5

Genius

Princeton, 1948–49

It is good that I did not let myself be influenced. — LUDWIG WITTGENSTEIN

KAI LAI CHUNG, a mathematics instructor who had survived the horrors of the Japanese conquest of his native China, was surprised to see the door of the Professors’ Room standing ajar.1 It was usually locked. Kai Lai liked to stop by on the rare occasions when it was open and nobody was about. It had the feel of an empty church, no longer imposing and intimidating as it was in the afternoons when it was crowded with mathematical luminaries, but simply a beautiful sanctuary.

The light in the west common room filtered through thick stained-glass windows inlaid with formulae: Newton’s law of gravity, Einstein’s theory of relativity, Heisenberg’s uncertainty principle of quantum mechanics. At the far end, like an altar, was a massive stone fireplace. On one side was a carving of a fly confronting the paradox of the Möbius band. Möbius had given a strip of paper a half twist and connected the ends, creating a seemingly impossible object: a surface with only one side. Kai Lai especially liked to read the whimsical inscription over the fireplace, Einstein’s expression of faith in science, “Der Herr Gott ist raffiniert aber Boshaft ist Er nicht,” which he took to mean that “the Lord is subtle but not malicious.”²

On this particular fall morning, as he reached the threshold of the half-open door, Kai Lai stopped abruptly. A few feet away, on the massive table that dominated the room, floating among a sea of papers, sprawled a beautiful dark-haired young man. He lay on his back staring up at the ceiling as if he were outside on a lawn under an elm looking up at the sky through the leaves, perfectly relaxed, motionless, obviously lost in thought, arms folded behind his head. He was whistling softly. Kai Lai recognized the distinctive profile immediately. It was the new graduate student from West Virginia. A trifle shocked and a little embarrassed, Kai Lai backed away from the door and hurried away before Nash could see or hear him.

•   •   •

The first-year students were an extremely cocky bunch, but Nash immediately struck everyone as a good deal cockier — and odder. His appearance helped create the impression.3 At twenty, Nash looked young, perhaps younger than he was, but he was no longer a gawky youngster who looked as if he’d just climbed off a tractor. Six foot one, he weighed nearly 170 pounds. He had broad shoulders, a heavily muscled chest, and a tapered waist. He had the build, if not the bearing, of an athlete, “a very strong, very masculine body,” one fellow graduate student recalled. He was, moreover, “handsome as a god,” according to another student. His high forehead, somewhat protruding ears, distinctive nose, fleshy lips, and small chin gave him the look of an English aristocrat. His hair flopped over his forehead; he was constantly brushing it away. He wore his fingernails very long, which drew attention to his rather limp and beautiful hands and long, delicate fingers. His voice, on the high, reedy side, was cool and southern and had a slightly ironic edge. His speech had an Olympian and ornamental quality that struck others as a bit stilted. Moreover, his expression was somewhat haughty and he smiled to himself in a superior way.

From the start, he was quite visible at teatime. He seemed eager to be noticed and seemed to want to establish that he was smarter than anyone else in the place. A fellow student, who had come to Princeton from the City College of New York, recalled, “He had a way of saying ‘trivial’ to anything you might have regarded as nontrivial. That could be taken as a put-down.” Nash would accuse people of burbling. If somebody was talking on and on, he was just burbling. “ALGEBRA IS BURBLE,” Nash once scrawled on a blackboard that another student, an algebraist, would pull down in the midst of a talk. “Hackers” was another favorite Nash term. A hacker was somebody who plodded along, somebody who was doing things not worth doing.⁴ As another student put it: “Nash was very interested that everyone would recognize how smart he was, not because he needed this admiration, but anybody who didn’t recognize it wasn’t on top of things. If anyone wasn’t aware, he would take a little trouble to make sure he found out.” Another student recalls, “He wanted to be noticed more than anything.”⁵

He seized opportunities to boast about his accomplishments. He would mention, out of the blue, that he’d discovered, as an undergraduate, an original proof of Gauss’s proof of the fundamental theorem of algebra, one of the great achievements of eighteenth-century mathematics, nowadays taught in advanced courses on the theory of complex variables.⁶

He was a self-declared free thinker. On his Princeton application, in answer to the question “What is your religion?” he wrote “Shinto.”⁷ He implied that his lineage was superior to that of his fellow students, especially Jewish students. Martin Davis, a fellow student who grew up in a poor family in the Bronx, recalled catching up with Nash when he was ruminating about blood lines and natural aristocracies one day as they were walking from the Graduate College to Fine Hall. “He definitely had a set of beliefs about the aristocracy,” said Davis. “He was opposed to racial mixing. He said that miscegenation would result in the deterioration of the racial line. Nash implied that his own blood lines were pretty good.”8 He once asked Davis whether Davis had grown up in a slum.

