A Century of Mathematics in America, Part III, History of Mathematics, Volume 3, P. Duren, Ed., pp.223-236, American Mathematical Society, 1989.
Gian-Carlo Rota was born in Italy, where he went to school through the ninth grade. He attended high school in Quito, Ecuador, and entered Princeton University as a freshman in 1950. Three years later, he graduated summa cum laude and went to Yale, where he received a Ph.D. in 1956 with a thesis in functional analysis under the direction ofJ. T. Schwartz. After a position at Harvard, in 1959 he moved to MIT, where he is now professor of mathematics and philosophy. He has made basic contributions to operator theory, ergodic theory, and combinatorics. The AMS recently honored him with a Steele Prize for his seminal work in algebraic combinatorics. He is a senior fellow of the Los Alamos National Laboratory and a member of the National Academy of Sciences.
GIAN-CARLO ROTA
[*The present article is a draft for a chapter of a book which the author is under contract to write for the Sloan science series. ]
Our faith in mathematics is not likely to wane if we openly acknowledge that the personalities of even the greatest mathematicians may be as flawed as those of anyone else. The greater a mathematician, the more important it is to bring out the contradictions in his or her personality. Psychologists of the future, if they should ever read such accounts, may better succeed in explaining what we, blinded by prejudice, would rather not face up to.
The biographer who frankly admits his bias is, in my opinion, more honest than the one who, appealing to objectivity, conceals his bias in the selection of facts to be told. Rather than attempting to be objective, I have chosen to transcribe as faithfully as I can the inextricable twine of fact, opinion and idealization that I have found in my memories of what happened thirty-five years ago. I hope thereby to have told the truth. Every sentence I have written should be prefixed by an "It is my opinion that ......
I apologize to those readers who may find themselves rudely deprived of the comforts of myth.
It cannot be a complete coincidence that several outstanding logicians of the twentieth century found shelter in asylums at some time in their lives: Cantor, Zermelo, Gödel, Peano, and Post are some. Alonzo Church was one of the saner among them, though in some ways his behavior must be classified as strange, even by mathematicians' standards.
He looked like a cross between a panda and a large owl. He spoke slowly in complete paragraphs which seemed to have been read out of a book, evenly and slowly enunciated, as by a talking machine. When interrupted, he would pause for an uncomfortably long period to recover the thread of the argument. He never made casual remarks: they did not belong in the baggage of formal logic. For example, he would not say: "It is raining." Such a statement, taken in isolation, makes no sense. (Whether it is actually raining or not does not matter; what matters is consistency.) He would say instead: "I must postpone my departure for Nassau Street, inasmuch as it is raining, a fact which I can verify by looking out the window." (These were not his exact words.) Gilbert Ryle has criticized philosophers for testing their theories of language with examples which are never used in ordinary speech. Church's discourse was precisely one such example.
He had unusual working habits. He could be seen in a corridor in Fine Hall at any time of day or night, rather like the Phantom of the Opera. Once, on Christmas day, I decided to go to the Fine Hall library (which was always open) to look up something. I met Church on the stairs. He greeted me without surprise.
He owned a sizable collection of science-fiction novels, most of which looked well thumbed. Each volume was mysteriously marked either with a circle or with a cross. Corrections to wrong page numberings in the table of contents had been penciled into several volumes.
