The Princeton
Mathematics Community in the 1930s
Transcript Number 38 (PMC38)
© The Trustees of Princeton University, 1985
CAREER, PART 1
This is an interview of Albert Tucker in September 1975 at the University of Western Australia. The interviewer is T.P. Speed.
Speed: Would you begin with your early schooling and with what led you to take up mathematics as a career?
Tucker: Well, I was born in Canada, in Ontario, not too far east of the city of Toronto. My father was a Methodist minister, and I was the only child of my parents. We lived in several small towns as I was growing up because Methodist ministers, then anyway, moved about a great deal. We were never longer than three years in any one place.
I was late in starting school, almost eight years of age when I first went to a school. I had learned to read and to do many things already at home. My parents were both very much interested in my education, and at a very early period my father bought a children's encyclopedia called The Book of Knowledge, which came from England. It was in that that I did my first reading. At a later time, but still when I was a school boy, he bought the eleventh edition of the Encyclopaedia Britannica. So I had pretty good facilities at home, and because I moved around from place to place I had a rather lonely childhood, didn't have much in the way of friends. I suppose that this had a certain influence in turning me at an earlier age than most children to things of the mind.
My mother tells that when I was four or five years of age the thing I used to ask for was, "Draw me a map." There was always a globe in the home, and that was practically a play thing of mine. When I was started school I moved ahead very rapidly. I think I went through what would have been three years, the first three years, in one year. I felt that I had very good teaching and did well in all my work. I don't know that I had any particular liking for mathematics over other things. The one thing I noticed about mathematics was that I could do it more easily than the other things. Latin I did well in, but with a great deal of hard work. Mathematics I did equally well but with no work at all.
The point where it became clear to my father at least that I had some mathematical gifts was when I was in my second year of high school. I was in high school for five years, the last year corresponding to grade 13. In other words, I did an extra year in high school in order to prepare for an honors course at the university, rather than the so-called general or pass course. In my second year of high school I had geometry. The mathematics was I think almost entirely geometry. And that was my first formal acquaintance with geometry. It was a small school, about 100 students., There were just three teachers, and the principal of the school, a man, was also the one who taught mathematics and science. He had observed that most students he was teaching could not do much with the original problems. So he had them learn the propositions by rote, if that was the only way the students could learn them, because we had to take examinations that were the same all over the province. He knew that if students could handle all the book propositions that were on the paper, then they would get a passing grade, not an honors grade but a passing grade. So he concentrated on teaching the propositions. There were these problems there in the book but nothing was said about them, so I was just concentrating on the propositions.
On the other hand I didn't particularly want to commit them to memory, so what I was in a way doing was really understanding the propositions. Well, about the middle of the year he gave us an examination. He just used an old provincial examination, and he forgot all about the fact that there were some originals on the examination.Well, I did the propositions and I did the originals. He came that night to see my father and said, "You have been coaching your son," because my father had actually planned to become a teacher of mathematics before he decided to go into the ministry and he had been through about two years at the university. He had taken calculus, and then he had switched to philosophy as a more appropriate background for a career as minister. So the high-school teacher, who knew something about my father's background, thought that my father had been coaching me at home and had me doing the originals.
My father said no, that he had not done this at all. The high-school principal said, "Well, it's just fantastic that he has done these originals although he has never been given that sort of thing to do in class. He has just done them." This so shamed him that for the rest of the year he really taught a bang up course. No more just teaching the propositions, he really threw himself into it and then was a very good teacher of geometry. He had a very good grasp of it, it was just that as principal of the school he had so many other things to do that he was just following the path of least resistance. He was so pleased with me that he told my father that he thought that I should aim to become an actuary. He felt that becoming an actuary was, so to speak, the top thing that one could do with mathematical skills. Anyway this then made my father very interested in my mathematical ability, and of course this rubbed off on me so that I was aware that I was pretty good at mathematics. And this I think stimulated me to take a greater interest in mathematics and to delve more deeply. Before that, because I could do it so easily and had no problem on examinations, I was inclined to spend my time mainly on other things that did require my effort. But this served to put back more of my interest in the mathematics.
In my fifth year in high school I had an introduction to analytic geometry, a very serious course in trigonometry, both plane and spherical, as well as a good deal of algebra and even a little bit of solid geometry. So I had a great deal more than students seem to get nowadays, even in Canada.
When I entered the university I enrolled in the honors course in mathematics and physics. This was a sort of basic program that branched in various ways later on, so there were about five or six different end possibilities. In my first year at the university I had about three courses in mathematics that went throughout the year, as well as courses in physics and chemistry. One of the mathematics courses was a course in conic sections which I had from someone who is quite well known in the mathematics world, John L. Synge of Dublin. He gave a very rigorous course following an old British classic book, C. Smith, Conic Sections. This was not just a book in analytic geometry, but also an introductory book to invariance and things that are now a part of ...
Speed: Algebraic geometry.
Tucker: Yes. We studied very thoroughly all the transformations that dealt with a general equation in the second degree, as well as even some of the projective properties. So that was a very demanding course, not at all modern, but one learned a tremendous amount in such a course.
Then I had a course which was called Introduction to Analysis which was entirely on limits. It was quite serious, you know. Besides doing series and that sort of thing we did the problem of how to define the area of a plane figure. In other words this was an introduction to calculus without ever any mention of derivative or integral. It was all set up in terms of limits.
Speed: Was that based on Hardy's book by any chance?
Tucker: There was no book. But it was very much like Hardy's Pure Mathematics, which I got to know later. I didn't know about it at that time. And then there was another course that was called Higher Algebra and it was based on another old British book, Hall and Knight. This was the complete Hall and Knight, so that it got into some elementary number theory. It was again a course that involved many of the ideas that are now regarded as part of abstract algebra.
Speed: Theory of equations.
