Even though membership in the Society did not require teaching of any kind, Professor Graustein asked me to teach an elementary undergraduate section of a freshman course called Math 1A. (It may even be that the late President Kennedy was for a while a student in this class. I remember a name like that and someone saying that the young man was a rather remarkable person. He left to go abroad in the middle of the term. Years later when I met President Kennedy I forgot to ask him whether he had really taken that course.)
I had given talks and seminars, but not yet taught a regular class, and I found this teaching interesting. The rule for young instructors was to follow very closely the prescribed textbook. Apparently I did not do too badly, for in an evaluation of teachers the student newspaper praised me as an interesting instructor. Soon after the beginning of the course, G. D. Birkhoff came to inspect my performance. Perhaps he wanted to check my English. He sat at the back of the room and watched as I explained to the students how to write equations of parallel lines in analytic geometry. Then I said that next we would study the formulae for perpendicular lines, which, I added, were "more difficult." After the lecture Birkhoff carne to me and commented, "You've done very well, but I would not have said that perpendicular lines are more difficult." I replied that I believed on the contrary that students would remember better this way than if I said everything was easy. Birkhoff smiled at this attempt at pedagogy on my part. I think he liked my independence and outspoken ways, and I saw him rather frequently.
Shortly after I arrived in Cambridge, he had invited me to dine at his house. It was my first introduction to strange dishes like pumpkin pie. After dinner, which was pleasant enough, I got ready to leave and G.D. took my overcoat to help me into it. This sort of courtesy was unheard of in Poland; an older man would never have helped a much younger one. I remember blushing crimson with embarrassment.
I frequently ate lunch with his son Garrett, and we often took walks together. We talked much about mathematics and also indulged in the usual gossip that mathematicians love. Surely it is a shallow theme to evaluate how good X or Y is, but it is a characteristic of our tribe. The reader may have noticed that I practice this, too. Mathematics being more in the nature of an art, values depend on personal tastes and feelings rather than on objective factual notions. Mathematicians tend to be rather vain — though less so than opera tenors or artists. But as every mathematician knows some special bit of math better than anyone else, and math is such a vast and now more and more specialized subject, some like to propose linear orders of "class" among the better-known ones and to comment on their relative merits. On the whole, it is a harmless if somewhat futile pastime.
I remember that at the age of eight or nine I tried to rate the fruits I liked in order of "goodness." I tried to say that a pear was better than an apple, which was better than a plum, which was better than an orange, until I discovered to my consternation that the relation was not transitive — namely, plums could be better than nuts which were better than apples, but apples were better than plums. I had fallen into a vicious circle, and this perplexed me at that age. Mathematicians' ratings are something like this.
Many mathematicians are also sensitive about what they consider their most beautiful mental offspring — results or theorems — and they tend to be possessive about them. Paradoxically, they also show a tendency to consider their own work as difficult and other work as easier. This is exactly opposite in other fields where the better acquainted one is with something the easier it seems.
Mathematicians are also prone to disputes, and personal animosities between them are not unknown. Many years later, when I became chairman of the mathematics department at the University of Colorado, I noticed that the difficulties of administering N people was not really proportional to N but to N2. This became my first "administrative theorem." With sixty professors there are roughly eighteen hundred pairs of' professors. Out of that many pairs it was not surprising that there were some whose members did not like one another.
Among the Harvard mathematicians I knew, I should mention Hassler Whitney, Marshall Stone, and Norbert Wiener. Whitney was a young assistant professor, interesting not only as a mathematician. He was friendly, but rather taciturn — psychologically of a type one encounters in this country more frequently than in central Europe — with wry humor, shyness but self-assurance, a probity which shines through, and a certain genius for persistent and deep follow-through in mathematics.
Marshall Stone, whom I had met when he came through Warsaw with von Neumann and Birkhoff in 1935 on the way back from the Moscow Congress, had had a meteoric career at the university, although he was only thirty-one years old. Already a full professor, he was quite influential in the affairs of the department and of the university for that matter. He wrote a classic work, a comprehensive and authoritative book on Hilbert space, an infinitely dimensional generalization of the three-dimensional or n-dimensional Euclidean space, mathematically basic to modern quantum theory in physics. He was the son of Harlan Stone, Chief Justice of the Supreme Court. It is said that his father proudly said of Marshall's mathematical achievements, "I am puzzled but happy that my son has written a book of which I understand nothing at all."
And there was Norbert Wiener! I met him at a colloquium talk I gave during my first year at Harvard. I was lecturing on some problems of topological groups, and mentioned a result I had obtained in Poland in 1930 on the impossibility of completely additive measure defined in all subsets of a given set. Wiener, who always sat at lectures in a semi-somnolent state except when he heard his name (at which he would suddenly jump up, then sit back in a very comical way) interrupted me to say, "Oh! Vitali has proved something like that already." I replied that I knew Vitali's result and that it was much weaker than mine because it required an additional property — namely equality for congruent sets — whereas my result did not make any such postulate and was a much stronger, purely set theoretical proof. After the lecture he came to me, apologized, and agreed with my statements. This was the beginning of our acquaintance.
I had heard of Wiener before this meeting, of course, not only about his mathematical wizardry, his work in number theory, his famous Tauberian theorems, and his work on Fourier Series, but also about his eccentricities. In Poland, I had heard through Jozef Marcinkiewicz about his book with Paley on the summability of Fourier transforms. Raymond Paley, one of the most promising and successful young English mathematicians, died in a mountaineering accident at a tragically young age. Marcinkiewicz was a student of Antoni Zygmund. He visited Lwów as a post-doctoral fellow and patronized the Scottish Café, where we discussed Wiener's work, since he had worked in trigonometric series, trigonometric transforms, and summability problems. Marcinkiewicz, like Paley, whom he resembled in genius and in mathematical interests and accomplishments, reportedly was killed while an officer in the Polish army in the 1939 campaign at the beginning of World War II.
Absentminded and otherworldly in appearance, Wiener nevertheless could make an intuitive appraisal of others, and he must have been interested in me. Great as the difference in age between us was (his forty to my twenty-six years), he would seek me out occasionally in my little apartment in Adams House, sometimes late in the evening, and propose a mathematical conversation. He would say, "Let's go to my office, where I can write on the blackboard." This suited me better than staying in my rooms, from which it would have been difficult to put him out without being rude. So he drove me in his car through darkened streets to MIT, opened the building doors, turned on the light, and he started talking. After an hour or so, although Wiener was always interesting, I would almost fall asleep and finally manage to suggest that it was time to go home.