Adventures of a Mathematician

Chapter 4. Princeton Days

I first heard about John von Neumann from my high school teacher Zawirski. Kuratowski also described von Neumann’s results and his personality. He told me how in a Berlin taxicab von Neumann had explained in a few sentences much more than he, Kuratowski, would have gotten by correspondence or conversation with other mathematicians about questions of set theory, measure theory, and real variables. Banach, too, talked about him. He told me how at the 1927 Lwów meeting he and other mathematicians, Stozek among them, had made von Neumann drunk at the congress banquet by plying him with vodka, to the extent that he had to leave the table to go to the toilet. He came back and continued the mathematical conversation without having lost his train of thought.

It was only toward the end of 1934 that I entered into correspondence with von Neumann. He was then in the United States, a very young professor at the Institute for Advanced Studies in Princeton. I wrote him about some problems in measure theory. He had heard about me from Bochner, and in his reply he invited me to come to Princeton for a few months, saying that the Institute could offer me a $300 stipend. I met him shortly after my return from England.

In the fall of 1935 a topology conference had been organized in Moscow. Alexandroff invited me to attend. At that time relations between Poland and Soviet Russia were strained. Passport applications in Poland for travel to Russia involved so much red tape that I did not receive my passport in time and thus missed going to the meeting. Von Neumann wrote me that he would be passing through Warsaw on his way back from Moscow, suggesting that we meet there. Samuel Eilenberg, a young Warsaw mathematician well known for his very ingenious topological results, and I went to meet the returning western group. At the station von Neumann (whom I was seeing for the first time) was accompanied by two American mathematicians, Garrett Birkhoff and Marshall Stone. We all conversed in English. Eilenberg spoke it a little; I spoke it adequately thanks to my Cambridge stay. Von Neumann would break into German occasionally.

From Kuratowski’s description, I had imagined him to be slim, as he apparently had been in 1927. He was instead rather plump, though not as corpulent as he was to become later. The first thing that struck me about him were his eyes — brown, large, vivacious, and full of expression. His head was impressively large. He had a sort of waddling walk. (This reminds me that when I first saw his grandson, Malcolm, the son of his daughter Marina, I found it uncanny to watch this little three-year-old perambulate down a long hotel corridor with his grandfather’s waddle, holding his hands behind his back exactly like Johnny. Since he had been born after his grandfather’s death, he could not possibly have been imitating him. It would seem that gestures, motions, and other time-dependent phenomena — not merely static characteristics or matters of spatial configuration — can be transmitted genetically.)

Von Neumann appeared quite young to me, although he was in his early thirties, some five or six years older than I. (I have always had mixed feelings toward people older than myself: on the one hand, something like respect; on the other, a slight feeling of superiority, of having a greater share in the future.) At once I found him congenial. His habit of intermingling funny remarks, jokes, and paradoxical anecdotes or observations of people into his conversation, made him far from remote or forbidding.

During this brief visit, Stone, von Neumann, and Birk-hoff gave a joint seminar at the Warsaw section of the Polish Mathematical Society. Their subject was the lattice theoretical foundations of quantum theory logic. Von Neumann gave most of the lecture, Birkhoff talked briefly, and Stone asked questions. Actually I had mixed impressions about this talk. I was not at all convinced that it had much to do with novel physical ideas. In fact, I thought the points were a bit stretched and the big notion of quantum theory logic a bit artificially contrived. I had a number of other conversations with Johnny, mainly on measure theory (about which I had sent him reprints of my earlier papers). We also talked a little about his recent work on the theory of Hilbert space operators, although I was not especially knowledgeable about or interested in it. Then he gave me some practical advice about my forthcoming trip to Princeton.

In connection with the Moscow topological meeting, several years after World War II, I received a letter from the French mathematician Leray, who with the Lwów mathematician Juliusz Schauder had written a celebrated paper on fixed points for transformations in function spaces and applications in the theory of differential equations. Schauder, our mutual friend, was murdered by the Nazis. Leray wanted to have a photograph of him for himself and for Schauder’s daughter who survived the war and lives in Italy. But he could not find any in Poland or anywhere and he wrote me asking whether I might have a snapshot. Some months after Johnny von Neumann’s death I was looking at some of the books in his library and a group photo of the participants in the Moscow conference fell out. Schauder was there, as were Alexandroff, Lefschetz, Borsuk, and some dozen other topologists. I sent this photograph to Leray. It has since been reproduced in several publications.

Just as in Lwów, the Warsaw mathematicians gathered in a pastry shop and discussed mathematics for hours. They also frequented the famous Fuker wine shop in the old town. This is where Eilenberg and I took Johnny and his companions to drink Fuker’s celebrated hydromel. Here he entertained us with the story of how at the request of Princeton friends he had bought several pounds of caviar in Moscow to bring back to the U.S. and had asked a steward to store it in the icebox of the restaurant car. In the morning when they woke up, in Poland, they discovered that the restaurant car had been uncoupled at the Polish-Russian border. They were returning to the States caviarless! He talked also about his own decision to emigrate to America and the general impracticality and lack of foresight of European scientists. In the German universities the number of existing and prospective vacancies for professorships was extremely small — something like two or three in the entire country for the next two years. Yet most of the two or three score docents counted on obtaining a professorship in the near future. With his typical rational approach, von Neumann computed that the expected number of professorial appointments within three years was three, whereas the number of docents was forty. This is what had made him decide to emigrate, not to mention the worsening political situation, which made him feel that unhampered intellectual pursuits would become difficult. In 1930, he accepted an offer of a visiting professorship at Princeton University, and, in 1933, shortly after the creation of the Institute, he was invited to become the youngest member of the permanent faculty of the Institute for Advanced Studies.

In December 1935 I sailed on the English ship Aquitania from Le Havre on my first transatlantic crossing. The weather was beautiful for the first two days; then a violent storm sprang up, and I became seasick. When the boat approached New York, the sea calmed down and my sea-sickness stopped.

After two days in New York, I tried to reach von Neu-mann in Princeton but got no answer, so I called the Institute. It was quite an experience for me to go into an American telephone booth for the first time. When the operator said, “Hold the wire!” I did not understand the expression and asked “Which wire should I hold?” I reached Solomon Lefschetz, a professor at the University, who told me how to get to Princeton from New York. He said it was very easy, that there were trains every hour. I could not understand that. Princeton I knew was a very small town, why should there be a train every hour? I did not know it was on the main line to Philadelphia and Washington.

In Princeton I went straight to register at the Institute which was housed in Fine Hall, a University building, for it did not have its own quarters yet. A young and pretty Miss Flemming and an older Miss Blake received me. I was greeted with smiles. It surprised me, and I wondered if there was something a little funny about the way I was dressed or whether my trousers were not properly buttoned (there were no zippers in those days).

I checked in at a boarding house and went directly to visit von Neumann in his large and impressive house. A black servant let me in, and there was Solomon Bochner in the living-room and a baby crawling on the floor. (The baby was von Neumann’s daughter Marina, six months old at the time.) Marietta, his first wife, also a Hungarian, greeted me. I knew of Bochner since we had corresponded about mathematics. Bochner and von Neumann were talking politics.

Von Neumann expressed great pessimism about the possibility of a war in Europe. (This was about three years before the actual outbreak.) Apparently he had a rather clear picture of the catastrophes to come. He saw Russia as the chief antagonist to Nazi Germany. Believing that the French army was strong, I asked, ’’What about France?” “Oh! France won’t matter,” he replied. It was really very prophetic.

My lodgings were in a boarding house on Vandevanter Street, if I remember correctly. There were six or eight other men there, not all students, and we ate together. I remember how the conversation was at first completely incomprehensible to me although I knew English. The American accent took me by surprise, and I missed most of what was being said. Then after a week I understood everything. This is a common experience, not only with languages but also with mathematics — a discontinuous process. Nothing, nothing, at first, and suddenly one gets the hang of it.

I became a frequent visitor at the von Neumanns who were very sociable and held parties two or three times a week. These were not completely carefree; the shadow of coming world events pervaded the social atmosphere. There I met the Alexanders, who were great friends of the von Neumanns. James Alexander, also a professor at the Institute, was an original topologist, the creator of novel problems and strange “pathological” examples of topological objects. He was the scion of a wealthy family and very eccentric.

At a party a few weeks later, I saw a man who must have been fifty but seemed to me infinitely old — I was twenty-six at the time. He was sitting in a big chair with a nice young lady on his knee. They were drinking champagne. I passed by Johnny and asked, “Who is this gentleman?” “Oh! don’t you know? He is von Kárman, the famous aerodynamicist.” Von Kárman was one of Johnny’s friends. And he added: “Don’t you know that he invented consulting?” Von Kárman was one of the first scientists who learned to fly a plane in the first World War. He told me that he had one of the low numbered international flying licenses. His flying experiences directly influenced his ideas about jet engines, which became so important in World War II developments. Much later I came to know him quite well. He used to say that engineers are people who perpetuate the mistakes made in the previous generation. In 1968 I found myself with him in Israel at a meeting on hydrodynamics. By then he was a rich old man. It was his first visit to that country, and he was so moved and impressed by what he saw that he gave away five- and ten-dollar bills as tips to waiters and taxi drivers, no matter what the size of the check.

Johnny was always impressed by people who were successful in political or organizational activities or in physical exploits, and lie cultivated them. It was Johnny who said as we walked past the elegant Gothic Princeton University Chapel: “This is our one-million-dollar protest against materialism.” I do not know whether this bon mot was his own or not, but it shows his sense of humor about money.

In those days he still called me “Mr. Ulam.” Once he said to me as we were driving in the rain and were caught in a traffic jam, “Mr. Ulam, cars are no good for transportation anymore, but they make marvelous umbrellas.” I often remember this when I am caught in the traffic jams of today. Johnny always loved cars but he drove somewhat carelessly.

Johnny lived rather sumptuously. The professors at the Institute were the highest paid academics in the United States — paid even more than at Harvard. This tended to create animosities between Institute and University professors. Also their compensation was in sharp contrast to the almost negligible stipends offered to the fellows and visitors of the Institute.

The great name, the great celebrity, the great light, so far as the general public was concerned, was, of course, Albert Einstein. I first met his assistant, Mayer, a mathematician and a strange person. Then I was introduced to Einstein himself and noticed his rather peculiar English. He would say: “He is a very good formula,” pointing at something on the blackboard!

A cousin of mine, Andrzej Ulam, a banker, came to New York on business about two months after my arrival, and I invited him to visit me in Princeton. It happened that during that week I was giving a talk in some seminar, and my name was listed on the same page of the Institute’s Bulletin as the announcement of Einstein’s regular weekly seminar. This impressed him enormously; he mentioned it in a letter home, and my reputation among friends and family in Poland was made.

Hermann Weyl was also a professor at the Institute. I met him in Princeton and went to his home several times. He was a legendary figure, much older than von Neumann. He had the same wide breadth of interests which impresses me so much. My friend Gian-Carlo Rota, now a professor at MIT, told me much later that he had heard Weyl’s original symmetry lectures and was enormously impressed. They were a bit heavy, but at the same time they gave a feeling of universal culture. More recently Weyl’s purely mathematical schemata or algebraic entities found essential applications as models for the properties of the mysterious neutrino particles and so-called weak interactions important in the beta decay of nuclei.

After Weyl’s first wife died he remarried and lived in Switzerland for a time. He was unaware of the rules and regulations governing the length of time a naturalized American citizen may live abroad without returning to this country and still retain his American citizenship. He lost his by negligence. When it happened, everyone was shocked. Members of the Mathematical Society and the National Academy of Sciences wanted to have him reinstated as a U.S. citizen. This required a special bill in Congress, and some friends asked me to intervene with Senator Anderson, whom I knew well, to help in this matter. In the meantime Weyl had collapsed in a street in Zürich while putting a letter in a mailbox and died of a heart attack.

Every day I went to the Institute for five or six hours. By that time I had quite a number of published papers, and people there knew some of them. I talked with Bochner a good deal and soon after my arrival communicated to him a problem about “Inverting the Bernouilli law of large numbers.” Bochner proved the theorem and published it in The Annals of Mathematics. (By the way, this problem is still solved in a simple case only. The inverse of the law of large numbers, those requiring a measure in a space of measures, has not yet been proved.)

I went to lectures and seminars, heard Morse, Veblen, Alexander, Einstein, and others, but was surprised how little people talked to each other compared to the endless hours in the coffee houses in Lwów. There the mathematicians were genuinely interested in each other’s work, they understood one another because their work revolved around the central theme of set theoretical mathematics. Here, in contrast, several small groups were working in separate areas, and I was somewhat disappointed at this lack of curiosity even though the Institute and the University had a veritable galaxy of celebrities, possibly constituting one of the greatest concentrations of brains in mathematics and physics ever to be assembled. Being a malicious young man, I told Johnny that this reminded me of the division of rackets among Chicago gangsters. The “topology racket” was probably worth five million dollars; the “calculus of variations racket,” mother five. Johnny laughed and added, “No! That is worth only one million.”

There was another way in which the Princeton atmosphere was entirely different from what I expected: it was fast becoming a way station for displaced European scientists. In addition, these were still depression days and the situation in universities in general and in mathematics in particular was very bad. People with impressive backgrounds and good credentials (not only visitors like me, but native-born Americans) were still without jobs several years after getting their doctorates. A very able mathematician and logician friend of mine, who is now a member of the National Academy of Sciences, was then on a miserable stipend at Princeton waiting for a position to open somewhere. One day a telegram came offering him an instructorship at twelve hundred dollars a year. He told me he thought he was dreaming and quickly accepted the job. There were many such cases. At that time I was told that three persons “owned” the American Mathematical Society: Oswald Veblen, G. D. Birkhoff, and Arthur B. Coble, from Illinois. Most academic positions were secured through the recommendations of these three. What a contrast to the vast number of university jobs in mathematics which exist today!

It was Veblen who was responsible for Johnny’s presence at the Institute. He had invited him for a semester’s stay at first, and later arranged for him to remain. He liked Johnny very much and considered him almost as a son.

