Excerpts from a fascinating (and lengthy!) interview with the great mathematician Michael Atiyah, which appeared in the 1984 Mathematical Intelligencer, reprinted in the book “Mathematical Conversations”, edited by Robin Wilson and Jeremy Gray. The interviewer is Roberto Minio.
Q: How do you select a problem to study?
A: I think that presupposes an answer. I don’t think that’s the way I work at all. Some people may sit back and say, “I want to solve this problem” and they sit down and say, “How do I solve this problem?” I don’t. I just move around in the mathematical waters, thinking about things, being curious, interested, talking to people, stirring up ideas; things emerge and I follow them up. Or I see something which connects up with something else I know about, and I try to put them together and things develop. I have practically never started off with any idea of what I’m going to be doing or where it’s going to go. I’m interested in mathematics; I talk, I learn, I discuss and then interesting questions simply emerge. I have never started off with a particular goal, except the goal of understanding mathematics.
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You can’t develop completely new ideas or theories by predicting them in advance. Inherently, they have to emerge by intelligently looking at a collection of problems. But different people must work in different ways. Some people decide that there is a fundamental problem that they want to solve, such as the resolution of singularities or the classificiation of finite simple groups. They spend a large part of their life devoted to working towards this end. I’ve never done that, partly because that requires a single-minded devotion to one topic which is a tremendous gamble.
It also requires a single-minded approach, by direct onslaught, which means you have to be tremendously expert at using technical tools. Now some people are very good at that; I’m not really. My expertise is to skirt the problem, to go around the problem, behind the problem … and so the problem disappears.
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Q: It’s clear that you have a strong feeling for the unity of mathematics. How much do you think that is a result of the way you work and your own personal involvement in mathematics?
A: It is very hard to separate your personality from what you think about mathematics. I believe that it is very important that mathematics should be though of as a unity. And the way I work reflects that; which comes first is difficult to say. I find the interactions between different parts of mathematics interesting. The richness of the subject comes from this complexity, not from the pure strand and isolated specialization.
But there are philosophical and social arguments as well. Why do we do mathematics? We mainly do mathematics because we enjoy doing mathematics. But in a deeper sense, why should we be paid to do mathematics? If one asks for the justification for that, then I think one has to take the view that mathematics is part of the general scientici culture. We are contributing to a whole, organic collection of ideas, even if the part of mathematics which I’m doing now is not of direct relevance and usefulness to other people. If mathematics is an integrated body of thought, and every part is potentially useful to every other part, then we are all contributing to a common objective.
If mathematics is to be thought of as fragmened specializations, all going off independently and justifying themselves, then it is very hard to argue why people should be paid to do this. We are not entertainers, like tennis players. The only justification is that it is a real contribution to human thought. Even if I’m not directly working in applied mathematics, I feel that I’m contributing to the sort of mathematics that can and will be useful for people who are interested in applying mathematics to other things.
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The more I’ve learned about physics, the more convinced I am that physics provides, in a sense, the deepest applications of mathematics. The mathematical problems that have been solved or techniques that have arisen out of physics in the past have been the lifeblood of mathematics. And it’s still true. The problems that physicists tackle are extremely interesting, difficult, challenging problems from a mathematical point of view. I think more mathematicians ought to be involved in and try to learn about some parts of physics; they should try to bring new mathematical techniques into conjuction with physical problems.
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Q: Do you think the Fields Medals serve a useful function?
A: Well, I suppose in some minor way. I think it’s a good thing that Fields Medals are not like the Nobel Prizes. The Nobel Prizes distort science very badly, especially physics. The prestige that goes with the Nobel prizes, and the hooplah that goes with them, and the way universities buy up Nobel prizemen — that is terribly discontinuous. The difference between someone getting a prize and not getting one is a toss-up — it is a very artificial distinction. Yet, if you get a Nobel prize and I don’t, then you get twice the salary and you university builds you a big lab; I think that is very unfortunate.
But in mathematics the Field Medals don’t have any effect at all, so they don’t have a negative effect. They are given to young people and are meant to be a form of encouragement to them and to the mathematical world as a whole.
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Q: When you’re working do you know if a result is true even if you don’t have a proof?
A: To answer that question I should first point out that I don’t work by trying to solve problems. If I’m interested in some topic, then I just try to understand it; I just go on thinking about it and trying to dig down deeper and deeper. If I understand it, then I know what is right and what is not right.
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Q: Where do you get your ideas for what you are doing? Do you just sit down and say, “All right, I’m going to do mathematics for two hours?”
A: […] There are occasions when you sit down in the morning and start to concentrate very hard on something. That kind of acute concentration is very difficult for a long period of time and not always very successful. Sometimes you will get past your problem with careful thought. But the really interesting ideas occur at times when you have a flash of inspiration. Those are more haphazard by their nature; they may occur just in casual conversation. You will be talking with somebody and he mentions something and you think, “Good God, yes, that is just what I need … it explains what I was thinking about last week.” And you put the two things together, you fuse them and something comes out of it. Putting two things together, like a jigsaw puzzle, is in some sense random. But you have to have these things constantly turning over in your mind so that you can maximize the possibilities for random interaction. I think Poincare said something like that. It is a kind of probabilistic effect: ideas spin around in your mind and the frutiful interactions arise out of some random, fortunate mutation. The skill is to maximize this degree of randomness so that you increase the chances of a fruitful interaction.
From my point of view, the more I talk with different types of people, the more I think about different bits of mathematics, the greater the chance that I am going to get a fresh idea from someone else that is going to connect up with something I know.