Before the end of the year Kuratowski mentioned in a lecture another problem in set theory. It was on the existence of set functions which are "subtractive" but not completely countably additive. I remember pondering the question for weeks. I can still feel the strain of thinking and the number of attempts I had to make. I gave myself an ultimatum. If I could solve this problem, I would continue as a mathematician. If not, I would change to electrical engineering.
After a few weeks I found a way to achieve a solution. I ran excitedly to Kuratowski and told him about my solution, which involved transfinite induction. Transfinite induction had been used by mathematical workers many times in other connections; however, I believe that the way in which I used it was novel.
I think Kuratowski took pleasure in my success, encouraging me to continue in mathematics. Before the end of my first college year I had written my second paper, which Kuratowski presented to Fundamenta. Now, the die was cast. I began to concentrate on the "impractical" possibilities of an academic career. Most of what people call decision making occurs for definite reasons. However, I feel that for most of us what is ultimately called a "decision" is a sort of vote taken in the subconscious, in which the majority of the reasons favoring the decision win out.
During the summer of 1928 when I took a trip to the Baltic coast of Poland, Kuratowski invited me to visit him on the way at his summer place near Warsaw. It was an elegant villa with a tennis court. Kuratowski was quite good at tennis in those days, and this surprised me since his figure was anything but athletic.
On the six-hour train ride from Lwów to Warsaw I thought almost without interruption about problems in set theory with the idea of presenting something that would interest him. I was thinking of ways to disprove the continuum hypothesis, a famous unsolved problem in foundations of set theory and mathematics formulated by Georg Cantor, the creator of set theory. My presentation was vague, and Kuratowski soon detected this. Nevertheless, we discussed its ramifications, and so I went on to Zoppot with my self-confidence intact.
Alfred Tarski, now a celebrated logician and professor at Berkeley, was a friend of Kuratowski from Warsaw, who occasionally visited Lwów. He was already known internationally as a logician, but his work in the foundations of mathematical logic and set theory was also important. He had been a candidate for a chair of philosophy that was vacant at the University of Lwów. The chair went instead to another logician, Leon Chwistek, an accomplished painter and author of philosophical treatises, a brother-in-law of Steinhaus, and well known for many eccentricities. (He died in Moscow during the war.) Years later in Cambridge, I happened to mention Chwistek to Alfred North Whitehead. In the course of the conversation I said, "Very strange, he was a painter too!" Whereupon Whitehead laughed out loud, clapped his hands and exclaimed: "How British of you to say that being a painter is strange." Mrs. Whitehead joined in the laughter. A very good biography of Chwistek by Estriecher has recently appeared in Poland. It is a fascinating account of the intellectual and artistic life of Cracow and Lwów from 1910 to 1946.
One of my early contacts with Tarski was a result of my second paper. In it I had proved a theorem on ideals of sets in set theory. (Marshall Stone later proved another version of this same theorem.) My note in Fundamenta also showed the possibility of defining a finitely additive measure with two values, 0 to 1, and established a maximum prime ideal for subsets in the infinite set. In a very long paper which appeared a year later, Tarski got the same result. After Kuratowski pointed out to him that it followed from my theorem, Tarski acknowledged this in a footnote. In view of my youth, this seemed to me a little victory — an acknowledgment of my mathematical presence.
There was a feeling among some mathematicians that logic is not "real" mathematics, but merely a preparatory and somewhat alien art. Today, this feeling is disappearing as a result of many concrete mathematical advances made by the methods of formal logic.
During the second year of studies I decided to audit a course in theoretical physics given by Professor Wojciech Rubinowicz, a leading Polish theoretician and a former student and collaborator of the famous Munich physicist Sommerfeld.
I attended his masterly lectures on electromagnetism and took part in a seminar he led on group theory and quantum theory for advanced students. We used Hermann Weyl's Gruppen Theorie und Quantum Mechanik. It was impressive to see the high level of mathematics involved in the study of Maxwell's equations and in the theory of electricity which made up its first part. Even though much of it was above my head technically, I managed to do a lot of reading on the side. I read popular accounts of theoretical physics in statistical mechanics, in the theory of gases and the theory of relativity, and on electricity and magnetism.
During the winter, Rubinowicz fell ill and asked me (although I was the youngest member of the class) to conduct a few sessions during his absence. I remember to this day how I struggled with the unfamiliar and difficult material of Weyl's book. This was my first active participation in the area of physics.
The mathematics offices of the Polytechnic Institute continued to be my hangout. I spent mornings there, every day of the week, including Saturdays. (Saturdays were not considered to be part of the weekend then; classes were held on Saturday mornings.)
Mazur appeared often, and we started our active collaboration on problems of function spaces. We found a solution to a problem involving infinitely dimensional vector spaces. The theorem we proved — that a transformation preserving distances is linear — is now part of the standard treatment of the geometry of function spaces. We wrote a paper which was published in the Compte-Rendus of the French Academy.
It was Mazur (along with Kuratowski and Banach) who introduced me to certain large phases of mathematical thinking and approaches. From him I learned much about the attitudes and psychology of research. Sometimes we would sit for hours in a coffee house. He would write just one symbol or a line like y = f(x) on a piece of paper, or on the marble table top. We would both stare at it as various thoughts were suggested and discussed. These symbols in front of us were like a crystal ball to help us focus our concentration. Years later in America, my friend Everett and I often had similar sessions, but instead of a coffee house they were held in an office with a blackboard.
Mazur's forte was making what he called "observations and remarks." These stated — usually in a concise and precise form — some properties of notions. Once made, they were perhaps not so difficult to verify, for sometimes they were peripheral to the usual formulations and had gone unnoticed. They were often decisive in solving problems.
In a conversation in the coffee house, Mazur proposed the first examples of infinite mathematical games. I remember also (it must have been sometime in 1929 or 1930) that he raised the question of the existence of automata which would be able to replicate themselves, given a supply of some inert material. We discussed this very abstractly, and some of the thoughts which we never recorded were actually precursors of theories like that of von Neumann on abstract automata. We speculated frequently about the possibility of building computers which could perform exploratory numerical operations and even formal algebraical work.
I have mentioned that I first saw Banach at a series of mathematics lectures when I was in high school. He was then in his middle thirties, but contrary to the impression given to very young people by men fifteen or twenty years their senior, to me he appeared to be very youthful. He was tall, blond, blue-eyed, and rather heavy-set. His manner of speaking struck me as direct, forceful, and perhaps too simple-minded (a trait which I later observed was to some extent consciously forced). His facial expression was usually one of good humor mixed with a certain skepticism.