Oswald Veblen, a nephew of Thorstein Veblen, author of The Theory of the Leisure Class, was a famous American mathematician, tall, slim, Scandinavian-looking, with a caustic sense of humor. He was well known for his work on the foundations of projective geometry and topology.

Veblen organized walks in the Princeton woods, and I was invited to join some of these expeditions during which there was lots of mathematical talk and gossip while he cut dead wood and tree branches to clear the paths.

To my mind the Princeton woods with their thin, spindly trees and marshes were not at all impressive compared to the Polish forests. But it was the first time I heard and saw what seemed like gigantic Kafkaesque frogs. The birds were also very different, and I felt truly on another continent, in a very exotic land.

Throughout these walks and discussions there always lurked in the back of my mind a question: would I receive an invitation from some American institution, which would enable me to remain? More subconsciously than consciously, I was eagerly looking for a way to stay, the reason being the critical political situation in Europe and the catastrophic job situation for mathematicians there, especially Jews. There was very little future for me in Poland and it was becoming increasingly evident that the country was in mortal danger. I also admired the freedom of expression, of work, the sense of initiative, the spirit which was in the air; here the future of the world was beckoning. Even though I did not mention any of this explicitly to Johnny, I became very eager to stay and to take work if it presented itself.

About that time, Kuratowski appeared in Princeton for a month’s visit. He arrived late in the spring, and wondered whether there was any chance that I might be invited to remain in the States for the next academic year. He went to Harvard to give a lecture, and several of the professors there, Birkhoff, Graustein, and others, asked him about me. He probably gave the best references. When he spoke to me about these possibilities, he had mixed feelings. He knew full well that there was very little chance of a professorship for me in Poland, and realized it was good for my future career to remain in the States a while longer, yet he was genuinely sorry at the thought that I might not return.

During this stay, he and Johnny obtained some very strong results on certain types of projective sets. This is a very elegant theory of operations in mathematical logic, going beyond the Aristotelian or Boolean ones. To this day the theory is full of mysterious situations very fundamental to problems in the foundations of mathematics and set theory. Much recent work concerns such projective operations, and some recent results certainly originate from this interesting paper. It is curious how this came about. With his technical virtuosity and depth of penetration, once Johnny had received the starting impulse he was able to find the decisive points. A good example of how collaboration in mathematics is very often fruitful!

Von Neumann invited me one day to give a talk in his seminar on my results in “semi-simple groups,” which was a subject I did not know very much about. I have often succeeded in obtaining rather original and not unimportant results in areas where I did not know the foundations or the details of a theory too well. At this seminar Johnny asked me some very searching and penetrating questions, and I had to think very hard to give satisfactory answers; I did not feel he was doing this to embarrass me, but only because of his overriding objectivity and desire to make things clear.

At some lectures von Neumann sometimes “snowed” the students by elaborating the easier points and quickly glossing over the difficulties, but he always demonstrated his fantastic and to some extent prophetic range of interests in mathematics and its applications and at the same time an objectivity which I admired enormously.

As a mathematician, von Neumann was quick, brilliant, efficient, and enormously broad in scientific interests beyond mathematics itself. He knew his technical abilities; his virtuosity in following complicated reasoning and his insights were supreme; yet he lacked absolute self-confidence. Perhaps he felt that he did not have the power to divine new truths intuitively at the highest levels or the gift for a seemingly irrational perception of proofs or formulation of new theorems. It is very hard for me to understand this. Perhaps it was because on a couple of occasions he had been anticipated, preceded, or even surpassed by others. For instance, he was disappointed that he had not first discovered Gödel’s undecidability theorems. He was more than capable of this, had he admitted to himself the possibility that Hilbert was wrong in his program. But it would have meant going against the prevailing thinking of the time. Another example is when G. D. Birkhoff proved the ergodic theorem. His proof was stronger, more interesting, and more self-contained than Johnny’s.

During my stay in Princeton I felt that there was some hesitation on Johnny’s part about his own work. He was immersed in his new work on continuous geometries and in the theory of classes of operators in Hilbert spaces. I myself was not so interested in problems concerning properties of Hilbert spaces. Johnny, I could feel, was not completely certain of the importance of this work, either. Only when he found, from time to time, some ingenious, technically elegant trick or a new approach did he seem visibly stimulated or relieved of his own internal doubts.

It was at that time that he began to think of problems away from pure mathematics, although this was not the first time in his life. (He had written his famous book on the mathematical foundations of quantum theory in 1929.) He was thinking now more about classical problems in physics. For example, he studied problems of turbulence in hydrodynamics. In his continuous geometries, their elements do not play the role of what we normally consider as “points” in Euclidean space; it is a creation of a “pointless” geometry, a name which lent itself to many an easy joke.

He came back again and again to the possibilities of reformulating the logic of quantum theory, the substance of the talk he gave at the Warsaw seminar. In Princeton he frequently worked on this topic. Listening to his conversation, I saw or felt his own hesitation, and I suffered from doubt, too, because there was no definite experimental possibility of verifying this — it seemed purely a question of logic. Purely “grammatical” approaches never interested me much. When something is merely convenient or typographically useful, it seems less interesting to me than when there is a more real physical base, or if abstract, still somehow palpable. I have to admit that there are cases where formalism by itself has great value — for example, the technique, or rather the notation of Feynman graphs in physics. It is a purely typographical idea, it does not bring in itself any tangible input into a physical picture, nevertheless, by being a good notation it can push thoughts in directions that may prove useful or even novel and decisive. Beyond this (and extremely important), there is the magic of “algorithms,” or symbolism in mathematics. Calculus itself shows the wonder of it. Various transforms, generating functions, and the like perform almost miraculously in mathematical applications.

Von Neumann was the master of, but also a little bit the slave to, his own technique. When he saw that something could be done, he let himself be carried away on tangents. My own feeling is that some of his mathematical work on classes of operators or on quasi-periodic functions, for example, is very interesting technically, but to my taste not terribly important; he could not resist doing it because of his facility.

How terribly important habit is. It may largely determine the characteristics or the nature of the brain itself. Habits influence or perhaps can largely determine the choice of trains of thought in one’s work. Once these are established (and in my opinion they may be established very quickly — sometimes after just a few trials), the “connections” or “programs” or “subroutines” become fixed. Von Neumann had this habit of considering the line of least resistance. Of course, with his powerful brain he could quickly vanquish all small obstacles or difficulties and then go on. But if the difficulty was great right from the start, he would not knock his head against the wall, nor would he — as I once expressed it to Schreier — walk around the fortress and knock here and there to find the weakest spots and try to break through. He would switch to another problem. On the whole in his work habits I would call Johnny more realistic than optimistic.

Johnny was always a hard worker; he had a great energy and toughness behind a physical appearance that was somewhat on the soft side. Each day he would start writing before breakfast. Even at parties in his house, he would occasionally leave the guests to go to his study for half an hour or so to record something that was on his mind.

He may not have been an easy person to live with — in the sense that he did not devote enough time to ordinary family affairs.

Some people, especially women, found him lacking in curiosity about subjective or personal feelings and perhaps deficient in emotional development. But in his conversations with me, I felt that only a certain shyness prevented him from having more explicit discussions along these lines. Such seeming diffidence is not uncommon among mathematicians. Non-mathematicians often reproach us for this and may resent this apparent emotional insensitivity and excessive quantitative and rational bent, especially in attitudes towards mundane matters outside science. Von Neumann was so busy with mathematics, physics, and with academic affairs, not to mention increasingly innumerable activities later on as a consultant to many projects and Government advisory work, he probably could not be a very attentive, “normal” husband. This might account in part for his not-too-smooth home life.

To be sure, he was interested in women, outwardly, in a peculiar way. He would always look at legs and the figure of a woman. Whenever a skirt passed by he would turn and stare — so much so that it was noticed by everyone. Yet this was absentmindedly mechanical and almost automatic. About women in general he once said to me, “They don’t do anything very much.” He meant, of course, nothing much of importance outside of their biological and physiological activities.

He did not show social prejudice and never concealed his Jewish origins (even though I think he had actually been baptized a Christian in his childhood). In fact, he was very proud of the birth of the state of Israel in 1948 and was pleased by the Jewish victories over the surrounding Arab countries — a sort of misplaced nationalism.

His father, a banker, had been titled “von.” In the Austro-Hungarian Empire people were rewarded with titles but these could also be obtained by gifts of money to the government. Johnny never used the full title (neither did von Kárman who was also of Jewish origin). He was ill at ease with people who were self-made or came from modest backgrounds. He felt most comfortable with third- or fourth-generation wealthy Jews. With someone like me, he would then often use Jewish expressions or jokes as a spice to conversation. He was a man of the world, not exactly snobbish but quite conscious of his position, who felt more at ease with people with the same background.

He was broadly educated and well versed in history, especially of the Roman Empire — its power and organization fascinated him. Perhaps part of this interest stemmed from a mathematician’s appreciation of the difference between variables involving individual points, or persons, and groups of such, or classes of things. He was given to finding analogies between political problems of the present and of the past. Sometimes, the analogy was genuinely there, but there were so many other different factors that I don’t think his conclusions were always justified.

In general, he tended not to disagree with people. He would not contradict or dissuade when asked for advice about things they were inclined to do. In matters of ordinary human affairs, his tendency was to go along, even to anticipate what people wanted to hear. He also had the innocent little trick of suggesting that things he wanted done originated with the persons he wanted to do them! I started using this ploy myself after I learned it from him. However, in scientific matters, he did defend the principles he believed in.

When it came to other scientists, the person for whom he had a deep admiration was Kurt Gödel. This was mingled with a feeling of disappointment at not having himself thought of ’’undecidability.” For years Gödel was not a professor at Princeton, merely a visiting fellow, I think it was called. Apparently there was someone on the faculty who was against him and managed to prevent his promotion to a professorship. Johnny would say to me, “How can any of us be called professor when Gödel is not?” When I asked him who it was who was unfriendly to Gödel, he would not tell me, even though we were close friends. I admired his discretion.

As for Gödel, he valued Johnny very highly and was much interested in his views. I believe knowing the importance of his own discovery did not prevent Gödel from a gnawing uncertainty that maybe all he had discovered was another paradox à la Burali Forte or Russell. But it is much, much more. It is a revolutionary discovery which changed both the philosophical and the technical aspects of mathematics.

When we talked about Einstein, Johnny would express the usual admiration for his epochal discoveries which had come to him so effortlessly, for the improbable luck of his formulations, and for his four papers on relativity, on the Brownian motion, and on the photo-electric quantum effect. How implausible it is that the velocity of light should be the same emanating from a moving object, whether it is coming toward you or whether it is receding. But his admiration seemed mixed with some reservations, as if he thought, “Well, here he is, so very great,” yet knowing his limitations. He was surprised at Einstein’s attitude in his debates with Niels Bohr — at his qualms about quantum theory in general. My own feeling has always been that the last word has not been said and that a new “super quantum theory” might reconcile the different premises.

I once asked Johnny whether he thought that Einstein might have developed a sort of contempt for other physicists, including even the best and most famous ones — that he had been deified and lionized too much. No one tried to go him one better by generalizing his theory of relativity, for example, or inventing something which would rival or change or improve it. Johnny agreed. “I think you are right,” he said, “he does not think too much of others as possible rivals in the history of physics of our epoch.”

Comparisons are invidious, and there is no question of any linear order of eminence or greatness in science. Much of it is a question of taste. It is probably as difficult to compare mathematicians linearly or otherwise as it would be to compare musicians, poets, or writers. There are, of course, large and obvious differences in “class.” One could safely say, I think, that Hilbert was probably a greater mathematician than some young teaching assistant chosen at random at a large university. I feel that some of the most permanent, most valuable, most interesting work of von Neumann came towards the end of his life, involving his ideas on computing, on the applications of computing, and on automata. Therefore, when it comes to lasting impact, I think in many ways it might be as great as that of Poincaré’s, who was, of course, quite theoretical and did not actually contribute directly to technology itself. Poincaré was one of the great figures in the history of mathematics. So was Hilbert. As mathematicians’ mathematicians, they are idolized, perhaps a little more than von Neumann. But final judgments have to be left to the future.

One of the luckiest accidents of my life happened the day G. D. Birkhoff came to tea at von Neumann’s house while I was visiting there. He seemed to have heard about me from his son Garrett, whom I had met in Warsaw. We talked and, after some discussion of mathematical problems, he turned to me and said, “There is an organization at Harvard called the Society of Fellows. It has a vacancy. There is about one chance in four that if you were interested and applied you might receive this appointment.” Johnny nodded eagerly in my direction, and I said, “Yes, I might be interested in spending some time at Harvard.” A month later, in April of 1936, I received an invitation to give a talk at the mathematics colloquium there. The talk was followed by an invitation to a dinner at the Society of Fellows. I suppose this was to look me over without my being aware of it.

At the colloquium, I talked about something which is still being worked on, the existence in many structures of a small number of elements which generate subgroups or subsystems dense in the whole structure. (Or, popularly speaking, out of an infinite variety of objects one can pick a few such that by combining them one can obtain, with only a small error, all the others.) The results were something that Jozef Schreier and I had proved a couple of years earlier. I talked with confidence — I don’t remember ever being very nervous about giving talks because I always felt I knew what I was talking about. It must have been well received, for when I returned to Princeton, I found a letter which gladdened me no end. It was from the Secretary of the Harvard Corporation, signed in the English manner “Your Obedient Servant.” It was a nomination to the position of Junior Fellow to begin the following autumn and to last for three years. The conditions were extremely attractive: fifteen hundred dollars a year plus free board and room together with some travel allowances. In those days it seemed a royal offer.

With this in my pocket, I happily began preparations to return to Poland for the summer. To make up for Johnny’s disaster of the summer before, my new Princeton acquaintances gave me an order to bring back a huge amount of caviar. Little did they realize that in Poland, which did not produce it, it was as expensive as in the West.

Chapter 5. Harvard Years

I came to the Society of Fellows during its first few years of existence. Garrett Birkhoff and B. F. Skinner, the psychologist, were among its original members. Most of the Junior Fellows, as we were called, were in their mid-twenties, mainly budding post-doctoral scholars.

