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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

#201

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…

There was a project long long ago where every piece of knowledge known was cross pollinated with every other piece of knowledge, creating a new and unique piece of knowledge, and it was intended to use that machine to invalidate the patent process - obviously everything had therefore been invented.

But that's not how new frontiers are conquered - there's a great deal of existing knowledge that is leveraged upon to get us into a position where we think we can succeed, yes, but there's also the recognition that there is knowledge we don't yet have that needs to be acquired in order for us to truly succeed.

THAT is where we (as humans) have excelled - we've taken natural processes, discovered their attributes and properties, and then understood how they can be applied to other domains.

Take fire, for example, it was in nature for billions of years before we as a species understood that it needed air, fuel, and heat in order for it to exist at all, and we then leveraged that knowledge into controlling fire - creating, growing, reducing, destroying it.

LLMs have ZERO ability (at this moment) to interact with, and discover on their own, those facts, nor does it appear to know how to leverage them.

edit: I am going to go further

We have only in the last couple of hundred years realised how to see things that are smaller than what our eye's can naturally see - we've used "glass" to see bacteria, and spores, and we've realised that we can use electrons to see even smaller

We're also realising that MUCH smaller things exist - atoms, and things that compose atoms, and things that compose things that compose atoms

That much is derived from previous knowledge

What isn't, and it's what LLMs cannot create - is tools by which we can detect or see these incredible small things

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

#202

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…

> the result has no meaning without the human intent behind the objective, human understanding to value the new pathway the AI used (more valuable than the result itself, by far) and the mathematical language (built by humans) to explore the concept.

Isn't this just anthropocentrism? Why is understanding only valid if a human does it? Why is knowledge only for humans? If another species resolved the contradictions between gravity and quantum mechanics, does that not have meaning unless they explain it to us and we understand it?

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

#203

Earlier quoted context omitted.

I absolutely believe that AI will supercharge science. I do not believe it will replace humans.

I absolutely believe that AI will supercharge science and replace humans. Why shouldn't it? Humans are poorly optimized for almost anything, and built on a substrate that's barely hanging together

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

#204

Earlier quoted context omitted.

I absolutely believe that AI will supercharge science. I do not believe it will replace humans.

I absolutely believe that AI will supercharge science and replace humans. Why shouldn't it? Humans are poorly optimized for almost anything, and built on a substrate that's barely hanging together

Well, for starters AI doesn't have goals. If there was a super intelligence with goals, why would they work for us?

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

#205

As I have stated before, AI will win a fields medal before it can manage a McDonald's A difficult part was constructing a chess board on which to play math (Lean). Now it's just pattern recognition and computation. LLMs are just the beginning, we'll see more specialized math AI resembling StockFish soon.

We're automating art and science so that we can flip burgers. This future sucks.

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

#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?

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

#208

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…

It’s easy to see that LLMs don’t merely recombine their training data. Claude can program in Arc, a mostly dead language. It can also make use of new language constructs. So either all programming language constructs are merely remixes of existing ideas, or LLMs are capable of working in domains where no training data exists.

LLMs ingest and output tokens, but they don’t compute with them. They have internal representations of concepts, so they have some capability to work with things which they didn’t see but can map onto what they know. The surprise and the whole revolution we’re going through is that it works so well.

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

#209

As I have stated before, AI will win a fields medal before it can manage a McDonald's A difficult part was constructing a chess board on which to play math (Lean). Now it's just pattern recognition and computation. LLMs are just the beginning, we'll see more specialized math AI resembling StockFish soon.

> 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

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

#210
post #30

Is there a reason why we only hear of Erdos problems being solved? I would imagine there are a myriad of other unsolved problems in math, but every single ChatGPT "breakthrough in math" I come across on r/singularity and r/accelerate are Erdos problems.

Erdős problems form a substantial fraction of all mathematical problems that have been explicitly stated but not solved; are sufficiently famous that people care about them; and are sufficiently uninteresting that people have not spent that much effort trying to solve them.

Solving problems people have already stated is a niche activity in mathematical research. More often, people study something they find interesting, try to frame it in a way that can be solved with the tools they have, and then try to come up with a solution. And in the ideal case, both the framing and the solution will be interesting on their own.

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