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By Gian-Carlo Rota

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, Lévy, 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. Nonetheless, 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 one 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

0 to 2p cos2 x dx

equals the integral

0 to 2p sin2 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 applicatons 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 gave 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 Collège 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.


For a more standard, but less lively, account of Feller's professional life read :

Feller was indeed all that Gian-Carlo Rota says, but his books are still wonderful half a century after having been written.