The Jacobian conjecture

41:59um which is like the synonymous the the the company of Claude the the company that wants AI safety AI safety >> regulatory capture >> right and it has apparently disproved something called the Jacobian conjecture which is a problem that has been at the center of algebraic geometry and the theory of polomial maps for about 90 years. It's insane. It's not proved it in the conventional sense because what it's actually done is disproved the conjecture. It has provided a counter example which is um a mathematical object that violates whatever the rule is. It's a bit easier to do albeit than

42:42actually proving something. Right? If you if you say a statement, proving the statement is a lot harder than just saying, "Hey, here's an example that shows it's not true." Because all you have to do is produce an example and boom, the debate is over. especially in mathematics. We're going to get into sort of how that works and why it's kind of easier than proving, but I think it's still an insane achievement that a machine effectively a neural network that is just doing multiplications of matrices at the end of the day is creating this kind of truth out of something that for 90 years the top mathematicians have been apparently wasting their time with. Yep. And trying

43:25to prove >> and this is the part of the ongoing debate about whether these AI systems are quote stochcastic parrots that >> don't have any original thought andor anything that they do that's original is just by gathering all the information on the internet and or brute forcing. And I understand that sentiment, but I think it is a little bit more nuanced than the way in which it's presented in this black and white way. >> Exactly. And I think like if you're not a mathematician, this should still matter because for one, it's showing the power of AI to like think complex

44:07things >> in a very in a very real sense, right? AI is restructuring the labor markets. Um, and now it's going after really creative. Mathematics is a creative discipline, right? And a lot of people say that AI can't be creative. It's, as you said, a stoastic parrot. Well, in this case, nobody had a nobody had a proof of this, >> right? >> And it has a novel proof that the smartest people on the planet couldn't come up with >> for almost a century. >> Yeah. Um, it's it's huge, right? So, mathematics, right, as I said, it's been the gold standard of human reasoning. Um it's the language of physics, the foundations of electrical engineering,

44:47every other kind of engineering, economics, medicine. When we say [laughter] >> yeah, >> literally everything. >> Literally everything. And when we say that something is mathematically proven, we mean it is literally true, >> right? It's not something out of physics where like you can do an experiment and then 400 years later somebody can disprove the experiment because they went farther in the decimal places to measure something, which has happened time and time again. In mathematics, the decimals are inherent in your proof, right? It's like it's there and it's true. And if I've proven that it's true, the the the logic is eternal. >> Would it be fair to say that mathematics is one of, if not the only thing that we have where you can say that there is an

45:28objective truth. Yeah. >> When you come to when it comes to like a mathematical proof, >> yes, it is it is it is it is I think the pedestal of objective truth, right? You go from there are integers and I can add and multiply them to then all of the stuff right Uklid's theorem that there are are an infinite number of prime numbers that still holds true today because no one can poke a hole into it. Why? Because it's a proof. They couldn't poke a hole back then because the logic was sound and logic is something that is eternal, right? And so it's it's really cool. And when when when we start thinking about AI starting to do mathematics, not just assisting and not just suggesting but actually proving and

46:10disproving things, right? Then we start asking philosophical existential questions and those become urgent, right? What is the role of the human mathematician? What is the role of the human mind? What does understanding mean? If a machine can generate a proof that is true that we can verify somehow through some kind of computation to be true but we can't even understand it right >> these are these are now becoming real questions it's kind of crazy so this is going to be a deep dive where I'm going to talk about the Jacobian conjecture which is the thing that this AI model has disproven going to get into the weeds of it and I think you're going to

46:50be able to understand exactly what it's done >> and then we're going to talk about the modern phenomenon of AI and mathematics and why everyone including mathematicians are very worried. >> Okay, so let's start with a Jacobian conjecture. It's something that was proposed by Hinrich Keller in 1939. So nearly 100 years ago. >> Um and it addresses a fundamental local to global paradigm. It has to do with maps. Okay. By maps I mean going from one set of coordinates to another set of coordinates. So in order