably never existed, given that both left the party soon after the end of the war. The graduate students and junior faculty in the department stood on the sidelines and watched lives and careers ruined and homes, even car insurance, lost. "By that time, young people had prospects, jobs, optimism," Mrs. Levinson recalled. "The younger people - Nash's group - didn't want to be too friendly. They were scared. They distanced themselves."11 Martin and several others named their former a.s.sociates. Norman Levinson refused to name anyone who had not been previously named. "Ted and Izzy Amadur hemmed and hawed. Norman knew that Ted Martin and Izzy would cooperate. They spilled all the names. Norman said he'd talk freely about the party but that he wouldn't name names. The lawyer told Norman, no you don't have to say any names. He'd cooperate, but he wouldn't give any names."12 Martin gave a pathetic, frightened performance. Levinson's testimony, by contrast, demonstrated the qualities of intellect and character that made him such a force in the mathematics community. In a series of forceful and eloquent answers to direct questioning, he managed at one and the same time to defend the youthful idealism that led Martin gave a pathetic, frightened performance. Levinson's testimony, by contrast, demonstrated the qualities of intellect and character that made him such a force in the mathematics community. In a series of forceful and eloquent answers to direct questioning, he managed at one and the same time to defend the youthful idealism that led him into the party, attack the intellectual poverty of communism, and, implicitly, call into question the committee's a.s.sumption that communism was a threat to the nation. He spoke out against the hounding of former party members and asked the committee to take a stand against the blacklisting of Browder's oldest son, Felix, who had finished his Ph.D. and was unable to obtain an academic post. him into the party, attack the intellectual poverty of communism, and, implicitly, call into question the committee's a.s.sumption that communism was a threat to the nation. He spoke out against the hounding of former party members and asked the committee to take a stand against the blacklisting of Browder's oldest son, Felix, who had finished his Ph.D. and was unable to obtain an academic post.

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 hara.s.sment 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 hara.s.sed by the FBI. 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 hara.s.sed 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. 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. 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

CHAPTER 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. C 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: 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 a.s.sistant Professors.h.i.+p (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 d.a.m.ned 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 s.p.a.ce, 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 a.n.a.log of a single ordinary differential equation it would be a d.a.m.ned 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 a.s.s 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 h.e.l.l, 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. 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. 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 cla.s.s one day with one shoelace tied and the other untied. 'Did you know your right shoelace is untied?' we asked. 'Oh, my G.o.d,' 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 t.i.t-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 a.s.sistants 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. 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. P.M. in the Common Room." Thursday at 2:00 in the Common Room." Thursday at 2:00 P.M. P.M. was the hour that Ambrose taught his graduate course in a.n.a.lysis. was the hour that Ambrose taught his graduate course in a.n.a.lysis.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. 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 "f.u.c.k 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. 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. 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, a.n.a.lyst, algebraist, geometer, topologist, or mathematical physicist. But he zeroed in on areas in these fields where essentially n.o.body had achieved anything. The thing was to find an interesting question that he could say something about. a.n.a.lyst, algebraist, geometer, topologist, or mathematical physicist. But he zeroed in on areas in these fields where essentially n.o.body 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." 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 s.p.a.ce?"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. 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 s.p.a.ces. But in the early 1950s interest s.h.i.+fted to manifolds partly because of the large role that distorted s.p.a.ce and time relations.h.i.+ps had in Einstein's theory of relativity. 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 s.p.a.ces. But in the early 1950s interest s.h.i.+fted to manifolds partly because of the large role that distorted s.p.a.ce and time relations.h.i.+ps had in Einstein's theory of relativity.