Tucker: Yes, theory of equations and so on. There was no calculus in the first year except a course that the professor in physics gave us one hour a week, based on the book of Sylvanus Thompson, Calculus Made Easy. This was so that we could use the calculus in the physics. At the end of that year Synge left to go back to Trinity College, Dublin. He left a note with the head of the department saying there's a student in the first year by the name of Tucker who should be watched. I didn't know about this until considerably afterwards. But the head of the department from that point on took a very fatherly interest in me. He was an old Irishman, a bachelor, very much a man of the world. But he had a very sharp knowledge of mathematics, not in the sense that he had produced mathematics, but, he was really a scholar in the best sense of the word. In my second year I had simultaneous courses in differential calculus and integral calculus. The professors giving these, one was fresh from Cambridge where he had stayed on as a fellow for a few years. Then he came out to Canada. He was English. His name was W.J. Webber. The other, a man by the name of Samuel Beatty, was a Canadian who had grown up in the University of Toronto.
They conspired very beautifully to teach their courses so that while Beatty in the differential calculus was teaching the usual thing about derivatives and such, Webber in the integral calculus was teaching essentially the Riemann integral with no mention at all of derivatives. Then suddenly after about six weeks the two courses came together in so called fundamental theorem of calculus. This was very spectacular. They also did an unusual thing that year. It didn't pan out for most of the students, but it was great for me. They used as textbook in both of these courses the Cours d'Analyse by de la Vallee Poussin, in the French of course. This really introduced me to rigorous presentation of mathematics.
I continued in the second year to follow physics as well as mathematics. Indeed, the physics attracted me more than the mathematics because physics has, so to speak, more glamour to it. I happened to have that year a course which was an introduction to the history of physics given by the chairman of the physics department. He had attended, during the summer of 1925, a conference in Italy in which there was discussion of the new results that were beginning to appear in quantum theory. So he said to the students in the course, " I don't know anything about this quantum theory. It's too mathematical for me. Who's the best mathematician in the course?" There were about 50 students, and they looked over towards me. "Well," he said, "I want you to make a report on quantum theory." You know, I had no background. There was nothing at that time in the way of a textbook. There were just a few papers by Dirac and Heisenberg, and I couldn't make very much of them.
Tucker: I had taken German in high school. This was very difficult. I was not able to get German until about my last year of high school because it was so soon after World War 1. In Canada they had dropped the teaching of German in World War 1, and it was only just starting to come back. But I was lucky that there was a man teaching German in the school that I attended in my fourth year of high school.
I actually attended four different schools in my five years of high school. In my fourth year I attended this school where there was this man by the name of Schultz, who was born in Canada but of German parentage, and who spoke German and really had a very good feeling for the language. He wasn't able to get very many students so he lavished a lot of attention on the students he did get. So I couldn't make my way easily with these papers on quantum mechanics. Anyhow, I gave this report, and I have no recollection of what I was able to say. But the chairman of the physics department was terribly impressed, so he right then and there said that if I would choose to major in physics that he would make sure that I had an opportunity to go on to post-graduate work. This was the first time that there had been any mention of post-graduate work. I had been thinking of getting my bachelor's degree and then probably teaching mathematics and physics in a high school, or perhaps becoming an actuary. That was also a possibility. But this raised the question then of post graduate work.
Well, I spent several weeks talking to all the people I could in mathematics and all the people I could in physics. I finally decided that physics was not for me, and that mathematics was. In the meantime the head of the mathematics department, when I had told him what Sir John McClennan, the head of the physics department, had said, "We can match it." But I really felt that physics was more glamorous. There's no doubt about that. It fascinated me more than mathematics, but when I talked to the physicists and the mathematicians I found somehow that when I talked to the mathematicians that we seemed to be talking the same language.
Most of the physicists there were experimental, and, you know, they would talk about magnetism in terms of a lot of little magnets that were in the middle of a big magnet. It was a very mechanistic view that they still had towards physics. And somehow or other this didn't satisfy me. I felt that they were not coming to grips with their subject. It was a descriptive rather than an analytical approach. Of course I was right, but I was very naive at the time. So I told people at the time that the reason I chose mathematics over physics was simply that I liked the mathematicians better than the physicists. So in my third and fourth years I dropped the physics and concentrated in the mathematics.
In my third year I had a very good course in differential geometry out of Eisenhart's book and a course in projective geometry and a course in differential equations. I had a very eminent man teaching me differential equations, J.C. Fields, after whom the Fields Medals were named when they were established by the International Mathematical Congress. But he was ill at that time and died shortly afterwards. He wasn't feeling well enough to do anything except go through the motions of teaching the differential equations course. Then in my fourth year I had, for some reason I don't know, two courses in complex variables. One entirely from the Weierstrass point of view, the power-series point of view, and the other pretty much from the Riemann mapping point of view.
I stayed on for an additional year to get my master's degree. This was largely because I had not been able to decide where I wanted to go for post-graduate work. The head of the department wanted me to go to Paris. He thought that was the place to go, but I was worried about having to study mathematics in a foreign language. And after Paris he suggested, because I said I was interested in geometry, Rome and Bologna because the Italians then were very active in geometry. He also mentioned Goettingen. He said, "All you've got to do is decide where you want to go, and I'll get you the money." He said that he knew various wealthy people who would simply set up a fellowship if he went to them and asked them to do it. Of course my father as a minister didn't have the resources to pay for my graduate education.
