The HLF Portraits: John Milnor
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The HLF Portraits14 / 66
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00:17
To begin at the beginning, what was your family like? Was it a family of intellectual interests?
00:24
I wouldn't say that. My father was an electrical engineer, so there was a certain amount of, some form of mathematics in the background. My oldest son is also an engineer, and my older brother is an engineer.
00:43
So you're the outlaw, but still within that universe, that broader universe. Were your parents ambitious for you, or did they pretty much leave you be to your interests? Oh, I think they probably were ambitious for me. I know they worked hard to get
01:04
me in school too early so that I was never very well socially adjusted to my classmates. Was that a part, a question of being younger than they are? Yes, younger. Younger, I see, I see. So the school you were in, and let us really speak sort of
01:25
at the high school level, was it a good school, one with a lot of good teachers and possibilities? Well, I think it was a quite good school, yes. But I was never very happy in school. It was a different
01:50
world when I came to Princeton as a fairly young undergraduate, and quickly made myself at home in the mathematics department there.
02:01
It was just the first time I'd been with so many people I could relate to. Even before that, I'm still wondering when the curiosity about mathematics begins. Well, as I said, my father was an engineer, so there were a few mathematical type books in the library. I remember studying one which was on calculus, and there was also a rather obscure translation from the German,
02:33
I can't remember the author, but it was on complex function theory, and I found it very mysterious and amazing.
02:44
When you decided to go to Princeton, was it in part because Princeton had a rather rich academic environment in this subject, or you weren't even sure yet what you would be interested in? I had no idea what I would be interested in, but it was close and seemed like a good choice.
03:11
You know, sometimes one can phrase it this way, that you found mathematics or mathematics found you, but how did you finally come to find that comfortable social but mostly intellectual environment?
03:25
Well, I was interested in many things, but somehow mathematics seemed much easier. For example, I fancied myself a poet, and when I took an English course, the professor gave my poem as a horrible example of what not to do.
03:50
So this was a negative push. Yes, and I was also interested in physics, but somehow I had two left hands when it came to the laboratory experiments, and just mathematics seemed easier and more natural to me.
04:07
Which by the way was a pretty good instinct. It turned out to be exactly so. So describe that common room and, if you will, the social beginnings of being a mathematician. The end of loneliness was one thing you've said.
04:24
People would gather there and somebody would pose a problem and another would solve it. Well, there were many things. There were certainly discussions of mathematics. There was a lot of game playing. Game theory was a big topic in Princeton in those days.
04:46
So it was just a general atmosphere in which you could talk with people and learn something and relax. One of the things that looks so interesting for those of us outside in other
05:03
academic fields is how young you can be to begin to make a contribution in mathematics. In history it takes quite a while. You were quite young when you began to address questions that were noticed. How young were you when you were already in a way contributing to the discourse?
05:26
Well, let's see, I suppose I was a freshman, so I must have been... That would make you 18, 19, something like that?
05:41
Perhaps 17, I'm not sure. Even 17? Well, it's not precise, but very young. Very young, yes. Very early in your college career, very early in mathematics. What had you begun to do that was getting noticed?
06:00
Well, there was an unsolved problem that Professor Tucker had described in a differential geometry class. I thought about it and was able to work out an answer. That was the first thing that happened. But you got published or did you get an award? Because this was...
06:23
It was eventually published, yes. It was eventually published. I mean a year or so later. Again, to pursue maybe the unanswerable question of youth and mathematical achievement, what do you think it is that somebody so fresh to the subject and so forth can begin to really participate in sophisticated discourse?
06:47
Is there something in the nature of mathematics that is not age-restrictive or the way you proceed? Well, I mean the popular belief that it is age-restricted in the opposite sense, that it's hard for older people to contribute.
07:03
Well, why would that assumption be? I mean, again... Well, it's just a matter of history that there have been many young mathematicians and it's fairly rare that older mathematicians make a real contribution. I see. One of your colleagues said to me at one point that he actually thought
07:21
it was naiveté, by which he meant that there were no assumptions to get in your way. Well, there are parts of mathematics which have very heavy baggage. You have to learn an awful lot before you can begin. But there are other areas which you can start fairly fresh.
07:47
Who is noticing you and the faculty at this point? And I'm really speaking of mentoring and inspiration. Well, Ralph Fox, I guess, was the one I worked with most, but there were many people that made an impression on me.
