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Did we invent number theory?


Created or discovered?
Michael Atiyah Mathematician
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[Q] Do you regard mathematics as creation or discovery? For example, did you create or discover the index theory?

Well that's the kind of question that most mathematicians try to understand most of their working lives, and you oscillate between one and the other. I think that the… in a sense you might say all the mathematical things that you can ever prove are… have been waiting there to be... to be identified. They are true theorems, they are true results which were there before you came along and... and spotted them. On the other hand you can say the... the totality of all true results is so vast, it is in picking out some rather than others, finding your way through the infinite number of possibilities, that is what a mathematician does.  You know, he’s exploring territory, the territory is all in darkness, it’s a beautiful landscape, he comes along with a powerful search light, or whatever it is, and in his jeep and he goes through and he finds an interesting... interesting route. In that sense he discovers what was there originally. He didn't create it, but he created the route to get there amongst the totality of all possibilities.

Well I think in some sense that has to be true, that... that you don't… but the degree to which you select, in other words what distinguishes identifying a concept, making it worthwhile and studying it, is what you... you… your invention. For example the terminology you use, you might give a word, name to something. I've certainly invented that name, because somebody else could have a given it… you know, ‘A rose by any other name could not smell as sweet’. So the names you give to things are certainly your invention. And the name quite often, and the sequence of names you put together, is also an invention because they could have been put together in a different way. So the way you organise things, I think, is your own invention because they could have been organised by somebody else differently. The totality of things is there, but how you see them, the way you order them and the way you connect them up is certainly your own imprint.

So I think it's a combination of imposing your own viewpoint, and the viewpoint is an intellectual viewpoint and it has more than just, you know, instantaneous flash.  It's, sort of, a dynamic one where all mutual relationships... and how you… what you decide is important to give names to, and big names and small names, that is certainly your own imposition on the background, which is somehow fundamentally there waiting to be discovered. So there're the two layers; there's the background and then what comes on top of it, the illumination, and you illuminate what was there. Something like that, I mean, of course if the answer was just yes or no, it would be a very dull question. All philosophical questions have two answers, you know, there's the truth of what is a deep, you know, definition of what is a deep truth… a deep truth is a truth which is… something which is both true and false, I mean, truth… something that is true and its opposite's also true. So these... these things I think don't have a simple answer, but that would be, I think, my way of trying to combine the two and synthesise them together. Obviously other people will have different answers, and if you're of a religious frame of mind you may say God created everything and you just are led there by the hand of God and that sort of thing. And there'd be other people who'd take a much different view and say everything is in the mind you know, the outside world has no reality; everything is a… comes from inside and we just build the world in our own image.

So there are extremes which philosophers have argued. Not only about mathematics, but about the entire nature of reality. So mathematics is part of that, it's an especially interesting part because in some sense it's concerned with intellectual concepts, and not on the face of it with the outside world. Although I think that's again an oversimplification. There are people who say mathematics is just an intellectual creation, we set down the rules, we work out the consequences and it has nothing to do with reality outside, and that I think is a gross oversimplification. We wouldn't have developed geometry if we didn't look round the world and see three dimensional objects, even though in some sense Euclidean geometry is a set of axiomatic subjects, is independent of the outside world. It's only a poor approximation to the outside world, but there's no question that if we didn't have eyesight and didn't see things, didn't live and move in a three dimensional world, we would not have developed geometry.

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: creation, discovery, theorem, God, philosophical questions, Euclidean geometry, axiomaticsubjects

Duration: 4 minutes, 27 seconds

Date story recorded: March 1997

Date story went live: 24 January 2008