A Beautiful Mind Part 9

A Slow Fire Burning

CHAPTER 30

Olden Lane and Was.h.i.+ngton Square 1956-57 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 reinfection of more or less directly empirical ideas.

- JOHN VON N NEUMANN

THE I INSt.i.tUTE FOR A ADVANCED S 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 Ca.n.a.l, 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. 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 Ca.n.a.l, 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 Inst.i.tute'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." 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 Inst.i.tute 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 gla.s.s in an artificial hand, or witness a very drunk French mathematician displaying his mountaineering skills by rope-climbing up and over the fireplace mantel. For young researchers, the Inst.i.tute 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 gla.s.s 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." 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, G.o.del 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 Inst.i.tute had become pure, very pure." By 1956, Einstein was dead, G.o.del 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 Inst.i.tute had become pure, very pure."5 Cathleen Morawetz, later president of the American Mathematical Society, put it more bluntly: "The Inst.i.tute was known to be about the dullest place you could find." Cathleen Morawetz, later president of the American Mathematical Society, put it more bluntly: "The Inst.i.tute was known to be about the dullest place you could find."6 By contrast, the Courant Inst.i.tute of Mathematical Sciences at New York University was "the national capital of applied mathematical a.n.a.lysis," as Fortune Fortune magazine was soon to inform its readers. 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 Was.h.i.+ngton 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-fas.h.i.+oned freight elevator - with a number of hat factories. Just a few years old and vibrant with energy, Courant occupied a nineteenth-century loft less than a block to the east of Was.h.i.+ngton 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-fas.h.i.+oned freight elevator - with a number of hat factories.8 Financing for the inst.i.tute 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 ma.s.s of vacuum tubes, with its armed guard, occupied 25 Waverly Place. Financing for the inst.i.tute 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 ma.s.s of vacuum tubes, with its armed guard, occupied 25 Waverly Place.9 The inst.i.tute was the creation of one of mathematics' great entrepreneurs, Richard Courant, a German Jewish professor of mathematics who had been driven out of Gottingen in the mid-1950s by the n.a.z.is.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 "a.s.sistants," 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. 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 "a.s.sistants," 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 inst.i.tute was formally founded, it was already rivaling more established mathematical centers like Princeton and Cambridge. The advent of World War II brought more money and more students, and by the mid-1950s, when the inst.i.tute 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, Juirgen Moser, and Louis Nirenberg, and among its stellar visitors were Lars Hormander, a future Fields medalist, and Shlomo Sternberg, who would soon move to Harvard. Its young stars included Peter Lax and his wife, Anneli, Cathleen Synge Morawetz, Juirgen Moser, and Louis Nirenberg, and among its stellar visitors were Lars Hormander, a future Fields medalist, and Shlomo Sternberg, who would soon move to Harvard.

The Courant Inst.i.tute 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 Inst.i.tute 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 He never came too early, for he liked to sleep late after working into the wee hours at the university library. 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. 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 compet.i.tiveness of MIT or the sn.o.bbery of the Inst.i.tute, 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, a.s.sumed Nash was just another postdoctoral fellow and found him "very charming," "an attractive fellow," "a lively conversationalist." Cathleen Morawetz, the daughter of John Synge, Nash's professor at Carnegie, a.s.sumed Nash was just another postdoctoral fellow and found him "very charming," "an attractive fellow," "a lively conversationalist."17 Hormander recalled his first impressions: "He wore a serious expression. Then he'd break out into a sudden smile. He was an enthusiast." Hormander 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." 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 - Na.s.ser nationalized the Suez Ca.n.a.l, 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 inst.i.tute 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, Na.s.ser 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 ...' " Another mathematician remembered a similar conversation in the inst.i.tute 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, Na.s.ser 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 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 By the mid-fifties, as Fortune 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 a.n.a.lytically," saying that they "defy even qualitative insights by present methods." 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 a.n.a.lytically," 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 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." 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 rus.h.i.+ng into a bay, heat or electrical charges traveling through metal, oil escaping from an underground pool, or clouds skimming over an air ma.s.s. It should be possible to model such motion mathematically. But it turns out to be extremely difficult. As Nash wrote: Turbulence refers to the flow of gas or liquid over any uneven surface, like water rus.h.i.+ng into a bay, heat or electrical charges traveling through metal, oil escaping from an underground pool, or clouds skimming over an air ma.s.s. 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 a priori estimate of continuity. estimate of continuity.26

It was Louis Nirenberg, a short, myopic, and sweet-natured young protege 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 a.n.a.lyst, 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]." Nirenberg, also in his twenties, and already a formidable a.n.a.lyst, 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. 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 a.s.sociated 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." 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." 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 Hormander, a tall, steely Swede who was already one of the top scholars in the field. Precise, careful, and immensely knowledgeable, Hormander 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," Hormander 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]." "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 Hormander, "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 circ.u.mstance."35 Nash's new result got far more immediate attention than his embedding theorem. It convinced Nirenberg, too, that Nash was a genius.36 Hormander's mentor at the University of Lund, Lars Grding, a world-cla.s.s specialist in partial differential equations, immediately declared, "You have to be a genius to do that." Hormander's mentor at the University of Lund, Lars Grding, a world-cla.s.s 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 Ma.s.sachusetts as opposed to New York. 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 Ma.s.sachusetts 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 attache in London. It was an Office of Naval Research sinecure.40 In January 1957, Garabedian took a long car 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. n.o.body 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." 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. n.o.body 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 relations.h.i.+ps, 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 G.o.d through mathematics. Later he would become an idol to the younger generation. He had no life outside mathematics, no family of his own or other close relations.h.i.+ps, 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 G.o.d 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 n.o.bel 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." 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 Inst.i.tute 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 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 att.i.tudes," complaining of their tendency to treat "anyone with any sort of questioning att.i.tude or a belief in 'hidden parameters'... as stupid or at best a quite ignorant person." who have studied Quantum Theory) ... quite too dogmatic in their att.i.tudes," complaining of their tendency to treat "anyone with any sort of questioning att.i.tude 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 Inst.i.tute 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 quant.i.ties," he wrote, adding that "I want to find a different and more satisfying under-picture of a non-observable reality." 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 quant.i.ties," 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

