Re:In Search of Proofs from “The Book”
by andrewla on Tuesday March 20, 2018 @08:49PM
(#16634266)
I feel like the article does a good job of addressing this when it talks about the role of "ugly" proofs. It might deserve more emphasis on this; that all pioneering mathematics starts with this, rather than assigning it a space as a "stepping stone" to the "book proof". But it is true that "cleaning up" after the pioneers is a difficult and interesting part of mathematics, that serves to make it easier for new pioneers to get to the end of the paved trail.
> Q: There’s a famous quote from the mathematician G. H. Hardy that says, “There is no permanent place in the world for ugly mathematics.” But ugly mathematics still has a role, right?
> A: You know, the first step is to establish the theorem, so that you can say, “I worked hard. I got the proof. It’s 20 pages. It’s ugly. It’s lots of calculations, but it’s correct and it’s complete and I’m proud of it.”
> ...
> To do these short and surprising proofs, you need a lot of confidence. And one way to get the confidence is if you know the thing is true. If you know that something is true because so-and-so proved it, then you might also dare to say, “What would be the really nice and short and elegant way to establish this?” So, I think, in that sense, the ugly proofs have their role.
> Q: There’s a famous quote from the mathematician G. H. Hardy that says, “There is no permanent place in the world for ugly mathematics.” But ugly mathematics still has a role, right?
> A: You know, the first step is to establish the theorem, so that you can say, “I worked hard. I got the proof. It’s 20 pages. It’s ugly. It’s lots of calculations, but it’s correct and it’s complete and I’m proud of it.”
> ...
> To do these short and surprising proofs, you need a lot of confidence. And one way to get the confidence is if you know the thing is true. If you know that something is true because so-and-so proved it, then you might also dare to say, “What would be the really nice and short and elegant way to establish this?” So, I think, in that sense, the ugly proofs have their role.