A Beautiful Mind

by Sylvia Nasar

  Nash started coming to Nirenberg’s office to discuss his progress. But it was weeks before Nirenberg got any real sense that Nash was getting anywhere. “We would meet often. Nash would say, ’I seem to need such and such an inequality. I think it’s true that…’” Very often, Nash’s speculations were far off the mark. “He was sort of groping. He gave that impression. I wasn’t very confident he was going to get through.”32

  Nirenberg sent Nash around to talk to Lars Hörmander, a tall, steely Swede who was already one of the top scholars in the field. Precise, careful, and immensely knowledgeable, Hörmander knew Nash by reputation but reacted even more skeptically than Nirenberg. “Nash had learned from Nirenberg the importance of extending the Holder estimates known for second-order elliptic equations with two variables and irregular coefficients to higher dimensions,” Hörmander recalled in 1997.33 “He came to see me several times, ’What did I think of such and such an inequality?’ At first, his conjectures were obviously false. [They were] easy to disprove by known facts on constant coefficient operators. He was rather inexperienced in these matters. Nash did things from scratch without using standard techniques. He was always trying to extract problems… [from conversations with others]. He had not the patience to [study them].”

  Nash continued to grope, but with more success. “After a couple more times,” said Hörmander, “he’d come up with things that were not so obviously wrong.”34

  By the spring, Nash was able to obtain basic existence, uniqueness, and continuity theorems once again using novel methods of his own invention. He had a theory that difficult problems couldn’t be attacked frontally. He approached the problem in an ingeniously roundabout manner, first transforming the nonlinear equations into linear equations and then attacking these by nonlinear means. “It was a stroke of genius,” said Lax, who followed the progress of Nash’s research closely. “I’ve never seen that done. I’ve always kept it in mind, thinking, maybe it will work in another circumstance.”35

  Nash’s new result got far more immediate attention than his embedding theorem. It convinced Nirenberg, too, that Nash was a genius.36 Hörmander’s mentor at the University of Lund, Lars Gårding, a world-class specialist in partial differential equations, immediately declared, “You have to be a genius to do that.”37

  Courant made Nash a handsome job offer.38 Nash’s reaction was a curious one. Cathleen Synge Morawetz recalled a long conversation with Nash, who couldn’t make up his mind whether to accept the offer or to go back to MIT. “He said he opted to go to MIT because of the tax advantage” of living in Massachusetts as opposed to New York.39

  Despite these successes, Nash was to look back on the year as one of cruel disappointment. In late spring, Nash discovered that a then-obscure young Italian, Ennio De Giorgi, had proven his continuity theorem a few months earlier. Paul Garabedian, a Stanford mathematician, was a naval attaché in London. It was an Office of Naval Research sinecure.40 In January 1957, Garabedian took a long car trip around Europe and looked up young mathematicians. “I saw some oldtimers in Rome,” he recalled. “It was a scene. You’d talk mathematics for half an hour. Then you’d have lunch for three hours. Then a siesta. Then dinner. Nobody mentioned De Giorgi.” But in Naples, someone did, and Garabedian looked De Giorgi up on his way back through Rome. “He was this bedraggled, skinny little starved-looking guy. But I found out he’d written this paper.”

  De Giorgi, who died in 1996, came from a very poor family in Lecce in southern Italy.41 Later he would become an idol to the younger generation. He had no life outside mathematics, no family of his own or other close relationships, and, even later, literally lived in his office. Despite occupying the most prestigious mathematical chair in Italy, he lived a life of ascetic poverty, completely devoted to his research, teaching, and, as time went on, a growing preoccupation with mysticism that led him to attempt to prove the existence of God through mathematics.

  De Giorgi’s paper had been published in the most obscure journal imaginable, the proceedings of a regional academy of sciences. Garabedian proceeded to report De Giorgi’s results in the Office of Naval Research’s European newsletter.