His one year course in mathematical logic was one of Princeton University's great offerings. It attracted as many as four students in 1951 (none of them were philosophy students, it must be added, to philosophy's discredit). Every lecture began with a ten-minute ceremony of erasing the blackboard until it was absolutely spotless. We tried to save him the effort by erasing the board before his arrival, but to no avail. The ritual could not be disposed of; often it required water, soap, and brush, and was followed by another ten minutes of total silence while the blackboard was drying. Perhaps he was preparing the lecture while erasing; I don't think so. His lectures hardly needed any preparation. They were a literal repetition of the typewritten text he had written over a period of twenty years, a copy of which was to be found upstairs in the Fine Hall library. (The manuscript's pages had yellowed with the years, and smelled foul. Church's definitive treatise was not published for another five years.) Occasionally, one of the sentences spoken in class would be at variance with the text upstairs, and he would warn us in advance of the discrepancy between oral and written presentation. For greater precision, everything he said (except some fascinating side excursions which he invariably prefixed by a sentence like: "I will now interrupt and make a metamathematical [sic] remark") was carefully written down on the blackboard, in large English-style handwriting, like that of a grade-school teacher, complete with punctuation and paragraphs. Occasionally, he carelessly skipped a letter in a word. At first we pointed out these oversights, but we quickly learned that they would create a slight panic, so we kept our mouths shut. Once he had to use a variant of a previously proved theorem, which differed only by a change of notation. After a moment of silence, he turned to the class and said: "I could simply say 'likewise', but I'd better prove it again."
It may be asked why anyone would bother to sit in a lecture which was the literal repetition of an available text. Such a question would betray an oversimplified view of what goes on in a classroom. What one really learns in class is what one does not know at the time one is learning. The person lecturing to us was logic incarnate. His pauses, hesitations, emphases, his betrayals of emotion (however rare), and sundry other nonverbal phenomena taught us a lot more logic than any written text could. We learned to think in unison with him as he spoke, as if following the demonstration of a calisthenics instructor. Church's course permanently improved the rigor of our reasoning.
The course began with the axioms for the propositional calculus (those of Russell and Whitehead's Principia Mathematica, I believe) that take material implication as the only primitive connective. The exercises at the end of the first chapter were mere translations of some identities of naive set theory in terms of material implication. It took me a tremendous effort to prove them, since I was unaware of the fact that one could start with an equivalent set of axioms using "and" and "or" (where the disjunctive normal form provides automatic proofs) and then translate each proof step by step in terms of implication. I went to see Church to discuss my difficulties, and far from giving away the easy solution, he spent hours with me devising direct proofs using implication only. Toward the end of the course I brought to him the sheaf of papers containing the solutions to the problems (all problems he assigned were optional, since they could not logically be made to fit into the formal text). He looked at them as if expecting them, and then pulled out of his drawer a note he had just published in Portugaliae Mathematica, where similar problems were posed for "conditional disjunction", a ternary connective he had introduced. Now that I was properly trained, he wanted me to repeat the work with conditional disjunction as the primitive connective. His graduate students had declined a similar request, no doubt because they considered it to be beneath them.
Mathematical logic has not been held in high regard at Princeton, then or now. Two minutes before the end of Church's lecture (the course met in the largest classroom in Fine Hall), Lefschetz would begin to peek through the door. He glared at me and at the spotless text on the blackboard; sometimes he shook his head to make it clear that he considered me a lost cause. The following class was taught by Kodaira, at that time a recent arrival from Japan, whose work in geometry was revered by everyone in the Princeton main line. The classroom was packed during Kodaira's lecture. Even though his English was atrocious, his lectures were crystal clear. (Among other things, he stuttered. Because of deep-seated prejudices of some of its members, the mathematics department refused to appoint him full-time to the Princeton faculty.)
I was too young and too shy to have an opinion of my own about Church and mathematical logic. I was in love with the subject, and his course was my first graduate course. I sensed disapproval all around me; only Roger Lyndon (the inventor of spectral sequences), who had been my freshman advisor, encouraged me. Shortly afterward he himself was encouraged to move to Michigan. Fortunately, I had met one of Church's most flamboyant former students, John Kemeny, who, having just finished his term as a mathematics instructor, was being eased—by Lefschetz's gentle hand—into the philosophy department. (The following year he left for Dartmouth, where he eventually became president.)
Kemeny's seminar in the philosophy of science (which that year attracted as many as six students, a record) was refreshing training in basic reasoning. Kemeny was not afraid to appear pedestrian, trivial, or stupid: what mattered was to respect the facts, to draw distinctions even when they clashed with our prejudices, and to avoid black and white oversimplifications. Mathematicians have always found Kemeny's common sense revolting.
