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Beauty in mathematics


Did we invent number theory?
Michael Atiyah Mathematician
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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... we have objects in the outside world, I have a glass, another glass and I can… I... 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... 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... it’s the most primitive thing there is. It's hard to imagine a world without individual objects, but ultimately it... 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... very abstract powerful mathematics.

And how is it that... 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... the causality and the interactions. So our human mind evolved in order to fit it to the outside world. Its... 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... you might argue that the outside world we only meet at... at the, kind of, level… macroscopic level. We don't see atoms and quarks and things, so how is it that the mind can... can somehow… which would be evolved at the level of macroscopic things, or things which are in the middle between the very large and the very small… can simultaneously develop a framework which enables it to tackle the very small atoms and the very large galaxies. That is a bit of a mystery.

All you can say is that we start with the... the framework we get from common garden experience, we build a logical framework which is well suited to that in order to survive, and then fortunately for us, the universe is reasonably consistent. The laws which we work out, the kind of thinking, the kind of structure which we'd… at that level of energy and scale, remarkably seems to persist at higher levels and at lower levels to a fantastic degree. Although it requires sometimes a great skill to… the quantum world is actually quite different from our present world and... and we have to build rather sophisticated mathematics on the basis of the initial mathematics to go down that level. It isn't that easy.

So we start off with ideas which are well suited to the immediate environment, as we dig further and further down we have to make it more and more elaborate. The fact that it succeeds is because, you know, God was kind to us, as Hermann Weyl said, you know, ‘God is not malicious’. And… it was Einstein said that, I think. And if he had been we would have been in the soup, you know, our... our ideas would have stopped at the sort of scale we live it, and of course we don't really know how successful we are. We... we, we’re not… we are reasonably successful, but we may soon… got a long way to go, we may still find surprises and things which we somehow are incapable of understanding. But anyway it's... it’s a fascinating story and everybody has their own interpretation of, you know, how the mind works, its relation to the outside world and the mathematics and the physics. And as you go through your life you sort of build up your own philosophical interpretation; I think everyone on his own.

Eminent British mathematician Sir Michael Atiyah (1929-2019) broke new ground in geometry and topology with his proof of the Atiyah-Singer Index Theorem in the 1960s. This proof led to new branches of mathematics being developed, including those needed to understand emerging theories like supergravity and string theory.

Listeners: Nigel Hitchin

Professor Nigel Hitchin, FRS, is the Rouse Ball Professor of Mathematics and Fellow of Gonville and Caius College, Cambridge, since 1994, and was appointed to the Savilian Professorship of Geometry in October 1997. He was made a Fellow of the Royal Society in 1991 and from 1994 until 1996 was President of the London Mathematical Society.

His research interests are in differential and algebraic geometry and its relationship with the equations of mathematical physics. He is particularly known for his work on instantons, magnetic monopoles, and integrable systems. In addition to numerous articles in academic journals, he has published "Monopoles, Minimal Surfaces and Algebraic Curves" (Presses de l'Universite de Montreal, 1987) and "The Geometry and Dynamics of Magnetic Monopoles" (Princeton University Press, 1988, with Michael Atiyah).

Tags: Alain Connes, Jean-Pierre Changeux, Hermann Weyl

Duration: 6 minutes, 1 second

Date story recorded: March 1997

Date story went live: 24 January 2008