Nash's own description of the embedding problem in his 1995 n.o.bel autobiography hints at the reason he wished to make sure that solving the problem would be worth the effort: "This problem, although cla.s.sical, was not much talked about as an outstanding 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 s.p.a.ce in some dimension. Take the surface of a balloon. You can't put it on a blackboard, which is a two-dimensional s.p.a.ce. But you can make it a subset of s.p.a.ces 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 s.p.a.ce, 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 s.p.a.ce of four dimensions and the thing no longer intersects itself. Like a ball embedded in three-s.p.a.ce, a Klein bottle in four-s.p.a.ce becomes a perfectly well-behaved manifold. 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 s.p.a.ce of four dimensions and the thing no longer intersects itself. Like a ball embedded in three-s.p.a.ce, a Klein bottle in four-s.p.a.ce 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 s.p.a.ce. He showed that you could fold the manifold like a silk handkerchief, without distorting it. n.o.body 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 Partial Differential Relations builds on Nash's work. He went on: 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 a.n.a.lysis in this century."20 It was also, one must add, a deliberate jab at then-fas.h.i.+onable 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 Jurgen 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 s.p.a.ce."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 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 a priori estimates lost derivatives. n.o.body 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. estimates lost derivatives. n.o.body 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 cla.s.sical 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. In this instance, Nash used differential calculus, not geometric pictures or algebraic manipulations, methods that were cla.s.sical 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 Jurgen Moser was to show how Nash's technique could be modified and applied to celestial mechanics - the movement of planets - especially for establis.h.i.+ng the stability of periodic orbits. Jurgen Moser was to show how Nash's technique could be modified and applied to celestial mechanics - the movement of planets - especially for establis.h.i.+ng 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 s.p.a.ce 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. 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. 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 s.p.a.ces provided that one would accept that the embedding would have only limited smoothness. And later, with "heavy a.n.a.lysis," 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. 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 cla.s.sical 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 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. 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." - 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. n.o.body 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

The editors of the Annals of Mathematics Annals of Mathematics hardly knew what to make of Nash's ma.n.u.script 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 n.a.z.ism 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 ma.n.u.scripts. hardly knew what to make of Nash's ma.n.u.script 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 n.a.z.ism 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 ma.n.u.scripts.36 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."37 Nash's ma.n.u.script presented Federer with a sensationally complicated puzzle and he attacked the task with relish. Nash's ma.n.u.script 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."38 Armand Borel, who was a visiting professor at Chicago when Nash gave a lecture on his embedding theorem, remembers the audience's shocked reaction. "n.o.body 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."39 (Characteristically, Nash's report to his parents merely said "talks went well.") (Characteristically, Nash's report to his parents merely said "talks went well.")40 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."41 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.42 A former MIT graduate student recalls a long and confused discussion between Ambrose, Singer, and Masatake Kuranis.h.i.+ (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. A former MIT graduate student recalls a long and confused discussion between Ambrose, Singer, and Masatake Kuranis.h.i.+ (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.43 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.44

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.45 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 G.o.d, Hopf was befuddled himself. He couldn't convey a complete picture. He was completely overwhelmed." 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 G.o.d, Hopf was befuddled himself. He couldn't convey a complete picture. He was completely overwhelmed."46 Several years later, Jurgen 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."47 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. 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.48 Levinson and Wiener were among Nash's strongest supporters. But others, like Warren Ambrose and George Whitehead, the distinguished topolo-gist, were opposed. Moore Instructors.h.i.+ps weren't meant to lead to tenure-track positions. More to the point, Nash had made plenty of enemies and few friends in Levinson and Wiener were among Nash's strongest supporters. But others, like Warren Ambrose and George Whitehead, the distinguished topolo-gist, were opposed. Moore Instructors.h.i.+ps 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. 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."49 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 pa.s.s judgment." 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 pa.s.s judgment."50 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.51

PART TWO

Separate Lives

CHAPTER 21

Singularity

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

- ARTHUR M MATTUCK, 1997

ALL 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 stars.h.i.+p Enterprise. 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. 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.

As these initial intimate connections multiplied and became ever-present elements in his consciousness, Nash's formerly solitary but coherent existence became at once richer and more discontinuous, separate and parallel existences that reflected an emerging adult but a fragmented and contradictory self. The others on whom he now depended occupied different compartments of his life and often, for long periods, knew nothing of one another or of the nature of the others' relation to Nash. Only Nash was in the know. His life resembled a play in which successive scenes are acted by only two characters. One character is in all of them while the second changes from scene to scene. The second character seems no longer to exist when he disappears from the boards.

