The Princeton Mathematics Community in
the 1930s
Transcript Number 22 (PMC22)
© The Trustees of Princeton University, 1985
(with ALBERT TUCKER)
This is an interview of John Kemeny at Bradley Hall, Dartmouth College, on 7 June 1984. The interviewer is Albert Tucker.
Tucker: Do you remember your first encounters with Fine Hall?
Kemeny: I remember many things about it. I entered Princeton as an undergraduate in February 1943, when conditions for universities were poor. My class entered in three instalments; I was in the tail end of it. People were being drafted almost as fast as they entered. I was sixteen and a half when I entered, so I would be there for almost two years before I got drafted. I had a rather unusual undergraduate education, as a result of the wartime conditions. One statistic is interesting: everyone I had as an undergraduate teacher was either a full professor or became a full professor before I graduated from Princeton, which is not typical of anyone's undergraduate education anywhere. That was because the younger people had been drafted.
Tucker: But it is still true of Princeton, as it is true of Dartmouth, that all professors engage in the full range of teaching.
Kemeny: Yes. I was unusual, however, in having only senior members of departments.
Tucker: Who were your undergraduate teachers?
Kemeny: In my first year Princeton had just made a decision that analytic geometry was not a prerequisite for calculus, so I signed up for calculus. I had someone named Al Tucker as my first college mathematics teacher. I got nervous as to what analytic geometry I had missed, so I took it as an extra course. I had Claude Chevalley for analytic geometry. It was one of the weirdest courses anyone could have had. He was superb for me and terrible for the rest of the class. He considered me the only one in that class who had an interest in mathematics.
I had Alonzo Church for both integral calculus and differential equations. I later wrote a junior paper and senior thesis and a Ph.D. thesis—none concerning calculus—under him. So I had a strange beginning relationship to a world-famous logician.
Tucker: Was he as thorough in teaching calculus as he was in a lot of other ways?
Kemeny: In some ways it was an unbearable course. I owe him a great deal, but I must say that the teaching style which is ideal for mathematical logic—particularly on an introductory level where every detail is covered with a thoroughness only Church is capable of—can be a bit boring in a calculus class. We learned an enormous amount about foundations and the basic concepts, but we covered probably only half of the material. He's a conscientious teacher, but ...
Tucker: Did you, at that time, make use of the Fine Hall library?
Kemeny: Yes, but not a great deal as an undergraduate. I occasionally went to look things up. If my memory serves me right, there were shelves at one end of the library that had books selected for undergraduates. I found that extremely helpful, and in a way the Mirkil Room here at Dartmouth College is patterned after that—a mathematics library made accessible to the undergraduate. As far as I know my high school didn't even have a library. I had no experience searching in a library, and in a large reference library you have to have some sophistication to find what you're looking for. So the biggest use I made of the Fine Hall library was my use of those shelves, where I could get books that were both good and at a level an undergraduate could understand.
Tucker: Did you have any contact with the people at the Institute?
Kemeny: Not at that time; I did when I came back from the war. My experience with upper-division courses was also unusual: so many undergraduates had been drafted that I had three courses with extremely small enrollments. I had Chevalley again, this time for complex variables; there were three of us in the course. I had a course where there were two of us. But the most remarkable was my modern-algebra course. It was announced that there would be offered either Eisenhart's differential-geometry course or Wedderburn's modern-algebra. I had had neither one, so I gleefully signed up. It turned out to be Wedderburn.'s course, and five of us were signed up. But two of the five had signed up hoping it would be differential geometry, and they dropped out. For reasons I don't know, the other two dropped out before the week was out. So I went to Professor Wedderburn and said, "Clearly you don't want to teach a course for only one student. " I offered to resign from the course. He wanted to know if I meant that I did not want to take the course. Of course I was dying to take it, so I was his only student in that course. As far as I know, I was his last algebra student.
Tucker: I think that's quite possible.
Kemeny: It was a remarkable experience. Wedderburn was a gentleman of the old school. It was funny—a course with one student. And it was a straight lecture course. At the beginning of class he would say, "Are there any questions?" More often than not, I had questions. It was an excellent way to teach a course; I learned a great deal.