CHAPTER 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 [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 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 disappointed in something big, it's frightening, and fright can precipitate - PAUL H HOWARD, M McLean H Hospital

JuRGEN M 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 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 Sputnik - a tiny silvery dot reflecting the sun's last rays - to pa.s.s slowly over Boston. tiny silvery dot reflecting the sun's last rays - to pa.s.s slowly over Boston.1 Having calculated the satellite's precise orbit, Moser always knew when it would appear on the horizon. 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 Jurgen. 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 Jurgen 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. 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. 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. 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. 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 relations.h.i.+p 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." Some of their acquaintances found their relations.h.i.+p 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." 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." "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. 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 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." 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 a.n.a.lyst. 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. 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." 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. 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. 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. 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. 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. 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 n.o.bel, Nash referred in his autobiographical essay, in his typically elliptical fas.h.i.+on, 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 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). 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 Medal 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 n.o.bel Prize of mathematics, the ultimate distinction that a mathematician can be granted by his peers, the trophy of trophies.24 There is no n.o.bel in mathematics, and mathematical discoveries, no matter how vital to n.o.bel disciplines such as physics or economics, do not in themselves qualify for a n.o.bel. The Fields is, if anything, rarer than the n.o.bel. In the fifties and early sixties, it was awarded once every four years and usually to just two recipients at a time. n.o.bels, 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." There is no n.o.bel in mathematics, and mathematical discoveries, no matter how vital to n.o.bel disciplines such as physics or economics, do not in themselves qualify for a n.o.bel. The Fields is, if anything, rarer than the n.o.bel. In the fifties and early sixties, it was awarded once every four years and usually to just two recipients at a time. n.o.bels, 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 n.o.bel, 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 incentive, incidentally, is of an intangible variety, as the cash involved, in contrast to the n.o.bel, 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 n.o.bel deliberations, the Fields selection process is shrouded in greatest secrecy. Like the n.o.bel deliberations, the Fields selection process is shrouded in greatest secrecy.

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 Gottingen, 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.) 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. 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 Rene 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. "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 partic.i.p.ants. 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 partic.i.p.ants.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." "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. 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 n.o.body 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 Bocher Prize, the only award remotely comparable in terms of prestige to the Fields. The Bocher 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. 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 ma.n.u.script to Acta Mathematica, Acta Mathematica, the Swedish mathematics journal, in the spring of 1958. 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 ma.n.u.script to Hormander to referee. Hormander 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. 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 ma.n.u.script to Hormander to referee. Hormander 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, American Journal of Mathematics, Hormander concluded that Nash had always intended to publish the paper there, since the Bocher restricted eligible papers to those published in American journals - or, worse, had submitted the paper to both journals, Hormander concluded that Nash had always intended to publish the paper there, since the Bocher 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 a clear-cut breach of professional ethics. "It turned out that Nash had just wanted to get a letter of acceptance from Acta Acta to be able to get fast publication in the to be able to get fast publication in the American Journal of Mathematics." American Journal of Mathematics."38 Hormander was angry at what he felt was "very improper and most unusual." Hormander 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 Acta before learning that doing so would exclude it from consideration for the Bocher, but that upon discovering this fact, he was willing to antagonize Carleson and Hormander in order to preserve his eligibility. He may therefore not have used before learning that doing so would exclude it from consideration for the Bocher, but that upon discovering this fact, he was willing to antagonize Carleson and Hormander in order to preserve his eligibility. He may therefore not have used Acta Acta quite so unscrupulously. Withdrawing the paper after it had been promised to quite so unscrupulously. Withdrawing the paper after it had been promised to Acta, Acta, and after it had been refereed, would have been unprofessional, but not as clear a violation of ethics as Hormander's scenario suggests. However, it still showed how very much winning a prize meant to Nash. and after it had been refereed, would have been unprofessional, but not as clear a violation of ethics as Hormander's scenario suggests. However, it still showed how very much winning a prize meant to Nash.

CHAPTER 32

Secrets Summer 1958 Summer 1958

It struck me that I knew everything: everything was revealed to me, all the secrets of the world were mine during those s.p.a.cious hours.

- GERARD DE N 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. 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 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. 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." 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." 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 compet.i.tion, a race. One never forgets the crowded field. And one's relative standing, vis-a-vis past and present compet.i.tors, 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 pa.s.sion 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." 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." 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. 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!" 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 sn.i.g.g.e.ring commentator inside Nash's head: "What, thirty already, and still no prizes, no offer from Harvard, no tenure even? And you thought you 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 antic.i.p.ation. Nash had a distinct feeling that he was on the brink of some revelation. And it was this sense of antic.i.p.ation, 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. 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. 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 Inst.i.tute for Advanced Study, said: "The Riemann Hypothesis is not just a problem. It is the 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." 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 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 n, less than n 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. 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 Gottingen 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." 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 Gottingen 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 Gottingen, 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 Mathematics 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 Theorie des Nombres Theorie des Nombres to read. to read.19 As Bell tells it, the young Riemann returned the 859page 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. As Bell tells it, the young Riemann returned the 859page 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" "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." ("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 whi