"There is no reason why a great mathematician should not also be a great bigot," he once said on concluding a discussion whose beginning I have by now forgotten. "Look at your teachers in Fine Hall, at how they treat one of the greatest living mathematicians, Alonzo Church."
I left literally speechless. What? These demi-gods of Fine Hall were not perfect beings? I had learned from Kemeny a basic lesson: a good mathematician is not necessarily a "nice guy."
His name was neither William nor Feller. He was named Willibold by his Catholic mother in Croatia, after his birthday saint; his original last name was a Slavic tongue twister, which he changed while still a student at Göttingen (probably on a suggestion of his teacher Courant). He did not like to be reminded of his Balkan origins, and I had the impression that in America he wanted to be taken for a German who had Anglicized his name. From the time he moved from Cornell to Princeton in 1950, his whole life revolved around a feeling of inferiority. He secretly considered himself to be one of the lowest ranking members of the Princeton mathematics department, probably the second lowest after the colleague who had brought him there, with whom he had promptly quarreled after arriving in Princeton.
In retrospect, nothing could be farther from the truth. Feller's treatise in probability is one of the great masterpieces of mathematics of all time. It has survived unscathed the onslaughts of successive waves of rewriting, and it is still secretly read by every probabilist, many of whom refuse to admit that they still constantly consult it, and refer to it as "trivial" (like high school students complaining that Shakespeare's dramas are full of platitudes). For a long time, Feller's treatise was the mathematics book most quoted by nonmathematicians.
But Feller would never have admitted to his success. He was one of the first generation who thought probabilistically (the others: Doob, Kac, Uvy, and Kolmogorov), but when it came to writing down any of his results for publication, he would chicken out and recast the mathematics in purely analytic terms. It took one more generation of mathematicians, the generation of Harris, McKean, Ray, Kesten, Spitzer, before probability came to be written the way it is practiced.
His lectures were loud and entertaining. He wrote very large on the blackboard, in a beautiful Italianate handwriting with lots of whirls. Sometimes only one huge formula appeared on the blackboard during the entire period; the rest was handwaving. His proof—insofar as one can speak of proofs—were often deficient. Nonethless, they were convincing, and the results became unforgettably clear after he had explained them. The main idea was never wrong.
He took umbrage when someone interrupted his lecturing by pointing out some glaring mistake. He became red in the face and raised his voice, often to full shouting range. It was reported that on occasion he had asked the objector to leave the classroom. The expression "proof by intimidation" was coined after Feller's lectures (by Mark Kac). During a Feller lecture, the hearer was made to feel privy to some wondrous secret, one that often vanished by magic as he walked out of the classroom at the end of the period. Like many great teachers, Feller was a bit of a con man.
I learned more from his rambling lectures than from those of anyone else at Princeton. I remember the first lecture of his I ever attended. It was also the first mathematics course I took at Princeton (a course in sophomore differential equations). The first impression he gave was one of exuberance, of great zest for living, as he rapidly wrote one formula after another on the blackboard while his white mane floated in the air. After the first lecture, I had learned two words which I had not previously heard: "lousy" and "nasty." I was also terribly impressed by a trick he explained: the integral
\int_0^{2\pi} cos^2 x dx
equals the integral
\int_0^{2\pi} sin^2 x dx
and therefore, since the sum of the two integrals equals 2 pi, each of them is easily computed.
He often interrupted his lectures with a tirade from the repertoire he had accumulated over the years. He believed these side shows to be a necessary complement to the standard undergraduate curriculum. Typical titles: "Gandhi was a phoney," "Velikovsky is not as wrong as you think," "Statisticians do not know their business," "ESP is a sinister plot against civilization," "The smoking and health report is all wrong." Such tirades, it must be said to his credit, were never repeated to the same class, though they were embellished with each performance. His theses, preposterous as they sounded, invariably carried more than an element of truth.