The only time I had a personal conversation with him was once, for some reason, we were moved to another room and had to walk across campus. He asked me what I might be interested in. I was just beginning to be interested in logic, so he said he would take up "the algebra of logic", now called Boolean algebra. Otherwise he knew absolutely nothing about me, including whether or not I had any mathematical talent. I found this out from the big work for the course, which was a set of problems that had to be done in ten days. You had to do four out of five. I went home and did four in two hours. I could have handed the exam in then, but of course I wanted to do the fifth one. But I absolutely couldn't do the fifth one, so I went back two days later horribly upset. It turned out the problem had a typographical error. So he gave what was for any good student an absolutely trivial exam. In that semester he did not find out that I really had some mathematical talent. I don't mean to criticize him, because I thoroughly enjoyed the course and learned a great deal from it.
I had Lefschetz for mathematical physics, except that Lefschetz decided that I shouldn't learn that. He didn't let me come to class. Instead, he made me work on some independent project. Actually as a result of that independent project I learned an enormous amount from him, but I missed out on mathematical physics. I did have Eisenhart's differential-geometry course later, and I had a real-variable course from Bohnenblust. And of course, logic courses from Church. It was a remarkable array of teachers. No one could have had a better undergraduate education.
Tucker: Let's talk about when you became a graduate student.
Kemeny: I went off for military service for a year and a half, and came back to finish my undergraduate work. Artin had come to Princeton, so I did my graduate algebra with him. I had Chevalley for point-set topology. The Chevalley course is the one I most enjoyed; in fact, I made point-set topology my specialty in the general exam. Of course I specialized in logic, so I took all the logic courses I could. I got to be close friends with Leon Henkin, who was slightly ahead of me in graduate school. We talked a great deal of logic to each other; you know how graduate students were with each other.
Kemeny: There was an atmosphere conducive to students talking with one another. As a matter of fact, much later I was thanked publicly by someone I had as a student, and I could have sworn I made no contribution to his education. This was when he introduced me publicly somewhere. He said, "I don't think I'll ever forget this. We were both standing by the bulletin board in Fine Hall, and I asked Professor Kemeny what a recursive function is." I had no reason to remember this two decades later. But the graduate student was Hartley Rogers, who later wrote a definitive book on recursive functions, and he remembered the incident vividly. That was what Fine Hall was like.
Tucker: You've mentioned Hartley Rogers and Leon Henkin. What other students do you remember?
Kemeny: I overlapped with Dick Bellman somewhere. I have trouble remembering exactly how we overlapped at Princeton, because I got to know him much better later on. Do you remember when Dick was at Princeton?
Tucker: It was right after the war.
Kemeny: That's what I suspected. We overlapped there.
Tucker: Did you know he died recently?
Kemeny: Yes, I heard.
Tucker: Another person you knew who died recently is Ulam.
Kemeny: I did not know Ulam well. I knew him from Los Alamos. He gave Peter Lax and me some private lessons at Los Alamos.
Tucker: He wasn't at Los Alamos during the war?
Kemeny: No, not during my time there.
Tucker: He was already working with von Neumann on things such as the Monte Carlo method.
Kemeny: Did you know that he had a serious brain problem? I got to Los Alamos in March of 1945. 1 remember him coming there about a year later; he had just recovered from serious brain surgery. I remember it vividly because of the story he told. He was about to undergo this operation; it was a very serious one. Paul Erdos came in to see him, and to reassure him Paul said, "Don't worry. If you don't survive the operation I'll finish your work." For some reason this did not reassure Ulam.
I'm glad I mentioned Paul Erdos. You do remember that Paul was continually in and out of Fine Hall; he was one of my unofficial teachers. Besides the fact that he liked young mathematicians—he would all his life—I was a fellow Hungarian, which was irresistable. So I learned a great deal. As a matter of fact, I have a lifelong interest in number theory. I never had a course in number theory, just contacts with Paul.
Tucker: Do you remember much contact, when you were a graduate student, between Fine Hall and Fuld Hall?
Kemeny: Yes. I had some personal contacts. In my last year as a graduate student, I became Einstein's research assistant, and for a year I was a member of the Institute for Advanced Study. I must say that during that entire year I did not get to know a single person I had not previously known. Somehow, unlike Fine Hall, the Institute was not conducive to getting to know new people. It was a wonderful year for me, because of Einstein.
I got to know von Neumann at Los Alamos, because I was assigned to the so-called computing division. Von Neumann had really set it up. He had figured out how to use bookkeeping machines to solve partial-differential equations. He would stop in periodically to see how things were going. Peter Lax and I became good friends there. We would occasionally corner von Neumann to chat, so that is where I got to know him. I saw him occasionally at the Institute after that, and also at least one summer at Rand—we overlapped for a while at Rand.