•   •   •

Nash appeared to be interested in almost everything mathematical — topology, algebraic geometry, logic, and game theory — and he seemed to absorb a tremendous amount about each of these during his first year.⁹ He himself recalled, without elaborating, having “studied mathematics fairly broadly” at Princeton.¹⁰ Yet he avoided attending classes. No one recalls sitting in a regular class with him.¹¹ He did, he later said, begin a course in algebraic topology offered by Steenrod, who essentially founded the field.¹² Steenrod and Samuel Eilenberg had just invented the axioms that were the foundation of homology theory. The stuff was very trendy and the course attracted many students, but Nash decided it was too formal for him and not geometric enough for his taste, so he stopped going.

Nobody remembers seeing Nash with a book during his graduate career either.¹³ In fact, he read astonishingly little. “Both Nash and I were dyslexic to some degree,” said Eugenio Calabi, a young Italian immigrant who entered Princeton the year before Nash. “I had great difficulty keeping my attention on reading that required great concentration. Then, I just thought of it as laziness. Nash, on the other hand, defended not reading, taking the attitude that learning too much secondhand would stifle creativity and originality. It was a dislike of passivity and giving up control.”¹⁴

Nash’s main mode of picking up information he deemed necessary consisted of quizzing various faculty members and fellow students.¹⁵ He carried around a clipboard and constantly made notes to himself. They were little hints to himself, ideas, facts, things he wanted to do, Calabi recalled. His handwriting was almost unreadable. He once explained to Lefschetz that he had to use ruled notebook paper even when writing a letter because without the lines his script formed a “very irregular wavy line.” As it was, his notes were full of crossouts and misspellings of even simple words like “InteresEted.”¹⁶

He compensated by learning through conversation in the common room and by attending lectures given by visiting mathematicians. According to Calabi, Nash “was quite systematic in asking shrewd questions and developing his own ideas from the answers. I’ve seen some of his results in the making.” Some of his best ideas came “from things learned only halfway, sometimes even wrongly, and trying to reconstruct them — even if he could not do so completely.”¹⁷

He was always asking probing questions. The questions, not only about game theory, but also about topology and geometry, often contained a kernel of speculation. John Milnor, who entered as a freshman that year, recalls one such question, posed in the common room: Let V₀ be a singular algebraic variety of dimension k, embedded in some smooth variety M₀ and let M₁ = G_k (M₀) be the Grassmann variety of tangent k-planes to M₀. Then V₀ lifts naturally to a k-dimensional variety . Continuing inductively, we obtain a sequence of k-dimensional varieties. . . . Do we eventually reach a variety V_q which is nonsingular? (As it turns out, Milnor adds, the conjecture has since been proven only in special cases.)18

•   •   •

Nash spent most of his time, it appears, simply thinking. He rode bicycles borrowed from the racks in front of the Graduate College in tight little figure eights or ever-smaller concentric circles.¹⁹ He paced around the interior quadrangle of the college. He glided along the gloomy second-floor hallway of Fine, his shoulder pressed firmly against the wall, like a trolley never losing contact with the dark paneled walls.²⁰ He would lie on a desk or table in the empty common room, or more frequently, in the third-floor library.²¹ Almost always, he whistled Bach, most often the Little Fugue.²² The whistling prompted the mathematics secretaries to complain about Nash to Lefschetz and Tucker.²³

Melvin Hausner recalled: “He was always buried in thought. He’d sit in the common room by himself. He could easily walk by you and not see you. He was always muttering to himself. Always whistling. Nash was always thinking. . . . If he was lying on a table, it was because he was thinking. Just thinking. You could see he was thinking.”²⁴

He seemed to be enjoying himself immensely. A profound dislike for merely absorbing knowledge and a strong compulsion to learn by doing is one of the most reliable signs of genius. In Princeton, Nash’s thinking began to take on an urgent, focused quality. He was obsessed with learning from scratch. Milnor recalled: “It was as if he wanted to rediscover, for himself, three hundred years of mathematics.”²⁵ Steenrod, who was to become Nash’s sounding board as the year wore on, wrote several years later, “More than any other student I have known, Nash believes in learning a subject by doing research in it.”²⁶

