From the first year of my employment I was given a very light teaching load — namely only eleven hours of elementary courses (recognizing the fact that I was doing research and writing a number of papers), while some of the other instructors taught thirteen or sixteen. Later, this was reduced to nine hours per week. These elementary courses required essentially no preparation on my part, aside from an occasional look at the sequence of topics in the book so that I would cover the prescribed material and not go too quickly or too slowly. But the very expression "teaching load" as used by almost everyone from famous scholars to administrators was not only repugnant to me but ridiculous. It implied physical effort and fatigue — two things I have always been afraid of, lest they interfere with my own thinking and research. I was grateful to Ingraham, the head of the department, for understanding this. He was a jovial and pleasant man, who used to come to the faculty club on weekends to watch movies of football games. Known for his fondness for apple pie with cheese, he introduced me to the cheeses of Wisconsin, one of the state's dairy specialties, before I made a later acquaintance with the infinite variety of those of France.
In general, teaching mathematics is different from teaching other subjects. I feel, as do most mathematicians, that one can teach mathematics without much preparation, since it is a subject in which one thing leads to another almost inevitably. In my own lectures to more advanced audiences, seminars, and societies, I discuss topics that currently preoccupy me; this is more of a stream of consciousness approach.
I am told that I teach rather well. This is possibly because I believe one should concentrate on the essence of the subject and also not teach all items at a uniform level. I like to stress the important and, for contrast, a few unimportant details. One remembers a proof by recalling, as it were, a sequence of pleasant and unpleasant points — that is, difficult and easy ones. First comes a difficulty on which one makes an effort, then things go automatically for a while, suddenly again there is a new special trick that one has to remember. It is somewhat like going through a labyrinth and trying to remember the turns.
When I was teaching calculus in Madison (and it is a marvelous thing to teach) and had worked out a problem on the board, I was amused when some student would say, "Do another one like it!" They didn't even have a name for "it." Needless to say, these students did not become professional mathematicians.
One may wonder whether teaching mathematics really makes much sense. If one has to explain things repeatedly to somebody and assist him constantly, chances are he is not cut out to do much in mathematics. On the other hand, if a student is good, he does not really need a teacher except as a model and perhaps to influence his tastes. A priori, I tend to be pessimistic about students, even the bright ones (though I remember some good students at Harvard with whom I could talk and feel that teaching was not merely an empty exercise).
Generally speaking, I don't mind teaching as such, although I do not like to spend too much time at it. What I dislike is the obligation to be at a given place at a given time — not being able to feel completely free. This is because one of my characteristics is a special kind of impatience. When I have a fixed date, even a pleasant dinner or party, I fret. And yet when I am completely free, I may become restless, not knowing what to do.
With my friend Gian-Carlo Rota I once computed that including seminars and talks to advanced audiences, we must have lectured for several thousand hours in our lives. If one recognizes that the average working year in this country amounts to around two thousand hours, this is a large proportion of one's waking time, even if it is not completely "waking" since teaching is sometimes done in a partial trancelike state.
It was in Madison that I met C. J. Everett, who was to become my close collaborator and good friend. Everett and I hit it off immediately. As a young man he was already eccentric, original, with an exquisite sense of humor, wry, concise, and caustic in his observations. He was totally devoted to mathematics — they were his only interest. I found in him much that resembled my friend Mazur in Poland, the same kind of epigrammatic comments and jokes. Physically, they had a certain similarity, both being thin, bony, and less than medium height. Even their handwriting was similar; they both wrote in very neat, almost microscopic little symbols. Everett is several years younger than I. We collaborated on difficult problems of "order" — the idea of order for elements in a group. In our mathematical conversations, as always, I was the optimist, and had some general, sometimes only vague ideas. He supplied the rigor, the ingenuities in the details of the proof, and the final constructions.
One paper we wrote on ordered groups caught the fancy of the woman who was head of one of the women's military organizations during the war. At a meeting we heard her describe the activities of the corps by calling the organization "ordered groups."
Later we wrote another joint paper on projective algebras. I think this was the first attempt to algebraize mathematical logic beyond the so-called Boolean or Aristotelian elementary operations to include the operations "there exists" and "for all," which are both vital and comprehensive for advanced mathematics.
We both taught courses for naval recruits in 1942 and 1943. Also, in order to earn some extra money, we corrected papers for the Army Correspondence School. Françoise helped in that, too — she could do it very well for elementary arithmetic and algebra exercises. The Correspondence School paid thirty-five cents per paper; this amounted to quite a bit of money and began to reach sums comparable to the university salary. At this point, the administration decided to step in and impose restrictions on the number of papers one person could correct. The Army correspondence work was administered by an older woman who was a member of the mathematics department; it was supervised by a professor, Herbert Evans, a very jovial and pleasant person with whom I became friendly. He was one of the most good-humored persons I have ever known anywhere.
Everett and I shared an office in North Hall, an old structure halfway up the hill which housed the mathematics department. Leon Cohen, a visiting professor from the University of Kentucky with whom we had published some joint work, was there with us. We spent hours in that office, the three of us; the entire building would resound with our frequent laughter. Before and after classes, we corrected student papers — an occupation I hated and always tried to put off. As a result my desk was piled high with stacks of uncorrected workbooks and as I deposited new ones at one end, the older ones at the other mercifully dropped into the waste basket. Sometimes the poor students wondered why I was not returning their work.
After lunch, we played billiards — or tried to. Hanley's lessons at the Faculty Club had very little effect on my game. Fun in this North Hall office and our frequent sessions in the Student Union, a luxurious building on the shore of the lake, were among the charms of life in Madison. This combination of leisure plus informal stimulation plays an important role in one's mental activity. Beyond the merely agreeable physical setting, it is often more valuable than the more formal gatherings at seminars and meetings which lead to discussions of a drier type. In its way, this somewhat replaced for me the old sessions in the coffee houses of Lwów for which I have had a longing ever since leaving Poland.