I knew so little about things that I didn't know that in many places you could apply to that place for admission and that perhaps they would furnish the financial support. I couldn't make up my mind, so I stayed on for a fifth year at Toronto working for my master's degree. Now I was a teaching fellow in that fifth year, and this was wonderful experience for me. There were actually three teaching fellows, but two of them dropped out very shortly after the year began. A very simple expedient was used: as they dropped out, their duties were given to me. So I ended up by teaching three courses, two of which were entirely my responsibility. One was a course in advanced calculus for aeronautical engineers. That was the first year at Toronto that there had been aeronautical engineers. They were a select group. Several were mechanical engineers at the end of their second year. This picked group started the program in aeronautical engineering, and it was decided that they ought to have more calculus than the civils and mechanicals got. So I was just assigned the job of teaching these 10 or 12 students the advanced calculus that would be appropriate to the needs of aeronautical engineering. And I did it. I don't know that it was as well done as it might have been, but I was asked to do it and so I did it.
Another was a course in mathematics for economists. There there were only about six students. They were all students who had started out in mathematics and then switched to economics. I had not at that point had a course in economics in my life. It was suggested to me that if I took a book by Alfred Marshall, Principles of Economics, I would find in that an appendix in which mathematical things such as marginal utility and marginal revenue and elasticity of demand are explained and the appropriate calculus notation introduced. There was nothing like R.G.D. Allen's book in mathematical economics available at that time. It came shortly after. I really had to find my own way and even improvise things to teach. I think I ended up most of the second half of the year in teaching straight statistics, which matters of course for the econometrics.
Well, the third course was to run a problems session for a course in actuarial science. I had been through that course, so this was a perfectly straightforward job. But for both the other ones I had a great deal of investigating to do.
Also in that fifth year I had the privilege of going into the stacks in the library. Before that I simply had to send in a card for any book that I wanted. So with that privilege I spent hours in the stacks just looking. Until that time I had no idea of the availability of mathematical books even at the textbook level, and I found out about journals for the first time that year. Another thing that I got out of being a teaching fellow is that I was given an office. It was a tiny little room up in the attic. The first time I entered this room there was a table and a chair and a naked lightbulb that hung down from the ceiling. That's all there was in the room, except on the table there was a Princeton catalog. So, of course, I studied that Princeton catalog.
As I said earlier, geometry was my favorite part of mathematics, and when I saw the courses that were described in this Princeton catalog it seemed as though about half of them were in geometry with the name of Eisenhart attached to some, and the names of Veblen, Alexander, and Lefschetz attached to others. Of course, later on I found that these courses were not given every year. Indeed it was sort of luck of the draw what courses would be given in any one year, but there was nothing said about this in the catalog. They were all there, and I just assumed that they were all available. So I decided almost instantly that Princeton was the place where I wanted to go. So I went in and told the head of the department that I wanted to go to Princeton. "Oh," he said, "I don't think you want to go to the United States." He said, "You should go to ..." He went through his list again, and I admitted that I was timid about going to a foreign country. "Well," he said, "then you should go to Cambridge." So I wrote away to Cambridge to get information about the courses at Cambridge. And the information came back and I compared it with the Princeton information, and it seemed very poor indeed.
At that time there was a 19th-century-style geometer by the name of H.F. Baker. He had written a many-volume book on geometry, which was a summary of things as they were at the turn of the century. I was very naive, but this didn't attract me in the same way that this variety of courses, analysis situs, Riemannian geometry, projective geometry, and so on did. And I knew that I had studied from Eisenhart's book on differential geometry. I also knew the book of Veblen on projective geometry. So I went back and said, "No, I want to go to Princeton." "Well," he said, "I don't know what we're going to do with you Mr. Tucker. You don't seem to be able to take advice." He said, "If you insist on going to the United States, there are only two places that are worth it, and these are Harvard and Chicago." So I wrote off to Harvard and Chicago and got catalogs back from there and made the comparison of the courses there with the Princeton courses. They were even worse than the Cambridge courses.
So I went back again. I at least was stubborn. I said, "Princeton is the place I want to go." And he really got angry with me. It was the only time in all the many years of excellent relations we had that he got angry, and I had to leave feeling this was an impasse. But there was a Frenchman by the name of [Jacques] Chapelon teaching then at Toronto. He had been a student of [Jacques] Hadamard in Paris, and he regularly returned to Paris because the Toronto academic year would finish about the first of May and didn't resume again until the beginning of October. In Canada the academic year was very short so that the students could go and work on the farms during the long summer. So he could actually spend five months a year in Paris and still have his job at the University of Toronto.
Well, he saw me and apparently perceived that I was depressed, and so he said, "What's the matter?" And I unburdened myself of the story. "Well," he said, "let me see these catalogs that you have." So I gave them to him. Well, he had gone to the same lycee in Paris that Lefschetz had gone to. They had known one another as schoolboys, and he had tremendous respect for Lefschetz. He felt that anywhere that Lefschetz was must be a good place. So he went to the head of the department and took up my cause. He said that he had looked into it and felt that I was right. The best place for me to go was Princeton. Well, the head of the department did a complete about face and said he would arrange for me to go to Princeton.
He wrote to Princeton to a man that he knew there, who was the chairman of the mathematics department, a man by the name of H.B. Fine, after whom Fine Hall is named. But Fine was already dead. Fine died in December, and the letter didn't arrive there until January. So the letter wasn't answered until along about April.
I thought that there was no hope, and I had just about reconciled myself to the idea of staying on at Toronto and getting a Ph.D. there when a letter came from Eisenhart saying that Dean Fine had died and the letter had found its way to him. He wrote that it was now too late for Mr. Tucker to apply for admission in normal course as a graduate student, but that they were short one person to serve as a half-time instructor, and that if Mr. Tucker had teaching experience and could be recommended highly in this respect they would consider him for this part-time instructorship. The appointment as part-time instructor automatically carried admission as a graduate student. Well, thanks to all this teaching that I was doing, there was no problem at all in recommending me as a teacher. So I was admitted at Princeton and went there in September of 1929 to start graduate work. And except for leaves-of-absence and such I've been at Princeton ever since.
Speed: You arrived in Princeton in September of 1929?