Like the nineteenth-century German mathematician Carl Friedrich Gauss, who complained that “such an overwhelming horde of ideas stormed my mind before I was twenty that I could hardly control them and had time but for a small fraction,”²⁷ Nash seemed to overflow with ideas. According to Steenrod, “During his first year of graduate work, he presented me with a characterization of a simple closed curve in the plane. This was essentially the same as one given by Wilder in 1932. Some time later he devised a system of axioms for topology based on the primitive concept of connectedness. I was able to refer him to papers by Wallace. During his second year, he showed me a definition of a new kind of homology group which proved to be the same as the Reidemeister group based on homotopy chains.”²⁸ What is striking about the ideas that Steenrod attributes to Nash as a first-year student is that they are not merely clever exercises designed to show off the brilliance of a precocious student, but mathematically interesting and important ideas.²⁹

Nash was always on the lookout for problems. “He was very much aware of unsolved problems,” said Milnor. “He really cross-examined people on what were the important problems. It showed a tremendous amount of ambition.”³⁰ In this search, as in so much else, Nash displayed an uncommon measure of self-confidence and self-importance. On one occasion, not long after his arrival at Princeton, he went to see Einstein and sketched some ideas he had for amending quantum theory.

•   •   •

That first fall in Princeton, Nash sometimes took a slight detour down busy Mercer Street in order to catch a glimpse of Princeton’s most remarkable resident.31 Most mornings between nine and ten, Einstein walked the mile or so from his white clapboard house at 112 Mercer Street to his office at the Institute. On several occasions, Nash managed to brush past the saintly scientist — wearing a baggy sweater, drooping trousers, sandals without socks, and an impassive expression — on the street.³² He imagined how he might strike up a conversation, stopping Einstein in his tracks with some startling observation.³³ But once when he passed him walking with Kurt Gödel, Nash caught snatches of German and sadly wondered whether his own lack of that language might constitute an insuperable barrier to communicating with the great man.³⁴

In 1948, Einstein had been a world cult figure for more than a quarter of a century.³⁵ His special theory of relativity was published in 1905, as was his assertion that light was propagated in space not as waves but as discrete particles. The general theory of relativity appeared in 1916. Astronomers’ confirmation in 1919 that light rays were bent by the sun’s gravity — as Einstein had predicted — brought him fame unrivaled by any scientist before or since. Einstein’s political activities — on behalf of the A-bomb and then for nuclear disarmament, world government, the state of Israel — added a saintly aura.

For decades, Einstein’s main scientific preoccupations had been two, one in which he achieved a measure of success, the other a complete failure.³⁶ He succeeded in casting doubt on some of the basic tenets of one of the most successful and widely accepted theories in physics — quantum theory — a theory first proposed by himself when he demonstrated the existence of light quanta in 1905, and subsequently developed by Niels Bohr and Werner Heisenberg, who insisted the act of observation changes the object being measured. Einstein’s 1935 attack on quantum theory produced a front-page headline in The New York Times and has never been satisfactorily refuted; indeed, as of the mid-1990s, the latest experimental evidence has breathed new life into his critique.

His greater preoccupation was the ultimate task of uniting the phenomena of light and gravity into a single theory. Einstein never was able, as one biographer put it, to “accept that the universe was fragmented into relativity’ on one side and quantum mechanics on the other.”³⁷ On the eve of his seventieth birthday, he was still searching for a single, consistent set of principles that applied to all of the universe’s diverse forces and particles and was, in fact, preparing what proved to be his final paper on so-called “unified field theory.”³⁸

It was a measure of Nash’s bravura and the power of his fantasy that he was not content merely to see Einstein but soon requested an audience with him. Just a few weeks into his first term at Princeton, Nash made an appointment to see Einstein in his office in Fuld Hall. He told Einstein’s assistant that he had an idea that he wished to discuss with Professor Einstein.39

Einstein’s office, a large airy room with a bay window that let in plenty of light, was messy. Einstein’s twenty-two-year-old Hungarian.assistant — an intense, chain-smoking logician named John Kemeny, who would later invent the computer language BASIC, become president of Dartmouth College, and head a commission to investigate Three Mile Island — ushered Nash in. Einstein’s handshake, which ended with a twist, was remarkably firm, and he showed Nash to a large wooden meeting table on the far side of the office.

