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Michael Atiyah - Did we invent number theory? (89/93)

3,609 views · Published 19 June 2017 · 6:00 · Indexed 10 November 2026

Channel: Web of Stories - Life Stories of Remarkable People · 2017 · Science & Technology

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British mathematician Michael Atiyah (1929-2019) studied in Cambridge where he became a Fellow of Trinity College and later held professorships at Princeton and Oxford. He is best known for his work on the K-theory and the Atiyah-Singer Index Theorem. [Listener: Nigel Hitchin date recorded: 1997]

TRANSCRIPT: There are those who argue, well for example, something like number theory, the theory about prime numbers, they say, ‘Ah! That's a mathematical invention, you know, prime numbers don't exist in the outside world. It's entirely a human creation. Even if geometry was borrowed, number theory is ours, we invented it’. I don't think that's right either and I have tried to argue in this review of this book by Connes and Changeux, they, we count.  Now counting, if you think about it very carefully, is actually an experience. We have objects in the outside world, I have a glass, another glass and I can I distinguish them and I start to count and enumerate. And when eventually I've enumerated them, I've put them into orders, rectangles, I can decide a multiplication, I can work out prime number properties. But the mere fact that you have discrete objects that you count is a consequence of the kind of world we live in.

And the example I gave was that supposing that in fact humanity had been quite different. Suppose we weren't as we are, but we were gigantic bits of protoplasm that floated around in some big sea. In other words that all we saw was the continuum, we never saw discrete things, we moved in some kind of continuous way and we didn't have single people, all the notion of counting wouldn’t appear. You would have notions of measurement and of volume and of pressure, and those notions, they would be fundamental. But the notion of counting is an experience that derives from the fact that we see individual objects. And if the universe was different and we didn't have individual objects around us, we just had a sea, a mass of continuity, well geometry would have been natural or mechanics or pressure, but counting would not be natural.

So I think counting still arises from the external world.  It's the most primitive thing there is. It's hard to imagine a world without individual objects, but ultimately it comes from the outside world. Once we've got it we can study it and develop beautiful theorems about prime numbers, but I don't think even I don't think I would let the number theorists get away with even that as being entirely the creation of the mind without the stimulus of the outside world. I think it's, you start with the outside world, you look at it from a particular point of view, you extract things you want to study further, and the human mind then takes it forward. Then you develop mathematics which you then feed back into the outside world, interlink the two; sometimes closely, sometimes not so closely. But I think it's the... it’s the human mind acting on the outside world.

But it is also the other side of the question as to whether the mathematics we develop, if we develop it, from a starting point in the outside world… you know, we might go off at a complete tangent, we might end up by doing things which have ultimately no bearing on the outside world even though they start from the outside world. And physicists have all been so mystified by the way in which mathematics turned out to be so powerful, as a way of understanding the most sophisticated bits of physics; quantum theory and all the sophistication, notion of relativity, use very abstract powerful mathematics.

And how is it that the mathematics that we develop in our mind seems to be so successful in solving the problems of physics? And again different people have different answers to that one. I suppose my own answer probably would be that mathematics is what the human mind creates, starting off with the raw material of the outside world. Now, it creates it of course using the logical processes that the mind has. Now those logical processes have been built up in evolutionary terms in order to allow human beings to survive and exist in the outside world, and it has to reflect therefore the... the laws of the outside world; the causality and the interactions. So our human mind evolved in order to fit it to the outside world. Its fundamental structure is reflects some fundamental aspects of the outside world; therefore it's not totally surprising if the machinery… intellectual machinery we develop within us is consistent with the outside world. I mean, that takes you so far, you might argue that the outside world we only meet at... at the, kind of, level… macroscopic level...
 
Read the full transcript at https://www.webofstories.com/play/michael.atiyah/89

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