  Nash’s own account, written after he had won the Nobel for his work in game theory, conveys the acute disappointment he felt:

  I ran into some bad luck since, without my being sufficiently informed on what other people were doing in the area, it happened that I was working in parallel with Ennio De Giorgi of Pisa, Italy. And De Giorgi was first actually to achieve the ascent of the summit (of the figuratively described problem) at least for the particularly interesting case of “elliptic equations.”42

  Nash’s view was perhaps overly subjective. Mathematics is not an intramural sport, and as important as being first is, how one gets to one’s destination is often as important as, if not more important than, the actual target. Nash’s work was almost universally regarded as a major breakthrough. But this was not how Nash saw it. Gian-Carlo Rota, a graduate student at Yale who spent that year at Courant, recalled in 1994: “When Nash learned about De Giorgi he was quite shocked. Some people even thought he cracked up because of that.”43 When De Giorgi came to Courant that summer and he and Nash met, Lax said later, “It was like Stanley meeting Livingstone.”44

  Nash left the Institute for Advanced Study on a fractious note. In early July he apparently had a serious argument with Oppenheimer about quantum theory — serious enough, at any rate, to warrant a lengthy letter of apology from Nash to Oppenheimer written around July 10, 1957: “First, please let me apologize for my manner of speaking when we discussed quantum theory recently. This manner is unjustifiably aggressive.”45 After calling his own behavior unjustified, Nash nonetheless immediately justified it by calling “most physicists (also some mathematicians who have studied Quantum Theory) … quite too dogmatic in their attitudes,” complaining of their tendency to treat “anyone with any sort of questioning attitude or a belief in ’hidden parameters’... as stupid or at best a quite ignorant person.”

  Nash’s letter to Oppenheimer shows that before leaving New York, Nash had begun to think seriously of attempting to address Einstein’s famous critique of Heisenberg’s uncertainty principle:

  Now I am making a concentrated study of Heisenberg’s original 1925 paper … This strikes me as a beautiful work and I am amazed at the great difference between expositions of “matrix mechanics,” a difference, which from my viewpoint, seems definitely in favor of the original.46

  “I embarked on [a project] to revise quantum theory,” Nash said in his 1996 Madrid lecture. “It was not a priori absurd for a non-physicist. Einstein had criticized the indeterminacy of the quantum mechanics of Heisenberg.”47

  He apparently had devoted what little time he spent at the Institute for Advanced Study that year talking with physicists and mathematicians about quantum theory. Whose brains he was picking is not clear: Freeman Dyson, Hans Lewy, and Abraham Pais were in residence at least one of the terms.48 Nash’s letter of apology to Oppenheimer provides the only record of what he was thinking at the time. Nash made his own agenda quite clear. “To me one of the best things about the Heisenberg paper is its restriction to the observable quantities,” he wrote, adding that “I want to find a different and more satisfying under-picture of a non-observable reality.”49

  It was this attempt that Nash would blame, decades later in a lecture to psychiatrists, for triggering his mental illness — calling his attempt to resolve the contradictions in quantum theory, on which he embarked in the summer of 1957, “possibly overreaching and psychologically destabilizing.”50

  31

  The Bomb Factory

  What’s the matter with being a loner and innovative? Isn’t that fine? But the [lone genius] has the same wishes as other people. If he were back in high school doing science projects, fine. But if he s too isolated and he’s disappointed in something big, it’s frightening, and fright can precipitate

  — PAUL HOWARD, McLean Hospital


  JÜRGEN MOSER had joined the M I T faculty in the fall of 1957 and was living with his wife, Gertrude, and his stepson, Richy, in a tiny rented house to the west of Boston in Needham near Wellesley College. Needham was then more exurb than suburb, still predominantly rural, a lovely place for walking, boating, and stargazing, all of which Moser, a nature lover, was fond. That October and November, Moser would go outside every evening at dusk with eleven-year-old Richy, climb a great dirt mound behind their house, and wait for Sputnik — a tiny silvery dot reflecting the sun’s last rays — to pass slowly over Boston.1 Having calculated the satellite’s precise orbit, Moser always knew when it would appear on the horizon.