Strangely enough, though I got to know von Neumann well, I got to know him best at Los Alamos and at Rand. At Princeton von Neumann gave the Vanuxem Lectures, which for complicated reasons he could not write up. I was picked to write up those lectures, and this led to a Scientific American article.
There were three people at the Institute I got to know well. One was von Neumann, one was Einstein, and one was Gödel. I met Gödel through a mutual friend, Paul Oppenheim. That's how I met Einstein also, rather than through some official machinery.
Tucker: Could you tell us a bit about Gödel?
Kemeny: I'd be happy to. Is Gödel still alive?
Tucker: He's been dead several years. I remember Steve Kleene came to Princeton for the memorial service. He had been appointed by the National Academy of Sciences to represent the Academy at Gödel's memorial service.
Kemeny: As everyone knows, Gödel was somewhat strange in personal habits. Paranoid, I think, is the right word. I was one of the few people he seemed to trust completely. Not because of who I was, but because he trusted Paul Oppenheim. Oppenheim was an old friend of Gödel's from way back, from Europe, and someone introduced by Oppenheim was okay. So I had many visits with Gödel and many conversations with him.
A strange thing happened years later. One of the young logicians at Princeton was dying to see Gödel. I was at Princeton myself giving the Vanuxem Lectures. He asked if I would write a letter to Gödel vouching for him. I said, "I'd be delighted, but this is crazy. Why don't you ask Church to do it? You're a student of Church's." It turned out that Church had written three times and Gödel hadn't replied. So I wrote a letter for him. A couple of weeks later he sent me a note saying that he had gotten a prompt answer from Gödel thanks to my letter. I mention this as an example of the strangeness of Gödel: I was okay, but Church wasn't. But to me, Gödel was always extremely pleasant. We had interesting conversations, mathematical and otherwise.
There was an incident my wife would want to recount. We were at someone's house. It was a pleasant social affair, and Gödel and his wife were present. In the middle of the evening, well before people were ready to leave, a little wrist-alarmclock went off and Gödel said, "I'm sorry, I have to leave." There was some incredibly trashy television show, and Gödel publicly announced that he had to leave to see this television show.
I should mention one more thing. When I really got to know Gödel was during the year that I was Einstein's assistant. They had gotten to be good friends and often walked home together from the Institute. I was with them on a number of these occasions. Incidentally, there was another sort of strangeness at the Institute you might be interested in. Do you know how Gödel and Einstein got to know each other?
Tucker: No.
Kemeny: It's a Paul Oppenheim story. Oppenheim was a great story-teller. It's the story of what he described as his only contribution to science—a typical Oppenheim statement.
When Gödel started working on the mathematics of general-relativity theory, Paul Oppenheim asked him, "What does Einstein think of your work?" Gödel said, "Unfortunately, I don't know Einstein." Paul was amazed at this: first of all because two such famous people at the Institute should know each other, and secondly because they were surely the only two people at the Institute working on relativity theory. Gödel said, "Yes, it strikes me as strange too, but I just have never met him. " Paul decided to do something about it, and went down to Fuld Hall the next day. It turned out that Gödel had been moved quite recently; he actually had the office across the hall from Einstein's. So Paul said his one contribution to science was to lift his two hands and knock simultaneously on two doors. The doors opened, and he said, "Einstein this is Gödel, Gödel this is Einstein." By the time I worked with Einstein they were close friends. But it took somebody not connected with the Institute to introduce the two of them to each other.
Tucker: Oskar Morgenstern was another person who was a close friend of Gödel's.
Kemeny: Yes, as a matter of fact, I think—I'm never certain of these things—it was a party at Oskar's house when the wristwatch incident occurred. I got to know Oskar very well. He would visit Dartmouth periodically. As a matter of fact, he made a major contribution to Finite Mathematics.
Tucker: Oskar used to talk to me about Gödel. Indeed, he made no bones about saying that Gödel was greater than Einstein. This was the time when Gödel was working on the unified theory. Oskar thought that Gödel was going to pull a coup and surpass Einstein.
Kemeny: I don't know if he ever completed that.
Tucker: I knew Gödel only as someone I saw and said good-day to, because I was never a logician.