Tucker: Yes, in September 1929, and I taught six hours a week, two sections of analytic geometry during the fall term, and differential calculus during the spring term. I took three courses, one from Eisenhart. He had written about three years before his book on Riemannian geometry, and in this course, which went throughout the year, we went through most of that book. It was his system that when he had taught a course once or twice he would then turn that into a book, and once he had the book available he let the book teach the course for him.
At the end of a class he would say, "Now for next time read such and such in the book and try such and such problems." And when he would come to class next time he would sit down and ask if there were any questions about the material in the book. Once those questions were disposed of, he would then ask members of the class to go to the board and do problems. He was very mild and jovial, but if he asked you to do a certain problem and you declined his disappointment was very hard to take. He didn't make any sarcastic remarks, but it was very clear that he was sadly disappointed. I don't think it would be good if all courses were like that, but an occasional course of that sort is, I think, a pleasant change.
I also had a course from Lefschetz. He was just finishing his first book on topology, and he was lecturing on the basis of his writing of that book. When I arrived at Princeton in 1929 'analysis situs' was the term used. But Veblen had written a book with the title Analysis Situs which had been published as one of the colloquium volumes of the American Mathematical Society. Well, Lefschetz wrote his first book on topology to be published in the colloquium series also, so he couldn't use the term analysis situs. He wanted, as he would say, a short snappy title. He didn't want any long-winded title.
So to get a title that would be different from 'analysis situs' he decided to use the word topology which had not been used prior to that in English. It had been used in German. Indeed, there was a book written about 1850 by a student of Gauss by the name [J.B.] Listing that had 'Topologie' in its title. But there was no precedent in English, the word 'topology' did not exist. Lefschetz invented it as the title of the book that he was writing. Well, once he decided to use that word, he mounted a campaign to get everybody to adopt it. He was very successful, mainly I think because the word topology lends itself to all sorts of derivative words, whereas analysis situs does not. So in one year analysis situs was dropped and topology was adopted.
The course that I had with Lefschetz based on his Topology was a rather poorly organized course because Lefschetz was always too restless to do things in a systematic fashion. When he would give a proof he would start at the beginning, get impatient and jump to the end, and come back from the end, and still usually leave a great big gap in the middle. He was notorious for his sloppiness in mathematical rigor. On the other hand he had just remarkable intuition. I don't think he ever published a result that was wrong. His proof may have been quite incomplete, but his results were always very sound.
I was more attracted to the Riemannian geometry of Eisenhart because it was more orderly than the disorderly topology of Lefschetz. Indeed, in the course of my first year at Princeton, I wrote my first paper. I didn't know I was doing it. I went to Eisenhart about the middle of the year to say that I felt there was something that he had developed in his book that could be done in a much simpler and neater way. Well, he was quite patient in listening to me, and he suggested that I write this down so that he could look it over. I did this. And he made certain criticisms, and I rewrote it two or three times. Finally, he bowled me over by saying, "Now, Mr. Tucker, I would like to submit this for publication in the Annals of Mathematics." Up to that point the idea of me writing a paper had not occurred to my mind. I was just doing this to make my point with Dean Eisenhart.
At this time he was Dean of the Faculty. He held the second most important position in the University administration, next to the president. So he was very heavily involved in administration. But he did take the time to spend with me about these ideas, which led to me writing my first paper without knowing I was writing a paper. I did submit it to the Annals of Mathematics, and it was published about a year later, published while I was still a graduate student.
I continued to be interested in both differential geometry and in topology. I also took a course in my first year from Hille, which was a course in complex variables. I had already had a considerable amount of complex variable in Canada, and this gave me a considerable advantage, with the result that I made a very good impression on Hille. So in my first year there I won three very important friends, Eisenhart and Lefschetz and Hille. In the spring term in my second year, Lefschetz exchanged with Alexandroff, so Lefschetz was in Moscow and Alexandroff was in Princeton. I took the course that Alexandroff gave which was a course leading up to dimension theory in the Alexandroff form rather than in the Menger-Vietoris form.
I had some topology also from J.W. Alexander. It was very difficult for him to give a course, because he was always wanting to work with the ideas that were then in his mind, but at the same time he was a perfectionist, so until he had these ideas in good form he didn't feel able to talk about them. The things I had from him were rather in bits and pieces. He would do something in, so to speak, seminar form that he had nicely worked out, and then he would get to a point where things didn't satisfy him and he would just call it off.
I had some influence from Veblen also. Veblen at that time held the research chair, so he did not give courses. He held only a seminar. He was working at that time on the foundations of differential geometry, something which he subsequently published as a Cambridge Tract with [J.H.C.] Whitehead. Whitehead was there as a graduate fellow; he took a Princeton Ph.D. along about 1931, about a year before I got my Ph.D.
Speed: He'd come from Cambridge?
Tucker: Yes. He was working very closely with Veblen. Veblen had spent a year, around 1927-28, in England at Oxford. He had traded positions with Hardy. This was when Hardy was still at Oxford. Veblen was very much impressed by Oxford and Cambridge, and he really tried when he came back to Princeton to copy in many ways what he had seen at Oxford. Particularly the idea of having afternoon tea at which everybody gets together. This was quite new in America, but succeeded so well at Fine Hall, which was built just shortly after that and had a room in it just for this purpose, that it has been copied very widely in other mathematics buildings throughout the United States. It was started by Veblen.
We had a little verse that we used to jokingly say about Veblen: "Here's to Cousin Oswald V.,/ Lover of England and her tea,/ He is that mathematician of note,/ Who uses four buttons to fasten his coat." He was very tall, particularly tall from the shoulders to the thighs, and he always wore a coat that had four buttons down the front, not just three, and these were always fully buttoned.
