Inclinations may be part of the inherited system of connections in the brain, a genetic trait that may not even depend on the physical arrangement of neurons. Apparently headaches are related to the ease with which blood circulates in the brain, which depends on whether the blood vessels are wide or narrow. Perhaps it is the "plumbing" that is important, rather than the arrangement of the neurons normally associated with the seat of thinking.
Another determining factor may be initial accidents of success or failure in a new pursuit. I believe that the quality of memory develops similarly as a result of initial accidents, random external influences, or a lucky combination of the two.
Consider the talent for chess, for example. José Capablanca learned the game at the age of six by watching his father and uncle play. He developed the ability to play naturally, effortlessly, the way a child learns to speak as compared with the struggles adults have in learning new subjects. Other famous chess players also first became interested by watching their relatives play. When they tried, perhaps a chance initial success encouraged them to pursue. Nothing succeeds like success, it is well known, especially in early youth.
I learned chess from my father. He had a little paper-bound book on the subject and used to tell me about some of the famous games it described. The moves of the knight fascinated me, especially the way two enemy pieces can be threatened simultaneously with one knight. Although it is a simple stratagem, I thought it was marvelous, and I have loved the game ever since.
Could the same process apply to the talent for mathematics? A child by chance has some satisfying experiences with numbers; then he experiments further and enlarges his memory by building up his store of experiences.
I had mathematical curiosity very early. My father had in his library a wonderful series of German paperback books — Reklam, they were called. One was Euler's Algebra. I looked at it when I was perhaps ten or eleven, and it gave me a mysterious feeling. The symbols looked like magic signs; I wondered whether one day I could understand them. This probably contributed to the development of my mathematical curiosity. I discovered by myself how to solve quadratic equations. I remember that I did this by an incredible concentration and almost painful and not-quite-conscious effort. What I did amounted to completing the square in my head without paper or pencil.
In high school, I was stimulated by the notion of the problem of the existence of odd perfect numbers. An integer is perfect if it is equal to the sum of all its divisors including one but not itself. For instance: 6 = 1 + 2 + 3 is perfect. So is 28 = 1 + 2 + 4 + 7 + 14. You may ask: does there exist a perfect number that is odd? The answer is unknown to this day.
In general, the mathematics classes did not satisfy me. They were dry, and I did not like to have to memorize certain formal procedures. I preferred reading on my own.
At about fifteen I came upon a treatise on the infinitesimal calculus in a book by Gerhardt Kowalevski. I did not have enough preparation in analytic geometry or even in trigonometry, but the idea of limits, the definitions of real numbers, the notion of derivatives and integration puzzled and excited me greatly. I decided to read a page or two a day and attempt to learn the necessary facts about trigonometry and analytic geometry from other books.
I found two other books in a secondhand bookstore.
These intrigued and fascinated me more than anything else for many years to come: Sierpinski's Theory of Sets and a monograph on number theory. At the age of seventeen I knew as much or more elementary number theory than I do now.
I also read a book by the mathematician Hugo Steinhaus entitled What Is and What Is Not Mathematics and in Polish translation Poincaré's wonderful La Science et l'Hypothèse, La Science et la Méthode, La Valeur de la Science, and his Dernières Pensées. Their literary quality, not to mention the science, was admirable. Poincaré molded portions of my scientific thinking. Reading one of' his books today demonstrates how many wonderful truths have remained, although everything in mathematics has changed almost beyond recognition and in physics perhaps even more so. I admired Steinhaus's book almost as much, for it gave many examples of actual mathematical problems.
The mathematics taught in school was limited to algebra, trigonometry, and the very beginning of analytic geometry. In the seventh and eighth classes, where the students were sixteen and seventeen, there was a course on elementary logic and a survey of history of philosophy. The teacher, Professor Zawirski, was a real scholar, a lecturer at the University and a very stimulating man. He gave us glimpses of recent developments in advanced modern logic. Having studied Sierpinski's books on the side, I was able to engage him in discussions of set theory during recess and in his office. I was working on some problems on transfinite numbers and on the problem of the continuum hypothesis.
I also engaged in wild mathematical discussions, formulating vast and new projects, new problems, theories and methods bordering on the fantastic, with a boy named Metzger, some three or four years my senior. He had been directed toward me by friends of' my father who knew that he too had a great interest in mathematics. Metzger was short, rotund, blondish, a typical liberated ghetto Jew. Later I saw a youthful portrait of Heine which reminded me of his face. People of his type can still be found occasionally. They exhibit amateurism, even about the very foundations of arithmetic. We discussed "an iterative calculus" on the basis of practically no knowledge of the existing mathematical material. He was "crazy" and full of the urge to innovate which is so Jewish. Stefan Banach once pointed out that it is characteristic of certain Jews always to try to change the established scheme of things — Jesus, Marx, Freud, Cantor. On a very small scale Metzger showed this tendency. Had he had a better education he might have done good things. He obviously came from a very poor family and his Polish had a strong, guttural accent. After a few months he abruptly vanished from my ken. This is the first time I have thought about him in all these years. Perhaps he is alive. This memory of Metzger and our discussions brings back the very smell and color of the "abstractions" we exchanged.
Strangely enough, at this youthful and immature age I was also occasionally trying to analyze my own thinking processes. I tried to make myself more aware of them by periodically going back every few seconds to see what it was that molded the train of thought. Needless to say, I was fully aware of the fact that there is a danger in indulging too much and too frequently in such introspection.
So far, the image I had formed of astronomers and scientists, and of mathematicians in particular, came almost exclusively from my reading. I got my first "live" impressions when I went to a series of popular mathematics lectures in 1926. On successive days there were talks by Hugo Steinhaus, Stanislaw Ruziewicz, Stefan Banach, and perhaps others. My first surprise was to discover how young they were. Having heard and read of their achievements I really expected bearded old scholars. I listened avidly to their talks. Young as I was, my impression of Banach was that here was a homespun genius. This first impression — deepened, enriched, and transformed, of course — remained during my subsequent long acquaintance, collaboration, and friendship with him.
Then in 1927, Zawirski told me a congress of mathematicians was to take place in Lwów and foreign scholars had been invited. He added that a youthful and extremely brilliant mathematician named John von Neumann was to give a lecture. This was the first time I heard the name. Unfortunately, I could not attend these lectures for I was in the midst of my own matriculation examinations at the Gymnasium.