Kemeny: One more Gödel incident. The only public lecture I heard by Gödel, I think, was during the Princeton Bicentennial. A horrible thing happened. The lecture notes were good, but Gödel walked in and faced the blackboard and delivered the half-hour lecture facing the blackboard without ever writing anything. It was the most uncomfortable thing I ever sat through in my life. I wished that he would pick up a piece of chalk and write one word on the blackboard just as an excuse. It was clear that he just could not face his audience. I heard a couple of lectures by von Neumann, which were, of course, brilliant. I heard one by Einstein, in Fine Hall, which was excellent. The content of Gödel's lecture was excellent, but the lecture itself was a disaster because of this peculiarity he had of not facing the audience.
Tucker: When did you get interested in computing?
Kemeny: I was forced into that at Los Alamos.
Tucker: That was the start?
Kemeny: Yes. During the war that was such a crazy operation there, and of course everybody was trying to think of ways to cut down the time it took.
Tucker: Did you have any contact with computing while you were a graduate student?
Kemeny: Absolutely none.
Tucker: You had no connection with von Neumann's work?
Kemeny: No. I might, though, mention one other connection I did have with von Neumann. He gave a lecture at Los Alamos, which for some reason hasn't been reported. To the best of my knowledge, the only place it appeared in print is in my little book Man and the Computer. I was there together with quite a few other people while von Neumann tried describing what he felt computers should be like. This was either late 1945 or early 1946. In effect he outlined in that one lecture what I consider to be all the fundamental principles of a modern computer. I had no idea at that time that he was actually planning to build such a thing himself. I wasn't aware he was doing it until I became a member of the Institute in '48-'49. But I didn't have any connection with it, nor did I get to see the product, till '53 at Rand.
The rekindling of my interest in computing came through Rand. As you know, at Rand von Neumann worked on some very interesting mathematical problems, many of which required a large amount of computing. So I had some connection with computing in '53, and in '56 I played a strange role. It resulted from one of those periodic federal-budget cuts. Rand had asked me to come in the summer of '55. 1 couldn't, so they asked if I could come the next summer. I said I could, but before the summer arrived, the budget had been cut way back. Consultants, of course, are one of the first things to be cut out. They found they could meet the commitment to me, but they had promised to have consulting in both mathematics and computing and they had money for only one consultant. So they asked me if I would divide my time between mathematics and computing. I told them, "Look, I haven't had all that much experience with computing, but I'm fascinated by it. I'd love to." So I spent half the summer working with the computer people. I would have to point to the summer of '56 as the beginning of my serious interest in computers.
Tucker: You once invited von Neumann to Dartmouth, didn't you?
Kemeny: Yes. We got the dean's permission for an annual man, what we called a "big-shot visitor". We asked von Neumann, and he accepted. He had to back off, though, because of illness, which turned out to be terminal. So we never did have him up.
Tucker: I was very briefly a member of the von Neumann project, just at the end of the war. The work that I had been doing had terminated and the University here was not ready to start, so I was temporarily unemployed. Von Neumann very kindly said he would be glad to have me work on his project.
Kemeny: When did you start the building of the computer?
Tucker: I had nothing at all to do with that. I was supposed to be working at a sort of topological problem, namely, 'what would be a good way to generalize finite-difference methods to higher dimensions?'. Of course, in higher dimensions you have many more ways of cutting things up, and that was as far as I got. I looked into other ways of doing it than rectangularly, doing it in terms of hexagons in the plane, or equilateral triangles for example. But before anything came to a head in this, I suddenly had to go back and teach fulltime. Some people who continued, at that time, to work with von Neumann were Valentine Bargmann and Deane Montgomery. Herman Goldstine was already there as the chief helper of von Neumann. That was, I would guess, in 1946. The answer is to be found in the book of Goldstine's book Computers from Pascal to von Neumann. Incidentally, have you seen the new book about Turing?
Kemeny: No. A couple of people have recommended it, but I haven't yet had the time. Is it as good as people say?
Tucker: Well, I plowed through it. I read it for the people. There is a thorough index; people are indexed every time they get mentioned. I was particularly interested in the topologists who were involved in the code-breaking during the war. The one who was in charge on the outfit Turing worked for was a well-known topologist, M.H.A. Newman. My first Ph.D. student, Shaun Wiley, was another member of that group. Peter Hilton was too, so there were a lot of familiar names. But a great deal of the book is devoted to the technical details of the those code-breaking machines and of the hardware of the computer that was started under Max Newman at Manchester. Turing was brought there and was working on that machine when he took his life. There's a lot in the book on his problems as a homosexual. I actually was a member of the generals committee for Turing, in about 1937 I think.