08:05
So he was my thesis advisor. Solomon Lefschetz was still around then. He was Fox's advisor and an amazing old gentleman. He'd lost both hands in industrial accidents.
08:22
Really? Yes. So he had sort of leather hands that could stick a pencil in to write, but it was... Extraordinary. And would you say that most of your education happened at that point in your life, in groups?
08:43
Or, I won't call it a lonely process, but more individually, you and a professor. How are ideas being formulated in your education? Well, classes were certainly important. This was a period shortly after the war where a number of the professors were refugees from Europe.
09:12
It was sometimes referred to as the Department of Broken English. So their English may not have been good, but certainly their mathematical skills were spectacular.
09:23
Yes. So it was an amazingly diverse group. I'd been in a very homogeneous society up to then. It was eye-opening to see people from so many different backgrounds. One can speak of the social loneliness or the companionability of mathematics, but as far as the ability to proceed and
09:53
making a contribution, is it necessary to have a community around you? Is that one of the ways that thinking happens?
10:03
Well, mathematicians are different. There are certainly cases of people who worked almost completely by themselves and made amazing contributions. To give some recent examples, there was Perelman in Russia, there was Andrew Wiles, who worked almost completely by themselves.
10:33
At Princeton at that time, the example was Christos Papakarikopoulos, a Greek mathematician who had very little contact with anyone else.
10:45
Fox had arranged some sort of stipend for him so that he could live adequately. I was there at the same time and hardly knew him. He didn't communicate much, but he made some amazing contributions.
11:03
Your work was, I won't call it social, but very much in the community of ideas that you were encountering and inspiring and being inspired. Can you mention some of the people who intrigued you as either colleagues or teachers at this point?
11:22
Well, I've mentioned Albert Tucker, who was a rather ponderous, spoke very slowly, but he was certainly an influence. He was particularly interested in game theory, and I was involved in game theory for a few years at that time.
11:41
Ralph Fox was my advisor, the one I was closest to. He gave a course in topology using the Moore method of teaching, which involves giving a list of theorems to prove and definitions and making the students do all the work.
12:07
That was wonderful for me. That's really what focused me on topology in my first years.
12:23
I'm also very interested, and you've spoken of this before, of the approach of different minds within mathematics, either habits or styles. I know that there's an algebraic mentality, if you will. Was that the kind you had?
12:44
Not especially, no. Well, speaking of algebra, Emile Artin was a very charismatic figure that I was certainly very impressed by. He had a completely different style. When Fox lectured, he was rather clumsy and often made mistakes.
13:03
One learned a lot, nonetheless. Artin gave totally polished lectures. Every word was precise and thought out. He attended an entire lecture for a year on algebraic number theory. He never actually mentioned algebraic
13:26
numbers, as far as I can remember. I didn't learn until later what an algebraic number was. But he was wonderful, also. What are you discovering about yourself and your mind in terms of the direction you're
13:44
most comfortable? Obviously, mathematics as a framework is very comfortable, but a way of proceeding. Well, my personal way of proceeding is just to think of some problem and concentrate on it. I'm not very good at having
14:02
a vision of where things are going. I just like to concentrate on solving a problem or concentrate on understanding someone else's work. So I've written a number of books explaining different theories, often just from the point of
14:22
view of trying to understand what other people were saying and working it out more clearly. And are you, it's a funny word to use, permitted to go where you want to go, both as a student and in mathematics? They basically trust once they know that you can do it. They'll just let you, so to speak, support thinking in wherever you want to go?
14:47
Well, that's certainly been my experience. I don't remember any pressure to do one thing. In the rest of academia, there's an old cliché, which is unfortunately very true about publish or perish, in mathematics. What does that take?
15:06
Well, it's very true. It's very hard on young people who have to produce. Well, it's hard at any age. So there's, it's not so much the mathematicians that enforce this as the universities that assume that publication is the criterion of doing something.
15:33
Mathematicians are not off the hook in that. I have sometimes an idea that as long as they know that you're going somewhere, they'll just let you go and wait until something emerges.
15:44
Well, this is certainly true to some extent, but if you want to be promoted or to have a larger salary, it's important.
16:05
Fair enough. It's a career as well as an intellectual excitement. Yes. Again, on the question of characterizing broadly mathematical approaches, one of the most obvious ones within and outside the field is the question of whether one would be described as a pure mathematician or an applied mathematician.