I was given a two-room suite in Adams House, next door to another new fellow in mathematics by the name of John Oxtoby. About my age, he did not have his doctor’s degree but was well known at the University of California — where he had done his graduate work — for his brilliance and promise. I took an instant liking to him. He was a tallish, blue-eyed redhead, with a constant good disposition. An attack of polio in his high-school years had severely crippled one leg, so that he had to walk with a crutch.

He was interested in some of the same mathematics I was: in set theoretical topology, analysis, and real function theory. Right off, we started to discuss problems concerning the idea of “category” of sets. “Category” is a notion in a way parallel to but less quantitative than the measure of sets — that is, length, area, volume, and their generalizations. We quickly established some new results, and the fruits of our conversations during the first few months of our acquaintance were published as two notes in Fundamenta. We followed this with an ambitious attack on the problem of the existence of ergodic transformations. The ideas and definitions connected with this had been initiated in the nineteenth century by Boltzmann; five years before work on this had culminated in von Neumann’s paper, followed (and in a way superseded) by G. D. Birkhoff’s more imposing result. Birkhoff, in his trail-breaking papers and in his book on dynamical systems, had defined the notion of ’’transitivity.” Oxtoby and I worked on the completion to the existence of limits in the ergodic theorem itself.

In order to complete the foundation of the ideas of statistical mechanics connected with the ergodic theorem, it was necessary to prove the existence, and what is more, the prevalence of ergodic transformations. G. D. Birkhoff himself had worked on special cases in dynamical problems, but there were no general results. We wanted to show that on every manifold (a space representing the possible states of a dynamical system) — the kind used in statistical mechanics — such ergodic behavior is the rule.

The nature, intensity and long duration of our daily conversations reminded me of the way work had been done in Poland. Oxtoby and I usually sat in my room, which was rather stark, although I had rented a couple of oriental rugs to furnish it, or in his own, which was even more spartan.

We discussed various approaches to a possible construction of these transformations. With my usual optimism, I was somehow sure of our ultimate success. We kept G. D. Birkhoff informed of the status of our attacks on the problem. He would smile when I talked to him at dinner at the Society of Fellows, partly amused, partly impressed by our single-minded persistence, and partly skeptical, though he really had an open mind about our chances. He would check what I told him with Oxtoby, a more cautious person. It took us more than two years to break through and to finish a long paper, which appeared in The Annals of Mathematics in 1941 and which I consider one of the more important results that I had a part in.

The chairman of the Society was L. J. Henderson, a famous biologist, author of a book, The Fitness of Environment, which enjoyed a great popularity at the time, not only among specialists, but quite generally. L.J., as he was called, was a great Francophile. Indeed, the Society was molded along the lines of the Fondation Thiers in Paris, rather than on the Cambridge or Oxford systems of fellows in the college.

The Society was composed of some five or six Senior Fellows and about twenty-two Junior Fellows.

The Senior Fellows were well-known distinguished professors, like John Livingston Lowes in literature, Samuel Eliot Morison, the historian, Henderson, and Alfred North Whitehead, the famous English philosopher, who had already retired from his professorship at Harvard when I entered the Society. I often had the pleasure of sitting next to him at the traditional Monday-night dinners of the Society.

Some of the Junior Fellows gave me the impression of being a somewhat precious group of young men, as far as manners were concerned. Oxtoby, Willard Quine (really a logician), and I were the only mathematicians among them. Among the physicists there were several who later became very well known, such as John Bardeen, Ivan Getting, and Jim Fisk. Among the biologists, I remember Robert B. Woodward, the chemist who first synthesized quinine and other important biological substances. Paul Samuelson, the economist who served as advisor to President Kennedy, was there; also Ivar Einerson, a great scholar in linguistics; Henry Guerlach, who became a historian of science; and Harry Levin, in English literature. Levin was rather proustian in his manner. He loved to engage in sophisti cated and what seemed to me occasionally rather precious discussions. Another foreign-born member was George Hanfmann, an archaeologist. Hanfmann was obviously a very learned person, and I appreciated his erudition. We shared the same fondness for Greek and Latin literature.

The logician Willard Quine was friendly and outgoing. He was interested in foreign countries, their culture and history, and knew a few words of Slavic languages, which he used on me with great gusto. He already had made a reputation in mathematical logic. I remember him as slim, dark-haired, dark-eyed — an intense person. During the presidential election of 1936 in which Franklin D. Roosevelt defeated Landon, I met him on the stairs of Widener Library at nine in the morning, after Roosevelt’s landslide victory. We stopped to chat and I asked him: “Well, what do you think of the results?” “What results?” he replied. ’’The presidential election, of course,” I said. “Who is President now?” he asked casually. This was characteristic of many in academe. I once heard that, during Charles W. Eliot’s presidency at Harvard, a visitor to his house was told, “The President is away in Washington to see Mr. Roosevelt”! (This was Theodore Roosevelt.)

I had my meals at Adams House, and the lunches there were particularly agreeable. We sat at a long table — young men and sometimes great professors; the conversations were very pleasant. But often, towards the end of a meal, one after the other would gulp his coffee and suddenly announce: “Excuse me, I’ve got to go to work!” Young as I was I could not understand why people wanted to show themselves to be such hard workers. I was surprised at this lack of self-assurance, even on the part of some famous scholars. Later I learned about the Puritan belief in hard work — or at least in appearing to be doing hard work. Students had to show that they were conscientious; the older professors did the same. This lack of’ self-confidence was strange to me, although it was less objectionable than the European arrogance. In Poland, people would also pretend and fabricate stories, but in the opposite sense. They might have been working frantically all night, but they pretended they never worked at all. This respect for work appeared to me as part of the Puritan emphasis on action versus thought, so different from the aristocratic traditions of Cambridge, England, for example.

The Society’s rooms were in Eliot House. We Junior Fellows would meet there on Mondays and Fridays for lunch, and for the famous Monday-night dinners which gathered Junior and Senior Fellows together around a long T-shaped table which was said to be the one featured in Oliver Wendell Holmes’ Autocrat of the Breakfast-Table. Henderson had secured it from some Harvard storeroom.

President Lowell attended almost every Monday dinner. He was fond of re-creating the Battle of Jutland of World War I, moving knives and forks and saltcellars around on the dinner table to show the positions of the British and German fleets. From time to time he would also betray his doubts and even remorse about the Sacco and Vanzetti case. He would recount it — not so much to defend but rather to restate the position of the court and the subsequent legal steps. He had been a member of one of the review committees.

Good French Burgundies or Alsatian wines accompanied the meals. These were the pride and joy of Henderson, who once told me that if he ever deserved a statue in Cambridge, he would like to be put in Harvard Square with a bottle of wine in his hands, in commemoration of his having been the first person to obtain University funds for a wine cellar. George Homans, one of the Junior Fellows, a descendant of President John Adams, was one of the young men entrusted with the selection and sampling of wines. I considered it a great distinction when I, too, was put on the wine-tasting committee of the Society. This was my very first administrative job in America! The Society is still very much alive today at Harvard, and it continues to hold its Monday-night dinners where former fellows are always welcome.

In 1936 the depression appeared to be ending. Harvard University seemed relatively untouched by this cataclysm. After the colloquium talk I gave there just before my appointment to the Society of Fellows, I remember Professor William Graustein telling me that at Harvard the professors had not felt the depression at all. This left me wondering at their lack of involvement in the general problems of the country or in the affairs of Massachusetts or even of Cambridge. It was evident that campus life in America meant at least partial isolation from the rest of society. Professors lived almost entirely among themselves and had very little contact with the rest of the professional or creative community as in Lwów. This had both good and bad effects: more time for scholarly work, but very little influence on the life of the country or vice versa. As everyone knows, things changed somewhat after World War II. In the Kennedy administration, for example, Harvardians had a great deal to do with the affairs of government and for a time the influence of scientists became even paramount.

Activities at the Society of Fellows were of course only one facet of my life at Harvard. I had many contacts with the younger members of the faculty at the university and quite often saw and talked with the senior professors and with G. D. Birkhoff himself. His son Garrett, a tall, good-looking, and brilliant mathematician, some two years younger than I, became a friend, and we saw each other nearly every day.

Even though membership in the Society did not require teaching of any kind, Professor Graustein asked me to teach an elementary undergraduate section of a freshman course called Math 1A. (It may even be that the late President Kennedy was for a while a student in this class. I remember a name like that and someone saying that the young man was a rather remarkable person. He left to go abroad in the middle of the term. Years later when I met President Kennedy I forgot to ask him whether he had really taken that course.)

I had given talks and seminars, but not yet taught a regular class, and I found this teaching interesting. The rule for young instructors was to follow very closely the prescribed textbook. Apparently I did not do too badly, for in an evaluation of teachers the student newspaper praised me as an interesting instructor. Soon after the beginning of the course, G. D. Birkhoff came to inspect my performance. Perhaps he wanted to check my English. He sat at the back of the room and watched as I explained to the students how to write equations of parallel lines in analytic geometry. Then I said that next we would study the formulae for perpendicular lines, which, I added, were “more difficult.” After the lecture Birkhoff carne to me and commented, “You’ve done very well, but I would not have said that perpendicular lines are more difficult.” I replied that I believed on the contrary that students would remember better this way than if I said everything was easy. Birkhoff smiled at this attempt at pedagogy on my part. I think he liked my independence and outspoken ways, and I saw him rather frequently.

Shortly after I arrived in Cambridge, he had invited me to dine at his house. It was my first introduction to strange dishes like pumpkin pie. After dinner, which was pleasant enough, I got ready to leave and G.D. took my overcoat to help me into it. This sort of courtesy was unheard of in Poland; an older man would never have helped a much younger one. I remember blushing crimson with embarrassment.

I frequently ate lunch with his son Garrett, and we often took walks together. We talked much about mathematics and also indulged in the usual gossip that mathematicians love. Surely it is a shallow theme to evaluate how good X or Y is, but it is a characteristic of our tribe. The reader may have noticed that I practice this, too. Mathematics being more in the nature of an art, values depend on personal tastes and feelings rather than on objective factual notions. Mathematicians tend to be rather vain — though less so than opera tenors or artists. But as every mathematician knows some special bit of math better than anyone else, and math is such a vast and now more and more specialized subject, some like to propose linear orders of “class” among the better-known ones and to comment on their relative merits. On the whole, it is a harmless if somewhat futile pastime.

I remember that at the age of eight or nine I tried to rate the fruits I liked in order of “goodness.” I tried to say that a pear was better than an apple, which was better than a plum, which was better than an orange, until I discovered to my consternation that the relation was not transitive — namely, plums could be better than nuts which were better than apples, but apples were better than plums. I had fallen into a vicious circle, and this perplexed me at that age. Mathematicians’ ratings are something like this.

Many mathematicians are also sensitive about what they consider their most beautiful mental offspring — results or theorems — and they tend to be possessive about them. Paradoxically, they also show a tendency to consider their own work as difficult and other work as easier. This is exactly opposite in other fields where the better acquainted one is with something the easier it seems.

Mathematicians are also prone to disputes, and personal animosities between them are not unknown. Many years later, when I became chairman of the mathematics department at the University of Colorado, I noticed that the difficulties of administering N people was not really proportional to N but to N². This became my first “administrative theorem.” With sixty professors there are roughly eighteen hundred pairs of’ professors. Out of that many pairs it was not surprising that there were some whose members did not like one another.

Among the Harvard mathematicians I knew, I should mention Hassler Whitney, Marshall Stone, and Norbert Wiener. Whitney was a young assistant professor, interesting not only as a mathematician. He was friendly, but rather taciturn — psychologically of a type one encounters in this country more frequently than in central Europe — with wry humor, shyness but self-assurance, a probity which shines through, and a certain genius for persistent and deep follow-through in mathematics.

Marshall Stone, whom I had met when he came through Warsaw with von Neumann and Birkhoff in 1935 on the way back from the Moscow Congress, had had a meteoric career at the university, although he was only thirty-one years old. Already a full professor, he was quite influential in the affairs of the department and of the university for that matter. He wrote a classic work, a comprehensive and authoritative book on Hilbert space, an infinitely dimensional generalization of the three-dimensional or n-dimensional Euclidean space, mathematically basic to modern quantum theory in physics. He was the son of Harlan Stone, Chief Justice of the Supreme Court. It is said that his father proudly said of Marshall’s mathematical achievements, “I am puzzled but happy that my son has written a book of which I understand nothing at all.”

And there was Norbert Wiener! I met him at a colloquium talk I gave during my first year at Harvard. I was lecturing on some problems of topological groups, and mentioned a result I had obtained in Poland in 1930 on the impossibility of completely additive measure defined in all subsets of a given set. Wiener, who always sat at lectures in a semi-somnolent state except when he heard his name (at which he would suddenly jump up, then sit back in a very comical way) interrupted me to say, “Oh! Vitali has proved something like that already.” I replied that I knew Vitali’s result and that it was much weaker than mine because it required an additional property — namely equality for congruent sets — whereas my result did not make any such postulate and was a much stronger, purely set theoretical proof. After the lecture he came to me, apologized, and agreed with my statements. This was the beginning of our acquaintance.

I had heard of Wiener before this meeting, of course, not only about his mathematical wizardry, his work in number theory, his famous Tauberian theorems, and his work on Fourier Series, but also about his eccentricities. In Poland, I had heard through Jozef Marcinkiewicz about his book with Paley on the summability of Fourier transforms. Raymond Paley, one of the most promising and successful young English mathematicians, died in a mountaineering accident at a tragically young age. Marcinkiewicz was a student of Antoni Zygmund. He visited Lwów as a post-doctoral fellow and patronized the Scottish Café, where we discussed Wiener’s work, since he had worked in trigonometric series, trigonometric transforms, and summability problems. Marcinkiewicz, like Paley, whom he resembled in genius and in mathematical interests and accomplishments, reportedly was killed while an officer in the Polish army in the 1939 campaign at the beginning of World War II.