He was Velikovsky's next-door neighbor on Random Road. They first met one day when Feller was working in his garden pruning some bushes, and Velikovsky rushed out of his house screaming: "Stop! You are killing your father!" Soon afterward they were close friends.
He became a crusader for any cause which he thought to be right, no matter how orthogonal to the facts. Of his tirades against statistics, I remember one suggestion he made in 1952, which still appears to me to be quite sensible: in multiple-choice exams, students should be asked to mark one wrong answer, rather than to guess the right one. He inveighed against American actuaries, pointing to Swedish actuaries (who gave him his first job after he graduated from Göttingen) as the paradigm. He was so vehemently opposed to ESP that his overkill (based on his own faulty statistical analyses of accurate data) actually helped the other side. He was, however, very sensitive to criticism, both of himself and of others. "You should always judge a mathematician by his best paper!," he once said, referring to Richard Bellman.
While he was writing the first volume of his book he would cross out entire chapters in response to the slightest critical remark. Later, while reading galleys, he would not hesitate to rewrite long passages several times, each time using different proofs; some students of his claim that the entire volume was rewritten in galleys, and that some beautiful chapters were left out for fear of criticism. The treatment of recurrent events was the one he rewrote most, and it is still, strictly speaking, wrong. Nevertheless, it is perhaps his greatest piece of work. We are by now so used to Feller's ideas that we tend to forget how much mathematics today goes back to his "recurrent events"; the theory of formal grammars is one outlandish example.
He had no firm judgment of his own, and his opinions of other mathematicians, even of his own students, oscillated wildly and frequently between extremes. You never knew how you stood with him. For example, his attitude toward me began very favorably when he realized I had already learned to differentiate and integrate before coming to Princeton. (In 1950, this was a rare occurrence.) He all but threw me out of his office when I failed to work on a problem on random walk he proposed to me as a sophomore; one year later, however, I did moderately well on the Putnam Exam, and he became friendly again, only to write me off completely when I went off to Yale to study functional analysis. The tables were turned again 1963 when he gave me a big hug at a meeting of the AMS in New York. (I learned shortly afterward that Doob had explained to him my 1963 limit theorem for positive operators. In fact, he liked the ideas of "strict sense spectral theory" so much that he invented the phrase "To get away with Hilbert space.") His benevolence, alas, proved to be short-lived: as soon as I started working in combinatorics, he stopped talking to me. But not, fortunately, for long: he listened to a lecture of mine on applications of exterior algebra to combinatorics and started again singing my praises to everyone. He had jumped to the conclusion that I was the inventor of exterior algebra. I never had the heart to tell him the truth. He died believing I was the latter-day Grassmann.
He never believed that what he was doing was going to last long, and he modestly enjoyed pointing out papers that made his own work obsolete. No doubt he was also secretly glad that his ideas were being kept alive. This happened with the Martin boundary ("It is so much better than my boundary!") and with the relationship between diffusion and semigroups of positive operators.
Like many of Courant's students, he had only the vaguest ideas of any mathematics that was not analysis, but he had a boundless admiration for Emil Artin and his algebra, for Otto Neugebauer and for German mathematics. Together with Emil Artin, he helped Neugebauer figure out the mathematics in cuneiform tablets. Their success give him a new harangue to add to his repertoire: "The Babylonians knew Fourier analysis." He was at first a strong Germanophile and Francophile. He would sing the praises of Göttingen and of the College de France in rapturous terms. (His fulsome encomia of Europe reminded me of the sickening old Göttingen custom of selling picture postcards of professors.) He would tell us bombastic stories of his days at Göttingen, of his having run away from home to study mathematics (I never believed that one), and of how, shortly after his arrival in Göttingen, Courant himself visited him in his quarters while the landlady watched in awe.
His views on European universities changed radically after he made a lecture tour in 1954; from that time on, he became a champion of American know-how.
