Continuity of mathematics

Mathematics has become so specialized, it seems, that one may fear that the subject will break up into separate areas. Is there a core holding things together?

ATIYAH: I like to think there is a core holding things together, and that the core is rather what I look at myself; but we tend to be rather egocentric. The traditional parts of mathematics, which evolved - geometry, calculus and algebra - all centre on certain notions. As mathematics develops, there are new ideas, which appear to be far from the centre going off in different directions, which I perhaps do not know much about. Sometimes they become rather important for the whole nature of the mathematical enterprise. It is a bit dangerous to restrict the definition to just whatever you happen to understand yourself or think about. For example, there are parts of mathematics that are very combinatorial. Sometimes they are very closely related to the continuous setting, and that is very good: we have interesting links between combinatorics and algebraic geometry and so on. They may also be related to e.g. statistics. I think that mathematics is very difficult to constrain; there are also all sorts of new applications in different directions.

It is nice to think of mathematics having a unity; however, you do not want it to be a straitjacket. The centre of gravity may change with time. It is not necessarily a fixed rigid object in that sense, I think it should develop and grow. I like to think of mathematics having a core, but I do not want it to be rigidly defined so that it excludes things, which might be interesting. You do not want to exclude somebody who has made a discovery saying: "You are outside, you are not doing mathematics, you are playing around". You never know! That particular discovery might be the mathematics of the next century; you have got to be careful. Very often, when new ideas come in, they are regarded as being a bit odd, not really central, because they look too abstract.

SINGER: Countries differ in their attitudes about the degree of specialization in mathematics and how to treat the problem of too much specialization. In the United States I observe a trend towards early specialization driven by economic considerations. You must show early promise to get good letters of recommendations to get good first jobs. You can't afford to branch out until you have established yourself and have a secure position. The realities of life force a narrowness in perspective that is not inherent to mathematics. We can counter too much specialization with new resources that would give young people more freedom than they presently have, freedom to explore mathematics more broadly, or to explore connections with other subjects, like biology these day where there is lots to be discovered.

When I was young the job market was good. It was important to be at a major university but you could still prosper at a smaller one. I am distressed by the coercive effect of today's job market. Young mathematicians should have the freedom of choice we had when we were young.

The next question concerns the continuity of mathematics. Rephrasing slightly a question that you, Prof. Atiyah are the origin of, let us make the following gedanken experiment: If, say, Newton or Gauss or Abel were to reappear in our midst, do you think they would understand the problems being tackled by the present generation of mathematicians - after they had been given a short refresher course? Or is present day mathematics too far removed from traditional mathematics?

ATIYAH: The point that I was trying to make there was that really important progress in mathematics is somewhat independent of technical jargon. Important ideas can be explained to a really good mathematician, like Newton or Gauss or Abel, in conceptual terms. They are in fact coordinate-free, more than that, technology-free and in a sense jargon-free. You don't have to talk of ideals, modules or whatever - you can talk in the common language of scientists and mathematicians. The really important progress mathematics has made within 200 years could easily be understood by people like Gauss and Newton and Abel. Only a small refresher course where they were told a few terms - and then they would immediately understand.

Actually, my pet aversion is that many mathematicians use too many technical terms when they write and talk. They were trained in a way that if you do not say it 100 percent correctly, like lawyers, you will be taken to court. Every statement has to be fully precise and correct. When talking to other people or scientists, I like to use words that are common to the scientific community, not necessarily just to mathematicians. And that is very often possible. If you explain ideas without a vast amount of technical jargon and formalism, I am sure it would not take Newton, Gauss and Abel long - they were bright guys, actually!

SINGER: One of my teachers at Chicago was André Weil, and I remember his saying: "If Riemann were here, I would put him in the library for a week, and when he came out he would tell us what to do next."

Next: Communication of mathematics

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