1956–57
Mathematical ideas originate in empirics. . . . But, once they are so conceived, the subject begins to live a peculiar life of its own and is better compared to a creative one, governed almost entirely by aesthetical motivations. . . . As a mathematical discipline travels, or after much “abstract” inbreeding, [it] is in danger of degeneration. . . . whenever this stage is reached, the only remedy seems to me to be the rejuvenating return to the source: the reinjection of more or less directly empirical ideas. — JOHN VON NEUMANN
THE INSTITUTE FOR ADVANCED STUDY, nestled on Princeton’s fringes on what had been a farm, was a scholar’s dream. It was bordered by woods and the Delaware-Raritan Canal, its lawns were immaculate, and one of its streets was Einstein Drive. It was also blessedly free of students. The atmosphere in the Fuld Hall common room resembled that of a venerable men’s club, with its newspaper racks and mingled scents of leather and pipe tobacco; its doors were never locked and its lights burned far into the night.
In 1956, the Institute’s permanent faculty were not many more than a dozen mathematicians and theoretical physicists.1 They were, however, outnumbered sixfold by a host of distinguished temporary visitors from around the globe, prompting Oppenheimer to call it “an intellectual hotel.”2 For young researchers, the Institute was a golden opportunity to escape the onerous demands of teaching and administration, and, indeed, the tasks of everyday life. Everything was provided the visitor: an apartment less than a few hundred yards from an office, an unending round of seminars, lectures, and, for those so inclined, parties where the booze was plentiful and where one could glimpse Lefschetz balancing a martini glass in an artificial hand, or witness a very drunk French mathematician displaying his mountaineering skills by rope-climbing up and over the fireplace mantel.3
Some found the idyllic setting, carefully designed to remove all impediments to creativity, vaguely disquieting. Paul Cohen, a mathematician at Stanford University, remarked, “It was such a great place that you had to stay at least two years. It took one year just to learn how to work under such ideal conditions.”4 By 1956, Einstein was dead, Gödel was no longer active, and von Neumann lay dying in Bethesda. Oppenheimer was still director, but much humbled by the McCarthyite inquisitions and increasingly isolated. As one mathematician said, “The Institute had become pure, very pure.”5 Cathleen Morawetz, later president of the American Mathematical Society, put it more bluntly: “The Institute was known to be about the dullest place you could find.”6
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By contrast, the Courant Institute of Mathematical Sciences at New York University was “the national capital of applied mathematical analysis,” as Fortune magazine was soon to inform its readers.7 Just a few years old and vibrant with energy, Courant occupied a nineteenth-century loft less than a block to the east of Washington Square in a neighborhood that, despite the university’s growing presence, was still dominated by small manufacturing concerns. Indeed, Courant initially shared the premises — with its fire escapes and creaky old-fashioned freight elevator — with a number of hat factories.8 Financing for the institute had come from the Atomic Energy Commission, which had been hunting for a home for its giant Univac 4 computer. At the time, this great mass of vacuum tubes, with its armed guard, occupied 25 Waverly Place.9
The institute was the creation of one of mathematics’ great entrepreneurs, Richard Courant, a German Jewish professor of mathematics who had been driven out of Göttingen in the mid-1930s by the Nazis.10 Short, rotund, autocratic, and irrepressible, Courant was famous for his fascination with the rich and powerful, his penchant for falling in love with his female “assistants,” and his unerring eye for young mathematical talent. When Courant arrived in 1937, New York University had no mathematics worth speaking of. Undaunted, Courant immediately set about raising funds. His own stellar reputation, the anti-Semitism of the American educational establishment, and New York’s “deep reservoir of talent,” enabled him to attract brilliant students, most of them New York City Jews who were shut out of the Harvards and Princetons.11 The advent of World War II brought more money and more students, and by the mid-1950s, when the institute was formally founded, it was already rivaling more established mathematical centers like Princeton and Cambridge.12 Its young stars included Peter Lax and his wife, Anneli, Cathleen Synge Morawetz, Jürgen Moser, and Louis Nirenberg, and among its stellar visitors were Lars Hörmander, a future Fields medalist, and Shlomo Sternberg, who would soon move to Harvard.
