Some Mathematical Reminiscences

by Morris Marden

This article originally appeared in the October, 1990 issue of Mathematics Magazine -- volume 63, number 4, pages 244-248 -- published by the Mathematical Association of America. It is posted here by kind permission of the publisher.

1. Introduction

It has been my good fortune, during over sixty years in mathematics, to have met and befriended many mathematicians (over thirty of them are pictured in the Pólya Picture Album published by Birkhäuser, Boston and Basel, 1987), some quite prominent in their day. In what follows I report not on their mathematical accomplishments and discoveries but relate, rather, my personal experiences with a few of them.

2. Cambridge, Massachusetts

Harvard is where I received my basic training: a bachelor's degree in 1925, two years as a part-time instructor in 1925-1927, and a Ph.D. in 1928. In those days the mathematics department was relatively small so that there was a close relationship between students and faculty. My first account concerns three Harvard faculty members, William F. Osgood, George D. Birkhoff and Julian Coolidge; and one visitor, Constantin Carathéodory.

Professor Osgood was a tall, black-bearded man, an 1886 graduate of Harvard College. In those days a young American desiring to pursue mathematics further had to go to Europe since mathematics in America was still in its infancy. Osgood chose Göttingen and Erlangen, Germany, where in 1890 he obtained a Ph.D. under the renowned Felix Klein. Returning to the United States and a position at Harvard, he made some distinguished contributions to complex variable theory, including results on conformal mapping and a two-volume treatise entitled, Lehrbuch der Funktionentheorie. By the way, this treatise was written in German, because English was not as yet regarded as an appropriate language for a mathematical treatise.

I met Osgood in 1920. I was a 15-year-old high school junior, eager to take advantage of the "anticipatory examinations" which a student could take if he were entering with more subjects than needed for regular admission to Harvard. The student could, thereby, earn in advance up to a year's college credit. I resolved to do just that, in particular to cover by myself the Harvard freshman course in analytic geometry and calculus. Osgood, then the mathematics department chairman, advised me as to the texts used in the course. I did well on the examination and was given an A. This success, plus Osgood's apparent interest in me, then persuaded me at age 16 to aspire to become a mathematician.

Osgood served as my college advisor, invited me to his home and visited me in the student infirmary. He was a superb teacher who struck a good balance between giving complete details and leaving matters to the student's initiative and intuition, frequently motivating a new subject through physical applications. My last meeting with him was during September 1932 in a Harvard Square cafeteria. Greeting me was a clean shaven man whose voice I recognized as Osgood's. Since that was only a few days before my marriage, I invited him and his new young wife to the wedding, and both came.

Professor Birkhoff had the distinction of being among the first American-trained mathematicians to receive international recognition. As a sophomore I was hired to be problem reader in his calculus course, and subsequently took his various courses on differential equations, including one on the three-body problem. His lecture style was quite different from Osgood's. He did not seem to come well prepared to class; however, when he got stuck on the proof of a difficult thorem, it was always interesting to watch this brilliant scholar work his way out. Every now and then he would propose what he alleged to be an unsolved problem. For one such problem, I developed a relatively simple solution. However, my hopes of thereby publishing my first mathematical article at age 19 were dashed when Birkhoff finally recalled that he had developed, several years earlier, a similar solution.

Professor Coolidge remains in my memory especially because of the following incident. When our small research club invited him to talk on his geometrical specialties, he accepted on the condition that we first have dinner at his home. Of course, we agreed. After seating us around his dining room table, each of us separated by our host's daughters or governesses, Collidge noticed that he was lacking one female. He then excused himself, only to return in a few minutes wearing a bathrobe and declaring that he would substitute for the missing female. After dinner, as we bid good night to the ladies, he stood in the corner of the room throwing peanuts into the air and catching them in his mouth -- quite a feat for a dignified Harvard professor.

Professor Carathéodory, a visiting professor at Harvard during 1926-27, was probably the most famous Greek mathematician since antiquity, yet he held a position not in Greece but in Munich. Physically, he was somewhat plump, of medium height, but cross-eyed so that it was hard to determine whether or not he was looking at you. After attending his lectures, I would often walk him home, choosing preferably a route passing gardens containing tulips, his favorite flower. I visited him around Christmas, 1929, at his home in Munich, but my final contact with him occurred during 1936-37 when he was serving as a visiting professor at the University of Wisconsin-Madison. During our walks, I tried to get his views on the Nazi regime but -- I suppose for his own protection -- he was completely silent about the matter. However, I could deduce his probable views from the fact that he had sent both his son and daughter back to Greece for their education.

