The seven-member prize committee for the 1958 Fields awards was headed by Heinz Hopf, the dapper, genial, cigar-smoking geometer from Zurich who showed so much interest in Nash’s embedding theorem, and included another prominent German mathematician, Kurt Friedrichs, formerly of Göttingen, and then at Courant.27 The deliberations got under way in late 1955 and were concluded early in 1958. (The medalists were informed, in strictest secrecy, in May 1958 and actually awarded their medals at the Edinburgh congress the following August.)
All prize deliberations involve elements of accident, the biggest one being the composition of the committee. As one mathematician who took part in a subsequent committee said, “People aren’t universalists. They’re horse trading.”28 In 1958, there were a total of thirty-six nominees, as Hopf was to say in his award ceremony speech, but the hot contenders numbered no more than five or six.29That year the deliberations were unusually contentious and the prizes, which ultimately went to René Thorn, a topologist, and Klaus F. Roth, a number theorist, were awarded on a four-three vote.’30 “There were lots of politics in that prize,” one person close to the deliberations said recently.31 Roth was a shoo-in; he had solved a fundamental problem in number theory that the most senior committee member, Carl Ludwig Siegel, had worked on early in his career. “It was a question of Thorn versus Nash,” said Moser, who heard reports of the deliberations from several of the participants.32 “Friedrichs fought very hard for Nash, but he didn’t succeed,” recalled Lax, who had been Friedrichs’s student and who heard Friedrichs’s account of the deliberations. “He was upset. As I look back, he should have insisted that a third prize be given.”33
Chances are that Nash did not make the final round. His work on partial differential equations, of which Friedrichs would have been aware, was not yet published or properly vetted. He was an outsider, which one person close to the deliberations thought “might have hurt him.” Moser said, “Nash was somebody who didn’t learn the stuff. He didn’t care. He wasn’t afraid of moving in and working on his own. That doesn’t get looked at so positively by other people.”34 Besides, there was no great urgency to recognize him at this juncture; he was just twenty-nine.
No one could know, of course, that 1958 would be Nash’s last chance. “By 1962, a Fields for Nash would have been out of the question,” Moser said recently. “It would never have happened. I’m sure nobody even thought about him anymore.”35
A measure of how badly Nash wanted to win the distinction conferred by such a prize is the extraordinary lengths to which he went to ensure that his paper would be eligible for the Bôcher Prize, the only award remotely comparable in terms of prestige to the Fields. The Bôcher is given by the American Mathematical Society only once every five years.36 It was due to be awarded in February 1959, which meant that the deliberations would take place in the latter part of 1958.
Nash submitted his manuscript to Acta Mathematica, the Swedish mathematics journal, in the spring of 1958.37 It was a natural choice, since Carleson was the editor and was convinced of the paper’s great importance. Nash let Carleson know he wanted the paper published as quickly as possible and urged Carleson to give it to a referee who could vet the paper in a minimum of time. Carleson gave the manuscript to Hörmander to referee. Hörmander spent two months studying it, verified all the theorems, and urged Carleson to get it into print as quickly as possible. But as soon as Carleson informed Nash of the formal acceptance, which was, in any case, largely a foregone conclusion, Nash withdrew his paper.
When the paper subsequently appeared in the fall issue of the American Journal of Mathematics, Hörmander concluded that Nash had always intended to publish the paper there, since the Bôcher restricted eligible papers to those published in American journals — or, worse, had submitted the paper to both journals, a clear-cut breach of professional ethics. “It turned out that Nash had just wanted to get a letter of acceptance from Acta to be able to get fast publication in the American Journal of Mathematics.”38 Hörmander was angry at what he felt was “very improper and most unusual.”39
It’s possible, though, that Nash had simply submitted the paper to Acta before learning that doing so would exclude it from consideration for the Bôcher, but that upon discovering this fact, he was willing to antagonize Carleson and Hörmander in order to preserve his eligibility. He may therefore not have used Acta quite so unscrupulously. Withdrawing the paper after it had been promised to Acta, and after it had been refereed, would have been unprofessional, but not as clear a violation of ethics as Hörmander’s scenario suggests. However, it still showed how very much winning a prize meant to Nash.