He related well to his superiors and to those whom he considered to be his inferiors (such as John Riordan, whom he used to patronize), but his relations with his equals were uneasy at best. He was particularly harsh with Mark Kac. Kitty Kac once related to me an astonishing episode. One summer evening at Cornell Mark and Kitt were sitting on the Fellers' back porch in the evening. At some point in the conversation, Feller began a critique of Kac's work, paper by paper, of Kac's working habits, and of his research program. He painted a grim picture of Kac's future, unless Mark followed Willy's advice to master more measure theory and to use almost-everywhere convergence rather than the trite (to Willy) convergence in distribution. As Kitty spoke to me—a few years after Mark's death, with tears in her eyes—I could picture Feller carried away by the sadistic streak that emerges in our worst moments, when we tear someone to shreds with the intention of forgiving him the moment he begs for mercy.
I reassured Kitty that the Feynman-Kac formula (as Jack Schwartz named it in 1955) will be remembered in science long after Feller's book is obsolete. I could almost hear a sigh of relief, forty-five years after the event.
Emil Artin came to Princeton from Indiana shortly after Wedderburn's death in 1946. Rumor had it (se non é vero é ben trovato) that Indiana University had decided not to match the Princeton offer, since during the ten years of his tenure he had published only one research paper, a short proof of the Krein-Milman theorem in "The Piccayune [sic] Sentinel," Max Zorn's samizdat magazine.
A few years later, Emil Artin had become the idol of Princeton mathematicians. His mannerisms did not discourage the cult of personality. His graduate students would imitate the way he spoke and walked, and they would even dress like him. They would wear the same kind of old black leather jacket he wore, like that of a Luftwaffe pilot in a war movie. As he walked, dressed in his too-long winter coat, with a belt tightened around his waist, with his light blue eyes and his gaunt fact, the image of a Wehrmacht officer came unmistakably to mind. (Such a military image is wrong, I learned years later from Jürgen Moser. Germans see the Emil Artin "type" as the epitome of a period of Viennese Kultur.)
He was also occasionally seen wearing sandals (like those worn by Franciscan friars), even in cold weather. His student Serge Lang tried to match eccentricities by never wearing a coat, although he would always wear heavy gloves every time he walked out of Fine Hall, to protect himself against the rigors of winter.
He would spend endless hours in conversation with his few protégés (at that time, Lang and Tate), in Fine Hall, at his home, during long walks, even via expensive long-distance telephone calls. He spared no effort to be a good tutor, and he succeeded beyond all expectations.
He was, on occasion, tough and rude to his students. There were embarrassing public scenes when he would all of a sudden, at the most unexpected times, lose his temper and burst into a loud and unseemly "I told you a hundred times that..." tirade directed at one of them. One of these outbursts occurred once when Lang loudly proclaimed that Pólya and Szegö's problems were bad for mathematical education. Emil Artin loved special functions and explicit computations, and he relished Pólya and Szegö's "Aufgaben und Lehrsätze," though his lectures were the negation of any anecdotal style.
He would also snap back at students in the honors freshman calculus class which he frequently taught. He might throw a piece of chalk or a coin at a student who had asked too silly a question ("What about the null set?"). A few weeks after the beginning of the fall term, only the bravest would dare ask any more questions, and the class listened in sepulchral silence to Emil Artin's spellbinding voice, like a congregation at a religious service.
He had definite (and definitive) views on the relative standing of most fields of mathematics. He correctly foresaw and encouraged the rebirth of interest in finite groups that was to begin a few years later with the work of Feit and Thompson, but he professed to dislike semigroups. Schützenberger's work, several years after Emil Artin's death, has proved him wrong: the free sernigroup is a far more interesting object than the free group, for example. He inherited his mathematical ideals from the other great German number theorists since Gauss and Dirichlet. To all of them, a piece of mathematics was the more highly thought of, the closer it came to Germanic number theory.