Veblen was trying very hard to come up with something which as of 1930 would do for geometry what the Erlangen Programme of Felix Klein had done around 1870. He was trying to define geometry in such a way that you would have differential geometry and topology included in a meaningful way. Projective geometry, non-Euclidean geometry, and so on are very well characterized by the group point of view. But to define topology as the study of the group of all homeomorphisms of a space onto itself is completely unsatisfactory. Nor is the group point of view satisfactory for differential geometry. Well, after putting a great deal of effort into this, Veblen finally came to the conclusion that any definition of geometry that would include all of the things that he wanted to have included would also include all of mathematics. I was right there to hear this conclusion when Veblen reached it and heard him tell the various things that he had tried and why they had failed. This has had actually a profound influence on my attitude towards mathematical education.
Now there are some parts of mathematics, algebra for example, that you can define in terms of the subject matter, the content. But geometry I feel you cannot. Geometry is a point of view. Even a somewhat emotional point of view. I can tell people how intensely I love geometry, but I can't tell them what geometry is in any way except simply by samples. Indeed Veblen somewhat jokingly proposed the following as a definition of geometry: something should be regarded as a part of geometry if at a given time there were people of taste who said that it was part of geometry. Well, I didn't ever work with Veblen in the way I did with Eisenhart and later with Lefschetz, but I was certainly very much influenced by him.
During the summer between my first and second years at Princeton, this would have been the summer of 1930, I was given by Lefschetz to take home to Canada some chapters of the manuscript of his book. I was to criticize these and find any slips that there were. But he didn't really define what he meant by criticism, so I really let myself go. In the fall I came back with a proposal for him to rewrite the book. I felt that he was changing his standpoint between the combinatorial view, where he was thinking of a manifold as made up out of cells, and the point-set view, where he was regarding a manifold as defined by point-set properties. He was using whichever of these points of view suited him at a particular point in the book. Well, my orderly attitude was that he should carry the purely combinatorial as far as he could and only bring in the point-set, which he needed for the final nature of a manifold, when he had done everything that he could do with the combinatorial.
Well, at first he just laughed, scoffed at my suggestion. But I persisted in arguing this, so he proposed that I should give a seminar in which I would present the way I thought the book should be written. He attended this seminar. Two of my close friends were R.J. Walker, later at Cornell and who wrote a book on algebraic geometry, and Nathan Jacobson, the algebraist who ended up at Yale University. These and some others as well as Lefschetz attended this seminar that I gave. At the start I had the plan to go as far as possible with the combinatorial. I had to implement that plan as I went along. Lefschetz was a critic. He never had the least bit of hesitation, right in the midst of anybody speaking, right in the midst of a sentence, of breaking in and, as we called it, heckling. So between Lefschetz heckling and my fellow students' criticism, I got, a good working over, which helped a great deal to sharpen things and push me on.
Well, that turned out to be my thesis. It was published in the Annals of Mathematics in 1933 under the title "An Abstract Approach to Manifolds." And in the end Lefschetz became very enthusiastic about it. Years afterwards he had a standard answer to people who would come to him with some combinatorial idea. He would say, "Have you looked in Tucker's thesis? It's probably in his thesis." So I got my Ph.D. in 1932.
Speed: So you were working on preliminary work over the summer?
Tucker: The summer of '30 and then throughout the next two academic years.
Speed: So two years on it.
Tucker: Yes.
Speed: And how did you find that period? Was it all straightforward, plain sailing?
Tucker: It was all plain sailing, but of course I didn't realize that this was going to be my thesis. It was like the thing with Eisenhart. I was doing it mainly to make my point with Lefschetz that one could do the combinatorial first and only then bring in the point-set. So in a sense I was doing the thing not so much as research as exposition. It was research only in the sense that it was something that had never been done before. But as far as I was concerned the efforts were largely expository efforts of putting things in the proper order and in making sure that there were no gaps in it. But during that period I was never held up by anything. It just moved right forward.
Speed: Has this been a common characteristic of your research?
Tucker: It has. Indeed, in retrospect I would say that most of the things of mine that have been published that would be called research have been done with the aim of trying to simplify and unify. There has been almost nothing of my work where a certain problem was presented and I set out to find a way of solving that problem. Instead in most cases it was creating a certain structure, and the structure just created itself if you pushed ahead and used good sense and judgment.
Speed: Yes, I was noticing the similarity between what you were saying about this work and the story about our involvement in linear programming.
Tucker: Yes.
Speed: So when you actually are confronted with a clear-cut problem, say that somebody just presented to you, what do you think about this? Are you attracted to that sort of thing?
Tucker: No. The one exception that I can think of is my combinatorial lemma for the n-cube. When I was a graduate student everybody was talking about [E.] Sperner's lemma for the n-simplex. This had been published in 1928, and I began my graduate work in 1929, so this was very fresh and very exciting. Well, in 1933 [Karol] Borsuk published a paper in English translation, "Three Theorems About the Sphere". These were antipodal-point theorems.
Well, shortly after this paper was published Lefschetz drew it to my attention and remarked that it should be possible to prove these theorems of Borsuk by Sperner's lemma. He remarked, "You're good at combinatorial things, Tucker. Why don't you try to get an elementary proof of the antipodal-point theorems?" Well, this attracted me. I worked at this on and off for quite a few years. I couldn't get anywhere with it, so I would drop it and then a few months later come back to it.
During World War II I had occasion to do a fair amount of traveling related to the war research in which I was engaged, and so I was often many hours in an airport between planes. This was all in the United States. I was not involved overseas in any way. And I would busy myself during these long waits by thinking about mathematics. And I thought about this Sperner's lemma and antipodal-point business, and in a flash the idea occurred to me that should have occurred to me years before. There is no chance of using Sperner's lemma because the n-simplex doesn't have antipodal symmetry. You do not have a vertex opposite a vertex, you have a face opposite a vertex. So if you are going to use simplicial methods to do antipodal-point theorems you need to work with something like the octahedron which has the antipodal symmetry.