Kemeny: Did he take his degree at Princeton?
Tucker: Yes, his Ph.D. was under Alonzo Church.
Kemeny: I did not know that. Some of Turing's great papers appeared at about the same time as some of Church's.
Tucker: Yes, Turing's great paper on computable numbers was written just before he left England to come to America.
Kemeny: So he had substantial publications before he took his degree at Princeton?
Tucker: Not exactly substantial, but one great paper. He actually learned mathematical logic from Newman, who was a combinatorial topologist and who occasionally taught something he didn't know well in order to learn it. He taught a course at Cambridge in mathematical logic, and that was what got Turing interested.
Kemeny: I understand these things, that graduate students sort of learn by lore. Kleene and Rosser are always mentioned as Church's students; I don't remember hearing Turing mentioned as Church's student.
Tucker: As in so many cases, his Ph.D. thesis was his own work. Church was supervisor only formally. Turing at that time had a fellowship at King's College, Cambridge. He came to study with Church at the suggestion of Max Newman. He was allowed to use his fellowship from King's for the first year. Then the Cambridge Procter Fellowship became available, so he was able to stay a second year, He decided to get a Princeton Ph.D. while he was around. Following that he could have stayed for a third year and worked with von Neumann on the computer project at the Institute, but King's College said he couldn't have a third year of leave from his fellowship. So he returned to England and resumed his fellowship.
He was a strange person. This is reflected in the title of the recent biography, Alan Turing: the Enigma, which refers both to Turing himself and to the code-breaking machine he was associated with. No, very few people realize that Turing is one of Church's Ph.D.'s.
Kemeny: As I said, the summer of '56 was the beginning of my serious involvement in computing. I might tell you an anecdote. At the end of that summer I was asked to write some recommendations for computing at Rand. Of course the memory is hazy after so many years, so I can't swear to the details, but I remember the point I stressed most was that it horrified me to see the well-known mathematicians at Rand, being paid large salaries, waiting around for hours to get a few seconds of computer time. I remember big shots fuming for hours waiting for one 5-second program. I suggested that that was quite unkind to the big shots and also economically unwise. I was trying to describe some sort of system by which mathematicians could have easy access for a few seconds of computing. I said there must be some way to "interrupt" the system. That was the beginning of the idea of time sharing, changing from batch processing.
Tucker: Anything else?
Kemeny: My most interesting experiences, except for getting to know many of the great figures at Princeton, were outside that period. Maybe I should say something: Lefschetz got absolutely furious at me when I decided to go into logic. You remember he was not very fond of logic.
Tucker: No.
Kemeny: Lefschetz liked me and took an interest in me as an undergraduate, and he got furious at me for picking mathematical logic as my specialty.
Tucker: Well, you'll agree that his personality wasn't suited to mathematical logic and vice versa.
Kemeny: No, he was at the opposite extreme. I remember a quest lecture in a course of mine. The freshman class was totally lost. Halfway through the lecture he still hadn't said anything that they understood. He mentioned invariance under translation, and there he noticed that the class was really lost. He said, "Oh, you know what translation is. You take a point, say three-quarters, and you move it three-quarters." Then he wrote that this equals seven-eighths. There was a stirring in the class. I was sitting at the back of the class. One of the better students, who was sitting next to me, asked me if I thought that was right. I said, "What do you think?" He said, "I think it's wrong." "Well, why don't you ask Professor Lefschetz?", I said. So he said, "Sir, I think that answer is wrong." Without hesitation Lefschetz erased his answer. He took it for granted that it was wrong. Then he stood there and thought about it for a minute, and then wrote down a second incorrect answer. Then half the class raised their hands. He went on, totally undisturbed by this, and finished what would have been an excellent lecture.
The whole next class we devoted to discussing how someone can get to be one of the world's great mathematicians and not know how to add two fractions. It was a fascinating classroom discussion. I took the occasion to explain that there are several different kinds of mathematics; numerical work underlies some parts of mathematics, but is essentially irrelevant to other parts. They got an enormous amount out of Lefschetz's lecture, but mostly because he made it clear that he couldn't add two fractions.