Absentminded and otherworldly in appearance, Wiener nevertheless could make an intuitive appraisal of others, and he must have been interested in me. Great as the difference in age between us was (his forty to my twenty-six years), he would seek me out occasionally in my little apartment in Adams House, sometimes late in the evening, and propose a mathematical conversation. He would say, “Let’s go to my office, where I can write on the blackboard.” This suited me better than staying in my rooms, from which it would have been difficult to put him out without being rude. So he drove me in his car through darkened streets to MIT, opened the building doors, turned on the light, and he started talking. After an hour or so, although Wiener was always interesting, I would almost fall asleep and finally manage to suggest that it was time to go home.

Wiener seemed childish in many ways. Being very ambitious about his place in the history of mathematics, he needed constant reassurance about his creative ability. I was almost stunned a few weeks after our first encounter when he asked me point blank: “Ulam! Do you think I am through in mathematics?” Mathematicians tend to worry about their diminishing power of concentration much as some men do about their sexual potency. Impudently, I felt a strong temptation to say “yes” as a joke, but refrained; he would not have understood. Speaking of that remark, “Am I through,” several years later at the first World Congress of Mathematicians held in Cambridge, I was walking on Massachusetts Avenue and saw Wiener in front of a bookstore. His face was glued to the window and when he saw me, he said, “Oh! Ulam! Look! There is my book!” Then he added, “Ulam, the work we two have done in probability theory has not been noticed much before, but see! Now, it is in the center of everything.” I found this disarmingly and blessedly naive.

Anecdotes about Wiener abound; every mathematician who knew him has his own collection. I will add my story of what happened when I came to MIT as a visiting professor in the fall of 1957. I was assigned an office across the hall from his. On the second day after my arrival, I met him in the corridor and he stopped me to say, “Ulam! I can’t tell you what I am working on now, you are in a position to put a secret stamp on it!” (This presumably because of my position in Los Alamos.) Needless to say, I could do no such thing.

Wiener always had a feeling of insecurity. Before the war he used to talk about his personal problems to J. D. Tamarkin, who was a great friend of his. When he was writing his autobiography, he showed a voluminous manuscript to Tamarkin. Tamarkin, whom I had met in 1936 and with whom I became quite friendly, told me about Wiener’s manuscript and how interesting it was. But he also expressed the opinion that Wiener might be sued for libel for many of his outspoken statements. He spoke almost with disbelief about Wiener’s text and how he tried to dissuade him from publishing the book in that form. What finally appeared apparently was considerably toned down from the original version.

Another memory I have of Wiener concerns his asking me to go with him to South Station in Boston to meet the English mathematician G. H. Hardy who was coming to the States for a visit. He knew I had met Hardy in England. We collected another mathematician, perhaps it was Norman Levinson, and picked up Hardy at the train. Wiener, who prided himself on his knowledge of the Chinese, their culture and even their language, invited everybody for lunch at a Chinese restaurant. Immediately he started talking Chinese to the waiter, who seemed not to understand a word. Wiener simply remarked, “He must be from the south and does not speak Mandarin.” (We were not quite convinced that this was the complete explanation.) It was a very pleasant lunch with much mathematical talk. And after lunch Wiener who had picked up the check discovered that he had no money. Fortunately we found the few necessary dollars in our pockets. Wiener scrupulously reimbursed us later.

It was said that Wiener, although he considered his professorship at MIT quite satisfactory, was very disappointed that Harvard never offered him a post. His father had been a professor at Harvard, and Norbert wanted very much to follow in his footsteps.

Although G. D. Birkhoff was at least ten years his senior, Wiener felt a rivalry with him and wanted to equal or surpass him in mathematical achievement and fame. When Birkhoff’s celebrated ergodic theorem proof was published, Wiener tried very hard to go him one better and prove an even stronger theorem. He did manage it, but the strengthening was not as simple or as fundamental as G.D.’s original proof. Here again is an example of the competitive nature of some mathematicians and the sources of their ambition.

I think Wiener had marvelous talents as a mathematician — that is perspicacity and technical genius. He had a supreme general intelligence but, in my opinion, not the spark of originality which does the unusual unrelated to what others have done. In mathematics, as in physics, so much depends on chance, on a propitious moment. Perhaps von Neumann also lacked some of the “irrational,” though with his wonderful creativity, he certainly went to and achieved the limits of the “reasonable.”

There are several ways in which Wiener and von Neumann intersected in their interests and in their feelings about what was important both in pure mathematics and its applications, but it is difficult to compare their personalities. Norbert Wiener was a true eccentric and von Neumann was, if anything, the opposite — a really solid person. Wiener had a sense of what is worth thinking about, and he understood the possibilities of using mathematics for seemingly more important and more visible applications in theoretical physics. He had a marvelous technique for using Fourier transforms, and it is amazing how much the power of algorithms or symbolism could accomplish. I am always amazed how much a certain facility with a special and apparently narrow technique can accomplish. Wiener was a master at this. I have seen other mathematicians who could do the same in a more modest way. For instance, Steinhaus obtained quite penetrating insights into other fields, and his student, Mark Kac, now at Rockefeller University, surpassed him. Antoni Zygmund in Chicago, another Pole, is a master of the great field of trigonometric series. Several of his students have obtained epoch-making results in other fields — for example, Paul Cohen, who did this in set theory, the most general and abstract part of mathematics.

I don’t think Wiener was particularly fond of combinatorial thinking or of working on foundations of mathematicological or set theoretical problems. At the beginning of his career, he may have gone in this direction, but later he applied himself to other fields and to number theory.

Von Neumann was different. He also had several quite independent techniques at his fingertips. (It is rare to have more than two or three.) These included a facility for symbolic manipulation of linear operators. He also had an undefinable “common sense” feeling for logical structure and for both the skeleton and the combinatorial superstructure in new mathematical theories. This stood him in good stead much later, when he became interested in the notion of a possible theory of automata, and when he undertook both the conception and the construction of electronic computing machines. He attempted to define and to pursue some of the formal analogies between the workings of the nervous system in general and of the human brain itself, and the operation of the newly developed electronic computers.

Wiener, somewhat hemmed in by the childishness and naiveté of his personality, was perhaps psychologically handicapped by the fact that, as a child, his father had pushed him as a prodigy. Von Neumann, who also began rather young, had a much wider knowledge of the world and more common sense outside the realm of pure intellect. Furthermore, Wiener was perhaps more in the tradition of talmudistic Judaic scholarship, even though his opinions and beliefs were very libertarian. This was quite conspicuously absent from von Neumann’s makeup.

Johnny’s overwhelming curiosity included many fields of theoretical physics, beginning with his pioneering work — his attempt to form a rigorous mathematical basis for quantum theory. His book, Die Mathematische Grundlagen der Quantum Mechanik, published over forty years ago, is not only a classic, but still the ’’bible” on the subject. He was especially fascinated by the puzzling role of the Reynolds number and the seeming mystery of sudden onsets of turbulence in the motions of fluids. He had discussions with Wiener on the perplexing values of this number which is “dimensionless” — a pure number expressing the ratio of the inertial forces to the viscous forces. It is of the order of two thousand, a large number. Why is this so and not around one, or ten, or fifty? At that time, Johnny and I came to the conclusion that actual detailed numerical computations of many special cases could help throw light on the reasons for the transition from a laminar (regular) to a turbulent flow.

He told me of another discussion he had with Wiener and their different points of view: Johnny advocated, in order to establish models for the working of the human brain, a numerical digital approach through a sequence of time steps, while Wiener imagined continuous or “hormonal” outlines. The dichotomy between these points of view is still of great interest and, of course, by now has been transformed and deepened by the greater knowledge of the anatomy of the brain and by more work in the theory of automata.

The relationship between G. D. Birkhoff and von Neumann was curious. Birkhoff did not really have complete admiration for or appreciation of von Neumann’s genius. He probably could not appreciate the many kinds of mathematics von Neumann was pursuing. He admired his technical brilliance, but G.D.’s tastes were more classical, in the tradition of Poincaré and the great French school of analysis. Von Neumann’s interests were different. Birkhoff had ambitions to produce something of great importance in physics, and he made a few, technically interesting but not conceptually important contributions to the general theory of relativity. He lectured several times on such subjects in Mexico, stimulating a small school of relativists there. Von Neumann’s interests lay in the foundations of the new quantum theory’s more recent developments. Theirs were differences of interests, of approaches, and of value systems. Birkhoff appreciated probing in depth more than exploring in breadth. Von Neumann, to some extent, did both. There was, of course, about a quarter-century’s difference in age between them, as well as in background and in upbringing. Also, von Neumann never quite forgave G.D. for having “scooped” him in the affair of’ the ergodic theorem: Von Neumann had been first in proving what is now called the weak ergodic theorem. By a sheer virtuoso kind of combinatorial thinking, Birkhoff managed to prove a stronger one, and — having more influence with the editors of the Proceedings of the National Academy of Sciences — he published his paper first. This was something Johnny could never forget. He sometimes complained about this to me, but always in a most indirect and oblique way.

In addition to the elementary mathematics courses which I taught during my first year in the Society, I was asked to add advanced courses gradually. I liked this, for the best way to learn a subject is to try to teach it systematically.

Then one gets the real points, the essentials. One was an important undergraduate course in classical mechanics, Math 4 if I remember its former name. Another was Math 9, a course on probability.

At the time I had no precise idea what grades meant: A, B, C, D, or F. But I had rigid standards. I remember an otherwise quite good student, who protested receiving the grade of “C.” Some other professors intervened, but I stubbornly, perhaps foolishly, stood my ground. Now I tend to be more lenient, and when I give a “C” or ’’D” the students really deserve an “F” or worse!

Tamarkin, who was a professor at Brown University, asked me to teach a graduate course in his place while he took his sabbatical leave for a term. I decided to give the course on the theory of functions of several real variables. It included a lot of new material — much of it my own recent work — and I was rather proud of it. Every Friday I went to Providence by train, taught the course, spent the weekend with Tamarkin at his home, returning to Cambridge on Sunday. When I mentioned the contents of the course to Mazur when I went home to Lwów for the last time during the summer of 1939, he liked it very much. He liked the material, the way it was organized, and said he would love to give such a course himself, all of which pleased me and encouraged me.

Tamarkin was a most interesting person. He was of medium height, very portly — I would say some thirty pounds overweight. He was quite nearsighted, a constant cigar-and-cigarette-smoker, and generally extremely jovial. As I got to know him better, I discovered the wonderful qualities of his mind and character.

Before World War I, he had written some mathematical research papers on the work of G. D. Birkhoff and even improved some of the latter’s results a bit, which led to a certain animosity in their relations. Yet when he came to the United States, Birkhoff helped him secure his position at Brown, which had a notable mathematics faculty, including James Richardson, Raymond C. Archibald, and others. Richardson was a gentleman of the old school. Archibald was an eminent historian of mathematics, who established the famous mathematical library of Brown, one of the best in the country.

Tamarkin was interested in Polish-style mathematics and had heard about some of my results in the theory of Banach spaces. He had a quality which perhaps only a small number of mathematicians possess: he was extremely interested in the works of others and less egocentric than most. He was also interested in what was going on in other fields besides his own, whereas most mathematician’s — even the best ones — are often deeply immersed in their own work and do not pay much attention to what others around them are doing. Tamarkin befriended me and encouraged me in my work.

He was Russian, not of Jewish origin exactly, but a Karaite. The Karaites were a sect of Semitic people not subject to the usual restrictions on Jews in Russia, the reason being that they claimed they were absent from Palestine when Jesus was condemned to death, and this exempted them. This claim was accepted by the Russian governors. They also had something in common with the ancient Khazars, people of a mysterious sixth- or seventh-century kingdom in southern Russia, a pagan tribe whose king decided to adopt a new religion. He selected Judaism after having asked Christian, Moslem, and Jewish representatives to explain their beliefs. Tamarkin believed he was one of their descendants. He had escaped from Leningrad after the Russian Revolution in a manner not unlike that of George Gamov some ten years later — over the ice of Lake Ladoga to Finland.

While I was at Harvard, Johnny came to see me a few times, and I invited him to dinner at the Society of Fellows. We would also take automobile drives and trips together during which we discussed everything from mathematics to literature and talked without interruption while still paying attention to our surroundings. Johnny liked this kind of travel very much.

Once at Christmas time in 1937, we drove from Princeton to Duke University to a meeting of the American Mathematical Society. On the way, among other things we discussed the effect that the arrival of increasingly large numbers of refugee European scientists would have on the American academic scene. We stopped at an inn where we found a folder describing a local Indian Chief, Tomo-Chee-Chee, who apparently had been unhappy about the arrival of white men. As an illustration of our frequently linguistic and philological jokes, I asked him why it was that the Pilgrims had “landed” while the present European immigrants and scientific refugees merely “arrived.” Johnny enjoyed the implied contrast and used this in other contexts as an example of an implied value judgment. We also likened G. D. Birkhoff’s increasing qualms about the foreign influence to the Indian Chief’s. Continuing our drive, we managed to lose our way a couple of times and joked that it was Chief Tomo-Chee-Chee who had magically assumed the shape of false road signs to lead us astray.

This was the first time I visited the South, and I was much taken by the difference in atmosphere between New York, New England, and the southern states. I remember a feeling of “déjà vu”: the more polished manners, the more leisurely pace of life, and the elegant estates. Something seemed familiar, and I wondered what it was. Suddenly I asked myself’ if it could be the remnants of the practice of slavery, which reminded me of the traces of feudalism still visible in the country life of Poland. I was also surprised to see so many black people, and their language intrigued me. At a gas station, one of the Negro attendants said, “What would you like now, Captain?” I asked Johnny, “Does he think I might be an officer and calls me Captain as a compliment?” Similarly, the first time I heard myself called “Doc,” I wondered how the porter knew that I had a Doctor’s degree!

As we passed the battlefields of the Civil War, Johnny recounted the smallest details of the battles. His knowledge of history was really encyclopedic, but what he liked and knew best was ancient history. He was a great admirer of the concise and wonderful way the Greek historians wrote. His knowledge of Greek enabled him to read Thucydides, Herodotus, and others in the original; his knowledge of Latin was even better.