Well, once this very simple idea occurred to me it didn't take me long to devise a lemma like the Sperner lemma for the octahedron that would prove the Borsuk theorems. This work did not get published in the ordinary way, but is published in Lefschetz' Introduction to Topology. There is a certain part of that book, about 10 pages, devoted to antipodal point theorems, and he says right at the beginning that this material was communicated to him by A.W. Tucker. The earlier part of the chapter deals with Sperner's lemma and things like the Brouwer fixed-point theorem that you can prove by means of Sperner's lemma. The end of the chapter deals with my octahedral lemma and proving the antipodal point theorem.
Incidentally I was supposed to have been his co-author in that book, but when we tried to work together, we just couldn't. I wanted to do everything in an orderly fashion, and he just didn't have the patience to do this. So we agreed to part company. But he did include this stuff of mine.
Speed: Is this his Princeton volume, or is it the colloquium volume?
Tucker: This is his Princeton volume called Introduction to Topology. I mention this as one instance of where I had a specific goal and didn't know how to accomplish it and spent many years of fruitless effort before I finally saw the clue to the thing. A very, very simple clue. Once I thought of that then everything just went quickly. Later in a paper published in the Proceedings of the First Canadian Mathematical Congress I developed another lemma, for the square in this case, or in general for the n-cube, which I will be talking about because I feel that it is much more promising for computational purposes than the octahedron, because a cube is so much more regular thing to apply to analytical matters than the octahedron.
Speed: Well, we've got roughly speaking to the end of your Ph.D., in the early '30s, but we haven't broached a topic of some interest to me, namely Princeton in the '30s.
Tucker: Well, in the fall of 1931 we moved into Fine Hall. It was built as a memorial to Henry Burchard Fine, who was the chairman of the mathematics department more or less from around 1900 until he died in 1928. Veblen was the one who took the lead in planning Fine Hall and in getting some wealthy alumni by the name of Jones, who had been students at Princeton at about the same time that Fine had been a student, to contribute. They had known Fine over the years, and they were persuaded to put up half a million dollars to build and endow Fine Hall.
It was built on a very lavish scale, at least for the United States. The principal offices, or studies as they were called, were paneled with oak that was brought from England, and they had fireplaces in them. This showed the effect of the year that Veblen had spent at Oxford in exchange with Hardy. Hardy came to Princeton for a year, and Veblen went to Oxford. Indeed Veblen was away at Oxford at the time that Dean Fine died. As soon as he came returned to Princeton he set about planning a memorial for Fine in the nature of a mathematics building and finding the funds for this. It was built to be about 50% bigger than seemed to be needed at the time. We moved in in the fall of 1931; this was the start of my final year as a graduate student.
I had the fellowship that year which was the regarded as the plum, the Procter Fellowship. I was called on by Veblen to organize the tea club. The building had a very nice lounge, or common room as we called it, and the plan was that that would be the focal point in the building. It proved to be exactly that. There was no morning tea (as is the Australian practice), but there was afternoon tea and coffee, usually around 4:00. The staff and graduate students gathered there for tea and to chat. Strangely enough, there we usually stood around as we were drinking our tea, not sit as you, do here. There is a certain advantage in sitting, but at the same time when you're standing you can move around. You're not anchored to one place as much. So if you hear snatches of a conversation over here that you want to participate in, you go over and join that group.
There were only about 25 graduate students in mathematics at the time that I went there. There would be about eight or ten new students coming in each year. Princeton had not yet, at the time I went there, started to produce Ph.D.s. There had been some, but these had been in a sort of hit and miss variety. But from the time that we went into the new Fine Hall there was a steady production of Ph.D.s. The year I got my Ph.D. there were, I think, three Ph.D.s awarded, one of whom was Banesh Hoffmann, the mathematical physicist.
That extra space that had been planned for in Fine Hall was immediately put to use, because in 1933-34 the Institute for Advanced Study began. It had no buildings, so the Institute simply rented space in Fine Hall for the School of Mathematics. So Fine Hall from '33 to '39 was the place where not only the University mathematicians were, but also those of the Institute for Advanced Study. There was no attempt to separate these groups. Indeed, no one paid very much attention to the question of "Am I paid by the University or am I paid by the Institute?" It was all one mathematical group. It is because of this that it became customary in Princeton, and also in other places, to refer to the "Fine Hall group" rather than Princeton University or the Institute for Advanced Study. So Fine Hall became a name for that group working there, and in the Institute this included Veblen and Alexander and von Neumann, who all had been previously at Princeton University. Indeed they continued to occupy the same offices that they had occupied, but they were paid by the Institute rather than by Princeton.
Speed: Could you speak a little about how the Institute came into being? The reasons behind that.
Tucker: The Institute was the conception of its first director, who was Abraham Flexner. Abraham Flexner and his brother Simon Flexner had been very prominent in criticizing the higher education in the United States for being too vocationally minded. With funds provided by the Rockefeller Foundation and by the Carnegie Foundation, both of them, I think, participated in making a study of medical education. They rated the various medical schools in terms of the criteria which they felt should be used. They concluded that there were only about five good medical schools in North America, and two of them were in Canada, McGill and the University of Toronto. In the United States they chose only Hopkins and Harvard Medical School and maybe that was all. I thought there were more.
Following this study on medical schools they did a study on graduate education, Ph.D. education, and they were very harsh in their criticisms. Well, some wealthy people in Newark, New Jersey, by the name of Bamberger, who had a big department store there, were friendly with the Flexners, and when one of the Bambergers died he left a large sum of money to establish a school of advanced study. His deed of gift essentially said that Abraham Flexner was to be the director of this school, and that the school was to be set up as Flexner wanted to design it. It was to have its own board of trustees and be completely independent from any other educational institution.
