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Created: 1999-06-08
Last Modified: 1999-06-08
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Graduate Math at Princeton in the 1940s

 

from Sylvia Nasar's magnificent biography of John Nash. Sylvia Nasar (1998), A Beautiful Mind: A Biography of John Forbes Nash, Jr. (New York: Simon and Schuster: 0684853701).


pp. 58-60: On [John] Nash's second afternoon in Princeton, Solomon Lefschetz rounded up the first-year graduate students in the West Common Room. 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. 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. 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), 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." 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 then educated in Paris. 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 Westinghous 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. 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 teachingposts 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 role 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."

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." Legend had it that he never wrote a correct proof or stated an incorrect theorem. His first comprehensive treatise on topology, a highly influential book in which he coined the term "algebraic topology," "harely contains one completely correct proof. It was rumored that it had been written during one of Lefschetz's sabbaticals... when his students did not have the opportunity to revise it."

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."

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. "Let a Princeton barber cut your hair," he said to another. 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.

There was only one requirement: come to tea. 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, Godel, or von Neumann. "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, and hence Princeton's, philosophy of graduate mathematics education had its roots in the great German and French research universities. The main idea was to plunge students, as quickly as possible, into their own research, an dto 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. 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 requirement. 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. 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. 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. 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 you were 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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Professor of Economics J. Bradford DeLong, 601 Evans Hall, #3880
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