More than a decade later, when he was already ill, Nash himself provided a metaphor for his life during the MIT years, a metaphor that he couched in his first language, the language of mathematics: B B squared -I- RTF = 0, a "very personal" equation Nash included in a 1968 postcard that begins, "Dear Mattuck, Thinking that you will understand this concept better than most I wish to explain ..." The equation represents a three-dimensional hypers.p.a.ce, which has a singularity at the origin, in four-dimensional s.p.a.ce. Nash is the singularity, the special point, and the other variables are people who affected him - in this instance, men with whom he had friends.h.i.+ps or relations.h.i.+ps. squared -I- RTF = 0, a "very personal" equation Nash included in a 1968 postcard that begins, "Dear Mattuck, Thinking that you will understand this concept better than most I wish to explain ..." The equation represents a three-dimensional hypers.p.a.ce, which has a singularity at the origin, in four-dimensional s.p.a.ce. Nash is the singularity, the special point, and the other variables are people who affected him - in this instance, men with whom he had friends.h.i.+ps or relations.h.i.+ps.1 Inevitably, the accretion of significant relations.h.i.+ps with others brings with it demands for integration - the necessity of having to choose. Nash had little desire to choose one emotional connection over another. By not choosing, he could avoid, or at least minimize, both dependence and demands. To satisfy his own emotional needs for connectedness meant he inevitably made others look to him to satisfy theirs. Yet while he was preoccupied with the effect of others on him, he mostly ignored - indeed, seemed unable to grasp - his effect on others. He had in fact no more sense of "the Other" than does a very young child. He wished the others to be satisfied with his genius -"I thought I was such a great mathematician," he was to say ruefully, looking back at this period - and, of course, to some extent they were satisfied. But when people inevitably wanted or needed more he found the strains unbearable.

CHAPTER 22

A Special Friends.h.i.+p Santa Monica, Summer 1952 Santa Monica, Summer 1952

Away from contact with a few special sorts of individuals I am lost, lost completely in the wilderness... so, so, so, it's been a hard life in many ways.

- JOHN F FORBES N NASH, JR., 1965

AFTER J JOHN N NASH LOST EVERYTHING - family, career, the ability to think about mathematics - he confided in a letter to his sister Martha that only three individuals in his life had ever brought him any real happiness: three "special sorts of individuals" with whom he had formed "special friends.h.i.+ps." - family, career, the ability to think about mathematics - he confided in a letter to his sister Martha that only three individuals in his life had ever brought him any real happiness: three "special sorts of individuals" with whom he had formed "special friends.h.i.+ps."1 Had Martha seen the Beatles' film A Hard Days Night? A Hard Days Night? "They seem very colorful and amusing," he wrote. "Of course they are much younger like the sort of person I've mentioned... . I feel often as if I were similar to the girls that love the Beatles so wildly since they seem so attractive and amusing to me." "They seem very colorful and amusing," he wrote. "Of course they are much younger like the sort of person I've mentioned... . I feel often as if I were similar to the girls that love the Beatles so wildly since they seem so attractive and amusing to me."2 Nash's first loves were one-sided and unrequited. "Nash was always forming intense friends.h.i.+ps with men that had a romantic quality," Donald Newman observed in 1996. "He was very adolescent, always with the boys."3 Some were inclined to see Nash's infatuations' as "experiments," or simple expressions of his immaturity - a view that he may well have held himself. "He played around with it because he liked to play around. He was very experimental, very try-outish," said Newman in 1996. "Mostly he just kissed." Some were inclined to see Nash's infatuations' as "experiments," or simple expressions of his immaturity - a view that he may well have held himself. "He played around with it because he liked to play around. He was very experimental, very try-outish," said Newman in 1996. "Mostly he just kissed."4 Newman, who liked to joke about his past and future female conquests,5 had firsthand knowledge because Nash was, for a time, infatuated with him - with predictable results. "He used to talk about how Donald looked all the time," Mrs. Newman said in 1996. had firsthand knowledge because Nash was, for a time, infatuated with him - with predictable results. "He used to talk about how Donald looked all the time," Mrs. Newman said in 1996.6 Newman recalled: "He tried fiddling around with me. I was driving my car when he came on to me." D.J. and Nash were cruising around in Newman's white Thunderbird when Nash kissed him on the mouth. DJ . just laughed it off. Newman recalled: "He tried fiddling around with me. I was driving my car when he came on to me." D.J. and Nash were cruising around in Newman's white Thunderbird when Nash kissed him on the mouth. DJ . just laughed it off.7 Nash's first experience of mutual attraction - "special friends.h.i.+ps," as he called them - occurred in Santa Monica.8 It was the very end of the summer of 1952, It was the very end of the summer of 1952, after Milnor had moved out and Martha had flown back home. The encounter must have been fleeting, coming in the last days of August, just before he was due to leave for Boston, and very furtive. But it was nonetheless decisive because for the first time he found not rejection but reciprocity. Thus it was the first real step out of his extreme emotional isolation and the world of relations.h.i.+ps that were purely imaginary, a first taste of intimacy, not entirely happy, no doubt, but suggestive of hitherto unsuspected satisfactions. after Milnor had moved out and Martha had flown back home. The encounter must have been fleeting, coming in the last days of August, just before he was due to leave for Boston, and very furtive. But it was nonetheless decisive because for the first time he found not rejection but reciprocity. Thus it was the first real step out of his extreme emotional isolation and the world of relations.h.i.+ps that were purely imaginary, a first taste of intimacy, not entirely happy, no doubt, but suggestive of hitherto unsuspected satisfactions.