3. Zürich, Switzerland

I was awarded a National Research Fellowship, which allowed me to do resarch under E. B. Van Vleck at Madison, Wisconsin, during the summer of 1928, under Einar Hille at Princeton during the academic year of 1928-29, and under George Pólya in Zürich during fall, spring and summer of 1929-30. While in Zürich, I renewed my acquaintance with the famous Hermann Weyl, whose lectures I had attended in Princeton.

Professor Pólya, a native of Hungary, had done postdoctoral work at Göttingen. Though he was 40 and I was only 24, we developed a friendship, taking walks and swims together. I remember, for instance, that when we walked on a hot summer day, he carried his hat in his hand, but would don it hastily when he sighted a friend. This he did so as to be able to conform to the custom of tipping one's hat to any passing acquaintance. I found he was quite different from typical European professors. For example, at picnics he would mix and chat with the students, whereas the other professors remained aloof.

During my stay in Zürich, Pólya was collaborating with Harday and Littlewood, the English mathematicians, to write the book Inequalities. Hardy would occasionally consult with Pólya in Zürich, following which Pólya would ask me to play tennis with Hardy (whose lectures I had attended at Princeton). I found Hardy to be quite a good player. In fact, he was an enthusiastic sports fan in general, including American baseball.

In 1940, when the Nazis seemed to be closing in on Switzerland, Pólya emigrated to the US, eventually taking a professorship at Stanford University. Among his colleagues there were three other refugees from Europe: Gábor Szego, Stefan Bergman, and Charles Loewner. The presence of these four men greatly enhanced the reputation of the Stanford Mathematics Department, especially in analysis. Pólya remained active almost to his death in 1985 at age 97. He wrote several bookd and hundreds of research papers, in recognition of which he received several honorary degrees including one from the University of Wisconsin-Milwaukee. Furthermore, Stanford named a building in his honor.

Professor Hermann Weyl was among several faculty in Zürich who were quite friendly to me, an American student. He used to invite me to his home for dinner and ping-pong. Later on, when he was appointed to the prestigious Hilbert chair in Göttingen, he gave a public lecture entitled "Die Stufen der Unendlichkeit" (the Levels of Infinity), which attracted a number of theologians who, however, were dismayed on finding that the lecture was on mathematics. Subsequently, a farewell picnic was given for him by the mathematics students and faculty at a Zürich open-air resort. On the grounds was a see-saw that Weyl seemed delighted to use, but when he was to receive a gift, he could not be found. Evidently, engrossed in conversation with someone, he had wondered off.

4. Paris

My visit to Zürich was interrupted by a stay in Paris, from January to April 1930, for study under Professor Paul Montel. However, before then, I did make short visits to Paris, and I begin by recounting one such visit.

I chose to return to Zürich indirectly by way of Strassbourg (France) and Fribourg (Germany). I reached Fribourg late one evening, quite tired. I asked the railroad stationmaster for the name of a small, quiet hotel. When I arrived long after regular dinner hours, I was at first alone in the dining room, then a small dog held on a leash by an elderly woman entered, followed soon by a tall man with a long gray beard. I was amazed to discover that he was none other than Ferdinand Lindemann, who made history about fifty years earlier as the first ot prove that pi is transcendental, that is, that pi is not the root of any algebraic equation with whole number coefficients. When he said he was about to take a walk, I asked (even though I was pretty tired) to accompany him. He took me through the darkened, local university campus, where he had once taught, pointing with his cane at the more important buildings. I finally got to ask him what he thought of the then modern mathematics and his answer was "Zehr compliziert". This is a reply that I must remember to use when some younger mathematician describes his research to me.

During a longer stay in Paris I made many friends among the Americans and French mathematicians. My closest friend was the eventually very well-known French mathematician, Jean Dieudonné, whom I had gotten to know during the period 1928-29 when we were both in residence at Princeton University. He and I visited the castles at Fontainebleu and Versailles and had a date every Thursday evening to have dinner at a provincial restaurant of Paris and then to attend a symphony. At dinner we always consumed two bottles of wine so that, though the evening began sedately, it was not so sedate when we headed for the concert hall.