The late-morning light streaming through the bay window produced a sort of aura around Einstein. Nash, however, quickly got into the substance of his idea while Einstein listened politely, twirled the curls on the back of his head with his finger, sucked on his tobaccoless pipe, and occasionally muttered a remark or asked a question. As he spoke, Nash became aware of a mild form of echolalia: deep, deep, interesting, interesting.⁴⁰

Nash had an idea about “gravity, friction, and radiation,” as he later recalled. The friction he was thinking of was the friction that a particle, say a photon, might encounter as it moved through space due to its fluctuating gravitational field interacting with other gravitational fields.⁴¹ Nash had given his hunch enough thought to spend much of the meeting at the blackboard scribbling equations. Soon, Einstein and Kemeny were standing at the blackboard as well.⁴² The discussion lasted the better part of an hour. But in the end all that Einstein said, with a kindly smile, was “You had better study some more physics, young man.” Nash did not immediately take Einstein’s advice and he never wrote a paper on his idea. His youthful foray into physics would become a lifetime interest — though, like Einstein’s search for the unified field, it would not be especially fruitful.⁴³ Many decades later, however, a German physicist published a similar idea.⁴⁴

•   •   •

Nash conspicuously avoided attaching himself to any particular faculty member, either in the department or at the institute. It was not a matter of shyness, his fellow students thought, but rather that he wished to preserve his independence. One mathematician who knew Nash at the time observed: “Nash was determined to keep his intellectual independence. He didn’t want to be unduly influenced. He’d talk freely with other students, but he was always worried about getting too close to other professors for fear that he’d be overwhelmed. He didn’t want to become dominated. He disliked the whole idea of being intellectually beholden.”⁴⁵

He did, however, use at least one faculty member, Steenrod, as a kind of sounding board. Temperamentally, Steenrod was an entirely different character from flamboyant, domineering types like Lefschetz and Bochner, whose lectures, it was said, were “exciting but 90 percent wrong.” Steenrod was a careful, methodical man who chose his suits and sports coats according to a mathematical formula and had a mania for thinking up highly logical, if impractical, solutions to social problems like crime.⁴⁶ Steenrod also happened to be friendly, helpful, and patient. He was immensely impressed by Nash, found him more charming than not, and treated the young man’s brashness and eccentricity with amused tolerance.47

Surrounded for the first time in his life by young men whom he regarded, if not exactly as his equals, at least as worth talking to, Nash preferred picking other students’ brains. “Some mathematicians work very much by themselves,” said one fellow student. “He liked to exchange ideas.”⁴⁸ One of the students he sought out was John Milnor, the first of a number of brilliant younger mathematicians to whom Nash was drawn. Tall, lithe, with a baby face and the body of a gymnast, Milnor was only a freshman but he was already the department’s golden boy.⁴⁹ In his freshman year, in a differential geometry course taught by Albert Tucker, he learned about an unproved conjecture of a Polish topologist, Karol Borsuk, concerning the total curvature of a knotted curve in space. The story goes that Milnor mistook the conjecture for a homework assignment.⁵⁰ Whatever the case, he arrived at Tucker’s door a few days after with a written proof and the request: “Would you be good enough to point out the flaw in this attempt. I’m sure there is one, but can’t find it.” Tucker studied it, showed it to Fox and to Shiing-shen Chern. No one could find anything wrong. Tucker encouraged Milnor to submit the proof as a Note to the Annals of Mathematics. A few months later Milnor turned in an exquisitely crafted paper with a full theory of the curvature of knotted curves in which the proof of the Borsuk conjecture was a mere by-product. The paper, more substantial than most doctoral dissertations, was published in the Annals in 1950. Milnor also dazzled the department — and Nash — by winning the Putnam competition in his second semester at Princeton (in fact, he went on to win it two more times and was offered a Harvard scholarship).⁵¹

Nash was choosy about whom he would talk mathematics with. Melvin Peisakoff, another student who would later overlap with Nash at the RAND Corporation, recalled: “You couldn’t engage him in a long conversation. He’d just walk off in the middle. Or he wouldn’t respond at all. I don’t remember Nash having a conversation that came to a nice soft landing. I also don’t remember him ever having a conversation about mathematics. Even the full professors would discuss problems they were working on with other people.”⁵²

On one occasion in the common room, however, Nash was sketching an idea when another graduate student got very interested in what he was saying and started to elaborate on the idea.⁵³ Nash said, “Well, maybe I ought to write a Note for the Proceedings of the National Academy on this.” The other student said, “Well, Nash, be sure to give me a credit.” Nash’s reply was, “All right, I’ll put in a footnote that So and So was in the room when I had the idea.”

