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  4

  School of Genius Princeton, Fall 1948

  Conversation enriches the understanding, but solitude is the school of genius.

  — EDWARD GIBBON

  ON NASH’S SECOND AFTERNOON in Princeton, Solomon Lefschetz rounded up the first-year graduate students in the West Common Room.1 He was there to tell them the facts of life, he said, in his French accent, fixing them with his fierce gaze. And for an hour Lefschetz glared, shouted, and pounded the table with his gloved, wooden hands, delivering something between a biblical sermon and a drill sergeant’s diatribe.

  They were the best, the very best. Each of them had been carefully hand-picked, like a diamond from a heap of coal. But this was Princeton, where real mathematicians did real mathematics. Compared to these men, the newcomers were babies, ignorant, pathetic babies, and Princeton was going to make them grow up, damn it!

  Entrepreneurial and energetic, Lefschetz was the supercharged human locomotive that had pulled the Princeton department out of genteel mediocrity right to the top.2 He recruited mathematicians with only one criterion in mind: research. His high-handed and idiosyncratic editorial policies made the Annals of Mathematics, Princeton’s once-tired quarterly, into the most revered mathematical journal in the world.3 He was sometimes accused of caving in to anti-Semitism for refusing to admit many Jewish students (his rationale being that nobody would hire them when they completed their degrees),4 but no one denies that he had brilliant snap judgment. He exhorted, bossed, and bullied, but with the aim of making the department great and turning his students into real mathematicians, tough like himself.

  When he came to Princeton in the 1920s, he often said, he was “an invisible man.”5 He was one of the first Jews on the faculty, loud, rude, and badly dressed to boot. People pretended not to see him in the hallways and gave him wide berth at faculty parties. But Lefschetz had overcome far more formidable obstacles in his life than a bunch of prissy Wasp snobs. He had been born in Moscow and been educated in France.6 In love with mathematics, but effectively barred from an academic career in France because he was not a citizen, he studied engineering and emigrated to the United States. At age twenty-three, a terrible accident altered the course of his life. Lefschetz was working for Westinghouse in Pittsburgh when a transformer explosion burned off his hands. His recovery took years, during which he suffered from deep depression, but the accident ultimately became the impetus to pursue his true love, mathematics.7 He enrolled in a Ph.D. program at Clark University, the university famous for Freud’s 1912 lectures on psychoanalysis, soon fell in love with and married another mathematics student, and spent nearly a decade in obscure teaching posts in Nebraska and Kansas. After days of backbreaking teaching, he wrote a series of brilliant, original, and highly influential papers that eventually resulted in a “call” from Princeton. “My years in the west with total hermetic isolation played in my development the role of ’a job in a lighthouse’ which Einstein would have every young scientist assume so that he may develop his own ideas in his own way.”8

  Lefschetz valued independent thinking and originality above everything. He was, in fact, contemptuous of elegant or rigorous proofs of what he considered obvious points. He once dismissed a clever new proof of one of his theorems by saying, “Don’t come to me with your pretty proofs. We don’t bother with that baby stuff around here.”9 Legend had it that he never wrote a correct proof or stated an incorrect theorem.10 His first comprehensive treatise on topology, a highly influential book in which he coined the term “algebraic topology,” “hardly contains one completely correct proof. It was rumored that it had been written during one of Lefschetz’ sabbaticals … when his students did not have the opportunity to revise it.”11

  He knew most areas of mathematics, but his lectures were usually incoherent. Gian-Carlo Rota, one of his students, describes the start of one lecture on geometry: “Well a Riemann surface is a certain kind of Hausdorff space. You know what a Hausdorff space is, don’t you? It’s also compact, ok. I guess it is also a manifold. Surely you know what a manifold is. Now let me tell you one non-trivial theorem, the Riemann-Roch theorem.”12

  On this particular afternoon in mid-September 1948, with the new graduate students, Lefschetz was just warming up. “It’s important to dress well. Get rid of that thing,” he said, pointing to a pen holder. “You look like a workman, not a mathematician,” he told one student.13 “Let a Princeton barber cut your hair,” he said to another.14 They could go to class or not go to class. He didn’t give a damn. Grades meant nothing. They were only recorded to please the “goddamn deans.” Only the “generals” counted.15