This prejudice gave him a particularly slanted view of algebra. He intensely disliked Anglo-American algebra, the kind one associates with the names of Boole, C. S. Peirce, Dickson, the late British invariant theorists (like D. E. Littlewood, whose proofs he would make fun of), and Garrett Birkhoff's universal algebra (the word "lattice" was expressly forbidden, as were several other words). He thought this kind of algebra was "no good"—rightly so, if your chief interests are confined to algebraic numbers and the Riemann hypothesis. He made an exception, however, for Wedderburn's theory of rings, to which he gave an exposition of as yet unparalleled beauty.
A great many mathematicians in Princeton, too awed or too weak to form opinions of their own, came to rely on Emil Artin's pronouncements like hermeneuts on the mutterings of the Sybil at Delphi. He would sit at teatime in one of the old leather chairs ("his" chair) in Fine Hall's common room, and deliver his opinions with the abrupt definitiveness of Wittgenstein's or Karl Kraus's aphorisms. A gaping crowd of admirers and worshippers, often literally sitting at his feet, would record them for posterity. Sample quips:
"If we knew what to prove in non-Abelian class field theory, we could prove it"; "Witt was a Nazi, the one example of a clever Nazi" (one of many exaggerations). Even the teaching of undergraduate linear algebra carried the imprint of Emil Artin's very visible hand: we were to stay away from any mention of bases and determinants (a strange injunction, considering how much he liked to compute). The alliance of Emil Artin, Claude Chevalley, and André Weil was out to expunge all traces of determinants and resultants from algebra. Two of them are now probably turning in their graves.
His lectures are best described as polished diamonds. They were delivered with the virtuoso's spontaneity that comes only after lengthy and excruciating rehearsal, always without notes. Very rarely did he make a mistake or forget a step in a proof. When absolutely lost, he would pull out of his pocket a tiny sheet of paper, glance at it quickly, and then turn to the blackboard, like a child caught cheating.
He would give as few examples as he could get away with. In a course in point-set topology, the only examples he gave right after defining the notion of a topological space were a discrete space and an infinite set with the finite cofinite topology. Not more than three or four more examples were given in the entire course.
His proofs were perfect but not enlightening. They were the end results of years of meditation, during which all previous proofs of his and of his predecessors were discarded one by one until he found the definitive proof. He did not want to admit (unlike a wine connoisseur, who teaches you to recognize vin ordinaire before allowing you the bonheur of a premier grand cru) that his proofs would best be appreciated if he gave the class some inkling of what they were intended to improve upon. He adamantly refused to give motivation of any kind in the classroom, and stuck to pure concepts, which he intended to communicate directly. Only the very best and the very worst responded to such shock treatment: the first because of their appreciation of superior exposition, and the second because of their infatuation with Emil Artin's style. Anyone who wanted to understand had to figure out later "what he had really meant."
His conversation was in stark contrast to the lectures: he would then give out plenty of relevant and enlightening examples, and freely reveal the hidden motivation of the material he had so stiffly presented in class.
It has been claimed that Emil Artin inherited his flair for public speaking from his mother, an opera singer. More likely, he was driven to perfection by a firm belief in axiomatic Selbsständigkeit. The axiomatic method was only two generations old in Emil Artin's time, and it still had the force of a magic ritual. In his day, the identification of mathematics with the axiomatic method for the presentation of mathematics was not yet thought to be a preposterous misunderstanding (only analytic philosophers pull such goofs today). To Emil Artin, axiomatics was a useful technique for disclosing hidden analogies (for example, the analogy between algebraic curves and algebraic number fields, and the analogy between the Riemannian hypothesis and the analogous hypothesis for infinite function fields, first explored in Emil Artin's thesis and later generalized into the "Weil conjectures"). To lesser minds, the axiomatic method was a way of grasping the "modern" algebra that Emmy Noether had promulgated, and that her student Emil Artin was the first to teach. The table of contents of every algebra textbook is still, with small variations, that which Emil Artin drafted and which van der Waerden was the first to develop. (How long will it take before the imbalance of such a table of contents—for example, the overemphasis on Galois theory at the expense of tensor algebra—will be recognized and corrected?)