The story of the Athenian expedition to the island of Melos, the atrocities and killings that followed, and the lengthy debates between the opposing parties fascinated him for reasons which I never quite understood. He seemed to take a perverse pleasure in the brutality of a civilized people like the ancient Greeks. For him, I think it threw a certain not-too-complimentary light on human nature in general. Perhaps he thought it illustrated the fact that once embarked on a certain course, it is fated that ambition and pride will prevent a people from swerving from a chosen course and inexorably it may lead to awful ends, as in the Greek tragedies. Needless to say this prophetically anticipated the vaster and more terrible madness of the Nazis. Johnny was very much aware of the worsening political situation. In a Pythian manner, he foresaw the coming catastrophe.

It was during this trip also that for the first time I sensed that he was having problems at home. He exhibited a certain restlessness and nervousness and would frequently stop to telephone to Princeton. Once he came back to the car very pale and obviously unhappy. I learned later that he had just found out that his marriage to Marietta was definitely breaking up. She would leave him shortly thereafter to marry a younger physicist, one of the frequent guests at the numerous parties which the von Neumanns gave in Princeton.

On the way back from the meeting I posed a mathematical problem about the relation between the topology and the purely algebraic properties of a structure like an abstract group: when is it possible to introduce in an abstract group a topology such that the group will become a continuous topological group and be separable? “Separable” means that there exists a countable number of elements dense in the whole group. (Namely, every element of the group can be approximated by elements of this countable set.) The group, of course, has to be of power continuum at most — obviously a necessary condition. It was one of the first questions which concern the relation between purely algebraic and purely geometric or topological notions, to see how they can influence or determine each other.

We both thought about ways to do it. Suddenly, while we were in a motel I found a combinatorial trick showing that it could not be done. It was, if I say so myself, rather ingenious. I explained it to Johnny. As we drove Johnny later simplified this proof in the sense that he found an example of a continuum group which is even Abelian (commutative) and yet unable to assume a separable topology. In other words, there exist abstract groups of power continuum in which there is no possible continuous separable topology. What is more, there exist such groups that are Abelian. Johnny, who liked verbal games and to play on words, asked me what to call such a group. I said, “nonseparabilizable.” It is a difficult word to pronounce; on and off during the car ride we played at repeating it.

Mathematicians have their own brand of “in” humor like this. Generally speaking, they are amused by stories involving triviality of identity of two definitions or “tautologies.” They also like jokes involving vacuous sets. If you say something which is true “in vacuo,” that is to say, the conditions of the statement are never satisfied, it will strike them as humorous. They appreciate a certain type of logical non sequitur or logical puzzle. For instance, the story of the Jewish mother who gives a present of two ties to her son-in-law.

The next time she sees him, he is wearing one of them, and she asks, “You don’t like the other one?”

Some of von Neumann’s remarks could be devastating, even though the sarcasm was of an abstract nature. Ed Condon told me in Boulder of a time he was sitting next to Johnny at a physics lecture in Princeton. The lecturer produced a slide with many experimental points and, although they were badly scattered, he showed how they lay on a curve. According to Condon, von Neumann murmured, “At least they lie on a plane.”

Some people exhibit an ability to recall stories and tell them to others on appropriate occasions. Others have the ability to invent them by recognizing analogies of situations or ideas. A third group has the ability to laugh and enjoy other people’s jokes. I sometimes wonder if types of humor could be classified according to personality. My friends and collaborators, C. J. Everett in the United States and Stanislaw Mazur in Poland, each had a wry sense of humor, and physically and in their handwriting they also resemble each other.

Generally von Neumann preferred to tell stories he had heard; I like to invent them. “I have some wit; it is a tremendous quality,” my wife says I once told her. When she pointed out that I was bragging, I promptly added, “True. My faults are infinite, but modesty prevents me from mentioning them all.”

In addition to “in” jokes, mathematicians also practice a form of “in” language. For example, they use the word “trivial.” It is an expression they are very fond of, but what does it really mean? Easy? Simple? Banal? A colleague of my friend Gian-Carlo Rota once told him that he did not like teaching calculus because it was so trivial. Yet, is it? Simple as it is, calculus is one of the great creations of the human mind, with beginnings dating back to Archimedes. It was “invented’’ by Newton and Leibnitz, and amplified by Euler, Lagrange, and others. It has a beauty and an importance going far beyond most of the mathematics of our present culture. So what is “trivial”? Certainly not Cantor’s great set theory, technically very simple, but deep and wonderful conceptually without being difficult or complicated.

I have heard mathematicians sneer at the special theory of relativity, calling it nothing but a technically trivial quadratic equation and a few consequences. Yet it is one of the monuments of human thought. So what is “trivial”? Simple arithmetic? It may be trivial to us, but is it to the third-grade child?

Let us consider some other words mathematicians use: what about the adjective “continuous”? Out of this one word came all of topology. Topology may be considered as a big essay on the word “continuous” in all its ramifications, generalizations, and applications. Try to define logically or combinatorially an adverb like “even” or “nevertheless.” Or take an ordinary word like ’’key,” a simple object. Yet it is an object far from easy to define quasi-mathematically. “Billowing” is a motion of smoke, for example, in which puffs are emitted from puffs. It is almost as common in nature as wave motion. Such a word may give rise to a whole theory of transformations and hydrodynamics. I once tried to write an essay on the mathematics of three-dimensional space that would imitate it.

Were I thirty years younger I might try to write a mathematical dictionary about the origins of mathematical expressions and concepts from commonly used words, imitating the manner of Voltaire’s Dictionnaire Philosophique.

Chapter 6. Transition and Crisis

Each summer between 1936 and 1939, I returned to Poland for a full three months. The first time, after only a few months’ stay in America, I was surprised that street cars ran, electricity and telephones worked. I had become imbued with the idea of America’s absolute technological superiority and unique “know-how.” My main emotional reactions were, of course, related to reunion with my family and friends, and the familiar scenes of Lwów, followed by a longing to return to the free and hopeful “open-ended” conditions of life in America. To simplify a description of these complicated feelings: in May I started counting the days and weeks left before returning to Europe, then after a few weeks in Poland I would count the days impatiently before I would return to America.

Most mathematicians remained in Lwów during the summer, and our sessions in the coffee houses and my own personal contacts with them continued until the outbreak of World War II. As before, I worked with Banach and Mazur. Twice, while Banach was spending a few days in Skole or in nearby villages in the Carpathian mountains, some seventy miles south of Lwów, I visited him. I knew these places from my childhood. Banach was working on some of his textbooks, but there was always lots of time to sit in a country inn and discuss mathematics and “the rest of the universe,” an expression which was dear to von Neumann. The last time I saw Banach was in late July of 1939 at the Scottish Café. We discussed the likelihood of war with Germany and inscribed a few more problems in the Scottish Book.

In the summer of 1937, Banach and Steinhaus asked me to invite von Neumann to come and give a lecture in Lwów. He arrived from Budapest and spent several days among us. He gave a nice lecture, and I brought him to the Café several times. He jotted some problems in the Scottish Book, and we had some very pleasant discussions with Banach and several others.

I told Banach about an expression Johnny had once used in conversation with me in Princeton before stating some non-Jewish mathematician’s result, “Die Goim haben den folgenden Satzbewiesen” (The goys have proved the following theorem). Banach, who was pure goy, thought it was one of the funniest sayings he had ever heard. He was enchanted by its implication that if the goys could do it, Johnny and I ought to be able to do it better. Johnny did not invent this joke, but he liked it and we started using it.

I showed Johnny the city. I had experience in showing it to foreign mathematicians; when I was only a freshman, because I could speak English, Kuratowski had assigned to me the job of showing the town to the American topologist Ayres. I also had to escort Edward Czech, G. T. Whyburn, and several others around Lwów during their Polish visits.

Johnny was much interested in Lwów and surprised at the nineteenth-century appearance of the center of town and its many relics of the fifteenth, sixteenth and seventeenth centuries. Both Hungary and Poland were still semi-feudal in some respects. There were many picturesque parts of town, where old houses leaned towards each other, and crooked narrow cobbled streets. In one little street in the Ghetto, black-market operations in currency were conducted openly. Butcher shops in the suburbs had sides of beef hanging exposed to full view. There were still horse-drawn carriages and electric street-car lines. Taxis were not numerous, and even in the late nineteen-thirties one could take a horse-drawn “fiacre,” usually pulled by two horses. When I first arrived in New York, I was surprised to see the shabby old fiacres in front of the best Fifth Avenue hotels with only one poor horse to pull them.

We visited an Armenian church with frescoes by Jan Henryk Rosen, a contemporary Polish artist now in the United States. We also went into a little Russian Orthodox church, where we were both shocked by the sight of a corpse in a half-open coffin about to be buried according to Russian ritual. It was the first time I had ever seen a dead person.

Johnny also came to our house. He met my parents — my mother, who was to die the following year, and my father, who had heard so much about him from me. I took him to see my father’s offices, which were in a different part of our big house on Kosciusko Street.

Johnny already knew some of my family. An aunt of mine, the widow of my father’s brother Michael, had married a Hungarian financier by the name of Arpad Plesch. Von Neumann knew the Plesches. Arpad’s brother Janos was Einstein’s physician in Berlin. Arpad was an immensely rich financier, but a rather controversial figure. My aunt was wealthy too, a remarkable woman, descended from a famous fifteenth-century Prague scholar named Caro. In Israel many years later, while I was visiting the town of Safed with von Kárman, an old Orthodox Jewish guide with earlocks showed me the tomb of Caro in an old graveyard. When I told him that I was related to a Caro, he fell on his knees — that cost me a triple tip. The Plesches traveled frequently and lived often in Paris. I visited them there on my trip in 1934. My aunt’s first husband, Michael Ulam, my uncle, was buried in Monte Carlo, and my aunt, who is now dead too, is also buried there in a fantastic marble mausoleum in the Catholic cemetery. Aunt Caro was directly related to the famous Rabbi Loew of sixteenth-century Prague, who, the legend says, made the Golem — the earthen giant who was protector of the Jews. (Once, when I mentioned this connection with the Golem to Norbert Wiener, he said, alluding to my involvement with Los Alamos and with the H-bomb, “It is still in the family!”) So much for the family’s rich connections.

A story which Johnny and I liked to tell each other, though I do not remember who proposed it first, was a quote from one of those rich uncles, who used to say: “Reich sein ist nicht genug, man musst auch Geld in der Schweiz haben!” (It is not enough to be rich, one must also have money in Switzerland.)

In the summer of 1938, it was von Neumann’s turn to invite me to Budapest. I traveled by train via Cracow, and went directly to his house. (I think the address was 16 Arany János Street.) He had reserved a room for me at the Hotel Hungaria, the best hotel in town at that time. It was at the end of a narrow little street, so narrow in fact, that there was a revolving platform at the end for cars to turn around on, as for locomotives in a roundhouse.

Johnny showed me Budapest. It was a beautiful city with the houses of Parliament and the bridges on the river. After dinner at his house, where I met his parents, we went to nightclubs and discussed mathematics! Johnny was alone that year, his marriage was breaking up and Marietta had remained in America.

The next day sitting in a “Konditorei,” talking, joking, and eating, we saw an elegantly dressed lady going by. Johnny recognized her. She entered, and they exchanged a few words. After she left, he explained that she was an old friend, recently divorced. I asked him, “Why don’t you marry the divorcée?” Perhaps this implanted the thought in his mind. The next year they indeed were married. Her name was Klara Dan. We became very good friends later on. Johnny and Klari, as she was known to her friends, were married in Budapest and she moved to Princeton in the late summer or fall of 1939. Klari was a moody person, extremely intelligent, very nervous, and I often had the feeling that she felt that people paid attention to her mostly because she was the wife of the famous von Neumann. This was not really the case, for she was a very interesting person in her own right. Nevertheless, she had these apprehensions, which made her even more nervous. She had been married twice before (and married a fourth time after von Neumann’s death). She died in 1963 in tragic and mysterious circumstances. After leaving a party given in honor of Nobel Prize-winner Maria Mayer, she was found drowned on the beach at La Jolla, California.

Johnny took me also to a pleasant mountain resort called Lillafüred to visit his former professors, Leopold Fejer and Frederick Riesz, who were both pioneer researchers in the theory of Fourier series. Lillafüred, about a hundred miles from Budapest, was a resort with luxurious castle-like big hotels. Fejer and Riesz were in the habit of summering there. Fejer had been Johnny’s teacher. Riesz was one of the most elegant mathematical writers in the world, known for his precise, concise, and clear expositions. He was one of the originators of the theory of function spaces — an analysis which is geometrical in nature. His book, Functions of Real Variables, is a classic. We were to walk in the forest, but in the morning before the walk Johnny said we had to wait until the master had his inspiration. By this he meant the daily physiological necessities — not spiritual ones, which had to be met with a pre-breakfast sip of brandy before the day could begin! We had a nice discussion. Of course the talk also concerned the world situation and the likelihood of war.

I returned to Poland by train from Lillafüred, traveling through the Carpathian foothills. I had to change trains several times and remember sitting for a while on an open flat car with my legs dangling over the side as we went through small villages named Satorolia-Ujhely and Munkaczewo. This whole region on both sides of the Carpathian Mountains, which was part of Hungary, Czechoslovakia, and Poland, was the home of many Jews. Johnny used to say that all the famous Jewish scientists, artists, and writers who emigrated from Hungary around the time of the first World War came, either directly or indirectly, from these little Carpathian communities, moving up to Budapest as their material conditions improved. The physicist I. I. Rabi was born in that region and brought to America as an infant. It will be left to historians of science to discover and explain the conditions which catalyzed the emergence of so many brilliant individuals from that area. Their names abound in the annals of mathematics and physics of today. Johnny used to say that it was a coincidence of some cultural factors which he could not make precise: an external pressure on the whole society of this part of Central Europe, a feeling of extreme insecurity in the individuals, and the necessity to produce the unusual or else face extinction. To me the picture was that of the Roman poet Virgil, describing the flood: “In the big whirlpool there appear only a few remaining swimmers,” surviving through intellectual frenzy and strenuous and vigorous work. A jocular version of survival is a story I told to Johnny who manufactured many variations. A little Jewish farm boy named Moyshe Wasserpiss emigrated to Vienna and became a successful businessman. He changed his name to Herr Wasserman. Going on to Berlin and to even greater success and fortune, he became Herr Wasserstrahl, then von Wasserstrahl. Now in Paris and still more prosperous, he is Baron Maurice de la Fontaine.