  There was only one requirement: come to tea.16 They were absolutely required to come to tea every afternoon. Where else would they meet the finest mathematics faculty in the world? Oh, and if they felt like it, they were free to visit that “embalming parlor,” as he liked to call the Institute of Advanced Study, to see if they could catch a glimpse of Einstein, Gödel, or von Neumann.17 “Remember,” he kept repeating, “we’re not here to baby you.” To Nash, Lefschetz’s opening spiel must have sounded as rousing as a Sousa march.

  Lefschetz’s, hence Princeton’s, philosophy of graduate mathematics education had its roots in the great German and French research universities.18 The main idea was to plunge students, as quickly as possible, into their own research, and to produce an acceptable dissertation quickly. The fact that Princeton’s small faculty was, to a man, actively engaged in research itself, was by and large on speaking terms, and was available to supervise students’ research, made this a practical approach.19 Lefschetz wasn’t aiming for perfectly polished diamonds and indeed regarded too much polish in a mathematician’s youth as antithetical to later creativity. The goal was not erudition, much as erudition might be admired, but turning out men who could make original and important discoveries.

  Princeton subjected its students to a maximum of pressure but a wonderful minimum of bureaucracy. Lefschetz was not exaggerating when he said that the department had no course requirements. The department offered courses, true, but enrollment was a fiction, as were grades. Some professors put down all As, others all Cs, on their grade reports, but both were completely arbitrary.20 You didn’t have to show up a single time to earn them and students’ transcripts were, more often than not, works of fiction “to satisfy the Philistines.” There were no course examinations. In the language examinations, given by members of the mathematics department, a student was asked to translate a passage of French or German mathematical text. But they were a joke.21 If you could make neither heads nor tails of the passage — unlikely, since the passages typically contained many mathematical symbols and precious few words — you could get a passing grade merely by promising to learn the passage later. The only test that counted was the general examination, a qualifying examination on five topics, three determined by the department, two by the candidate, at the end of the first, or at latest, second year. However, even the generals were sometimes tailored to the strengths and weaknesses of a student.22 If, for example, it was known that a student really knew one article well, but only one, the examiners, if they were so moved, might restrict themselves to that paper. The only other hurdle, before beginning the all-important thesis, was to find a senior member of the faculty to sponsor it.

  If the faculty, which got to know every student well, decided that so-and-so wasn’t going to make it, Lefschetz wasn’t shy about not renewing the student’s support or simply telling him to leave. You were either succeeding or on your way out. As a result, Princeton students who made it past the generals wound up with doctorates after just two or three years at a time when Harvard students were taking six, seven, or eight years.23 Harvard, where Nash had yearned to go for the prestige and magic of its name, was at that time a nightmare of bureaucratic red tape, fiefdoms, and faculty with relatively little time to devote to students. Nash could not possibly have realized it fully that first day, but he was lucky to have chosen Princeton over H
arvard.

  That genius will emerge regardless of circumstance is a widely held belief. The biographer of the great Indian mathematician Ramanujan, for example, claims that the five years that the young Ramanujan spent in complete isolation from other mathematicians, having failed out of school and unable to get as much as a tutoring position, were the key to his stunning discoveries.24 But when writing Ramanujan’s obituary, G. H. Hardy, the Cambridge mathematician who knew him best, called that view, held earlier by himself, “ridiculous sentimentalism.” After Ramanujan’s death at thirty-three, Hardy wrote that the “the tragedy of Ramanujan was not that he died young, but that, during his five unfortunate years, his genius was misdirected, side-tracked, and to a certain extent distorted.”25

  As was to become increasingly obvious over the months that followed, Princeton’s approach to its graduate students, with its combination of complete freedom and relentless pressure to produce, could not have been better suited to someone of Nash’s temperament and style as a mathematician, nor more happily designed to elicit the first real proofs of his genius. Nash’s great luck, if you want to call it luck, was that he came onto the mathematical scene at a time and to a place tailor-made for his particular needs. He came away with his independence, ambition, and originality intact, having been allowed to acquire a truly first-class training that was to serve him brilliantly.

