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When Math Is Beautiful

Question: What’s the most elegant proof you’ve ever encountered?

Peter Woit: Elegant proof. That's a good question. It's hard to say. There's a lot of - the problem with - elegance is actually an interesting question maybe to say what is elegance and what is beauty, I guess. So, one thing that's come up within this String Theory issue is this question of is this a beautiful idea, or beautiful theory. As my colleague, Brian Green has written a very good book about this called The Elegant Universe. And so, within in physics, this is our - we're often very struck by some of these basic ideas are very striking in the way in which we like to describe as being elegant, or beautiful. But the problem with that terminology, especially with the terminology of beauty is that beauty means a lot of different things to a lot of different people. A lot of different ways in which things can be beautiful. But this really has a very specific meaning and which is more along the lines of elegance which is that we say an idea is beautiful or elegant in mathematics or physics if a very simple principle or a very simple idea, or simple set of ideas, turns out to be very powerful and leads to all sort of unexpected structure and unexpected predictions.

I think maybe what I find is the most beautiful and most elegant thing about this whole subject is a little bit about what I was trying to explain earlier. If for instance this idea that the fact that the laws of physics are the same as you move in time means that is has very unexpected significance. It means that there is this thing called energy and you can associate a number called energy to physical systems and it's not going to change it as time evolves. So, in classical physics, there's actually a theorem called Noether's Theorem after a mathematician called Emmy Noether. So, there is a little slightly **** trivial theorem and that's certainly a beautiful theorem. What fascinates me more is that in quantum mechanics, Noether's Theorem isn't even a theorem, it's just implicit in the definition. Just the way you set things up is automatically true, so it's - anyway, there's a theorem and a proven **** and you start to think about quantum mechanics and you think about it, and it's just kind of automatically true. There's not even the Theorem any more. And that's just beautiful.

Recorded on December 16, 2009
Interviewed by Austin Allen

Peter Woit believes that mathematicians and scientists are led astray by seeking "elegant" solutions. But there is one theorem that makes him wax poetic.

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