A Beautiful Mind

by Sylvia Nasar

  The students who gathered at teatime were as remarkable, in a way, as the faculty. Poor Jews, new immigrants, wealthy foreigners, sons of the working classes, veterans in their twenties, and teenagers, the students were a diverse as well as brilliant group, among them John Tate, Serge Lang, Gerard Washnitzer, Harold Kuhn, David Gale, Leon Henkin, and Eugenio Calabi.45 The teas were heaven for the shy, friendless, and socially awkward, a category in which many of these young men belonged. John Milnor, the most brilliant freshman in the history of the Princeton mathematics department, described it this way: “Everything was new to me. I was awkward socially, shy and isolated. Everything was wonderful. This was a whole new world. Here was a whole community in which I felt very much at home.”46

  The atmosphere was, however, as competitive as it was friendly.47 Insults and one-upmanship were always major ingredients in teatime banter. The common room was where the young bucks warily sized each other up, bluffed and postured, and locked horns. No culture was more hierarchical than mathematical culture in its precise ranking of individual merit and prestige, yet it was a ranking always in a state of suspense and flux, in which new challenges and scuffles erupted almost daily. Back in their undergraduate colleges, most of these young men had gotten used to being the brightest and best, but now they were bumping up against the brightest and best from other schools. One of the graduate students who entered with Nash admitted, “Competitiveness, it was sort of like breathing. We thrived on it. We were nasty. This guy, he’s dumb, we’d say. Therefore he no longer existed.”48

  There were cliques, mostly based on fields. The clique at the top of the hierarchy was the topology clique, which clustered around Lefschetz, Fox, and Steenrod. Then came analysis, grouped around Lefschetz’s archrival in the department, a civilized and erudite lover of music and art named Bochner. Then came algebra, which consisted of Emil Artin and a handful of anointed followers. Logic, for some reason, was not highly regarded, despite Church’s towering reputation among early pioneers of computer theory. The game theory clique around Tucker was considered quite déclassé, an anomaly in this ivory tower of pure mathematics. Each clique had its own thoughts about the importance of its subject and its own way of putting the others down.

  Nash had never in his life encountered anything like this exotic little mathematical hothouse. It would soon provide him with the emotional and intellectual context he so much needed to express himself.

  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, h
e 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 lie 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 = Gk (M0) be the Grassmann variety of tangent k-planes to M0. Then V0 lifts naturally to a k-dimensional variety V1 ⊂ M1 Continuing inductively, we obtain a sequence of k-dimensional varieties. … Do we eventually reach a variety Vq 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 selfconfidence 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 o
n 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