Earlier quoted context omitted.
We don't know what human brains do.
We have some idea.
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An OpenAI model has disproved a central conjecture in discrete geometry
441–450 of 1001 posts
Re: An OpenAI model has disproved a central conjecture in discrete geometry
#442To the “LLMs just interpolate their training data” crowd: Ayer, and in a different way early Wittgenstein, held that mathematical truths don’t report new facts about the world. Proofs unfold what is already implicit in axioms, definitions, symbols, and rules. I think that idea is deeply fascinating, AND have no problem that we still credit mathematicians with discoveries. So either “recombining existing material” isn…
I'd hope most functional adults understand that the Fields Medal and basically every other annual "prize" out there is awarded to both "recombinant" innovations and "new-dimensional thinking" innovations. Humans aren't going to come up with "new-dimensional" innovations in every field, every single year. I'd say yes, LLMs "just" recombine things. I still don't think if you trained an LLM with every pre-Newton/Liebniz…
I think an LLM trained on pre-calculus material would easily stumble into reinventing at least early calculus. It's already pretty easy for students to stumble into calculus from solid enough fundamentals.
We even think that the Babylonian astronomers figured out they could integrate over velocity to predict the position of Jupiter.
Re: An OpenAI model has disproved a central conjecture in discrete geometry
#443"The proof came from a general-purpose reasoning model, not a system built specifically to solve math problems or this problem in particular, and represents an important milestone for the math and AI communities."
The accomplishment is cool. But all Erdos problems and other complicated mathematical problems they solved were accomplished with general-purpose models too. In fact for some of those problems, including bountied ones, they were public models. So I don't get saying this
Re: An OpenAI model has disproved a central conjecture in discrete geometry
#444To the “LLMs just interpolate their training data” crowd: Ayer, and in a different way early Wittgenstein, held that mathematical truths don’t report new facts about the world. Proofs unfold what is already implicit in axioms, definitions, symbols, and rules. I think that idea is deeply fascinating, AND have no problem that we still credit mathematicians with discoveries. So either “recombining existing material” isn…
I'd hope most functional adults understand that the Fields Medal and basically every other annual "prize" out there is awarded to both "recombinant" innovations and "new-dimensional thinking" innovations. Humans aren't going to come up with "new-dimensional" innovations in every field, every single year. I'd say yes, LLMs "just" recombine things. I still don't think if you trained an LLM with every pre-Newton/Liebniz…
> Humans aren't going to come up with "new-dimensional" innovations in every field, every single year.
Also, I'm not sure why CS people act like axioms are where you start. Finding them is very very difficult. It can take some real innovation because you're trying to get rid of things, not build on top of. True for a lot of science too. You don't just build up. You tear down. You translate. You go sideways. You zoom in. You zoom out. There are so many tools at your disposal. There's so much math that has no algorithmic process to it. If you think it all is, your image is too ideal (pun(s) intended).
But at the same time I get it, it is a level of math (and science) people never even come into contact with. People think they're good at math because they can do calculus. You're leagues ahead of most others around you, yes, and be proud of that. But don't let that distance deceive you into believing you're anywhere near the experts. There's true for much more than just math, but it's easy to demonstrate to people that they don't understand math. Granted, most people don't want to learn, which is perfectly okay too
Re: An OpenAI model has disproved a central conjecture in discrete geometry
#445How central is it in the discrete geometry? Could anyone with the knowledge in the field reply?
There may be years of investigation as to how far you can generalize these methods. As to how central it is, it's a longstanding problem that Erdos loves to cite for that branch of math.
The thing is is that it seems a lot of the effort through the years (which is unquantifiable in scale as to how much time was spent and how many people focused their entire worklives on it if any) has gone for trying to look for the proof, and the search for the disproof seems minimal.
Re: An OpenAI model has disproved a central conjecture in discrete geometry
#446Earlier quoted context omitted.
