I always preferred to try to imagine new possibilities rather than merely to follow specific lines of reasoning or make concrete calculations. Some mathematicians have this trait to a greater extent than others. But imagining new possibilities is more trying than pursuing mathematical calculations and cannot be continued for too long a time.
An individual's output is, of course, conditioned by what he can accomplish most easily and this perhaps restricts its scope. In myself I notice a habit of twisting a problem around, seeking the point where the difficulty may lie. Most mathematicians begin to worry when there are no more difficulties or obstacles for "new troubles." Needless to say, some do it more imaginatively than others. Paul Erdös concentrates all the time, but usually on lines which are already begun or which are connected to what he was thinking about earlier. He doesn't wipe his memory clean like a tape recorder to start something new.
Banach used to say, "Hope is the mother of fools," a Polish proverb. Nevertheless, it is good to be hopeful and believe that with luck one will succeed. If one insists only on complete solutions to problems, this is less rewarding than repeated tries which result in partial answers or at least in some experience. It is analogous to exploring an unknown country where one does not immediately have to reach the end of the trail or all the summits to discover new realms.
It is most important in creative science not to give up. If you are an optimist you will be willing to "try" more than if you are a pessimist. It is the same in games like chess. A really good chess player tends to believe (sometimes mistakenly) that he holds a better position than his opponent. This, of course, helps to keep the game moving and does not increase the fatigue that self-doubt engenders. Physical and mental stamina are of crucial importance in chess and also in creative scientific work. It is easier to avoid mistakes in the latter, in that one can come back to rethinking; in chess one is not allowed to reconsider moves once they have been made.
The ability to concentrate and the decrease in awareness of one's surroundings come more naturally to the young. Mathematicians can start very young, in some cases in their teens. In Europe, even more than in America, mathematicians exhibit precocity, education in European high schools having been several years ahead of the more theoretical education in the United States. It is not unusual for mathematicians to achieve their best results at an early age. There are some exceptions; for instance, Weierstrass, who was a high-school teacher, achieved his best results when he was forty. More recently, Norman Levinson proved a very beautiful theorem when he was sixty-one or sixty-two.
At twenty-five, I had established some results in measure theory which soon became well known. These solved certain set theoretical problems attacked earlier by Hausdorff, Banach, Kuratowski, and others. These measure problems again became significant years later in connection with the work of Gödel and more recently with that of Paul Cohen. I was also working in topology, group theory, and probability theory. From the beginning I did not become too specialized. Although I was doing a lot of mathematics, I never really considered myself as only a mathematician. This may be one reason why in later life I became involved in other sciences.
In 1934, the international situation was becoming ominous. Hitler had come to power in Germany. His influence was felt indirectly in Poland. There were increasing displays of inflamed nationalism, extreme rightist outbreaks and anti-Semitic demonstrations.
I did not consciously recognize these portents of things to come, but felt vaguely that if I was going to earn a living by myself and not continue indefinitely to be supported by my father, I must go abroad. For years my uncle Karol Auerbach had been telling me: "Learn foreign languages!" Another uncle, Michael Ulam, an architect, urged me to try a career abroad. For myself, unconscious as I was of the realities of the situation in Europe, I was prompted to arrange a longish trip abroad mainly by an urge to meet other mathematicians, to discuss problems with them and, in my extreme self-confidence, try to impress the world with some new results. My parents were willing to finance the trip.
My plans were to go west (go west, young man!); first I wanted to spend a few weeks in Vienna to see Karl Menger, a famous geometer and topologist, whom I had met in Poland through Kuratowski. This was the fall of 1934, right after the assassination of the Austrian Premier Dollfuss. Vienna was in a state of upheaval, but I was so absorbed and almost perpetually drunk with mathematics that I was not really aware of it.
After a couple of days in a Vienna hotel, I moved to a private boarding house near the University, where a widowed lady rented rooms to students. This was quite a common arrangement in those days. The house was on a little street named after Boltzmann, a great physicist of the nineteenth century, one of the principal creators of the kinetic theory of gases and of thermodynamics.
I visited Menger and at his house met a brilliant young Spanish topologist named Jimenez y Flores, who had already some nice results to his credit. We talked mathematics a good deal. He seemed very well known in night clubs and introduced me to the life of a young man about town.
From Vienna I traveled to Zürich to meet Heinz Hopf, the topologist. He was a professor at the famous Technische Hochschule, with whom I had corresponded. Hopf knew something about my topological results and invited me to visit the institute to give two lectures. One was about work I had done jointly with Borsuk on the "antipodal theorem," a topological problem. I spoke in German, in a lecture room of the department of agriculture. I recall there were many pictures of prize cows along the walls, which seemed to look at me with sadness and commiseration.
Nevertheless, this visit to Zürich was quite fruitful. I also met a physicist named Grossman, who was a few years my senior and widely traveled. He recommended hotels in France and in England to suit my purse. We discussed philosophy and the role of mathematics in physics.
After two weeks in Zürich, I went to Paris for five weeks, and that was sheer delight. I had been in France before, but this was my first visit to Paris. My uncle Michael's wife happened to live there at the time and she kindly offered to receive me and to send to my modest hotel her chauffeured limousine to take me sightseeing. I was so embarrassed at the thought of being seen arriving in a Rolls Royce or a Dusenberg at the Louvre or some other museum, it felt so incongruous, that I declined her offer.
I went to the Institut Poincaré with a letter of introduction from one of my professors to the famous old mathematician Elie Cartan. When I entered his office I plunged directly into a mathematical discussion, telling him how I had an idea for a simple and general proof for solving Hilbert's fifth problem on continuous groups. At first he said he did not quite follow my reasoning, but then he added, "Ah! I see now what you want to do." Cartan's little white goatee, vivacious smile, and sparkling eyes gave him an appearance which somehow fitted my mental image of all French mathematicians. He was remarkable for many reasons, not the least because he had done some of his best work in his fifties, when the creativity of most mathematicians is on the decline.