16:27
Yes. Can you give me a sense of which you thought you were or going to be and really what that distinction means? Well, I'm certainly primarily pure mathematician, but, you know, if there happen to be applications, that's wonderful. Now,
16:46
well, I was involved with game theory for some years and there, the applications were sort of evident. But they were incidental to your interest? No.
17:01
Well, the object was to solve something of practical interest, whereas often in mathematics, it's a pure matter of intellectual curiosity, understanding the way the world of mathematics should be organized or is organized.
17:24
Is there a slumbery associated with the two approaches? I mean, some notion of pure mathematics being a higher pursuit? Sure, there are many pure mathematicians who feel that way. There's also a difference in the way things operate.
17:45
Applied mathematics, one really is, one has to produce useful applications. The funding comes from specific problems, so it's much more, the goals are given to you rather than you being able to select them.
18:04
But there's, I mean, there's always been a big interchange between pure and applied mathematics and also between mathematics and mathematical physics. These interchanges always enrich both subjects.
18:23
And did the environments in which you found yourself, academic environments, encourage those interchanges? Well, no, I think there was a lot of tension between the pure and applied mathematicians and there often is. I mean, one example of the effect, one of the most amazing applied mathematicians was Claude Shannon at Bell Laboratories.
18:49
He was concerned with mathematical questions which were directly concerned with communication, what later became computer technology and so on.
19:02
And yet his ideas also transformed mathematics that became very important as a part of pure mathematics. The interaction between mathematics and theoretical physics has always been amazing. Ideas which arose in one becomes
19:25
very important. It goes in both directions, so people who can operate in both fields have that. Which is the ideal intellectual context for really anything. Yes, yes. As I continue to try to understand approaches within mathematics broadly, I remember
19:43
reading about you that you said that your approach had a lot of visualization. That's true. Can you explain that to me as a way of both thinking and transmitting ideas? Well, I think just different minds work differently. I tend to be happier when
20:03
I can see pictures. I enjoy working in fields where pictures play an important role. And I have a lot of difficulty in, say, understanding someone when they're just talking and I don't see anything written down. So it's just my eyes are more important to me than my ears.
20:28
Well said. But, I mean, that's my peculiarity. It's not typical. Oh, yes. I know it doesn't have to be one way or the other, but it is you I'm interested in.
20:41
Now, visualization, both in terms of how it really means, I think, in mathematics, literally the writing down the symbolic language that you use. I mean, is that…? Well, we often have to work in higher dimensions, so it's a
21:03
question of what visualization really means when you're talking about Fort dimensional space. Well, what does it mean when you're talking? Well, it's a good question. But it still is, in a way, the habit of how your mind will take it in. Yes, yes. I spoke to one mathematician who said, until I can write it down, which is
21:26
also a kind of transmission to the specific, I don't think I really understand it. Yes, well, I agree with that, because it's very easy to have vague ideas and think you understand something, and
21:41
then when you actually try to write down the details, you find you're assuming something which is not so obvious. So you have to check all of the details to make sure you understand what is going on. I think you've been often complimented on the lucidity of your presentation of information, which
22:06
is probably not inevitable among all mathematicians. Is it part of this process that you're describing? Yes, well, when I have trouble understanding something, then I have to try to write it out clearly, and if I
22:27
can really write it out so that I understand it, there's a much better chance that somebody else will understand it. So you're your toughest editor, so to speak. I rewrite a great deal, yes. I used to drive secretaries crazy. I would handwrite a manuscript, and they would type it
22:45
up beautifully, and then I would change this and this and paste this and that, and they'd have to start over again. The poor ladies had a hard time. It was wonderful when computers made it possible to avoid this.
23:01
Any other profound implication of the application of computerization to solving mathematical problems? Well, I make a great deal of use of computers, but in a very limited sense. I don't use very fancy techniques or high-speed computers.
23:23
But just being able to make computations which are easy now and which would have been a nightmare 30 years ago, you can see what you're talking about and see what things are reasonable to prove and what things turn out to be nonsense.
23:45
And I think one of the characteristics of your achievement, if I'm right, and I really ask you if I am, is that you found assumptions not to be valid. You would assume something because everyone did, but then you would take it somewhere else and find out that it was wrong.
24:08
Well, I think mathematicians are very well aware of the difference between something which is known and something which is unknown.
24:21
They make guesses as to what the true state of affairs is, but I was involved in this, I guess, particularly in the question of differentiable structures on manifolds where it was more or less assumed by most people that
24:42
a manifold had a differentiable structure, it was unique, but certainly people were aware that this was just a guess, that it hadn't been proved. I certainly wasn't expecting to disprove it. I was just studying some examples and arriving at a contradiction which I couldn't understand.
