Grant Sanderson: belief

Transcript context

…It’s very bad. If you’re a young genius, don’t work on the quintic. He asks his brother and his close friend to get his notes to Gauss, to get these notes to the important mathematicians of the day, because he thinks there’s something there. Even then, it didn’t really take. His brother and his friend tried to get them out, but it was another twenty years until Liouville sees these notes, sees that maybe there’s something in them, and tries to clean them up and understand what Galois was getting at. Even then, it was another twenty years or so until Jordan actually puts together something like a modern treatment of group theory that they attributed to Galois. You could easily imagine history turning differently, where these ideas were coming about from other points in math, and Galois could have been forgotten in history if he was a less florid character. But between the time of Lagrange having this inkling that maybe symmetries of roots is the right way to go, to where it all looks like modern group theory, you’ve got this long span. A lot of the time, it’s not even passing the verified reward of human reviewers. It gets on someone’s desk and they say, “I don’t really know if there’s anything here.” You have to have this one person recognize it. Even then, it’s not really solving practical problems at that point. You pointed out cryptography and physics and things like that. You have to get into the twentieth century before you have Gell-Mann thinking that maybe understanding the nature of how certain groups break down has a relationship with what particles are made out of. He anticipates quarks based on a purely group-theoretic question. That’s one of the more interesting applications of group theory: to even predict the existence of quarks is a group-theoretic question. That’s so long after Lagrange before you have anything like that. So you have to ask, what is the way of measuring progress that’s not based on solving a problem, but that is somehow capturing the instinct inside Galois’s mind when he says, “I think there’s something here”? What’s the instinct inside Lagrange’s mind when he says, “I think this is the right way to think about it”? What’s the instinct inside Liouville’s mind when he says, “These scattered notes from this long-dead youngster might have something to them”? It’s so hard to put a finger on that. A different series of videos I’m making right now is about the whole “compression is intelligence” idea. Even though this isn’t really the angle I’m taking, there is something to the idea that the smaller expression that’s more predictive feels more intelligent. So I wonder the extent to which you can give some kind of verifiable reward around not just whether you solved it or what it is solving, but around the smallness of the concepts required to do it. Going back to Riemann hypothesis solutions, what would that look like if an AI solves it? I think a third way it could happen is it just straight-up works harder. ss of the concepts required to do it. Going back to Riemann hypothesis solutions, what would that look like if an AI solves it? I think a third way it could happen is it just straight-up works harder. In the same way, you could maybe have an elementary proof of Fermat’s Last Theorem that’s just spelled out over thousands of pages that would be incoherent. But the cleaner way to view it is with elliptic curves and all that. Maybe there’s some thousand-page proof of the Riemann hypothesis that no one’s really getting anything out of, and what you actually want are the succinct, compressed versions of those ideas that would then lend themselves to human understanding. Maybe you throw Kolmogorov complexity into your attempt to quantify what you mean by elegance. I don’t think it’s easy, but I do think it’s something you would have to do in order to reward the Galois-like instinct, rather than just rewarding whether you solved a problem. It’s very hard to come up with the heuristic for science. But it’s clear humans have been doing this somehow, and obviously, AIs will do it at some point.…

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