At Princeton, Emil Artin and Alonzo Church inspired more loyalty in their students than Bochner or Lefschetz. It is easy to see why. Both of them were prophets of new faiths, of two conflicting philosophies of algebra that are still vying with each other for mastery.
Emil Artin's mannerisms have been carried far and wide by his students and his students' students, and are now an everyday occurrence (whose origin will soon be forgotten) whenever an algebra course is taught. Some of his quirks have been overcompensated: Serge Lang will make a volte-face on any subject, given adequate evidence to the contrary; Tate makes a point of being equally fair to all his doctoral students; and Arthur Mattuck's lectures are an exercise in high motivation. Even his famous tantrums still occasionally occur. A few older mathematicians still recognize in the outbursts of the students the gestures of the master.
No one who talked to Lefschetz failed to be struck by his rudeness. I met him one afternoon at tea, in the fall term of my first year at Princeton, in the Fine Hall common room. He asked me if I was a graduate student: after I answered in the negative, he turned his back and behaved as if I did not exist. In the spring term, he suddenly began to notice my presence. He even remembered my name, to my astonishment. At first, I felt flattered until (perhaps a year later) I realized that what he remembered was not me, but the fact that that I had an Italian name. He had the highest regard for the great Italian algebraic geometers, for Castelnuovo, Enriques, and Severi, who were slightly older than he was, and who were his equals in depth of thought as well as in sloppiness of argument. "You should have gone to school in Rome in the twenties. That was the Princeton of its time!" he told me.
He was rude to everyone, even to the people who doled out funds in Washington and to mathematicians who were his equals. I recall, Lefschetz meeting Zariski, probably in 1957 (while Hironaka was already working on the proof of the resolution of singularities for algebraic varieties). After exchanging with Zariski warm and loud Jewish greetings (in Russian), he proceeded to proclaim loudly (in English) his skepticism on the possibility of resolving singularities for all algebraic varieties. "Ninety percent proved is zero percent proved!," he retorted to Zariski's protestations, as a conversation stopper. He had reacted similarly to several other previous attempts that he had to shoot down. Two years later he was proved wrong. However, he had the satisfaction of having been wrong only once.
He rightly calculated that skepticism is always a more prudent policy when a major mathematical problem is at stake, though it did not occur to him that he might express his objections in less obnoxious language. When news first came to him from England of Hodge's work on harmonic integrals and their relation to homology, he dismissed it the work of a crackpot, in a sentence that has become a proverbial mathematical gaffe. After that débacle, he became slightly more cautious.
Solomon Lefschetz was an electrical engineer trained at the École Centrale, one of the lesser of the French grandes écoles. He came to America probably because, as a Russian-Jewish refugee, he had trouble finding work in France. A few years after arriving in America, an accident deprived him of the use of both hands. He went back to school and got a quick Ph.D. in mathematics at Clark University (which at that time had a livelier graduate school than it has now). He then accepted instructorships at the Universities of Nebraska and Kansas, the only means he had to survive. For a few harrowing years he worked night and day, publishing several substantial papers a year in topology and algebraic geometry. Most of the ideas of present-day algebraic topology were either invented or developed (following Poincaré's lead) by Lefschetz in these papers; his discovery that the work of the Italian algebraic geometers could be recast in topological terms is only slightly less dramatic.
To no one's surprise (except that of the anti-Semites who still ruled over some of the Ivy League universities) he received an offer to join the Princeton mathematics department from Luther Pfahler Eisenhart, the chairman, an astute mathematician whose contributions to the well-being of mathematics have never been properly appreciated (to his credit, his books, carefully and courteously written as few mathematics books are, are still in print today).
His colleagues must have been surprised when Lefschetz himself started to develop anti-Semitic feelings which were still lingering when I was there. One of the first questions he asked me after I met him was whether I was Jewish. In the late thirties and forties, he refused to admit any Jewish graduate students in mathematics. He claimed that, because of the Depression, it was too difficult to get them jobs after they earned their Ph.D.s. He liked and favored red-blooded American boyish Wasp types (like Ralph Gomory), especially those who came from the sticks, from the Midwest, or from the South.