32
Secrets Summer 1958
It struck me that I knew everything: everything was revealed to me, all the secrets of the world were mine during those spacious hours.
— GERARD DE NERVAL
NASH TURNED THIRTY that June. For most people, thirty is simply the dividing line between youth and adulthood, but mathematicians consider their calling a young man’s game, so thirty signals something far more gloomy. Looking back at this time in his life, Nash would refer to a sudden onset of anxiety, “a fear” that the best years of his creative life were over.1
What an irony that mathematicians, who live so much more in their minds than most of humanity, should feel so much more trapped by their bodies! An ambitious young mathematician watches the calendar with a sense of trepidation and foreboding equal to or greater than that of any model, actor, or athlete. The Mathematician’s Apology by G. H. Hardy sets the standard for all laments of lost youth. Hardy wrote that he knew of no single piece of first-rate mathematics done by a mathematician over fifty.2 But the age anxiety is most intense, mathematicians say, as thirty draws near. “People say that for better or worse you will probably do your best work by the time you are thirty,” said one genius. “I tend to think that you are at your peak around thirty. I’m not saying you won’t equal it. I would like to think that you could. But I don’t think you will ever do better. That’s my gut feeling.”3 Von Neumann used to say that “the primary mathematical powers decline at about twenty-six,” after which the mathematician must rely on “a certain more prosaic shrewdness.”4
Compounding the irony is that the act of creating new mathematics, which appears so solitary from the outside, feels from the inside like an intramural competition, a race. One never forgets the crowded field. And one’s relative standing, vis-à-vis past and present competitors, is what counts. Again, Hardy best conveyed what motivates many mathematicians, including himself. He wrote that he could not recall ever wanting to be anything but a mathematician, but also that he could not remember feeling any passion for mathematics as a boy. “I wanted to beat other boys, and this seemed to be the way in which I could do so most decisively.”5 More ambitious than most, Nash was also more age-conscious than most — or perhaps simply more frank about it. “John was the most age-conscious person I’ve ever met,” recalled Felix Browder in 1995. “He would tell me every week my age relative to his and everybody else’s.”6 His determination to avoid the draft during the Korean War suggested not just a desire to avoid regimentation, but also an unwillingness to take time out of the race.
The most successful are the most vulnerable to the feeling that time is running out. Such fears may be exaggerated, but they are quite capable of producing real crises, as the history of mathematics amply attests. Artin, for example, switched frantically from field to field trying to catch hold of something that would equal his early accomplishments.7 Steenrod slipped into a deep depression. When one of his students published a note on “Steenrod’s Reduced Powers” — the reference was, of course, mathematical, not personal — other mathematicians smirked and said, “Oh, yes, Steenrod’s reduced powers!”8
Nash’s thirtieth birthday produced a kind of cognitive dissonance. One can almost imagine a sniggering commentator inside Nash’s head: “What, thirty already, and still no prizes, no offer from Harvard, no tenure even? And you thought y
ou were such a great mathematician? A genius? Ha, ha, ha!”
Nash’s mood was odd. Periods of gnawing self-doubt and dissatisfaction alternated with periods of heady anticipation. Nash had a distinct feeling that he was on the brink of some revelation. And it was this sense of anticipation, as much as his fear, as he put it, of “descending to a professional level of comparative mediocrity and routine publication,” that spurred him to begin working on two great problems.9
Sometime during the spring of 1958, Nash had confided to Eli Stein that he had “an idea of an idea” about how to solve the Riemann Hypothesis.10 That summer, he wrote letters to Albert E. Ingham, Atle Selberg, and other experts in number theory sketching his idea and asking their opinion.11 He worked in his office in Building Two for hours, night after night.
