Mathematics

“A lot of these behaviors that we're describing mathematically, like backtracking, or kind of starting again, I mean, these are not really specific to mathematics. I mean, we're seeing them specifically in mathematics in these examples, but kind of, they're gen…”

“Along, I mean, of course models are going to help us produce exponentially more mathematics, but they also make it much easier to absorb it and right now, okay, it's still a bit of a challenge back and forth, but I think it's, for me at least much, much faster…”

“The ability of someone who's not working on math is, like, their literal job all the time to, like, understand what's going on and, like, you know, learn about some of the mysteries they might have wondered about will go up quite a lot. Also, you know, if you'…”

“So I think some of the results we've seen have had kind of the flavor of, like, you know, you kind of take some known techniques and apply them in maybe a clever way, or you I don't know, they've kind of been some kind of results I would characterize as, like,…”

“At least for me the point of an open problem is it's, like, supposed to measure your failure to understand something. So it's kind of like a benchmark.”

“So there's definitely also, like, slop coming from non-experts, but that I kind of actually don't see as a net negative. Like, ok, there's a lot of, now there's a lot of, like, documents on the internet one might have to comb through to figure out if a problem…”

“One thing I always say about a model result is, like, you cannot evaluate it except in retrospect, and like, this is also true of human mathematics. Like, sometimes a problem we thought was really important or would require really deep new ideas does not, and …”

“I think a lot of what we educate people for is, like, pretty robust changes in the nature of the world. Like I think the reason to learn math has always been, like, to think clearly and, like, better understand the world, and, like, presumably that's something…”

“That math is very much a leading edge indicator of what the market might be interested in.”

“And then I read a lot of these discourses on the math solutions, and there's kind of these claims where if it can solve all math, you can predict anything. And I just think that that's a huge, huge logical leap, which is not clear to me that is, is, is, is, is…”

“Coding in general has nice spillover effects in terms of improving reasoning capabilities of models as a whole. Whereas you know, let us say the model companies prioritized something like the models getting better at medicine. I'm not too sure if that would he…”

“I believe, for example, in areas like mathematics, because it's a closed system you don't need data. You can just make conjectures and see if you can prove them and keep on like that. In that sense, it's a bit like AlphaGo where you can play against yourself.”

“I think mathematics is also, it's also primed to see a revolution in the next few years for the same reasons, again, because The domain just gives you verifiable rewards.”

“In the mathematics I think we will see the models getting better on proving technical results, technical lemmas maybe including formalization and like things like lean the formal theory, improving language. I think the models, it's easy to imagine the models b…”

“I think you know, a third area is the next set of foundation models are going to come, and by that I don't mean the Neolabs and the Next Gen LLM, which of course will happen, but I mean, physics, materials, science progress by models, math progress. And I thin…”

“If you do have one-on-one tutoring and that tutor is able to identify what your gaps are, and fill in those gaps, especially in subjects like mathematics, that most kids can actually, probably all kids could be accelerated dramatically”

“It definitely seems plausible, but there's a lot of other concerns there where it needs to It needs to be able to, like, actually do experiments, and get data, and interact with the real world for a lot of these in a way that does not need to happen at all for…”

“I just don't think you're going to train the AI so much on math that is going to learn how to do Henry Kissinger level diplomacy. I do think skills are somewhat more self-contained.”

“When there's very high performance on, on that benchmark that would be suggestive of something that has, say, in the ballpark of superhuman mathematician capabilities. And so I think that would revolutionize the, academy quite substantially. Because all the th…”

“But if you instead relied on a formal system to check everyone else's work, then You could do a little bit like in astronomy where you could have an amateur kind of living in the middle of maybe nowhere and you, you've never met. And then you wouldn't have to …”

“Depending how you attack the problem, it's definitely not at that end. And I think even because it's so hard to have a perfect reward when you interleave language, I think by, even by accident, the field will move forward as we get to discover these more gener…”

“There's actually a lot of research to back this up, that if you do have one-on-one tutoring and that tutor is able to identify what your gaps are and fill in those gaps, especially in subjects like mathematics, that most kids can actually, probably all kids co…”

“If you're ever going to be able to reach the most beautiful elements of the subject, if you're ever really going to be able to properly to put it to use, you can't have your working memory being swamped by remembering all of these rules and remembering these r…”

“And I think for me, that's what makes the maths come to light because when you realize how important this stuff is to people, that they know that they're going to their death and still the only thing they want to do, Is finish their mass. I think that's the st…”

“The reality is that math is a web, not a tower. That you can jump in at different points in the network of ideas, And then find your way around from there.”

“Before he started trading full-time in 1978, he was one of the most acclaimed mathematicians, especially as a geometer over the past 50, maybe even a hundred years.”

“The last, I think, 20, 25 years, almost all of the major breakthroughs have occurred by people coming in with ideas from psychology or from mathematics. And basically introducing these into economics.”

“Well, I think it's, you know a strong contender for the most important unsolved problem in math, you know, of this century.”

“That that's how you've made your decision, or that that's how, now, what happens is, often that's how a good way to justify a decision, but it's imagination that gets you to the hypothesis in the first place. So in mathematics, someone instinctively believes s…”

“There's actually an awful lot of form in music. It's like maths. It really is like maths. That's why a lot of mathematicians love music as well, because it's all about structures and forms and about relationships.”