He considered Princeton to be a just reward for his hard work in Kansas, as well as a comfortable, though only partial, retirement home. After his move he did little new work of his own in mathematics, though he did write several books, among them the first comprehensive treatise on topology. This book, whose influence on the further development of the subject was decisive, hardly contains one completely correct proof. It was rumored that it had been written during one of Lefschetz's sabbaticals away from Princeton, when his students did not have the opportunity to revise it and eliminate the numerous errors, as they did with all of their teacher's other writings.
He despised mathematicians who spent their time giving rigorous or elegant proofs for arguments which he considered obvious. Once, Spencer and Kodaira, still associate professors, proudly explained to him a clever new proof they had found of one of Lefschetz's deeper theorems. "Don't come to me with your pretty proofs. We don't bother with that baby stuff around here!" was his reaction. Nonetheless, from that moment on he held Spencer and Kodaira in high esteem. He liked to repeat, as an example of mathematical pedantry, the story of one of E. H. Moore's visits to Princeton, when Moore started a lecture by saying: "Let a be a point and let b be a point." "But why don't you just say 'Let a and b be points'!" asked Lefschetz. "Because a may equal b," answered Moore. Lefschetz got up and left the lecture room.
Lefschetz was a purely intuitive mathematician. It was said of him that he had never given a completely correct proof, but had never made a wrong guess either. The diplomatic expression "open reasoning" was invented to justify his always deficient proofs. His lectures came close to incoherence. In a course on Riemann surfaces, he started with a string of statements in rapid succession, without writing on the blackboard: "Well, a Riemann surface is a certain kind of Hausdorff space. You know what a Hausdorff space is, don't you? It is also compact, o.k.? I guess it is also a manifold. Surely you know what a manifold is. Now let me tell you one nontrivial theorem: the Riemann-Roch Theorem." And so on until all but the most faithful students dropped out of the course.
I listened to a few of his lectures, curious to find out what he might be saying in a course on ordinary differential equations he had decided to teach on the spur of the moment. He would be holding a piece of chalk with his artificial hands, and write enormous letters on the blackboard, like a child learning how to write. I could not make out the sense of anything he was saying, nor whether what he was saying was gibberish to me alone or to everyone else as well. After one lecture, I asked a rather senior-looking mathematician who had been religiously attending Lefschetz's lectures whether he understood what the lecturer was talking about. I received a vague and evasive answer. After that moment, I knew.
When he was forced to relinquish the chairmanship of the Princeton mathematics department for reasons of age, he decided to promote Mexican mathematics. His love/hate of the Mexicans occasionally got him into trouble. Once, in a Mexican train station, he spotted a charro dressed in full regalia, complete with a pair of pistols and rows of cartridges across his chest. He started making fun of the charro's attire, adding some deliberate slurs in his excellent Spanish. His companions feared that the charro might react the way Mexicans traditionally react to insult. In fact, the charro eventually stood up and reached for his pistols. Lefschetz looked at him straight in the face and did not back off. There were a few seconds of tense silence. "Gringo loco!" said the charro finally, and walked away. When Lefschetz decided to leave Mexico and come back to the United States, the Mexicans awarded him the Order of the Aztec Eagle.
During Lefschetz's tenure as chairman of the mathematics department, Princeton became the world center of mathematics. He had an uncanny instinct for sizing up mathematicians' abilities, and he was invariably right when sizing up someone in a field where he knew next to nothing. In topology, however, his judgment would occasionally slip, probably because he became partial to work that he half understood.
His standards of accomplishment in mathematics were so high that they spread by contagion to his successors, who maintain them to this day. When addressing an entering class of twelve graduate students, he told them in no uncertain terms: "Since you have been carefully chosen among the most promising undergraduates in mathematics in the country, I expect that you will all receive your Ph.D.s rather sooner than later. Maybe one or two of you will go on to become mathematicians."