As to the location of this, the Bamberger will said that it was to be located in New Jersey or in adjacent area, which could have allowed it to be in New York City or in Philadelphia because these are both across rivers from New Jersey. But Flexner decided to locate it in Princeton because he wanted a small town atmosphere rather than a big city atmosphere. It was also necessary to locate it somewhere where there would be a good library already in existence. Over the years since then, the Institute has built up a library of its own, but it still depends on Princeton University library for general library purposes. It has its specialized library, probably as far as mathematics is concerned it's got all it needs. But if somebody wants to look up something in, say, geology, then they have to go over to the University libraries. So it was important to set it up someplace where there was already a good library. So Princeton was the natural choice for this.
Flexner's idea was that you should have an institution where there would be no degrees given, no examinations, that the people who came there to study were people who had already passed the student stage. So anyone that comes there is supposed to have a Ph.D. or the equivalent of it. Also there were to be professors, but the professors had no duties except to be scholars and to be residence and be available for properly qualified people to consult them.
The decision was made to start with a school of mathematics. I've heard the story—and it seems quite plausible—that, in the year before they were going to try to begin, Flexner traveled around the world, and everywhere he went he asked people in various fields "Who are the leading people in the world in your field?" He found that he got the best general agreement in mathematics. There were other fields where there was almost complete disagreement, but in mathematics he got pretty good agreement. Also, of course, mathematics doesn't require very much investment in the way of laboratory and so on.
So he decided to start with mathematics. He knew Veblen, and undoubtedly Veblen had had some influence on his deciding to start with mathematics. He appointed Veblen as the first professor of mathematics and the first professor in the Institute for Advanced Study. He then asked Veblen to put together a group, of course consulting with Flexner all of the time. Veblen started with von Neumann, who was an obvious choice I think. He chose Alexander, who was not an obvious choice, but Alexander had been a protege of Veblen. Alexander is a topologist, who worked in knot theory and such. A very able person, but at the same time such a perfectionist that he published very little, and students found it very hard to consult with him because of his perfectionism. Then Hermann Weyl, Albert Einstein, ...
Speed: Gödel.
Tucker: Gödel, and then Morse. Marston Morse came somewhat later, and Hassler Whitney still later. The original ones were Veblen, Alexander, von Neumann, Weyl, Einstein, and Gödel. Gödel did not hold a professorship until much later. He held just a so-called permanent membership.
Then people came for a year or two; sometimes the Institute provided the funds, sometimes they came on fellowships of various sorts from all over the world. Particularly in the very early days, due to the situation in Germany, many of the people at the Institute for Advanced Study, in mathematics at least, were refugees. And each one of the professors had an assistant of his choosing, and this assistant was reasonably well paid. So Hermann Weyl, for example, took Richard Brauer to be his assistant as a way of giving him a position while it would be possible to look around and find a position for him. After being Weyl's assistant for about two years, he went to the University of Toronto. Indeed it happened that I served as the negotiator with the University of Toronto in bringing this about. Because Hermann Weyl and von Neumann were accustomed to giving lectures, they continued to give lectures, even though that was not part of their duties. Several of these courses of lectures at that time led to books of one sort or another.
Fine Hall was just a remarkable place to be. The various seminars that were going on, the courses, the informal atmosphere. It was really just a sort of a mathematical club. Because this was the Depression, the young people were usually trying to live quite frugally and would live in just a rented room in town and eat meals at an inexpensive restaurant. They were mainly single, so spent most of their time, except for eating and sleeping, in Fine Hall. There was the library on the third floor, the top floor, that was never locked. If you could get into the building, you could get into the library, and it was a very pleasant place to read and study. Of course there were lots of offices, studies, in the building, and there was the common room. You could go into the common room anytime between 9:00 in the morning until 3:00 the next morning, and there'd probably be somebody there, playing chess. Later on, go took over, but in the beginning it was chess, particularly a form of chess called Kriegspiel. Or there would be bridge played there. I was a bachelor until 1938, and So during those years I spent most of my time at Fine Hall.
Speed: Did you find yourself picking up large amounts of mathematics in many areas? Was it difficult to remain interested or specialize in your own area? How did it affect the young person?
Tucker: Well, I felt that a large part of the learning that I did was from my fellow students. We would get together and discuss things. These things were referred to as "baby seminars", where there would be no member of the staff present. We were encouraged to have these. And I felt that a large part of what I learned was from these. The courses, particularly at the beginning, were important. My first year at Princeton I had courses from Eisenhart in Riemannian geometry and from Hille in analysis and from Lefschetz in topology, or analysis situs as it was still called at that time.
Speed: What about logic? Would you take the course by von Neumann on functional operators?
Tucker: No. I could have, but it was very clear that if you went to everything, you were going to have no time at all to think. If you went to everything, all you could do was be a piece of blotting paper. So at the beginning of the year I would shop around. This was the customary thing to do. Then after a week or two, I would single out three or four and concentrate on them. But of course some seminars I would attend also.
Veblen had a weekly seminar that was mainly in geometry, but it was a seminar that he could make whatever he wanted to. Then there was a topology seminar that was run jointly by Lefschetz and Alexander. My interest was in geometry; that's why I had chosen to go to Princeton. I was interested both in the differential geometry sort of thing with Eisenhart, and in the topological sort of thing with Lefschetz and Alexander. I had quite a bit of difficulty making up my mind, when it came time to write a thesis, as to whether I would do it in differential geometry or whether I would do it in topology. My first published paper, which I did with Eisenhart during the first year that I was at Princeton, largely by accident, was in differential geometry. My thesis was with Lefschetz in topology.
I took only enough of other things to pass the oral examination, which included real variable, complex variable, modern algebra, and two topics of the student's own choice, which for me were topology and Riemannian geometry. One thing that I do very much regret is that I never took Wedderburn's course, because he was in a sense the last of the classical algebraists. And yet it was things that he did that provided the groundwork for things in modern algebra. But I was repelled by his style of lecturing. He wrote everything out before he lectured and more or less memorized it, and then wrote it on the board. No questions allowed.