  Like nearly all the other graduate students at Princeton, Nash lived in the Graduate College. The College was a gorgeous, faux-English edifice of dark gray stone surrounding an interior courtyard that sat on a crest overlooking a golf course and lake. It was located about a mile from Fine Hall on the far side of Alexander Road, about halfway between Fine and the Institute for Advanced Study. Especially in winter, when it was dark by the time the afternoon seminar ended, it was a good long walk, and once you were there, you didn’t feel like going out again. Its location was the outcome of a fight between Woodrow Wilson and Dean Andrew West.26 Wilson had wanted the graduate students to mix and mingle with the undergraduates. West wanted to re-create the atmosphere of one of the Oxbridge colleges, far removed from the rowdy, snobbish undergraduate eating clubs on Prospect Street.

  In 1948, there were about six hundred graduate students, their ranks swelled by the numbers of returning veterans whose undergraduate or graduate careers had been interrupted by the war.27 The College, a bit shabbier than before the war and in need of sprucing up, was full, overflowing really, and a good many less lucky first-year students had been turned away and were being forced to lodge in rented rooms in the village. Almost everyone else had to share rooms. Nash, who lived in Pyne Tower, was lucky to get a private room, one of the perks of his fellowship.28 About fifteen or twenty of the mathematics students, second- and third-year as well as first-year students, and a couple of instructors lived in the college at the time.

  Life was masculine, monastic, and scholarly, exactly as Dean West had envisioned.29 The graduate students ate breakfast, lunch, and dinner together at the cost of fourteen dollars a week. Breakfast and lunch were served in the “breakfast” room, hurried meals that were taken on the run. But dinner, served in Procter Hall, a refectory very much in the English style, was a more leisurely affair. There were tall windows, long wooden tables, and formal portraits of eminent Princetonians on the walls; the evening prayer was led by Sir Hugh Taylor, the college’s dean, or his second in command, the college’s master. There were no candles and no wine, but the food was excellent. Gowns were no longer required as before the war (they were reinstated in the early 1950s, and did not disappear for good until the 1970s), but jackets and ties were required.

  The atmosphere at dinner was a combination of male debating society, locker room, and seminary. Though historians, English scholars, physicists, and economists all lived cheek by jowl with the mathematicians, the mathematicians segregated themselves as strictly as if they were living under some legal system of apartheid, always occupying a table by themselves.30 The older, more sophisticated students, namely Harold Kuhn, Leon Henkin, and David Gale, met for sherry in Kuhn’s rooms before dinner. Conversation at dinner, sometimes but not always mathematical, was more expansive than at teatime. The talk, one former student recalls, frequently revolved around “politics, music, and girls.” Political debate resembled discussions about sports, with more calculation of odds and betting than ideology. In that early fall, the Truman-Dewey race provided a great deal of entertainment. Being a more diverse group, the graduate students were more evenly split between the candidates than the Princeton undergraduates; 98 percent of the undergraduates at Princeton, it turned out, were Dewey supporters. One graduate student even wore a Wallace button for Henry Wallace, the candidate supported by the American Labor party, a Communist front organization.31

  Girls, or rather the absence of girls, the difficulty of meeting girls, the real or imagined exploits of certain older and more worldly students, were also hot subjects.32 Very few of the students dated. Women were not allowed in the main dining hall, and, of course, there were no female students. “We are all homosexuals here” was a famous remark made by a resident to fluster the dean’s wife.33 Isolation made the real prospects of meeting a girl remote. A few venturesome souls, organized by a young instructor named John Tukey, went to Thursday night folk dances at the local high school.34 But most were too shy and self-conscious to do even that. Sir Hugh, a stuffed shirt roundly disliked by the mathematicians, did his best to discourage what little socializing there was. One student was called into the dean’s office because a pair of women’s panties had been found in his room; it turned out his sister had been visiting and he, to preserve appearances, had moved out for the night. At one point, a seemingly unnecessary rule was handed down that residents of the Graduate College were not allowed to entertain a woman past midnight. The very few students who actually had girlfriends interpreted the rule literally to mean that a woman could be in the room, but couldn’t be entertained. Harold Kuhn spent his honeymoon there.35 The only time and place that women were allowed to join the larger group was Saturday lunch in the Breakfast Room.

