Lex Fridman: belief

Transcript context

…I’m much more comfortable with the fox paradigm. Yeah. So yeah, I like looking for analogies, narratives. I spend a lot of time… If there’s a result, I see it in one field, and I like the result, it’s a cool result, but I don’t like the proof, it uses types of mathematics that I’m not super familiar with, I often try to re-prove it myself using the tools that I favor. Often, my proof is worse, but by the exercise they’re doing, so I can say, “Oh, now I can see what the other proof was trying to do,” and from that, I can get some understanding of the tools that are used in that field. So it’s very exploratory, very… Doing crazy things in crazy fields and reinventing the wheel a lot, whereas the hedgehog style is, I think, much more scholarly. You’re very knowledge-based. You stay up to speed on all the developments in this field, you know all the history, you have a very good understanding of exactly the strengths and weaknesses of each particular technique. I think you rely a lot more on calculation than sort of trying to find narratives. So yeah, I can do that too, but other people are extremely good at that. Let’s step back and maybe look at a bit of a romanticized version of mathematics. So I think you’ve said that early on in your life, math was more like a puzzle-solving activity when you were young. When did you first encounter a problem or proof where you realized math can have a kind of elegance and beauty to it? That’s a good question. 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 proofs themselves as occupying some sort of space. So if you want to prove something, let’s say that there’s infinitely many primes, you have all 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, and so there’s this cloud, so the space of all proofs itself has some sort of shape, and so he was interested in extreme points of this shape. Out of all these proofs, what is one of those, the shortest, at the expense of everything else, or the most elementary or whatever? So 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, that it’s not just getting a proof for a result that was interesting, but once you have that proof, trying to optimize it in various ways, that proofing itself had some craftsmanship to it. It’s certainly informed my writing style, like when you do your math assignments and as you’re an undergraduate, your homework and so forth, you’re sort of encouraged to just write down any proof that works and hand it in, and 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. It should also be a pleasure to read, motivated, be adaptable to generalize to other things. It’s the same in many other disciplines, like coding. There’s a lot of analogies between math and coding. I like analogies, if you haven’t noticed. You can code something, spaghetti code, that works for a certain task, and it’s quick and dirty and it works, but there’s lots of good principles for writing code well so that other people can use it, build upon it so it has fewer bugs and whatever, and there’s similar things with mathematics.…

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