Mathematical Conversations

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I strongly disagree with the view that mathematics is simply a collection of separate subjects, that you can invent a new branch of mathematics by writing down axioms 1,2,3 and going away and working on your own. Mathematics is much more of an organic development...

Hardcore mathematics is, in some sense, the same as it has always been. It is concerned with problems that have arisen from the actual physical world and other problems inside mathematics having to do with numbers and basic calculations, solving equations. This has always been the main part of mathematics. Any development that sheds light on these topics is an important part of mathematics.
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Memory is important in mathematics in a different way. I will be thinking about something and suddenly it will dawn on me that this is related to something else I heard about last week, last month, talking to somebody. Much of my work has come that way. I go around shopping, talking to people, I get their ideas, half understood, pigeon-holed in the back of my mind, I have this vast card-index of bits of mathematics from all of those areas. So I think memory plays a role in mathematics, but it is a different sort of memory than you would use in other areas.
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Broadly speaking, once you really feel that you understand something and you have enough experience with that type of questions through lots of examples and through connections with other things, you get a feeling for what is going on and what ought to be right. And then the questions is: How do you actually prove it? That may take a long time.

The Index Theorem, for example, was formulated and we knew it should be true. But it took us a couple of years to get a proof... I don't pay very much attention to the importance of proofs. I think it is more important to understand something.
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Q: How could we encourage young people to take up mathematics, especially in the schools?

A: I have a theory on this, which is that one should first discourage people from doing mathematics; there is no need for too many mathematicians. But, if after that, they still insist on doing mathematics, then one should indeed encourage them, and help them.

As for high school students, the main point is to make them understand that mathematics exists, that it is not dead(they have a tendency to believe that only physics, or biology, has open questions)... There are many such, for instance in number theory, that teenagers could very well understand: Fermat of course, but also Goldbach, and the existence of infinitely many primes of the form n^2 + 1.
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Q: Do you think that a beginning graduate student could absorb this large amount of mathematics in four, five, or six years and begin original work immediately?

A: Why not? For a given problem, you don't need to know that much, usually--and, besides, very simple ideas will often work.

Some theories get simplified. Some just drop out of sight. For instance, in 1949, I remember I was depressed because every issue of the Annals of Mathematics would contain another paper on topology which was more difficult to understand than the previous ones. But nobody looks at these papers any more; they are forgotten(and deservedly so; I don't think they contained anything deep..) Forgetting is a very healthy activity.

Still, it is true that some topics need much more training than some others, because of the heavy technique which is used. Algebraic geometry is such a case; and also representation theory.

Anyway, it is not obvious that one should say "I am going to work in algebraic geometry," or anything like that. For some people, it is better to just follow seminars, read things, and ask questions to oneself; and then learn the amount of theory which is needed for these questions.
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Q: What do you think of life after the classification of finite simple groups?

A: You are alluding to the fact that some finite group theorists were demoralized by the classification; they said(or so I was told) "there will be nothing more to do after that." I find this ridiculous. Of course there would be plenty to do! First, of course, simplifying the proof(that's what Gorenstein calls "revisionism"). But also finding applications to other parts of mathematics; for instance, there have been very curious discoveries relating the Griess-Fischer monster group to modular forms(the so-called "Moonshine").
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