Grant Sanderson: belief

Transcript context

…I thought you were going to say we all learn about group theory in school, but I missed that class. We all learn about group theory… No, the quadratic formula. This was known. In some sense, the Greeks could solve quadratics, but they didn’t really write things in algebra. It’s really the Arabs who wrote down that formula. There’s this delightful story about dueling Italian mathematicians—not real duels, just intellectual challenges—who secretly found a formula for the cubic, and then very shortly thereafter found a formula for degree-four polynomials. So a natural open question for mathematicians is, can you find a formula that solves degree-five equations? The degree-four formula is a monster. It would be wild to write it down. You usually don’t write it down in full. You break it up as a procedural thing. You might believe these things have this exponentially increasing complexity. So for many hundreds of years, nobody was really answering that question. Usually, we say Abel was the first to prove it. He was this young, precocious Norwegian mathematician. He showed it’s simply impossible. It’s not that you can find a quintic formula. He thought he found one initially, but he showed it’s impossible. I think the real credit though, you have to back up a bit and talk about Lagrange. He found the right kind of question to ask about this. I’ll give it at a very high level. He was studying the question and recognized that being able to solve these polynomials is very related to understanding the way certain algebraic expressions are symmetric. If I write down a + b + c + d, just adding four variables, and I permute those, it doesn’t change the value of the expression. Whereas if I write a + b * c + d, some of the permutations don’t change it, but some of them do. He had this really nice insight about how if you can find expressions that have four free variables, but all the permutations take on three distinct values, that has this unexpected relationship with being able to reduce degree four into degree three. He started approaching the question of whether we can find a quintic polynomial by wondering if he could extend that method. To extend that method, you would have to have an expression that has five free variables such that as you permute them over all the five factorial permutations, it takes on only four values or fewer. You could put that in a puzzle book. You could put that in a brain teaser that a twelve-year-old could engage with. It’s not too hard to find yourself feeling like that’s an impossible task. Lagrange is sitting there saying, “Here is a strategy to solve this problem of finding a quintic polynomial. It seems like it might be impossible, at least from this strategy.” But that was the first time in history that people had the instinct that some kind of question about symmetry was the right way to study these polynomials. In his mind, it was just a way. It had yet to be discovered that there was actually a tighter connection. stinct that some kind of question about symmetry was the right way to study these polynomials. In his mind, it was just a way. It had yet to be discovered that there was actually a tighter connection. Also maybe rather than searching for the formula, we should be asking the opposite question: can you prove that it’s impossible? He sort of planted that seed. Around fifty years later, Abel definitely read Lagrange and was influenced by it. We know that Galois loved Lagrange when he was falling in love with math. It’s very hard to imagine that these two young geniuses coming up with pretty similar insights around that problem wasn’t born from Lagrange. But to your question on whether you are able to verify that this was a good idea, there wasn’t any result that Lagrange came to. He didn’t solve the problem, so it wasn’t a case of knowing it was the right question to ask based on a solution. He just asked it. There’s some intrinsically interesting thing about it. It also wasn’t very important for math at the time. Most people were more interested in the applications to physics. This was almost a side, recreational, hobbyist-type thing. Abel started working on quintic stuff, but then he was advised to spend more of his efforts studying elliptic functions, so more of his work was on that before he died young. He died at twenty-six from tuberculosis. And then Galois pushed both of those ideas in the right direction, where he really understood the nature of abstraction. He had this really nice piece that he wrote while he was in prison. We could talk all about his life story. It’s pretty wild. But he’s this teenager, he’s in prison, and he had tried to submit his math papers and they had been rejected. So again, thinking about verifiable reward, the verifier function that is the academy at that time is rejecting what he wrote. Frankly, it was not very coherent. It wasn’t a complete proof. He wasn’t giving a clear thought of what the theory actually was. He was just a young fledgling mathematician getting his bearings. The verified reward there is, “No good.” But he has some instinct that there’s something there. So he’s writing this diatribe on the nature of math being something that undergoes these shifts over time. He talks about the advent of algebra itself and going from just thinking in terms of numbers to having a certain fluency with pure algebraic expressions, where you’re not tied to interpreting those expressions. He has this instinct that there seems to be another layer of abstraction that we should be doing, where rather than thinking about the formulas themselves, we’re thinking about what symmetries underlie those formulas. But it was still a pretty ill-defined theory. If you’re trying to say the verified reward is that he solved a problem that other people haven’t, well, Abel already proved that quintics are unsolvable. So what was Galois doing? In principle, Galois theory lets you take a specific polynomial, and it gives you the rules to say whether that specific polynomial has roots that you could write down. For example, with x5 - 1, you know that a solution is 1. Or x5 - 2, you can write down the fifth root of two.…

Stored transcript either side of the excerpt. The highlighted words are the published quote; the surrounding text is unedited source, never generated.