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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

#361

Post by School-Cotton »

trostaft wrote: post #232

Speaking as a postdoc in math, I must say that this is rather exciting. This is outside of my field, but the companion remarks document is quite digestible. It appears as though the proof here fairly inspired by results in literature, but the tweaks are non-trivial. Or, at least to me, they appear to be substantial to where I would consider the entire publication novel and exciting. Many of my colleagues and I have b…

Why would it excite you, rather than terrifying you? The better LLMs get at math, the closer the expertise you spent your whole life building is to being worthless.

Along with all the rest of what humans find meaningful and fulfilling.

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

#362

Post by tacitusarc »

ykl wrote: post #334
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 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…

I like this construction, but I don’t think you take it far enough.

If you have a multi dimensional space, and you are trying to compute which points lie “inside” some boundary, there are large areas that will be bounded by some dimensions but not others. This is interesting because it means if you have a section bounded by dimensions A, B, and C but not D, you could still place a point in D, and doing so then changes your overall bounds.

I think this is how much of human knowledge has progressed (maybe all non-observational knowledge). We make observations that create points, and then we derive points within the created space, and that changes the derivable space, and we derive more points.

I don’t see why AI could do the same (other than technical limitations related to learning and memory).

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

#363

Post by thechao »

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…

You can build a census of all gen-2, degree-2 formal products of polynomial like terms. If you insist on instituting your own rewrite rules and identity tables, it is straightforward — maybe an 15 minutes of compute time — to perform a complete census of all of the algebraic structures that naturally emerge. Every even vaguely studied algebra that fits in the space is covered by the census (you've got to pick a broad enough set of rewrite- and identity- operations). There's even a couple of "unstudied" objects (just 2 of the billion or so objects); for instance:

    (uv)(vu) = (uu)(vv)
Shows up as a primitive structure, quite often.

If you switch to degree-3 or generator-3 then the coverage is, essentially, empty: mathematics has analyzed only a few of the hundreds (thousands? it's hard to enumerate) naturally occurring algebraic structures in that census.

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

#365

Post by davebren »

atleastoptimal wrote: post #206

To all AI skeptics: What is preventing AI from continuing to improve until it is absolutely better than humans at any mental task? If we compare AI now vs 2022 the difference is outstandingly stark. Do you believe this improvement will just stop before it eclipses all humans in everything we care about?

> What is preventing AI from continuing to improve until it is absolutely better than humans at any mental task?

No matter how much compute time it's given to combine training samples with each other and run through a validation engine it will still be missing some chunk of the "long tail". To make progress in the long tail it would need to have understanding, and not just a mimicry of understanding. Unless that happens they will always be dependent on the humans that they are mimicking in order to improve.

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

#366

Post by mooreat »

davebren wrote: post #323
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…

> 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 better and better approximations.

The vibe I get though is that we aren't more powerful than regular languages, cause human beings feel computationally bounded. So I could see given enough "human signal" these things could learn to imitate us precisely.

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

#367

Post by atleastoptimal »

davebren wrote: post #365
atleastoptimal wrote: post #206

To all AI skeptics: What is preventing AI from continuing to improve until it is absolutely better than humans at any mental task? If we compare AI now vs 2022 the difference is outstandingly stark. Do you believe this improvement will just stop before it eclipses all humans in everything we care about?

> What is preventing AI from continuing to improve until it is absolutely better than humans at any mental task? No matter how much compute time it's given to combine training samples with each other and run through a validation engine it will still be missing some chunk of the "long tail". To make progress in the long tail it would need to have understanding, and not just a mimicry of understanding. Unless that happ…

What is the difference between what LLM's do and "true" understanding?

I feel like people grasping straws on the shrinking limitations of AI systems are just copying the "god of the gaps" fallacy

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

#368

Post by red75prime »

vatsachak wrote: post #209
volkercraig wrote: post #179

Earlier quoted context omitted.

> we'll see more specialized math AI resembling StockFish soon Heuristically weighted directed graphs? Wow amazing I'm sure nobody has done that before.

My claim is that LLMs waste a lot of time training on all available data. Math is a sequence of formal rules applied to construct a proof tree. Therefore an AI trained on these rules could be far more efficient, and search far deeper into proof space

It has been tried. Lenat's Automated Mathematician, for example. The problem is that the system succumbs to combinatorial explosion, not knowing which directions are interesting/promising/productive. LLMs seem to pick up some kind of intuition from the data they are fed. The generated data might not have the needed "human touch" or whatever it is.

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

#370

Post by davebren »

atleastoptimal wrote: post #367
davebren wrote: post #365

Earlier quoted context omitted.

> What is preventing AI from continuing to improve until it is absolutely better than humans at any mental task? No matter how much compute time it's given to combine training samples with each other and run through a validation engine it will still be missing some chunk of the "long tail". To make progress in the long tail it would need to have understanding, and not just a mimicry of understanding. Unless that happ…

What is the difference between what LLM's do and "true" understanding? I feel like people grasping straws on the shrinking limitations of AI systems are just copying the "god of the gaps" fallacy

> What is the difference between what LLM's do and "true" understanding?

The thing where you can understand the meaning of this sentence without first compiling a statistical representation of a 10 trillion line corpus of training data.

Unless you're an NPC of course.

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