Even when a genius makes such an announcement, the rational response is skepticism. The Riemann Hypothesis is the holy grail of pure mathematics. “Whoever proves or disproves it will cover himself with glory,” wrote E. T. Bell in 1939. “A decision one way or the other disposing of Riemann’s conjecture would probably be of greater interest to mathematicians than a proof or disproof of Fermat’s Last Theorem.”12
Enrico Bombieri, at the Institute for Advanced Study, said: “The Riemann Hypothesis is not just a problem. It is the problem. It is the most important problem in pure mathematics. It’s an indication of something extremely deep and fundamental that we cannot grasp.”13
Whole numbers that are evenly divisible only by themselves and one — socalled prime numbers — have exerted a fascination for mathematicians for two thousand years or more. The Greek mathematician Euclid proved that there were infinitely many primes. The great European mathematicians of the eighteenth century — Euler, Legendre, and Gauss — began a quest, still under way, to estimate how many primes there are, given a whole number n, less than n.14 And since 1859 a string of mathematical giants — G. H. Hardy, Norman Levinson, Atle Selberg, Paul Cohen, and Bombieri, among others — have attempted, unsuccessfully, to prove the Riemann Hypothesis.15 George Polya once gave a young mathematician who had confided in him that he was working on the Riemann Hypothesis a reprint of a faulty proof of the conjecture by a Göttingen mathematician who thought he’d solved the problem. “I think about it every day when I wake up in the morning,” the young mathematician had said, and Polya delivered the reprint the following morning with a note: “If you want to climb the Matterhorn you might first wish to go to Zermatt where those who have tried are buried.”16
Before World War I, a German banker endowed a prize, lodged in Göttingen, for whoever proved or disproved the hypothesis. The prize was never awarded and, indeed, vanished in the inflation of the 1920s.17
Nash’s first encounter with Georg Friedrich Bernhard Riemann and his famous conjecture took place when Nash was fourteen, probably lying on the den floor in front of the radio, reading Bell’s Men of Mathematics18
Riemann, the sickly son of an impoverished Lutheran minister, was also fourteen and preparing to follow in his father’s footsteps when a sympathetic headmaster, who sensed that the boy was more suited to mathematics than the ministry, gave him a copy of Legendre’s Théorie des Nombres to read.19 As Bell tells it, the young Riemann returned the 859–page work six days later, saying, “That is certainly a wonderful book. I have mastered it.” This episode, which took place in 1840, was likely the origin of Riemann’s lifelong interest in the riddle of prime numbers and, as Bell theorizes, Riemann’s Hypothesis may have originated in his later attempt to improve upon Legendre.
In 1859, at the age of thirty-three, Riemann wrote an eight-page paper, “Ueber die Anzahl der Primzahlen unter einer gegebenen Groesse” (“On the number of prime numbers under a given magnitude”), in which he laid out his famous conjecture —“one of the outstanding challenges, if not the outstanding challenge to pure mathematics.”
Here is how Bell explains the conjecture:
The problem concerned is to give a formula which will state how many primes there are less than any given number n. In attempting to solve this Riemann was driven to an investigation of the infinite series 1 + 1/2S + 1/3S + 1/4S + … in which s is a complex number, say where u and v are real numbers, so chosen that the series converges. With this proviso the infinite series is a definite function of s, say zeta(s) (the Greek zeta is always used to denote this function, which is called “Riemann’s zeta function”); and as s varies zeta(s) continuously takes on different values. For what values of s will zeta(s) be zero? Riemann conjectured that all such values of s for which u lies between 0 and 1 are of the form l/2 + iv, namely, all have their real part equal to l/220
When Riemann died of tuberculosis at thirty-nine, he left behind a vast legacy, including the abstract, four-dimensional geometry that Einstein would employ in formulating his general theory of relativity. Just as geographers had to go from two-dimensional plane geometry to three-dimensional solid geometry to create an undistorted map of the earth, Einstein, to map the cosmos, went from three-dimensional to four-dimensional geometry. But it was for his tantalizing conjecture that Riemann is best remembered. Proving or disproving it would settle many extremely difficult questions in the theory of numbers and in some fields of analysis. As Bell put it, “Expert opinion favors the truth of the hypothesis.”21
It is impossible to say how long Nash had been contemplating his own attempt, but it seems likely that his interest crystallized sometime toward the end of his year in New York. Jack Schwartz recalled conversations with Nash on the subject in the Courant common room.22 Jerome Neuwirth, a second-year graduate student at MIT in 1957–58, remembered that Nash had developed a very proprietary feeling about the problem around that time.23 Neuwirth recalled that Newman, perhaps to tease Nash, told Nash that Neuwirth, too, was working on the Riemann Hypothesis. Nash came roaring into Neuwirth’s office. “How dare you?” he said. “What’s a guy like you doing?” It quickly became a running joke. Every time Nash saw Neuwirth he’d say, “Well, did you get anywhere yet?” And Neuwirth would answer, “Almost got it. I’d tell you about it, but I’ve got to run.”