Speed: Did he have graduate students at that time?
Tucker: Oh yes.
Speed: So he was still an active researcher?
Tucker: Yes. A graduate student of his at that time and a very good friend of mine was Nathan Jacobson. Another friend of mine who did his Ph.D. with Wedderburn is Merrill Flood. He hasn't gone on in algebra, though, the way Jacobson did.
Speed: What happened to you after you finished your Ph.D.? Was it soon afterwards that you were taken onto the faculty of Princeton University?
Tucker: I had gone to Princeton with the expectation that if I succeeded in getting a Ph.D. there, I would return to the University of Toronto. When I saw that my Ph.D. was certain, I wrote to the University of Toronto and was told, "There is a moratorium on any appointments, and there is a 10% cut in salary for the regular staff. Can't you find something to do for a year or two?" So I applied and was awarded a post-doctoral fellowship of ihe National Research Council of the United States. I was away from Princeton, then, in the year '32-'33.
I spent the fall term at Cambridge, England. I wanted to attend the International Congress that was held at Zurich in August, 1932, so I combined going to the Congress with starting my fellowship at Cambridge, where my supervisor was M.H.A. Newman, then a fellow of Johns College. Then I returned to America in December, to Harvard, where my supervisor was Marston Morse. I was at Harvard from December until June. Then because the fellowship was a 12-month fellowship I went to the University of Chicago for the summer quarter. The University of Chicago at that time was the only place where there was mathematical work going on officially during the summer period.
At Harvard I helped Morse with certain topological tools that he needed for the book he was writing. He was writing a book called Calculus of Variations in the Large and so he really picked my brains, because he needed the sort of topology that was done at Princeton for the purposes of his book. Also at that time I wrote a paper in the calculus of variations. Initially I wrote it for him, to explain certain things that I thought would be useful to him. Then he proposed that I write it as a paper. Of course when I got to the University of Chicago I was also there somewhat involved in the calculus of variations, because that was the big thing with [G.A.] Bliss and [L.M.] Graves and the people at the University of Chicago. So although my objective had been for that year to do certain things in topology, the only place that I had been able to do any of this was in working with Newman at Cambridge. Marston Morse was such a strong personality he immediately annexed me. I don't know that he ever asked me what I was interested in or wanted to do. He just essentially took me over.
Well, in the spring of 1933 I was offered an instructorship at Harvard which would have been for two years beginning in the fall of 1933. This so-called Peirce instructorship was arranged by Morse because he was anxious to keep me around there as a helper. So I wrote to Toronto again and said, "I have this offer at Harvard, but my aim is really to go back to Toronto. Do you want me?" And the word came back, "No, we have no opening for you, and Harvard's a pretty good place. You'll get some more seasoning." But I then consulted Lefschetz and Eisenhart at Princeton. And they said, "Well, why don't you come back to Princeton?" So I said fine, and I went back. They sort of matched the Harvard offer of an instructorship at Princeton, and I much preferred Princeton to Harvard because Harvard had nothing like Fine Hall.
At Harvard there was no place where mathematicians got together, except in formal courses and meetings of colloquium and so on. There was no informal life there, and the year I was away from Princeton I missed so much the informal life at Fine Hall. So when I had a chance to go back there, I jumped at it. The following year, after I had returned to Princeton, I got an offer from Yale. This was arranged by Hille, who had moved from Princeton to Yale. He felt that Yale ought to have something in the way of topology, so he persuaded them to make me an offer. I went there and was interviewed by Oystein Ore and gave a talk and so on, and they offered me an assistant professorship. I went back to Princeton, and Princeton met the offer. So I stayed only one year as an instructor, and then I moved up to assistant professor.
What had made positions possible at Princeton was the Institute for Advanced Study. Veblen, Alexander, and von Neumann were all taken off the Princeton payroll, and they were getting top salaries. The two principal people left in the department, Eisenhart and Lefschetz, decided that the thing to do was to bring in young mathematicians. So over a period of about a year there were five people who were appointed there as assistant professors in mathematics: Bohnenblust, Bochner, McShane, Wilks, and myself.
It was a very difficult time to get a job, and I had been very lucky to get these offers from Harvard and from Yale. This was largely because I was in the field of topology, which was beginning to be recognized. Harvard, because of Morse, and Yale, because of Hille, recognized this; they arranged these, but they were only temporary positions, no guarantee of them continuing beyond the initial appointment. But because of the Fine Hall atmosphere I chose the Princeton opportunity over both the Harvard and Yale, even though people pointed out to me that I wouldn't have so much competition in these other places as I would at Princeton. Nevertheless, I didn't think of it from that point of view, because I still thought that I was just putting in time until I could go back to Canada. So the question of whether I had a better long-range future at Harvard or Yale or Princeton just wasn't an issue with me.
It was 1938 when I had my first real offer from Canada. Oh, at the University of Toronto by this time they had sort of said, "Well, now if you insist on our finding you a position, we will." But they didn't make it sound as though they really wanted to have me. It was sort of, "Well, if you've got nothing else, we'll find something for you." The first really enthusiastic offer I got was from the University of British Columbia in 1938. And I went there, I flew there—it was the first time I ever flew in my life—and spent a couple of days seeing the place and being interviewed. I was very favorably attracted to it and came back to Princeton intending to accept the appointment. But then Eisenhart and Lefschetz went to work. They said, "You have your roots here now, and you've become a key member of the department. We want, you to stay, and we'll be very hurt, if you leave." So I stayed.
At that point I was given a so-called tenured appointment as associate professor. I realized then that I was not going back to Canada. But until that point I had always thought that ultimately I was going back to Canada. So at that point I took out my first citizenship papers and set about becoming an American citizen. That was the story.