The Courant Institute was practically on Nash’s doorstep and, given its lively atmosphere, it was not surprising that Nash was soon spending at least as much time there as at the Institute for Advanced Study. At first Nash would stop by for an hour or two before driving down to Princeton, but he soon found himself staying the whole day.13 He never came too early, for he liked to sleep late after working into the wee hours at the university library.14 But he was almost always there for teatime in the lounge on the building’s penultimate floor.15
As for the Courant crowd, a friendly, open group with little taste for the competitiveness of MIT or the snobbery of the Institute, it was happy to have him. Tilla Weinstein, a mathematician at Rutgers, who recalled that Nash liked to pace around on one of the building’s fire escapes, said, “He was just a delight. There was a wit and humor about him that was thoroughly unstandard. There was a wonderful playful quality, a lightness.”16 Cathleen Morawetz, the daughter of John Synge, Nash’s professor at Carnegie, assumed Nash was just another postdoctoral fellow and found him “very charming,” “an attractive fellow,” “a lively conversationalist.”17 Hörmander recalled his first impressions: “He wore a serious expression. Then he’d break out into a sudden smile. He was an enthusiast.”18 Peter Lax, who had spent the war at Los Alamos, was interested in Nash’s research and “his own way of looking at things.”19
At first, Nash seemed more interested in the political cataclysms of that fall — Nasser nationalized the Suez Canal, prompting an invasion by England, France, and Israel, the Russians crushed the Hungarian uprising, and Eisenhower and Stevenson were again battling for the presidency — than in pursuing mathematical conversations. “He’d be in the common room,” one Courant visitor recalled, “talking and talking of his views of the political situation. From the afternoon teas, I remember him as voicing very strong opinions on the Suez crisis, which was going on at that time.”20 Another mathematician remembered a similar conversation in the institute dining room: “When the British and their allies were trying to grab Suez, and Eisenhower had not made his position unmistakably clear (if he ever did), one day at lunch Nash started in on Suez. Of course, Nasser wasn’t black, but he was dark enough for Nash. ‘What you have to do with these people is to take a firm hand, and then once they realize you mean it . . .’ ”21
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The leading lights at Courant were very much at the forefront of rapid progress, stimulated by World War II, in certain kinds of differential equations that serve as mathematical models for an immense variety of physical phenomena involving some sort of change.22 By the mid-fifties, as Fortune noted, mathematicians knew relatively simple routines for solving ordinary differential equations using computers. But there were no straightforward methods for solving most nonlinear partial differential equations that crop up when large or abrupt changes occur — such as equations that describe the aerodynamic shock waves produced when a jet accelerates past the speed of sound. In his 1958 obituary of von Neumann, who did important work in this field in the thirties, Stanislaw Ulam called such systems of equations “baffling analytically,” saying that they “defy even qualitative insights by present methods.”23 As Nash was to write that same year, “The open problems in the area of non-linear partial differential equations are very relevant to applied mathematics and science as a whole, perhaps more so than the open problems in any other area of mathematics, and this field seems poised for rapid development. It seems clear however that fresh methods must be employed.”24
Nash, partly because of his contact with Wiener and perhaps his earlier interaction with Weinstein at Carnegie, was already interested in the problem of turbulence.25 Turbulence refers to the flow of gas or liquid over any uneven surface, like water rushing into a bay, heat or electrical charges traveling through metal, oil escaping from an underground pool, or clouds skimming over an air mass. It should be possible to model such motion mathematically. But it turns out to be extremely difficult. As Nash wrote:
Little is known about the existence, uniqueness and smoothness of solutions of the general equations of flow for a viscous, compressible, and heat conducting fluid. These are a non-linear parabolic system of equations. An interest in these questions led us to undertake this work. It became clear that nothing could be done about the continuum description of general fluid flow without the ability to handle non-linear parabolic equations and that this in turn required an a priori estimate of continuity.26
It was Louis Nirenberg, a short, myopic, and sweet-natured young protégé of Courant’s, who handed Nash a major unsolved problem in the then fairly new field of nonlinear theory.27 Nirenberg, also in his twenties, and already a formidable analyst, found Nash a bit strange. “He’d often seemed to have an internal smile, as if he was thinking of a private joke, as if he was laughing at a private joke that he never [told anyone about].”28 But he was extremely impressed with the technique Nash had invented for solving his embedding theorem and sensed that Nash might be the man to crack an extremely difficult outstanding problem that had been open since the late 1930s.
He recalled:
I worked in partial differential equations. I also worked in geometry. The problem had to do with certain kinds of inequalities associated with elliptic partial differential equations. The problem had been around in the field for some time and a number of people had worked on it. Someone had obtained such estimates much earlier, in the 1930s in two dimensions. But the problem was open for [almost] thirty years in higher dimensions.29
Nash began working on the problem almost as soon as Nirenberg suggested it, although he knocked on doors until he was satisfied that the problem was as important as Nirenberg claimed.30 Lax, who was one of those he consulted, commented recently: “In physics everybody knows the most important problems. They are well defined. Not so in mathematics. People are more introspective. For Nash, though, it had to be important in the opinion of others.”31
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
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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