In mathematics, beauty is a very important ingredient. Beauty exists in mathematics as in architecture and other things. It is a difficult thing to define but it is something you recognise when you see it. It certainly has to have elegance, simplicity, structure and form. All sorts of things make up real beauty. There are many different kinds of beauty and the same is true of mathematical theorems. Beauty is an important criterion in mathematics because basically there is a lot of choice in what you can do in mathematics and science. It determines what you regard as important and what is not.
I think the aim of a mathematician is to capture as much truth as you possibly can in small packages, a high density of truth per unit word, and beauty is a criterion. If you've got a beautiful result, really it means that you've got an awful lot identified as a very small compass as opposed to a rather large waffle, a woolly, woolly result which is rather dilute. So I think beauty is a measure of significance. It is absent in experimental science where your result is tested by experiment but a mathematician has to test significance with some other criteria. Some of these criteria are simplicity, beauty and elegance; these show the mathematician he's on the right track and he's been truthful and so on. It appeals to you and you like it - but it is more than just the way you like a pretty picture - beauty had a significance over and above just the immediate attraction of an elegant result. At least there are different kinds of beauty and some beauty is more superficial than others - just little tricks which don't go very far. It is like some pictures which are superficially attractive and others which are much deeper - the same is true of mathematical theorems. You can have beautiful results and beautiful proofs.
But of course with a beautiful result it can be very simple to state yet proving it can be very complicated like Fermat's Last Theorem. What happens is that you try then to evolve variations of the proof. The first proof you get is maybe not the best way to get there. If its a beautiful result, like the top of a mountain, and you got there by a very roundabout method, you see this beautiful result ought to be proved in a better way. You go back and you try, and again try, and a lot of the time you find a host of other results which reflect the beauty of the endpoint by building up beautiful steps in between and you may end up with a whole theory which is a beautiful road with marvellous scenic views all the way up to the top. But of course that takes time and the first shot at a beautiful result may have a big ugly proof but you're very happy to have it at all. I'm not sure you can have a beautiful proof and an ugly result, that is not consistent, you have a beautiful road ending up at a dump. I don't think that is going to happen very often - you don't go down that road.