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.