Eugene Wigner is one of the famous scientists from Budapest. He and Johnny were school friends and studied together for a time in Zürich. Johnny told me a nice story from those days: Eugene and Johnny wanted to learn to play billiards. They went to a café where billiards were played and asked an expert waiter there if he would give them lessons. The waiter said, “Are you interested in your studies? Are you interested in girls? If you really want to learn billiards, you will have to give up both.” Johnny and Wigner held a short consultation and decided that they could give up one or the other but not both. They did not learn to play billiards.

Von Neumann was primarily a mathematician. Wigner is primarily a physicist, but also half a mathematician and brilliant user of mathematics, a virtuoso of mathematical techniques in physics. I would add here that he recently published an interesting article on the a priori unexpected effectiveness of mathematics in physics. Von Neumann’s book on the foundations of quantum theory had more philosophical and psychological meaning rather than direct applications in theoretical physics. Wigner made many concrete contributions to physics, perhaps nothing quite as overwhelming as Einstein’s ideas of relativity, but many important specific technical achievements and also something rather general — namely, the very fundamental role of group theoretical principles in the physics of quantum theory and the physics of elementary particles.

When von Neumann died, Wigner wrote a beautiful obituary article in which he described the deep despair that came upon Johnny when he knew that he was dying, for it was impossible for him to imagine that he would stop thinking. For Wigner, von Neumann and thinking were synonymous.

After this visit to Budapest it was time to prepare my return to Harvard. I had to go to the American consulate in Warsaw each summer I was in Poland to apply for a new visitor’s visa in order to return to the United States. Finally, the consul said to me, “Instead of coming here every summer for a new visa, why don’t you get an immigration visa?” It was lucky that I did, for just a few months later these became almost impossible to obtain.

Twice I made plans to travel across the Atlantic with Johnny. The year his marriage with Marietta broke up, we went to Europe together. I joined him on the Georgic, a small Cunard boat which took a week to cross. Johnny always traveled first class, so I went first class also, although usually I traveled tourist. As always, we discussed a lot of mathematics. We flirted with a young woman named Flatau, whom we found rather attractive. The day after we met her, I asked him: “Have you solved the problem of Flatau?” He liked the play on words. In mathematics there is the famous problem of Plateau: given a curve or a wire in space, the problem is to find a surface such that the wire is its boundary and of minimal surface area. It can be demonstrated with soap bubbles. If you immerse a closed curved wire in a soap solution, you get some nice surfaces spanning them. The man who first formulated and studied this mathematically was Plateau.

In 1939, my three-year appointment at the Society of Fellows was expiring. Unfortunately, it was not renewable because I had passed the upper age limit. Thanks to G. D. Birkhoff, I received an extension of the stay at Harvard in the form of a lectureship in the mathematics department. More permanent prospects were not promising; it seemed there were no vacant assistant professorships. Because of the influx of large numbers of German and Central European scientists, and in spite of Johnny’s efforts on my behalf, things did not look much better in Princeton. Thus, I returned to America with an assured position for only one more year, accompanied this time by my younger brother Adam, who was not quite seventeen. With Adam in my charge, a sense of responsibility came over me. Our mother had died the year before, and the feeling of impending crisis had convinced my father that Adam would be safer in America with me. When he tried to apply for a visa, the American consul in Warsaw seemed to have reservations. It is only when I proved that I was living and teaching in the United States that he agreed to issue him a student visa.

Our father and uncle Szymon accompanied us to Gdynia, a Polish port on the Baltic Sea, to see us off on the Polish liner Batory. This was the last time we were to see either of them.

We were at sea when the announcement of the pact between Russia and Germany came over the ship’s radio. In a state of strange agitation, I told Adam upon hearing the news, “This is the end of Poland.” On a map in the ship’s salon, I drew a line through the middle of Poland, saying, Cassandra-like, “It will be divided like that.” We were, to say the least, shaken and worried.

At dinner on the first night, I suddenly noticed Alfred Tarski in the dining room. I had no idea he was on the boat. Tarski, famous logician and lecturer in Warsaw, told us he was on his way to a Congress on the Unity of Philosophy and Science to be held in Cambridge. It was his first trip to America. We ate at the same table and spent a good deal of time together. I still have an old shipboard photograph, which shows Adam, Tarski, and me dressed in dinner jackets ready for the gay American social life. He intended to stay only a couple of weeks and was traveling with a small suitcase of summer clothes. Because the war broke out shortly after we landed, he found himself stranded in the United States without money, without a job, and with his family — a wife and two small children — in Warsaw. For some time he was in the most precarious and terrible situation.

Adam was frightened and nervous when we landed, a young boy abroad for the first time and away from familiar surroundings. Johnny had come to meet us at the boat; on seeing Adam, he asked, out of his hearing, “Who is this fellow?” He had not heard my introduction and was surprised. There is a difference of thirteen years between my brother and me, and we do not look at all alike. Adam is taller than I, straight, blond, and with a pink complexion. I am somewhat dark and stockier. In appearance he resembles some of our uncles, while I look more like our mother. At the pier, Johnny appeared very agitated. People in the United States had a much clearer and more realistic view of events than we had had in Poland. For example, when it was time to obtain an exit visa from Poland, because I was in the Polish army reserve I had to first secure the permission of the army to leave the country. The officer in charge asked casually why I wanted to go abroad and raised no further question when I told him of my lecturing engagement in America. As a rule people in Poland had not felt the imminence of war, but rather a continuation of the state of crisis, similar to the one in Munich the year before.

We stayed in New York a few days, visiting cousins — the painter Zygmund Menkès and his wife — and a friend of the family who had given additional financial guarantees for Adam, who had a student visa. Actually, the plan was that my brother would receive monthly checks from home via our uncle’s bank in England. We also saw young Mr. Loeb, an acquaintance of my cousin Andrzej, and when I talked to him on the telephone, he asked: ’’Will Poland give in?” I replied that I was pretty sure it would never surrender, and there was going to be a war.

I left Adam in New York to go with Johnny to Veblen’s summer place in Maine. Though we were gone only two or three days, Adam was very unhappy with me for having left him. On the way to Veblen’s, we discussed some mathematics as usual but mostly talked about what was going to happen in Europe. We were both nervous and worried; we examined all possible courses which a war could take, how it could start, when. And we drove back to New York. These were the last days of August.

Adam and I were staying in a hotel on Columbus Circle. It was a very hot, humid, New York night. I could not sleep very well. It must have been around one or two in the morning when the telephone rang. Dazed and perspiring, very uncomfortable, I picked up the receiver and the somber, throaty voice of my friend the topologist Witold Hurewicz began to recite the horrible tale of the start of war: “Warsaw has been bombed, the war has begun,” he said. This is how I learned about the beginning of World War II. He kept describing what he had heard on the radio. I turned on my own. Adam was asleep; I did not wake him. There would be time to tell him the news in the morning. Our father and sister were in Poland, so were many other relatives. At that moment, I suddenly felt as if a curtain had fallen on my past life, cutting it off from my future. There has been a different color and meaning to everything ever since.

On the way to Cambridge, I accompanied Adam to Brown University in Providence, registered him as a freshman, and introduced him to a few of my friends, including Tamarkin and his son. His English was quite good, and he did not seem to mind being left alone in college.

I became a compulsive buyer of newspapers, all the extra editions every hour, and easily went through eight or ten papers a day looking for news of Lwów, of the military situation, and of the progress of battles. Early in September I saw in the Boston Globe a large photograph of Adam surrounded by other young freshmen at Brown. It was captioned, “Wonders whether his home was bombed.”

From the start, Adam did very well in school; a few months later, he was able to obtain a tuition waiver. Nevertheless we were finding ourselves in severe financial straits. The proposed income from Britain had been frozen — the English government stopped all outgoing money, and my salary as a lecturer at Harvard was hardly sufficient to put a younger brother (who was not allowed to work because he was on a student visa) through college. On previous trips, I had never thought of transferring funds or property from Poland. Now it could no longer be done. I went to see a dean of the College and explained my situation. His name was Ferguson, and I feared that with such a Scottish name he would be rather parsimonious. Fortunately he was not. I told him that if I could not get a little more assistance from the University, I would be forced to leave my academic career and look for some other means of support. He was sympathetic and was able to find two or three hundred dollars more for the year, which was a sizable help in those days.

From indications at department meetings, I gathered that my chances of staying on at Harvard on a permanent basis were poor, so I began to make inquiries about another position for 1940. An assistant professorship became vacant at Lehigh University in Bethlehem, Pennsylvania, and I received a letter inviting me for an interview. I was not much interested in going to Bethlehem, but G. D. Birkhoff told me, “Stan, you must know that it is impossible in this country to get any advancement or raises in salary without offers from elsewhere. Yes, go have the interview at Lehigh.” I replied, “Who will teach my class that day?” “I will,” he said. I felt both embarrassed and honored that the great Professor Birkhoff would condescend to teach my class in undergraduate mechanics. Indeed, charmingly childish as he often was, and wanting to show the young students who he was, he gave a complicated and advanced lecture which, I later learned, they did not understand very well.

Bethlehem was covered with a yellow pall of acrid smog when I arrived for the interview — an inauspicious beginning. The chairman took me around the department and introduced me to a young professor there. It happened to be the number theorist D. H. Lehmer. As we entered his office, he was correcting a large pile of blue books and said to me, right in front of the chairman, “See what we have to do here!” This contributed to a negative impression and brought instantly to my mind a similar situation in Poland. When a chambermaid or some other servant girl was leaving a place of employment, she would take the new prospective maid aside and show her the less favorable aspects of the job.

This was the period of my life when I was perhaps in the worst state, mentally, nervously, and materially. My world had collapsed. Prospects for the reconstitution of Poland in any recognizable form were dim indeed. There was a terrible anxiety about the fate of all those whom we had left behind — family and friends. Adam was also in a very depressed state, and this contributed to my worries. L. G. Henderson, who was always friendly and helpful, gave me all the moral support he could. When France collapsed in the spring of 1940, the situation became so dark and seemingly hopeless that despair gripped all the European émigrés on this side of the ocean. There was the added worry that should German ideas prevail along with their military successes, life in America would become quite different, and xenophobia and anti-Semitism might grow here too.

During that period I lived in a little room on the fourth floor of the Ambassador Hotel. On the fifth, the Alfred North Whiteheads lived in a large apartment whose walls were curiously painted in black. I knew Whitehead from the Society of Fellows dinners. He and Mrs. Whitehead held a weekly “at home” evening to which they invited me. He was already quite old, but his mind was crystal clear, sharp, incisive, with a better memory than many of his juniors. I remember their fortitude and courage at the time of the bombing of London. Whitehead never seemed to give up hope that the war would be won in the end, and he lived to see the defeat of Germany.

Conversation at the Whiteheads’ was extremely varied. Besides the war, one discussed philosophy, science, literature, and people. The subject of Bertrand Russell came up once. He was having great troubles in this country. He had quarreled with Barnes, the Philadelphia millionaire who employed him at the time, and he was in difficulty at City College because of his views on sex and lectures on free love. Harvard tried to get him, but the invitation did not go through because of a wave of protests from proper Bostonians. I remember Whitehead talking about all this and Mrs. Whitehead exclaiming, “Oh, poor Bertie!” There was also talk of mathematics. Once someone asked, “Professor Whitehead, which is more important: ideas or things?” “Why, I would say ideas about things,” was his instant reply.

I spent much of my time with the other Poles who had found their way to Cambridge — Tarski, Stefan Bergman and Alexander Wundheiler. They were all terribly unhappy, Wundheiler most of all. He always had some kind of “Weltschmerz.” We would sit in front of my little radio which I left on all day long, and listen to the war news. He would stay for hours in my room, and we drank brandy from toothbrush glasses. He was a talented mathematician, an extremely nice, pleasant, and intelligent person, with a mind rather hard to describe — that of an intelligent critic, but somewhat lacking the “it” of mathematical invention. Genius is not the word I mean. It is hard to describe the talent for innovation, even on a modest scale; besides, it exists in a continuous spectrum and is largely influenced by “luck.” There may be such a thing as habitual luck. People who are said to be lucky at cards probably have certain hidden talents for those games in which skill plays a role. It is like hidden parameters in physics, this ability that does not surface and that I like to call ’’habitual luck.” It is often remarked that in science there are people who have so much luck that one begins to feel it is something else. Wundheiler lacked this special spark.

I don’t remember when and how he first appeared in the States. He had a temporary job at Tufts College in Boston. He had the usual impressions, complaints, appreciation and admiration for the United States; we talked about that at great length. Attitudes of students amused and shocked him. Being used to the more formal Polish manners, he was quite put out when in class one day a student called out to him, “Psst! The window is open. Can you close it?” One did not talk to professors that way in Poland.

He was interested in the geometry of the Dutch mathematician Schouten, which to me was too formal and symbolic. The notations were so complicated, I made fun of the formulae saying that they considered a geometric object a mere symbol, a letter around which indices were hung right, left, up, and down — like decorations on a Christmas tree.

We gradually lost contact after I left Cambridge. I learned later that he had committed suicide. I had a premonition of this because of a poem he would recite about a man who hanged himself with his tie. He was lonely, and many times he had told me of his unhappiness because of his looks. He was a very short man with an intelligent face, but not one that was considered appealing to women. He thought of himself as ugly, which bothered him.

In many cases, mathematics is an escape from reality. The mathematician finds his own monastic niche and happiness in pursuits that are disconnected from external affairs. Some practice it as if using a drug. Chess sometimes plays a similar role. In their unhappiness over the events of this world, some immerse themselves in a kind of self-sufficiency in mathematics. (Some have engaged in it for this reason alone.) Yet one cannot be sure that this is the sole reason; for others, mathematics is what they can do better than anything else.