The only traces of Nash's friends.h.i.+p with Ervin Thorson that remain are his description of him as a "special" friend in his 1965 letter and a series of elliptical references to "T" in letters in the late 1960s.9 Few if any of Nash's acquaintances met him; Martha recalled a friend of Nash's who once spent the night on the couch of their Georgina Avenue apartment, but not his name. Few if any of Nash's acquaintances met him; Martha recalled a friend of Nash's who once spent the night on the couch of their Georgina Avenue apartment, but not his name.10 Thorson, who died in 1992, was thirty years old in 1952.11 He was a native Californian of Scandinavian extraction. Nash described him to Martha as an aeros.p.a.ce engineer, but he may in fact have been an applied mathematician. He had been a meteorologist in the Army Air Corps during the war. Afterward, he earned a master's degree in mathematics at UCLA and went to Douglas Aircraft in 1951, just a few years after Douglas had spun off its R&D division to form the RAND Corporation. He was a native Californian of Scandinavian extraction. Nash described him to Martha as an aeros.p.a.ce engineer, but he may in fact have been an applied mathematician. He had been a meteorologist in the Army Air Corps during the war. Afterward, he earned a master's degree in mathematics at UCLA and went to Douglas Aircraft in 1951, just a few years after Douglas had spun off its R&D division to form the RAND Corporation.12 At that time, Douglas was mapping the future of interplanetary travel for the Pentagon, and Thorson, who eventually led a research team, was very likely involved in these efforts. At that time, Douglas was mapping the future of interplanetary travel for the Pentagon, and Thorson, who eventually led a research team, was very likely involved in these efforts.13 His great pa.s.sion, conceived twenty years before the United States launched His great pa.s.sion, conceived twenty years before the United States launched Viking, Viking, was the dream of exploring Mars, his sister Nelda Troutman recalled in 1997. was the dream of exploring Mars, his sister Nelda Troutman recalled in 1997.

Thorson was, his sister said, "very high strung, not a social person at all, very bright, knew a lot, very very academic."14 Nash could easily have met him - given the close ties between Douglas and RAND, which was also heavily involved in studies of s.p.a.ce exploration - at a talk or seminar, or perhaps even at one of the parties that John Williams, the head of RAND's mathematics department, gave. Nash could easily have met him - given the close ties between Douglas and RAND, which was also heavily involved in studies of s.p.a.ce exploration - at a talk or seminar, or perhaps even at one of the parties that John Williams, the head of RAND's mathematics department, gave.

If Thorson, who never married, was a h.o.m.os.e.xual, his surviving sister did not know it.15 With his family, at any rate, he was unusually closemouthed, not just about his work, which was highly cla.s.sified, but about all aspects of his personal life. With his family, at any rate, he was unusually closemouthed, not just about his work, which was highly cla.s.sified, but about all aspects of his personal life.16 Given the mounting pressure to root out h.o.m.os.e.xuals in the defense industry during the McCarthy era, Thorson would have had to practice great discretion in any case; his career at Douglas was to last for another fifteen years. Given the mounting pressure to root out h.o.m.os.e.xuals in the defense industry during the McCarthy era, Thorson would have had to practice great discretion in any case; his career at Douglas was to last for another fifteen years.17 When he abruptly resigned from Douglas in 1968, he apparently did so at the age of forty-seven because he feared dying. Several of his colleagues had recently died of heart attacks and Thorson, who had some sort of mild heart condition, decided he couldn't cope with the stress and overwork anymore. He moved back to his hometown of Pomona and became a virtual recluse except for an active involvement in the Lutheran church, living with his parents for the next twenty-five years until his death. When he abruptly resigned from Douglas in 1968, he apparently did so at the age of forty-seven because he feared dying. Several of his colleagues had recently died of heart attacks and Thorson, who had some sort of mild heart condition, decided he couldn't cope with the stress and overwork anymore. He moved back to his hometown of Pomona and became a virtual recluse except for an active involvement in the Lutheran church, living with his parents for the next twenty-five years until his death.

Whether Nash and Thorson saw each other again when Nash returned to Santa Monica for a third summer two years later or on one of his trips to Santa Monica during his illness in the early and mid-1960s is not known. But Nash continued to think of Thorson and to refer to him obliquely until at least 1968.