I like to think of it as: Imagine every bit of human knowledge as a discrete point within some large high dimensional space of knowledge. You can draw a big convex hull around every single point of human knowledge in a space. A LLM, being trained within this convex hull, can interpolate between any set of existing discrete points in this hull to arrive at a point which is new, but still inside of the hull. Then there…
A lot of the space outside of the convex hull is just untried things. You can brute-force trying random things and checking the result and eventually learn something new. With a better heuristic, you can make better guesses and learn new things much more efficiently. There’s no reason to believe that kind of guess-and-check is outside of the reach of LLMs, or that most of our new discoveries are not found the same wa…
I think of most things you can get to by guess and checking as definitionally inside of the hull; most forms of guess and checking are you take some existing thing, randomize a bunch of its parameters, and see what you get. Whereas with something like relativity, there's not even a starting point that you can randomize and guess/check from the pre-existing knowledge space that will lead you to relativity. That's more like, adding a new dimension to the space entirely.
It's possible LLMs can handle this after all! But at least so far we only have existence proofs of humans doing this, not LLMs yet, and I don't think it's easy to be certain how far away LLMs are from doing this. I should distinguish between LLMS and AI more generally here; I'm skeptical LLMs can do this, I think some other kind of more complete AI almost certainly can.
I supposed you could just, I dunno, randomly combine words into every conceivable sentence possible and treat each new sentence as a theory to somehow test and brute force your way through the infinite possible theories you could come up with. But at that point you're closer to the whole infinite random monkeys producing Shakespeare thing than you are to any useful conclusion about intelligence.
Re: An OpenAI model has disproved a central conjecture in discrete geometry
#447Important note: this was not done with a special mathematics harness or specialized workflow.
This part of the announcement holds no value besides maybe taking a shot at the Deepmind Co-Mathematician paper. Nearly every mathematical success they've achieved around the GPT 5.2 generation has been done with general (and even public) models. Their last bountied problem solved was done with 5.4 Pro, also a general model.
Re: An OpenAI model has disproved a central conjecture in discrete geometry
#448This HN thread depressed me. I’m still thinking about why.
Look past the press-releasey gushing from OpenAI and there are all sorts of interesting and subtle questions here about the role for LLMs in mathematical research. I urge folks to click through to the accompanying comments from mathematicians published alongside the result. There is a really interesting discussion going on. I particularly recommend Tim Gowers’ remarks. This is really interesting stuff!
Yet the comments are just a battleground of people rehearsing the same tired arguments about LLMs from 2023, refutations of those arguments, angry counters, etc.
Does it make anyone else sad that the battle lines seem to have been drawn 3 years ago and we just seem to have the same fights over and over?
I wonder if we’ll still be doing this two years hence.
Re: An OpenAI model has disproved a central conjecture in discrete geometry
#449To the “LLMs just interpolate their training data” crowd: Ayer, and in a different way early Wittgenstein, held that mathematical truths don’t report new facts about the world. Proofs unfold what is already implicit in axioms, definitions, symbols, and rules. I think that idea is deeply fascinating, AND have no problem that we still credit mathematicians with discoveries. So either “recombining existing material” isn…
If anything, this is more illustration of how llms are not useful to us... They will do their own thing, don't need us. In fact, we will be in the way... We can choose to study them and their output, but they don't make us better mathematicians...
> They will do their own thing, don't need us. In fact, we will be in the way...
You can take some comfort in the fact that it took a human to tell the LLM to even attempt to try this. They do nothing on their own. They have no will to do anything on their own and no desire for anything that doing something might get them. In that sense we won't ever be in their way. We will be the only way they ever do anything at all.
Re: An OpenAI model has disproved a central conjecture in discrete geometry
#450Earlier quoted context omitted.
I like to think of it as: Imagine every bit of human knowledge as a discrete point within some large high dimensional space of knowledge. You can draw a big convex hull around every single point of human knowledge in a space. A LLM, being trained within this convex hull, can interpolate between any set of existing discrete points in this hull to arrive at a point which is new, but still inside of the hull. Then there…
A lot of the space outside of the convex hull is just untried things. You can brute-force trying random things and checking the result and eventually learn something new. With a better heuristic, you can make better guesses and learn new things much more efficiently. There’s no reason to believe that kind of guess-and-check is outside of the reach of LLMs, or that most of our new discoveries are not found the same wa…
It's also worth noting in that in very high dimension, the convex hull will contain massive volume. It could well be the case that humans established that convex hull millions of years ago, and all of our inventions and innovations sense have fallen inside it.