25:05
I studied the same example from two different points of view and got different answers, and eventually the only way to resolve the problem was to realize that I had two manifolds which were topologically the same but differentially different.
25:24
But this was certainly not a conscious effort on my part to solve the problem. I was just pushing ahead a little bit at a time trying to understand a class of objects and it turned out to be more complicated than I expected.
25:41
You are beginning to develop a rather good career in mathematics. How do you decide where to locate? I think really so much of your work was done at Princeton. Yes. So you did graduate work there as well as undergraduate and then stayed on the faculty. How difficult a career decision is that?
26:05
Is that natural? Are you challenged by other places or you decide you've got a good thing going? Why did you stay at Princeton? Well, it was really just for family reasons. My wife was at Stony Brook and I think we were all right commuting back and
26:27
forth until our son got old enough to talk and began to complain about it. So it seemed better to locate in one place. Women in mathematics in general. When you were a young man at Princeton, were there
26:43
women mathematicians around in the community or was this a difficult place for them to be? Well, I remember while I was at Princeton, they accepted a Japanese person as a
27:03
graduate student and were amazed when she arrived and turned out to be a woman. So unusual was that? Well, it was... Well, you didn't know the name was a female name, but... Exactly. Well, no, Princeton was an all-male school at that time. It was all-male? Yeah. I see.
27:21
So I think... She might not have been admitted had they known. I'm sure, surely not. Yes, I think so. I think, well, I don't remember exactly what happened. I think they made the best of it. She eventually left for one reason or another, but it was much later that Princeton actually became a good university.
27:44
That makes me ask about your community. How similar are you all fundamentally? Are you relatively from similar backgrounds or is it really, aside from the question of gender and so forth, is it, and perhaps even ethnicity, are people mostly just alike or is it that they speak a common language that is the...
28:08
They certainly don't seem alike to me. They seem very different. On the other hand, the community of mathematicians has, it's an international community. We have perhaps more in common with each other than other people of our nationality.
28:28
Political background is irrelevant in terms of...
28:43
Well, just in the matter of history, for example, it's certainly a fact that there were some great mathematicians who were ardent Nazis, and I suppose this is... Well, we just have to be aware of it. We have to enjoy their mathematics and yet not approve of their politics.
29:07
Right. It's not only in mathematics that people can't have uncomfortable political positions and still contribute to the field. But also yours is the generation of the Cold War.
29:22
Yes. And how much is the Cold War relevant or irrelevant to the pursuit of mathematics at that time? Well, I think there was very little communication with Russia, so I remember often getting messages from Vladimir Arnold in
29:42
Russia telling me that, yes, what I was doing was very nice, but this or that Russian had already done it. So it was a valuable cross-communication to compare approaches of different people at the time.
30:02
Now, of course, after more than half of the Russian mathematicians left Russia, the situation became very different. And that's got to roll over. We certainly have many here in Stony Brook. Did you, in fact, in full throttle in your career travel to the Soviet Union or were they to you?
30:25
Yes, I've certainly made several trips to the Soviet Union. It was a little scary in a way. People would not be willing to talk to me unless we were out in the park or somewhere.
30:44
Really? Yeah. You actually found that? Yes. So that's a dramatic example of the intrusion of the world. Yes. Is there certain students that you had, you may say no, it's not true, but I'm interested, whose contact with you was more
31:01
almost collegial, that their minds, their interests were so compatible with yours that in a way you were together discovering rather than teaching? That sounds like it should always be true. But favorite students or ones whose work fascinated you particularly?
31:32
I suppose my early students, Larry Siebenman was the one who has made the biggest name for himself and produced many students of his own.
31:46
More recently, I guess the two students I've been closest to have been from Chile and from Iran.
32:06
But well, mathematical community can be a very close one. People from many different backgrounds can.
32:24
I understand. It was just an opportunity in case somebody particularly popped mine. I suppose the person I was closest to in recent years was Adrien Doherty from France, who died in an unfortunate accident a few years ago.
32:45
He was a very impetuous life to the full. He died in southern France enjoying himself by jumping off a cliff into the Mediterranean where he had a heart attack.
33:07
If you were a mathematician today, where would you expect some of the exciting developments to be? I feel that I'm no good as a predictor. I've seen many changes in mathematics during my lifetime. They were all a surprise to me.
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