Toward the end of the 1940 academic year Birkhoff intimated to me that there might be a vacancy at the University of Wisconsin. He added, “You should not be like the other European refugees who try at all costs to stay on the East Coast. Do as I have done, try to get a job in Madison. It is a good university; I was there as a young man.” Taking his advice, I went to a meeting of the American Mathematical Society at Dartmouth to meet Professor Mark Ingraham, who was chairman of the mathematics department in Madison. In those days, mathematics meetings were job fairs at which one could see the department chairmen — almost like hereditary caciques, each surrounded by supplicants, groups of younger men looking for jobs. The situation was completely reversed in the late 1950s and early 1960s, when a lone young man fresh from school, with a brand-new Ph.D., would be surrounded by chairmen looking for young professors.

At the Dartmouth meeting I had a comical adventure. Late in the evening I walked into my dormitory room. It was dark, and I tried to get into bed without putting the light on. As I sat down, I heard a squeak and a groan. My bed had another occupant. I groped for the other bed. The occupant of my bed said, “Dr. Ulam?” I answered, “Yes.” Immediately he said, “Given a group which is such and such, does it have this and that property?” I thought a moment and answered, ’’Yes,” proceeding to outline a reason. “If it is compact, then it is true.” “But if it isn’t compact,” he tried to continue. It was late, I was tired, and felt like saying, “If it isn’t compact, to hell with it.” I let the conversation drop and fell asleep.

G. D. Birkhoff seemed to like and appreciate my work. I think I know a possible reason. He liked my self-confidence and my near impudence in defending the point of view of modern mathematics based on set theory against his more classical approach. He admired the so-to-speak hormonal, emotional side of mathematical creativity. I probably reminded him of the way he felt when he was young. He liked the way I got almost furious when — in order to draw me out — he attacked his son Garrett’s research on generalized algebras and more formal abstract studies of structures. I defended it violently. His smile told me that he was pleased that the worth and originality of his son’s work was appreciated.

In discussing the general job situation, he would often make skeptical remarks about foreigners. I think he was afraid that his position as the unquestioned leader of American mathematics would be weakened by the presence of such luminaries as Hermann Weyl, Jacques Hadamard, and others. He was also afraid that the explosion of refugees from Europe would fill the important academic positions, at least on the Eastern seaboard. He was quoted as having said, “If American mathematicians don’t watch out, they may become hewers of wood and carriers of water.” He never said that to me, but he did often make slighting remarks about the originality of some foreigners. He also maintained that they ought to be content with more modest positions; objectively speaking, this was understandable and even fair. Just the same, I would sometimes get angry. Perhaps because my family had been so well off and that until 1940 I never had to give much thought to my financial situation, I could be independent and said what I thought openly. Once I countered one of his attacks on foreigners with, “What pleasure do you find in playing a game where the outcome does not depend on the skill of the opponent but on some external circumstance? What pleasure is there in winning a game of chess from a player who is forced to make poor moves because he needs help from his opponent?” He was quite taken back by this remark.

But Birkhoff helped me to secure the job in Madison. He spoke to Ingraham on my behalf; after the Dartmouth meeting, I received an offer of an instructorship in Madison. At thirty, and with a certain reputation among mathematicians in Poland and America, I felt I could have been offered at least an assistant professorship. But the circumstances were so far from normal, and with someone like Jacques Hadamard, the most celebrated French mathematician, having been offered a lectureship in New York, or Tarski having been taken on as an instructor in Berkeley, I swallowed my pride and accepted the offer. Financially, it was not bad — something like twenty-three hundred dollars a year. Nevertheless, I felt sad at leaving Harvard and the “cultural East” for what I believed to be a more primitive and intellectually barren Middle West. On the East Coast it was implied that Harvard and perhaps Yale and Princeton were the only places with “culture.” I was sure that Madison, about which I knew nothing, would be like Siberia and that I was being exiled. But since there were no alternatives, I prepared to leave Cambridge at the end of the summer, grimly determined to live through the years of exile and to await the outcome of the war.

Chapter 7. The University of Wisconsin

I traveled to Madison by way of Chicago, where I changed to a smaller train that made several whistle stops, one at a town named Harvard. The irony did not escape me, and I felt fate was playing a cruel joke on me. It did not take long for me to change my outlook completely. I promptly found that the state of Wisconsin had important liberal political traditions, that the famous La Follettes had left their imprint not only on the state capital but on the University as well. The entire physical impression, the landscape, the lakes, the woods, the houses, and the size of the city were most agreeable. Living conditions were a pleasant surprise. I was given a room at the University Club and almost at once met there congenial, intelligent people not only in mathematics and science, but also in the humanities and arts. The rooms were small, with bath, bed, desk, and chairs. (I remembered how in one of Anatole France’s novels, the hero, Father Coignard, says, “All one needs is a table and a bed. A table on which to have in turn learned books and delicious meals, a bed for sweet repose and ferocious love!”) Downstairs there were comfortable common rooms, a library, dining room, even a game room with billiard tables.

The university was adequately supported by state funds and had extra income from a former professor’s discovery of a special treatment of milk, the patent rights for which belonged to the University.

Johnny’s friend Eugene Wigner was a physics professor there. I had a letter of introduction to another eminent physicist, Gregory Breit, from Harlow Shapley, the Harvard astronomer with whom I had had pleasant scientific and social contacts during my stay in Cambridge. It was Shapley who discovered the “scale of the universe” by using the luminosity period of the cepheid stars as markers. I quickly made friends with most of the mathematicians — many of them my contemporaries, Steve Kleene the logician, C. J. Everett, Donald Hyers, and others. Being by nature gregarious, I liked living at the University Club, meeting and taking meals with interesting colleagues.

One of them was Vassilief, a Russian émigré, a great expert on Byzantine history, and almost a character out of Nabokov’s book Pnin. At dinner, he always ordered a second bowl of soup and would say to me, “Americans are funny; even when the soup is excellent, they never think of ordering a second bowl.” Like many Russians, he liked to drink and carried a small flask of vodka in his coat pocket. He must have been in his sixties at the time. Some two years later, when the U.S. Army took over the Faculty Club for its quarters, Vassilief and the other occupants had to find other housing. Vassilief was given a two-room suite in a private house. He was enchanted with this new spaciousness. “It’s wonderful,” he explained. ’’You can sleep in one room and work in the other.” And just like Pnin he threw what he called “a house-heating party to celebrate.”

Another interesting person was a professor of English literature, a bachelor, Professor Hanley. Thanks to my memory, which enabled me to quote Latin and to discuss Greek and Roman civilization, it became obvious to some of my colleagues in other fields that I was interested in things outside mathematics. This led quickly to very pleasant relationships. Hanley was a good billiard player. He insisted on teaching me how to play, though he was rather appalled by my ineptitude. This, I found, is a very nice and very American trait — the desire to coach and instruct.

So I found Madison not at all the intellectual desert I had feared it would be. The university had a tradition of excellence in several fields of the natural sciences. It had great expertise in limnology. Limnology, the science of lakes, was developed by an old professor whose name I cannot recall, but who used to say, I was told, that every time he remembered the name of a student he forgot the name of a fish. Biology, too, was strong at the University of Wisconsin, as well as economics and political science. The economist Selig Perlman, and Nathan Feinsinger, who later became nationally known as a labor relations expert, were there at the time among many other eminent professors.

It also seemed that foreigners like myself who were, so to speak, not unpresentable, were well received into the academic community’s social life and quickly established good relations with many professors in various fields. The intellectual atmosphere was lively. On the whole the professors did not put on airs as a few have at Harvard. On the contrary, perhaps in order to bear comparison with the famous older universities, they worked more energetically, but the “Oh, excuse me I’ve got to get to work” syndrome was not as evident as at Harvard.

Something else happened to make Madison most important to me. It was there that I married a French girl, who was an exchange student at Mount Holyoke College and whom I had met in Cambridge, Françoise Aron. Marriage, of course, changed my way of life, greatly influencing my daily mode of work, my outlook on the world, and my plans for the future.

The poet William Ellery Leonard, a tall man with a very large head and a mane of white hair, was one of the interesting and colorful members of the faculty, a great eccentric. The author of the book The Locomotive God, he was reputed to have an intense, neurotic fear of trains. This prevented him from ever leaving the University in Madison; the story was that his salary (which was very low for a full professor) never was increased, because he would never leave anyway. I found this reason rather humorous.

At that time, at many universities deans and chairmen often ran their departments not so much for excellence in scholarly or educational pursuits, but for good economy and efficiency — rather like a business. Indeed, shortly after I arrived, someone pointed out that the marvelous physical location of the campus on the shores of Lake Mendota constituted a part of our salaries, making them a little lower than at other comparable state universities. This made me joke with some of my younger colleagues: every time we looked at the beautiful lake it was costing us about two dollars. At one of the first faculty meetings I attended, Clarence A. Dykstra, the President, very imposing in appearance (and actually a very good man), started his speech with, “All of us face a challenge this year.” At this I nudged my neighbor and whispered, “Watch out! This means no raises for the faculty. Sure enough, ten minutes later Dykstra said something to this effect, and my neighbor laughed out loud.

The astronomer Joel Stebbins was a professor in Madison. I enjoyed meeting him and talking to him in the observatory. He had an excellent sense of humor and was a great practical joker. Once on a clear, cold sunny winter Sunday, he drove to our apartment and honked the car horn. I looked out and there he was, saying, “Would you like to come with me to the Yerkes Observatory? There is a meeting of the Astronomical Society there.” Yerkes was not far from Madison — about a two-hour drive. I dressed warmly, hurried down and on the way we discussed all kinds of problems. Then teasingly he said: “Would you like to give a talk?” To answer a joke with a joke, I responded, “Yes, for five or ten minutes.” Quickly I started to think about what I could say to astronomers in a few minutes. I remembered that once I had been thinking about the mathematics of the way trajectories of celestial bodies might look from a moving system of coordinates and how, by a suitable motion of the observer, one could make complicated-looking orbits appear simpler if one assumed that the observer was himself moving. I called this general question ’’The Copernican Problem” and spoke for a few minutes on that. Indeed, it is a worthwhile subject to consider, and it does give rise to some bona fide topological and metric questions in which I had obtained a few simple results.

From the first year of my employment I was given a very light teaching load — namely only eleven hours of elementary courses (recognizing the fact that I was doing research and writing a number of papers), while some of the other instructors taught thirteen or sixteen. Later, this was reduced to nine hours per week. These elementary courses required essentially no preparation on my part, aside from an occasional look at the sequence of topics in the book so that I would cover the prescribed material and not go too quickly or too slowly. But the very expression “teaching load” as used by almost everyone from famous scholars to administrators was not only repugnant to me but ridiculous. It implied physical effort and fatigue — two things I have always been afraid of, lest they interfere with my own thinking and research. I was grateful to Ingraham, the head of the department, for understanding this. He was a jovial and pleasant man, who used to come to the faculty club on weekends to watch movies of football games. Known for his fondness for apple pie with cheese, he introduced me to the cheeses of Wisconsin, one of the state’s dairy specialties, before I made a later acquaintance with the infinite variety of those of France.

In general, teaching mathematics is different from teaching other subjects. I feel, as do most mathematicians, that one can teach mathematics without much preparation, since it is a subject in which one thing leads to another almost inevitably. In my own lectures to more advanced audiences, seminars, and societies, I discuss topics that currently preoccupy me; this is more of a stream of consciousness approach.

I am told that I teach rather well. This is possibly because I believe one should concentrate on the essence of the subject and also not teach all items at a uniform level. I like to stress the important and, for contrast, a few unimportant details. One remembers a proof by recalling, as it were, a sequence of pleasant and unpleasant points — that is, difficult and easy ones. First comes a difficulty on which one makes an effort, then things go automatically for a while, suddenly again there is a new special trick that one has to remember. It is somewhat like going through a labyrinth and trying to remember the turns.

When I was teaching calculus in Madison (and it is a marvelous thing to teach) and had worked out a problem on the board, I was amused when some student would say, “Do another one like it!” They didn’t even have a name for “it.” Needless to say, these students did not become professional mathematicians.

One may wonder whether teaching mathematics really makes much sense. If one has to explain things repeatedly to somebody and assist him constantly, chances are he is not cut out to do much in mathematics. On the other hand, if a student is good, he does not really need a teacher except as a model and perhaps to influence his tastes. A priori, I tend to be pessimistic about students, even the bright ones (though I remember some good students at Harvard with whom I could talk and feel that teaching was not merely an empty exercise).

Generally speaking, I don’t mind teaching as such, although I do not like to spend too much time at it. What I dislike is the obligation to be at a given place at a given time — not being able to feel completely free. This is because one of my characteristics is a special kind of impatience. When I have a fixed date, even a pleasant dinner or party, I fret. And yet when I am completely free, I may become restless, not knowing what to do.

With my friend Gian-Carlo Rota I once computed that including seminars and talks to advanced audiences, we must have lectured for several thousand hours in our lives. If one recognizes that the average working year in this country amounts to around two thousand hours, this is a large proportion of one’s waking time, even if it is not completely “waking” since teaching is sometimes done in a partial trancelike state.

It was in Madison that I met C. J. Everett, who was to become my close collaborator and good friend. Everett and I hit it off immediately. As a young man he was already eccentric, original, with an exquisite sense of humor, wry, concise, and caustic in his observations. He was totally devoted to mathematics — they were his only interest. I found in him much that resembled my friend Mazur in Poland, the same kind of epigrammatic comments and jokes. Physically, they had a certain similarity, both being thin, bony, and less than medium height. Even their handwriting was similar; they both wrote in very neat, almost microscopic little symbols. Everett is several years younger than I. We collaborated on difficult problems of “order” — the idea of order for elements in a group. In our mathematical conversations, as always, I was the optimist, and had some general, sometimes only vague ideas. He supplied the rigor, the ingenuities in the details of the proof, and the final constructions.

One paper we wrote on ordered groups caught the fancy of the woman who was head of one of the women’s military organizations during the war. At a meeting we heard her describe the activities of the corps by calling the organization “ordered groups.”