  In short, social life was rather enveloping — it would be hard to become really lonely — and at the same time limited to other men, in Nash’s case specifically to other mathematicians. The parties held in student rooms were thus mostly all-male affairs. Such evenings, as often as not, were devoted to mathematical parties organized by one of the graduate students at Lefschetz’s request to entertain some visitor but actually to get his students much-needed job contacts.36

  The quality, diversity, and sheer volume of mathematics talked about in Princeton every day, by professors, Institute professors, and a steady stream of visitors from all over the world, not to mention the students themselves, were unlike anything Nash had ever imagined, much less experienced. A revolution was taking place in mathematics and Princeton was the center of the action. Topology. Logic. Game theory. There were not only lectures, colloquia, seminars, classes, and weekly meetings at the institute that Einstein and von Neumann occasionally attended, but there were breakfasts, lunches, dinners, and after-dinner parties at the Graduate College, where most of the mathematicians lived, as well as the daily afternoon teas in the common room. Martin Shubik, a young economist studying at Princeton at that time, later wrote that the mathematics department was “electric with ideas and the sheer joy of the hunt. If a stray ten-year-old with bare feet, no tie, torn blue jeans, and an interesting theorem had walked into Fine Hall at tea times, someone would have listened.”37

  Tea was the high point of every day.38 It was held in Fine Hall between three and four between the last class and the four-thirty seminar that went until five-thirty or six. On Wednesdays it was held in the west common room, or the professor’s room as it was also called, and was a far more formal affair, where the self-effacing Mrs. Lefschetz and the other wives of the senior faculty, wearing long gowns and white gloves, poured the tea and passed the cookies. Heavy silver teapots and dainty English bone c
hina were brought out.

  On other days, tea was held in the east common room, also known as the students’ room, a much-lived-in, funky place full of overstuffed leather armchairs and low tables. The janitor would bring in the tea and cookies a few minutes before three o’clock and the mathematicians, tired from a day of working alone or lecturing or attending seminars, would start drifting in, one by one or in groups. The faculty almost always came, as did most of the graduate students and a sprinkling of more precocious undergraduates. It was very much a family gathering, small and intimate. It is hard to think where a student could get to know as many other mathematicians as well as at Princeton teatime.

  The talk was by no means purely formal. Mathematical gossip abounded — who was working on what, who had a nibble from what department, who had run into trouble on his generals. Melvin Hausner, a former Princeton graduate student, later recalled, “You went there to discuss math. To do your own version of gossiping. To meet faculty. To meet friends. We discussed math problems. We shared our readings of recent math papers.”39

  The professors felt it their duty to come, not only to get to know the students but to chat with one another. The great logician Alonzo Church, who looked “like a cross between a panda and an owl,” never spoke unless spoken to, and rarely then, would head straight for the cookies, placing one between the fingers of his splayed hand, and munch away.40 The charismatic algebraist Emil Artin, son of a German opera singer, would fling his gaunt, elegant body into one of the leather armchairs, light a Camel, and opine on Wittgenstein and the like to his disciples, huddled, more or less literally, at his feet.41 The topologist Ralph Fox, a go master, almost always made a beeline for a game board, motioning some student to join him.42 Another topologist, Norman Steenrod, a good-looking, friendly midwesterner who had just created a sensation with his now classic exposition of fiber bundles, usually stopped in for a game of chess.43 Albert Tucker, Lefschetz’s righthand man, was the straitlaced son of a Canadian Methodist minister and Nash’s eventual thesis adviser. Tucker always surveyed the room before he came in and would make fussy little adjustments — such as straightening the curtain weights if the drapes happened to be awry, or issuing a word-to-the-wise to a student who was taking too many cookies.44 More often than not, a few visitors, often from the Institute for Advanced Study, would turn up as well.