  Very often, he would still be thinking of the afternoon’s conversation with Nash. Nash drove out to Needham often. Despite their very different temperaments, Nash and Moser had great respect for each other. Moser, who thought Nash’s implicit function theorem might be generalized and applied to celestial mechanics, was eager to learn more of Nash’s thinking. Nash, in turn, was interested in Moser’s ideas about nonlinear equations. Richard Emery recalled in 1996: “I remember Nash being very much a part of our life. He used to come to the house and talk with Jürgen. They would walk and talk together and spend time in the study. The intensity of it was unimaginable. There could be no interruptions. An interruption was an absolute sin, a violation most serious. It was met with real wrath. When Jürgen and Nash met, it was very intense. I always had to be quiet.”2

  Returning to Cambridge in late summer, Nash and Alicia found an apartment with some difficulty.3 They each paid half the rent, for they had decided not to pool their funds.4 Alicia got a job as a physics researcher at Technical Operations, one of the small high-tech companies that were springing up along Route 128.5 She also enrolled in a course on quantum theory taught by J. C . Slater.

  They quickly settled into the pleasant private and social rituals of a newly married academic couple. Alicia almost never cooked. She would meet Nash on the campus after work, they would eat out with one or more of Nash’s mathematics friends, and often spend the evening at a lecture, concert, or some social gathering.6 Alicia made sure that they were always surrounded by amusing people, sometimes Nash’s old graduate-student friends, including Mattuck and Bricker, sometimes Emma Duchane and whomever Emma happened to be dating, and, increasingly, other young couples like themselves, including the Mosers, the Minskys, Hartley Rogers and his wife, Adrienne, and Gian-Carlo Rota and his wife, Terry.

  When they were with other people, Nash talked to the mathematicians, Alicia to the wives or Emma. Yet her attention was always focused on Nash: what he was saying, how he looked, how others reacted to him. He too, seemed always aware of her, even when he appeared to be ignoring her. That he wasn’t especially nice to her, or generous, mattered less than that he was interesting and made things happen.

  Their friends accepted Nash’s new status as a married man with more or less good grace. Some found Alicia “ambitious, strong-willed,” others quite the opposite. Rogers recalled in 1996 that “Alicia subordinated herself to John. She wasn’t there to compete with him. She was totally dedicated to his support.”7 Some of their acquaintances found their relationship oddly cool, but others came away with the impression that marriage suited Nash well and that Alicia was having a good effect on him. “Somehow, he was relating a little better,” Rogers recalled. Zipporah Levinson agreed: “John was awkward. Alicia made him behave.”8 Photographs of Alicia taken in those months show a radiant young woman. It was, as Alicia would say many years later, “a very nice time of my life.”9

  Nash continued to work on the problem he had solved the previous year at Courant. There were some small gaps in the proof, and the paper Nash had begun to write, laying out a full account of what he had done, was in very rough shape.10 “It was,” a colleague said in 1996, “as if he were a composer and could hear the music, but he didn’t know how to write it down or exactly how to orchestrate it.”11 As it turned out, it would take most of the year, and a collective effort, before the final product — which some mathematicians regard as Nash’s most important work — was finally ready to be submitted to a journal.

  To complete, it, Nash came as close as he ever had or would to an active collaboration with other mathematicians. “It was like building the atom bomb,” recalled Lennart Carleson, a young professor from the University of Uppsala who was visiting MIT that term. “This was the beginning of nonlinear theory. It was very difficult.”12 Nash knocked on doors, asked questions, speculated out loud, fished for ideas, and at the end of the day, got a dozen or so mathematicians around Cambridge interested enough in his problem to drop their own research long enough to solve little pieces of his puzzle. “It was a kind of factory,” Carleson, who contributed a neat little theorem on entropy to Nash’s paper, said. “He wouldn’t tell us what he was after, his grand design. It was amusing to watch how he got all these great egos to cooperate.”13

  Besides Moser and Carleson, Nash also turned to Eli Stein, now a professor of mathematics at Princeton University but then an MIT instructor. “He wasn’t interested in what I was doing,” recalled Stein. “He’d say, You’re an analyst. You ought to be interested in this.’ ”14

  Stein was intrigued by Nash’s enthusiasm and his constant supply of ideas. He said, “We were like Yankees fans getting together and talking about great games and great players. It was very emotional. Nash knew exactly what he wanted to do. With his great intuition, he saw that certain things ought to be true. He’d come into my office and say, This inequality must be true.’ His arguments were plausible but he didn’t have proofs for the individual lemmas — building blocks for the main proof.”15 He challenged Stein to prove the lemmas.