Later we wrote another joint paper on projective algebras. I think this was the first attempt to algebraize mathematical logic beyond the so-called Boolean or Aristotelian elementary operations to include the operations “there exists” and “for all,” which are both vital and comprehensive for advanced mathematics.

We both taught courses for naval recruits in 1942 and 1943. Also, in order to earn some extra money, we corrected papers for the Army Correspondence School. Françoise helped in that, too — she could do it very well for elementary arithmetic and algebra exercises. The Correspondence School paid thirty-five cents per paper; this amounted to quite a bit of money and began to reach sums comparable to the university salary. At this point, the administration decided to step in and impose restrictions on the number of papers one person could correct. The Army correspondence work was administered by an older woman who was a member of the mathematics department; it was supervised by a professor, Herbert Evans, a very jovial and pleasant person with whom I became friendly. He was one of the most good-humored persons I have ever known anywhere.

Everett and I shared an office in North Hall, an old structure halfway up the hill which housed the mathematics department. Leon Cohen, a visiting professor from the University of Kentucky with whom we had published some joint work, was there with us. We spent hours in that office, the three of us; the entire building would resound with our frequent laughter. Before and after classes, we corrected student papers — an occupation I hated and always tried to put off. As a result my desk was piled high with stacks of uncorrected workbooks and as I deposited new ones at one end, the older ones at the other mercifully dropped into the waste basket. Sometimes the poor students wondered why I was not returning their work.

After lunch, we played billiards — or tried to. Hanley’s lessons at the Faculty Club had very little effect on my game. Fun in this North Hall office and our frequent sessions in the Student Union, a luxurious building on the shore of the lake, were among the charms of life in Madison. This combination of leisure plus informal stimulation plays an important role in one’s mental activity. Beyond the merely agreeable physical setting, it is often more valuable than the more formal gatherings at seminars and meetings which lead to discussions of a drier type. In its way, this somewhat replaced for me the old sessions in the coffee houses of Lwów for which I have had a longing ever since leaving Poland.

Everett stayed in Madison throughout the war. Later he joined me at Los Alamos, where we did much more work together including our now well-known collaboration and work on the H-bomb.

Everett exhibited a trait of mind whose effects are, so to speak, non-additive: persistence in thinking. Thinking continuously or almost continuously for an hour, is at least for me — and I think for many mathematicians — more effective than doing it in two half-hour periods. It is like climbing a slippery slope. If one stops, one tends to slide back. Both Everett and Erdös have this characteristic of long-distance stamina.

There were also Donald Hyers and Dorothy Bernstein. Hyers also had persistence in thinking about problems and an ability to continue to push the train of thought on a specific problem; we wrote several joint papers together. Dorothy Bernstein was a graduate student in my class. She was an interested, enthusiastic, and faithful taker of notes and organizer of material from a course I taught on measure theory. She collected much material, and we intended to write a book together, but our work was interrupted by my departure from Madison in 1944 and our plans were abandoned.

One day in my office, a young and brilliant graduate student named Richard Bellman appeared and expressed a desire to work with me. We discussed not only mathematics but the methodology of science. When the United States entered the war, he wanted to go back East — I believe to New York where he came from — and asked me to help him obtain a fellowship or a stipend so he could continue working after he left Madison. I remembered that in Princeton Lefschetz had some new scientifico-technological enterprise connected with the war effort. I wrote to him about Bellman in a sort of Machiavellian way, saying that I had a very able student who was so good that he deserved considerable financial support. I added that I doubted that Princeton could afford it. This, of course, immediately challenged Lefschetz and he offered Bellman a position. Two years later, Dick Bellman appeared suddenly in Los Alamos in uniform as a member of the SEDs, a special engineering detachment of bright and scientifically talented draftees who had been assigned to help with the technical work.

Through my connections with physicists and a seminar I taught in physics in the absence of Gregory Breit, I heard of the recent arrival in Madison of a very famous French physicist, Léon Brillouin. I called on him and found that his wife, Stéfa, was Polish, born in the city of Lódz, a large textile-manufacturing town. Stéfa and Léon had met when she was a young student in Paris, and they were married sometime before World War I. When World War II broke out he was, I think, a director of the French broadcasting service, with all the military responsibilities that entailed. After the collapse of France and the installation of the Vichy régime, he managed to escape from France at the first opportunity. He was internationally known for his work in quantum theory, statistical mechanics, and also in solid state physics. In fact, he was one of the pioneers in the theory of solid state. (The idea of “Brillouin” zones and other important notions are due to him.) He was also a very prolific and successful writer of physics textbooks and monographs.

Mrs. Brillouin had a great artistic flair. In the early nineteen-twenties she acquired, for modest sums, many works by Modigliani, Utrillo, Vlaminck, and other painters. In Madison she herself started to paint flower compositions in oil made tip of thick layers — a style all her own. The Brillouins invited us to stop at their apartment the day Françoise and I were married. They held a small surprise reception for us, where we drank French champagne and partook of a memorable cake by Stéfa. Stéfa Brillouin spoke hardly any English, but a few weeks after her arrival, in shopping for various objects she discovered that “le centimètre d’ici” as she called our inch, was about two and a half times the “centimètre de France.” This almost exact estimate, the inch being 2.54 centimeters, was obtained solely by looking at the sizes of materials, curtains, and rugs. In Madison a close association began, which continued long after the war and lasted until their deaths a few years ago.

Before my second year in Madison, I was promoted to an assistant professorship — a step which gave me hope and some confidence in the material aspects of the future. To start a home and at the same time help support my brother on my modest salary (of twenty-six hundred dollars a year) was difficult. Often to make ends meet I visited the Faculty Credit Union, where a sympathetic officer made me loans of up to one hundred dollars, which had to be repaid in a few months’ time.

I was asked to run the mathematics colloquium, which took place every two weeks and involved both local and visiting mathematicians. I might add that the payments to speakers were ridiculously small; even for those times, they amounted to about twenty-five dollars — and this included traveling expenses.

The colloquium was run differently from what I had known in Poland, where speakers gave ten- or twenty-minute informal talks. At Madison they were one-hour lectures. There is quite a difference between short seminar talks like those at our math society in Lwów, and the type of lecture which necessitates talking about major efforts. The latter were better prepared, of course, but their greater formality removed some of the spontaneity and stimulation of the shorter exchanges. In this connection I met André Weil, the talented French mathematician, who had gone to South America at the beginning of the war. He disliked conditions there and came to the United States, where he had gotten a job at Lehigh. Weil was already well known internationally for his important work in algebraic geometry and in general algebra. His colloquitim talk was on one of his most important results on the Rieman hypothesis for fields of finite characteristic. The Rieman hypothesis is a statement that is not easy to explain to laymen. It is important because of its numerous applications in number theory. It has challenged many of the greatest mathematicians for about a hundred years. It is still unproved although considerable progress has been made toward a possible solution.

Dean Montgomery, a friend I had met at Harvard, came at my invitation and gave a colloquium talk. There was a vacancy in the department, and I tried to interest him in coming to our university, where Ingraham and Langer, the two most senior professors, were both very much in favor of his appointment; instead he went to Yale. Later he told me stories about the atmosphere at Yale, which at that time in some circles was ultra-conservative. In his interview he was asked whether he was for or against Jews in the academic profession and also whether he was a liberal. Even though he answered both questions “wrongly” front the interlocutor’s point of view, he nevertheless received an offer. He left Yale a few years later to join the Institute in Princeton.

Eilenberg and Erdös were also among the speakers invited to the colloquia. Erdös was one of the few mathematicians younger than I at that stage of my life. He had been a true child prodigy, publishing his first results at the age of eighteen in number theory and in combinatorial analysis.

Being Jewish he had to leave Hungary, and as it turned out, this saved his life. In 1941 he was twenty-seven years old, homesick, unhappy, and constantly worried about the fate of his mother who had remained in Hungary.

His visit to Madison became the beginning of our long, intense — albeit intermittent — friendship. Being hard-up financially — “poor,” as he used to say — he tended to extend his visits to the limits of welcome. By 1943, he had a fellowship at Purdue and was no longer entirely penniless — “even out of debts,” as he called it. During this visit and a subsequent one, we did an enormous amount of work together — our mathematical discussions being interrupted only by reading newspapers and listening to radio accounts of the war or political analyses. Before going to Purdue, he was at the Institute in Princeton for more than a year with only a pittance for support that was to end later.

Erdös is somewhat below medium height, an extremely nervous and agitated person. At that time he was even more in perpetual motion than now — almost constantly jumping up and down or flapping his arms. His eyes indicated he was always thinking about mathematics, a process interrupted only by his rather pessimistic statements on world affairs, politics, or human affairs in general, which he viewed darkly. If some amusing thought occurred to him, he would jump up, flap his hands, and sit down again. In the intensity of his devotion to mathematics and constant thinking about problems, he was like some of my Polish friends — if possible, even more so. His peculiarities are so numerous it is impossible to describe them all. One of them was (and still is) his own very peculiar language. Such expressions as “epsilon” meaning a child, “slave” and “boss” for husband and wife, “capture” for marriage, “preaching’’ for lecturing, and a host of others are now well known throughout the mathematical world. Of all the results we obtained jointly, many have still not been published to this day.

Erdös has not changed much as the years have gone by. He is still completely absorbed by mathematics and mathematicians. Now over sixty, he has more than seven hundred papers to his credit. Among the many sayings about him one goes, “You are not a real mathematician if you don’t know Paul Erdös.” There is also the well-known Erdös number — the number of steps it takes any mathematician to connect with Erdös in a chain of collaborators. “Number two,” for example, is to have a joint paper with someone who has written a paper with Erdös. Most mathematicians can usually find a link with him, if not in one then in two stages.

Erdös continues to write short handwritten letters beginning, “Suppose that x is thus and so…” or “Suppose I have a sequence of numbers…” Toward the end, he adds a few personal remarks, mainly about getting old (this started when he was thirty) or with hypochondriac or pessimistic observations about the fate of our aging friends. His letters are nevertheless charming and always contain new mathematical information. Our correspondence started before the Madison years, containing many discussions of the hardships of young mathematicians who could not find jobs or of how to deal with officials and administrators. He used the expression, “Oh, he is a big shot” about young American assistant professors, and when he called me one, I introduced a subclassification of “big shots, small shots, big fry and small fry’’—four orders of status. In 1941, as an assistant professor I told him I was at best a “small shot.” This amused him, and he would allot one of these four grades to our friends in conversations or correspondence.

Through thick and thin, Johnny von Neumann and I also continued our correspondence, which included a little mathematics even in those days, and a lot about the tragic happenings in the world. There was much isolationism in the United States, and the obvious and widespread disinclination to enter the war created in me a feeling of despair mixed with resentment. On the whole, Johnny was more optimistic and knew better than I the power of the United States and the long-range goals of U.S. policies. He was already an American citizen engaged (though I did not know it at the time) in the war effort the country was preparing.

The tone of our mathematical correspondence and of our conversations when we met at mathematics meetings changed from the abstract to more applied, physics-related topics. He was now writing about problems of turbulence in hydrodynamics, aerodynamics, shocks, and explosives.

Johnny held discussions with many scientists, among them Norbert Wiener. Although Norbert was a pacifist, he badly wanted to contribute something important to the American war effort. Wiener felt as Russell did that this was a “just war,” a necessary war, and the only hope for mankind lay in U.S. intervention and victory. But Norbert was difficult in his dealings with the military, whereas Johnny always got along with them.

Wiener wrote in his autobiography that he had ideas similar to the ones I later proposed as the Monte Carlo method. He says vaguely that he found no response when he talked to someone and so dropped the matter, in the same way that he lost interest in the idea of geometry of vector spaces and function spaces à la Banach. In fact, in one of his books he called these vector spaces (which are associated with Banach’s name alone) Banach-Wiener spaces. This nomenclature did not “take” at all.

In the first World War mathematicians had done much work in classical mechanics, calculations of trajectories, and external and internal ballistics. This work was resumed at the beginning of the second World War, although it soon turned out not to be the main thrust of the scientific applications. Hydrodynamic and aerodynamic questions became more detailed and urgent, particularly because they were directly connected with special war problems. Early in 1940 I took from the library a German textbook on ballistics and studied it, but noticed that there was not much in it of importance to the military technology of the forties. At the beginning of the war electronic computing machines did not exist. There were only the beginnings of the mechanical relay machines constructed at Harvard, at IBM, and one or two other places.

As soon as the prescribed time had elapsed I applied for and received American citizenship in Madison in 1941. I hoped that this would make it easier for me to enroll in the war effort. To pass the examination I studied the history of the United States, the essence of the Constitution, the names of Presidents, and the other topics one was likely to be asked about at the examination. I don’t remember why, but instead of my having to go to Chicago, an examiner came to Madison to our apartment. After a few words I noticed that he must have been an immigrant himself or the son of an immigrant. His appearance was quite Jewish, and perhaps impudently I asked him about his own origins and background. He did not seem to mind and replied that his parents had come from the Ukraine. Soon I realized with embarrassment that it was I who was examining him.

Immediately after I received the citizenship papers, I tried to volunteer in the Air Force. At thirty I was considered too old to become a combat pilot, but with my mathematical background I thought I could receive training as a navigator, because the University had received a notice that the Air Force was looking for volunteers. I went to a recruiting center not far from Madison for a physical examination. It was given by West Coast Japanese medics who had been relocated in this Middle Western camp. Because the physical tests involved the taking of blood samples, I told myself jokingly that I was losing some blood to the Japanese in defense of my new country. I was disappointed when my application was rejected because of my peculiar eyesight.

Teaching army math courses did not seem sufficiently relevant; I wanted to do something more immediately useful, something that would contribute more directly. I thought of going to Canada to enlist there, and I remembered a conversation in Cambridge in 1940 with Whitehead, who had relatives who were officers in the Royal Canadian Air Force. So I wrote him asking whether he could help me to become involved in the war effort in Canada. He sent back a letter which I treasure for all the things it said. Even though he said that he had written to someone in Canada on my behalf, nothing came of this.

Then Los Alamos entered the picture.