As Stein recalled it, Nash’s idea was “to try to prove the hypothesis by logic, by internal consistency of the system. Some proofs are based on analogies, on rules of logic whereby something is proved [indirectly]. If one could show that the structure of two problems was in some sense identical, one could show that the logic of one proof had to apply to the other. It’s a proof by logic and it doesn’t relate to the real context. It’s not proving that one object is related to another object.”24
Stein was dubious. “He told me this very sketchy thing. It was an idea of an idea about how he was going to prove this thing. He was going to find another number system in which it was true. I thought, ’It’s wild, it doesn’t hang together.’ This struck me as simply unbelievable. This was as opposed to my earlier conversations with him about parabolic equations, which struck me as daring but probably right.”25
Richard Palais, a professor of mathematics at Brandeis University, recalls some particulars: “Nash was considering so-called pseudoprime sequences, i.e., increasing sequences p1, p2, p3, … of integers that have many of the same distribution properties as the sequence 2, 3,5,7,… of prime numbers. For each of these one can associate in a natural way a ’zeta function,’ which for the case of the true primes reduces to the Riemann zeta function. As I recall, Nash claimed to be able to show that for ’almost all’ of these pseudoprime sequences the corresponding zeta function satisfied the Riemann Hypothesis.”26
Bell warned that “Riemann’s Hypothesis is not the sort of problem that can be attacked by elementary methods. It had already given rise to an extensive and thorny literature.”27 By the time Nash turned to it seriously, that literature had grown several-fold. Both Ingham and Selberg, possibly others as well, warned Nash that his ideas had been tried before and hadn’t led anywhere.28 Eugenio Calabi, who was in touch with Nash in this period, said: “For a person who is not a library hound,
it’s a very dangerous area to go into. If you have a flash of an idea with a scenario and think you may get a result, in the first flash of illumination you think you have a revelation. But that’s very dangerous.”29
There was, as Nash suggested, nothing absurd in his attempting to solve the outstanding problems in pure mathematics and theoretical physics. The skepticism with which his early formulations were greeted was, after all, merely a replay of the skepticism voiced by experts toward his earlier efforts, and has no doubt been exaggerated in hindsight. When those problems are solved it will be by a young mathematician who attacks them with the hubris, originality, raw mental power, and sheer tenacity that Nash brought to bear on his greatest work.
Yet the timing of Nash’s decision to pursue these problems, just as he turned thirty and while he was licking various wounds to what he would later call his “merciless superego,”30 suggests that a fear of failure lay behind his willingness to take unusual risks. Stein’s impression of Nash during their conversations about the Riemann problem is interesting: “He was a little .. . on the wild side. There was something exaggerated about his actions. There was a flamboyance in the way he talked. Mathematicians are usually more careful about what they will assert to be true.”31 But, of course, hubris is not exactly uncommon. As Hörmander, who went on to win a Fields Medal in 1962, put it: “It’s part of life that not all things one works on work out. You overestimate your own abilities. After solving a big problem, nothing smaller is good enough. It’s very dangerous.”32 Later, quite possibly because of the effects of shock treatments, Nash had absolutely no recall of his attempt to solve Riemann’s conjecture.33 But, as it was, Nash’s compulsion to scale this most difficult, most dangerous peak proved central to his undoing.