  “You don’t accept arguments based on plausibility,” said Stein in 1995. “If you build an edifice based on one plausible proposition after another, the whole thing is liable to collapse after a few steps. But somehow he knew it wouldn’t. And it didn’t.”16

  Nash’s thirtieth year was thus looking very bright. He had scored a major success. He was adulated and lionized as never before.17Fortune magazine was about to feature him as one of the brightest young stars of mathematics in an upcoming series on the “New Math.”18 And he had returned to Cambridge as a married man with a beautiful and adoring young wife. Yet his good fortune seemed at times only to highlight the gap between his ambitions and what he had achieved. If anything, he felt more frustrated and dissatisfied than ever. He had hoped for an appointment at Harvard or Princeton.19 As it was, he was not yet a full professor at MIT, nor did he have tenure. He had expected that his latest result, along with the offer from Courant, would convince the department to award him both that winter.20 Getting these things after only five years would be unusual, but Nash felt that he deserved nothing less.21 But Martin had already made it clear to Nash that he was unwilling to put him up for promotion so soon. Nash’s candidacy was controversial, Martin had told him, just as his initial appointment had been.22 A number of people in the department felt he was a poor teacher and an even worse colleague. Martin felt Nash’s case would be stronger once the full version of the parabolic equations paper appeared in print. Nash, however, was furious.

  Nash continued to brood over the De Giorgi fiasco. The real blow of discovering that De Giorgi had beaten him to the punch was to him not just having to share the credit for his monumental discovery, but his strong belief that the sudden appearance of a coinventor would rob him of the thing he most coveted: a Fields Medal.

  Forty years later, after winning a Nobel, Nash referred in his autobiographical essay, in his typically elliptical fashion, to his dashed hopes:

  It seems conceivable that if either De Giorgi or Nash had failed in the attack on this problem (or a priori estimates of Holder continuity) then that the lone climber reaching the peak would have been recognized with the mathematics’ Fields medal (which has traditionally been restricted to persons less than 40 years old).23

  The next Fields Meda
l would be awarded in August 1958, and as everyone knew, the deliberations had long been under way.

  To understand how deep the disappointment was, one must know that the Fields Medal is the Nobel Prize of mathematics, the ultimate distinction that a mathematician can be granted by his peers, the trophy of trophies.24 There is no Nobel in mathematics, and mathematical discoveries, no matter how vital to Nobel disciplines such as physics or economics, do not in themselves qualify for a Nobel. The Fields is, if anything, rarer than the Nobel. In the fifties and early sixties, it was awarded once every four years and usually to just two recipients at a time. Nobels, by contrast, are awarded annually, with as many as three winners sharing each prize. Tradition demands that recipients of the Fields be under forty years of age, a practice designed to honor the spirit of the prize charter, which stipulates that the purpose of the honor is “to encourage young mathematicians” and “future work.”25 The incentive, incidentally, is of an intangible variety, as the cash involved, in contrast to the Nobel, is negligible, a few hundred dollars. Yet since the Fields is an instant ticket in midcareer to endowed chairs at top universities, ample research funds, and star salaries, this seeming disadvantage is more apparent than real.

  The prize is administered by the International Mathematical Union, the same organization that organizes the quadrennial world mathematical congresses, and the selection of Fields medalists is, as one recent president of the organization put it, “one of the most important tasks, one of the most taxing responsibilities.”26 Like the Nobel deliberations, the Fields selection process is shrouded in greatest secrecy.