A Beautiful Mind (12 page)

The most dramatic contributions were in the areas of weaponry: radar, infrared detection devices, bomber aircraft, long-range rockets, and torpedoes with depth charges.
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The new weapons were extremely costly, and the military needed mathematicians to devise new methods for assessing their effectiveness and the most efficient way to use them. Operations research was a systematic way of coming up with the numbers the military wanted. How many tons of explosive force must a bomb release to do a certain amount of damage? Should airplanes be heavily armored or stripped of defenses to fly faster? Should the Ruhr be bombed, and how many bombs should be used? All these questions required mathematical talent.

The ultimate contribution was, of course, the A-bomb.
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Wigner at Princeton and Leo Szilard at Columbia composed a letter, which they brought to Einstein to sign, warning President Roosevelt that a German physicist, Otto Hahn, at the Kaiser Friedrich Institute in Berlin had succeeded in splitting the uranium atom. Lise Meitner, an Austrian Jew who was smuggled into Denmark, performed the mathematical calculations on how an atomic bomb could be constructed from these findings. Niels Bohr, the Danish physicist, visited Princeton in 1939 and transmitted the news. “It was they rather than their American born colleagues who sensed the military implications of the new knowledge,” wrote Davies. Roosevelt responded by appointing an advisory committee on uranium in October 1939, two months into the war, which eventually became the Manhattan Project.

The war enriched and invigorated American mathematics, vindicated those who had championed the émigrés, and gave the mathematical community a claim on the fruits of the postwar prosperity that was to follow. The war demonstrated not only the power of the new theories but the superiority of sophisticated mathematical analysis over educated guesses. The bomb gave enormous prestige to Einstein’s relativity theory, which before then had been seen as a small correction of the still-valuable Newtonian mechanics.

Princeton rode high on the newfound status of mathematics in American society. It found itself on the leading edge not just of topology, algebra, and number theory, but also of computer theory, operations research, and the new theory of games.
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In 1948, everyone was back and the anxieties and frustrations of the 1930s had been swept away by a feeling of expansiveness and optimism. Science and mathematics were seen as the key to a better postwar world. Suddenly the government, particularly the military, wanted to spend money on pure research. Journals
started up. Plans were made for another world mathematical congress, the first since the dark days before the war.

A new generation was crowding in, eager to drink up the wisdom of the older generation, yet full of ideas and attitudes of its own. There were no women yet, of course — with the exception of Oxford’s Mary Cartwright, who was in Princeton that year — but Princeton was opening up. Suddenly, being a Jew or a foreigner, having a working-class accent, or graduating from a college that wasn’t on the East Coast were no longer automatic bars to a bright young mathematician. The biggest divide on campus was suddenly between “the kids” and the war veterans, who, in their mid-to-late twenties, were starting graduate school alongside twenty-year-olds like Nash. Mathematics was no longer a gentlemen’s profession, but a wonderfully dynamic enterprise. “The notion was that the human mind could accomplish anything with mathematical ideas,” a Princeton student of that era later recalled. He added: “The postwar years had their threats — the Korean War, the Cold War, China going to the commies — but in fact, in terms of science, there was this tremendous optimism. The sense at Princeton wasn’t just that you were close to a great intellectual revolution, but that you were part of it.”
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4
School of Genius
Princeton, Fall 1948

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

—
E
DWARD
G
IBBON

O
N
N
ASH’S SECOND AFTERNOON
in Princeton, Solomon Lefschetz rounded up the first-year graduate students in the West Common Room.
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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.
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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.
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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),
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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.”
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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.
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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.
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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.”
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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.”
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Legend had it that he never wrote a correct proof or stated an incorrect theorem.
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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.”
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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.”
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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.
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“Let a Princeton barber cut your hair,” he said to another.
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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.
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There was only one requirement: come to tea.
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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.
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“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.
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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.
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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.
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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.
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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.
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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.
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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 Harvard.

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.
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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.”
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