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An OpenAI model has disproved a central conjecture in discrete geometry

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Re: An OpenAI model has disproved a central conjecture in discrete geometry

#391

Post by toraway »

cubefox wrote: post #116
lubujackson wrote: post #60

For anyone using LLMs heavily for coding, this shouldn't be too surprising. It was just a matter of time. Mathematicians make new discoveries by building and applying mathematical tools in new ways. It is tons of iterative work, following hunches and exploring connections. While true that LLMs can't truly "make discoveries" since they have no sense of what that would mean, they can Monte Carlo every mathematical tool…

There is a long and interesting recent essay on that topic by a mathematician: https://davidbessis.substack.com/p/the-fall-of-the-theorem-e...

Thank you for sharing, that was one of the most insightful long form pieces I've read in a long time! And the writing was enjoyable to read even as a math layperson.

I was going to say you should submit it but I saw you did a few days ago but it only got a few votes... If Dang sees this IMO it would be extremely deserving of the second chance pool as I wouldn't be surprised to see easily jump to the front page with a different roll of the dice.

Re: An OpenAI model has disproved a central conjecture in discrete geometry

#392

Post by mooreat »

gus_massa wrote: post #328
mooreat wrote: post #164

I think one interesting thing to point out is that the proof (disproof) was done by finding a counterexample of Erdős' original conjecture. I agree with one of the mathematician's responses in the linked PDF that this is somewhat less interesting than proving the actual conjecture was true. In my eyes proving the conjecture true requires a bit more theory crafting. You have to explain why the conjecture is correct by…

Searching for a proof and disproof are sometimes not so different. In most cases, you nibble the borders to simplify the problem. For example, to prove something is impossible let's say you first prove that there are only 5 families, and 4 of them are impossible. So now 80% of the problem is solved! :) If you are looking for counterexamples, the search is reduced 80% too. In both cases it may be useful In counterexam…

I agree there can be some theory crafting in the search for a counterexample, but in general I think it is easier to search for.

For proving a proposition P I have to show for all x P(x), but for contradiction I only have to show that there exists an x such that not P(x).

While I agree there could be a lot of theory crafting to reduce the search space of possible x's to find not P(x), but with for all x P(x) you have to be able to produce a larger framework that explains why no counter example exists.

Re: An OpenAI model has disproved a central conjecture in discrete geometry

#394

Post by Ar-Curunir »

cpard wrote: post #301
doubledamio wrote: post #238

Earlier quoted context omitted.

I’ve always been skeptical about the role of LLMs in mathematics, but this is the first time I’ve seen this argument, and I actually find it very compelling. Maybe LLMs will help us develop more horizontal understanding of the field.

It's up to us I think. We can use LLMs to generate web pages in candy crash style and end up dumper by outsourcing thinking to the machines or we can use it to expand our cognitive capabilities. What makes me more of an optimist in this case is that people who today decide to go into these sciences are mostly people who are driven by intellectual activity so I feel they are the right ones to figure this out, probably…

Unfortunately, LLMs might lead to the demise of the primary institution that allows for people that are in it for the love of intellectual activity to do that activity, namely research universities. Certainly the people proposing the tech are quite opposed to the modern university.

Re: An OpenAI model has disproved a central conjecture in discrete geometry

#395

Post by amelius »

pseudocomposer wrote: post #222
m-hodges wrote: post #40

To 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 still don't think if you trained an LLM with every pre-Newton/Liebniz algebra/geometry/trig text available, it could create calculus.

Yes but that is because there was not enough text available to create an intelligent LLM to begin with.

Re: An OpenAI model has disproved a central conjecture in discrete geometry

#396

Post by neuroelectron »

I wonder if it has anything to do with the fact that AI is a grid of grid-calculating grids. It seems like it would be especially well suited to finding solutions about grids. That is until you consider the fact that even 1 trillion billion grids is still not anywhere close to an infinite grid. So, probably slop.

Re: An OpenAI model has disproved a central conjecture in discrete geometry

#397

Post by zeofig »

johnfn wrote: post #349
zeofig wrote: post #284

Earlier quoted context omitted.

I still laugh.

Have you updated your priors after this announcement? If not, why not?

I don't have enough information about the announcement for it to mean much to me. I don't know much about this field of maths. I don't know how many mathematicians were actively working on this problem. It could be zero, which would indicate it's not really that interesting. The article gushes about how it's a Very Important Problem, but it's not even mentioned on https://en.wikipedia.org/wiki/List_of_conjectures_by_Paul_Er.... I'm sure the busy folk at openAI will fix that soon however. Furthermore the extensive dishonesty of companies like openAI makes me suspicious of just how this was achieved. Overall the announcement is of little interest to my "priors", although I don't typically think in such terms.

Re: An OpenAI model has disproved a central conjecture in discrete geometry

#398

Post by davebren »

mooreat wrote: post #366
davebren wrote: post #323

Earlier quoted context omitted.

> I think that's just a matter of having them able to work on longer and longer time horizons. No this will never do the kind of math that humans did when coming up with complex numbers, or hell just regular numbers ex nihilo. No matter how long it's given to combine things in its training data.

I currently operate under the assumption that humans are at most as powerful as Turing Machines. And from what I understand these models internally are modeling increasingly harder and larger DFAs, so they're at least as powerful as regular languages. Assuming humans are more powerful than regular languages I could maybe agree that these methods may not eventually yield entirely human like intelligence, but just bett…

Well yeah there is likely an equivalence between computability and epistemology, but I'm not sure it matters when comparing LLM intelligence to human intelligence. There is clearly a missing link that prevents the LLM from reaching beyond its training data the way humans do.

Re: An OpenAI model has disproved a central conjecture in discrete geometry

#400

Post by woah »

lovecg wrote: post #337
CGMthrowaway wrote: post #199

How do you even get an LLM to try to solve one of these problems? When I ask it just comes back with the name of the problem and saying "it can't be done"

By making it think for 100+ pages https://cdn.openai.com/pdf/1625eff6-5ac1-40d8-b1db-5d5cf925d... Regular ChatGPT users don’t have a way to do that, this is something they do internally only. edit: apparently that’s only the _condensed summary_ of the chain of thought.

you can do this easily with the api or with codex
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