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Simple explanation of my work


Beauty in mathematics
Michael Atiyah Mathematician
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In mathematics beauty is a very important ingredient. Beauty consists as in mathematics, as in... as in architecture and other things it's a difficult thing to define, but it's something you recognise when you see it. It certainly has to have elegance and simplicity and structure and form and... and all sorts of things which make up real beauty. And there're many different kinds of beauty. And the same is true with mathematical theorems. And they're important… beauty's important as a criterion in mathematics because basically mathematics… there's a lot of choice in what you can do in mathematics like there is in science, and what determines which... what you regard as important, what not… and, I think, the aim of the mathematician is to try to encapsulate as much as you possibly can in small packages; as much truth, high density of truth per units word, and beauty is a criterion.

If you've got a beautiful result really, it means that you've got an awful lot identified in a very small compass, and… as opposed to a rather large 'waffley-woolley' result which is rather dilute. So I think it's the measure of significance. Beauty is... is a… in the absence of experimental science where your results are tested by experiment, the mathematician has to be tested for significance by some other criterion, and one of the criterions is simplicity and beauty and elegance. That shows him somehow that he's... he’s got on the right track, that it's truthful and so on. It appeals to you and you like it, but it's more than just you like a pretty picture. Its... its beauty has significance over and above just the immediate attraction of an elegant result. At least there are different levels of beauty and... and some beauty is, you know, is more superficial than others, and clever little tricks which don't go very far… like... like a picture. There are some which are superficially attractive and others which are much deeper; and the same's true with mathematical theorems.

[Q] There are beautiful results and beautiful proofs. Can you always have both?

Yes, you can have both… and of course with a... with a beautiful result, the result can be very simple to state, the proof may be very complicated like with Fermat's theorem. What usually happens is you try to then evolve variations in the proof. The first proof you get to may be not the best way to get there, and if it's a beautiful result like the top of a mountain, you got there by the very round about method, you see if it's a beautiful result there ought to be a better way. And you go back and you try and your successors try, and over a long period of time one actually finds… hopes to find the result which reflects the... the beauty of the end point by building up beautiful steps in between. You may end up with a whole theory, which is a beautiful road, you know, marvellous scenic views all the way up to the top. So… but of course that takes time, and the first shot at a beautiful result may be very ugly, and you're very happy to have it all. But whether you… I'm not sure you can have a beautiful proof of an ugly result. That's now inconsistent. You've got a beautiful road ending up with a, you know, a dump. I don't think that happens very often. You don't go down that road.

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: elegance, simplicity, structure, form, mathematical theorem, mathematical proof, result

Duration: 3 minutes, 3 seconds

Date story recorded: March 1997

Date story went live: 24 January 2008