Terence Tao

When I came to graduate school in Princeton, so John Conway was there at the time.

He passed away a few years ago.

But I remember one of the very first research talks I went to was a talk by Conway on what he called extreme proof.

So Conway just had this amazing way of thinking about all kinds of things in a way that you wouldn't normally think of.

So

He thought of proofs themselves as occupying some sort of space.

If you want to prove something, let's say that there's infinitely many primes, there will be different proofs, but you could rank them in different axes.

Some proofs are elegant, some proofs are long, some proofs are elementary, and so forth.

The space of all proofs itself has some sort of shape.

He was interested in extreme points of this shape.

Out of all these proofs, what is the shortest at the expense of everything else, or the most elementary, or whatever?

He gave some examples of well-known theorems, and then he would give what he thought was the extreme proof.

in these different aspects.

I just found that really eye-opening.

It's not just getting a proof for a result was interesting, but once you have that proof, trying to

to optimize it in various ways.

That proofing itself had some craftsmanship to it.

It's something for my writing style that, you know, like when you do your math assignments as an undergraduate, your homework and so forth, you're sort of encouraged to just write down any proof that works and hand it in.

As long as it gets a tick mark, you move on.

But if you want your results to actually be influential and be read by people, it can't just be correct.