•   •   •

Nash was respected but not well liked. He wasn’t invited to Kuhn’s room for sherry or out with the others when they went to Nassau Street to drink beer. “He wasn’t somebody you’d want as a close friend,” Calabi recalled. “I don’t know many people who felt any warmth for him.”⁵⁴ Most of the graduate students were slightly odd ducks themselves, beset by shyness, awkwardness, strange mannerisms, and all kinds of physical and psychological tics, but they collectively felt that Nash was even odder. “Nash was out of the ordinary,” said a former graduate student from his time. “If he was in a room with twenty people, and they were talking, if you asked an observer who struck you as odd, it would have been Nash. It wasn’t anything he consciously did. It was his bearing. His aloofness.”55

Another recalled, “Nash was totally spooky. He wouldn’t look at you. He’d take a lot of time answering a question. If he thought the question was foolish, he wouldn’t answer at all. He had no affect. It was a mixture of pride and something else. He was so isolated but there really was underneath it all a warmth and appreciation [for other people].”⁵⁶

When Nash did engage in one of his flights of garrulity, he often seemed to be simply thinking out loud. Hausner remembered, “A lot of us would discount a lot of what Nash said. A lot of the things he said were so far out, you didn’t want to engage him. ‘What was happening on earth when the Martians took over and there was a period of violence and why such and such.’ You wouldn’t know what he was talking about. Nash came out with things. They were unfinished and we weren’t ready to hear them. I wouldn’t want to listen. You didn’t feel comfortable with the person.”⁵⁷

His sense of humor was not only childish but odd. One former student recalled that Nash was personally responsible for getting the much-despised gown requirement at meals temporarily restored. “First,” recounted Felix Browder, who left Princeton in the fall of 1948, “he wrote a letter to Hugh Taylor, a pompous ass who was looking for an excuse, demanding that the custom be restored. After it was, nobody ate in the hall. It didn’t make John popular.”⁵⁸

He was also capable of frightening people when provoked. Occasionally, the teasing and needling would spill over into a sudden eruption of violence. On one occasion, Nash was baiting one of Artin’s students by telling him that the best way into Artin’s graces was to catch his beautiful daughter Karin.⁵⁹ The student, Serge Lang, who everyone knew was painfully obsessed by his shyness around girls, threw a cup of hot tea in Nash’s face. Nash chased him around the table, threw him to the ground, and stuffed ice cubes down the back of his shirt. Another time, Nash picked up a metal ashtray stand — the kind that supports a heavy glass ashtray — and brought it down on Melvin Peisakoff’s shins, hard enough to cause considerable pain for a number of weeks.⁶⁰

•   •   •

In the spring of 1949, Nash ran into some trouble.⁶¹ He had acquired some strong supporters on the faculty, namely Steenrod, Lefschetz, and Tucker. Tucker was among those who believed that Nash was “very brilliant and original but rather eccentric,” arguing that “his creative ability . . . should make one tolerate his queerness.”⁶² But not everyone in the department felt that way. Some felt that Nash didn’t belong at Princeton at all. Among them was Artin.

Slender, handsome, with ice-blue eyes and a spellbinding voice, Artin looked like a 1920s German matinee idol.⁶³ He wore a black leather trench coat and sandals throughout the academic year, wore his hair long, and smoked incessantly. The representative of “modern” algebra, Artin, who had been recommended by Weyl for the appointment at the institute that von Neumann eventually got, was a wonderful lecturer who admired polish and scholarship, but was famously intolerant of those who did not meet his rather fastidious standards. He was well known for screaming and throwing chalk at students who asked obtuse questions in his classes.

Artin and Nash had clashed a number of times in the common room. Artin was always interested in talking with talented students. Yet he apparently found Nash not only irritatingly brash but also shockingly ignorant.64 At a faculty meeting in the spring, Artin commented that he could see no way for Nash to pass his generals, which the better students were expected to take at the end of their first year. When Lefschetz proposed an Atomic Energy Commission fellowship for Nash for the following year, Artin opposed it and made it clear he thought it would be better if Nash left Princeton.

Lefschetz and Tucker overruled Artin on the subject of the fellowship.⁶⁵ But they dissuaded Nash from sitting for the generals that spring and suggested that he take them in the fall instead. He was safe for the time being, but his unpopularity among some faculty members was to crop up again when he sought, two years later, to join the department as an assistant professor.