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.