5. Providence, Rhode Island

During the summers of 1943 and 1944, I attended lectures at Brown University, which had received a large government grant to convert pure mathematicians into the applied mathematicians needed for the war effort. The staff at Brown was augmented by such European mathematicians at Stefan Bergman, Lipman Bers, Stefan Warshawski, and J. D. Tamarkin.

Whereas the first three men were refugees from Nazi-held Europe, the fourth, Tamarkin, had fled from the Soviet Union by tramping across the muddy fields of Russia and Poland. When he and his companion reached the German border, they were so covered with mud as to be unrecognizable. Doubtful of their identity, the border guard called the local American consul, who proceeded to test Tamarkin's knowledge of mathematics. However, the consul had gone only as far as calculus in college so that Tamarkin had no difficulty in passing the test. Tamarkin was a somewhat stout man who walked slowly, often taking with him his little Scotch terrier. He was very near-sighted so that when Arnold Ross and I invited him and his visitor, Antoni Zygmund, to the movies, we all had to sit in the front row.

6. Milwaukee

I close my account by describing three distinguished visitors to the University of Wisconsin-Milwaukee during the postwar period: the Polish mathematician, Casimir Kuratowski; the Japanese mathematician, Akitsugu Kawaguchi, and the English mathematician, J. E. Littlewood.

Professor Kuratowski held two official positions: university professor and vice-president of the Polish Academy of Science. I met him in 1958 while attending the meetings of the Mathematics Institute of the Polish Academy, held in Lublin. After we gave our scheduled talks, he invited me to ride back to Warsaw in his chauffeured car. On the way, besides viewing the ruins of old palaces, we visited a former Nazi extermination camp maintained as a museum by the Polish government in memory of the thousands of innocent men, women and children who were murdered there.

Kuratowski described his narrow escape from the Gestapo. One day when the Gestapo visited him in his Warsaw apartment, he told them his profession. They replied that he was no longer a professor. He showed the Gestapo a letter he had just received from the editor of the Mathematische Zeitschrift asking him to collaborate on the editorial work of the journal. The word "collaborate" seemed to appease the Gestapo men, who then left whereupon Kuratowski immediately went into hiding for the rest of the war.

I last saw Kuratowski when he gave a colloquium in Milwaukee. On the way back to the airport, I drove him along a route which passed many stores with good Polish nameplates. He was amazed to learn that more people of Polish ancestry lived in Milwaukee than in most Polish cities.

Professor Kawaguchi served as chairman of two large mathematics departments, one in Tokyo and the other in Sapporo, many miles from Tokyo. I met him too in 1958 at the meetings of the Polish Academy. As we both had gone to the meetings alone, we took walks together and otherwise provided mutual companionship. Several years later, he invited me to give a lecture at the meetings of the Japanese Mathematics Society in Tokyou and then to give several lectures at his university in Sapporo. While in Sapporo, my wife and I were entertained in a geisha house. Because my wife was accompanying me, Mrs. Kawaguchi went along with her husband. It was her first visit ever to a geisha house. Other mathematiciians in our party had left their wives at home, as was the Japanese tradition. The geisha girls, attractive and talented, provided conversation at the table and then enacted a short play for our entertainment.

In appreciation of this hospitality, I invited Professor and Mrs. Kawaguchi to Milwaukee, where he gave a colloquium. We saw the Kawaguchis once more at the International Mathematical Congress in August 1970 in Nice. Thereafter we continued to exchange New Year cards until their deaths.

Whereas Professors Kawaguchi and Kuratowski remained in Milwaukee only for a day or two, the third visitor, Professor Littlewood, spent nearly a semester there. Initially declining an invitation to spend September to mid-January in Milwaukee, he agreed to come provided he could leave by Christmas for the skiing season in Switzerland. My last contact with Littlewood as at Trinity College, Cambridge University. I understand that the University guaranteed him occupancy of his rooms after his retirement, along with an adequate supply of his favorite sherry. (In fact, soon after arriving in Milwaukee he made a deal for a case of sherry with the manager of his hotel.) His quarters at Trinity included a large living room overfilled with books and papers as well as a small piano and an open fireplace.

During my visit Littlewood took me on a tour of the college library. He seemed especially proud of the large collection of beautifully bound volumes of mathematics dating over several centuries. He would have been equally proud of the inscription on a plaque on the wall of the college chapel. It listed the many distinguished mathematicians (including Newton) who had been associated with Trinity College. Eventually this plaque would list his name alongside that of Godfrey H. Hardy, his close friend and colleague.