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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

#451

Post by nrightnour »

daishi55 wrote: post #202
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…

> 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 be…

anthropocentrism? An interesting thought, I don't think that word applies with computers.

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

#452

Post by astrange »

FloorEgg wrote: post #13
aurareturn wrote: post #2

One thing seems for certain is that OpenAI models hold a distinct lead in academics over Anthropic and Google models. For those in academics, is OpenAI the vendor of choice?

Gemini seems better trained for learning and I think Google has made a more deliberate effort to optimize for pedagoical best practices. (E.g. tutoring, formative feedback, cognitive load optimization) As far as academic research is concerned (e.g. this threads topic), I can't say.

Gemini the chatbot has a very strange personality that intensely overindexes on your user profile and absolutely loves insane mixed metaphors.

Its explanations are quite good but they're also hard to understand because it keeps trying to relate everything back to programming metaphors or what it thinks it knows about the streets in the neighborhood I live in.

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

#453

Post by bsder »

beering wrote: post #353
ykl wrote: post #334

Earlier 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…

> You can brute-force trying random things and checking the result and eventually learn something new.

And most of the mathematicians seem to welcome this "brute forcing" by the LLMs. It connects pieces that people didn't realize could be connected. That opens up a lot of avenues for further exploration.

Now, if the LLMs could just do something like ingesting the Mochizuki stuff and give us a decent confirmation or disproof ...

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

#454

Post by [deleted] »

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.

[deleted]

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

#455

Post by cm2012 »

missyougowers wrote: post #348
ks2048 wrote: post #133

Timothy Gowers' tweet about this: "If you are a mathematician, then you may want to make sure you are sitting down before reading futher.". woah.

Unfortunately Gowers has taken Tao's lead on this one. Gowers has one of my favourite video series about how he approaches a problem he is unfamiliar with: https://www.youtube.com/watch?v=byjhpzEoXFs It is disheartening to see him jump into this GenAI puffery. I hope these GenAI labs are paying Tao handsomely for legitimizing their slop, but more likely he's feeling pressure from his University to promote and work wi…

If seems like you have an axe to grind about AI capabilities that is making you think irrationally

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

#456

Post by godelski »

Nevermark wrote: post #302
symfrog wrote: post #260

Earlier quoted context omitted.

We have had LLMs for much longer than 3 years.

I took humans thousands of years, then hundreds of years, to come to terms with very basic concepts about numbers. Its amazing to me when people talk about recombining things, or following up on things as somehow lesser work. People can't separate the perspective they were given when they learned the concepts, that those who developed the concepts didn't have because they didn't exist. Simple things are hard, or ever…

It's why the invention of teaching has been so important. Took a long time for humans to develop calculus. A long time to then refine it and make it much more useful. But then in a year or two an average person can learn what took hundreds of years to invent. It's crazy to equate these tasks as being the same. Even incremental innovation is difficult. You have to see something billions of people haven't. But there's also paradigm shifts and well... if you're not considered crazy at first then did you really shift a paradigm?

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

#457

Post by jryan49 »

libraryofbabel wrote: post #448

This 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…

People are afraid for their livelihood. What do you expect?

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

#458

Post by efavdb »

cpard wrote: post #224

The proof brings unexpected, sophisticated ideas from algebraic number theory to bear on an elementary geometric question. The more I read about these achievements the more I get a feeling that a lot of the power of these models comes from having prior knowledge on every possible field and having zero problems transferring to new domains. To me the potential beauty of this is that these tools might help us break thro…

It’s as if the body of human knowledge is our I’ve mind. It used to be expensive to access that, but no more.

Cool thing is now when someone contributes something to the hive mind, it can instantly be applied to any other problem people are working on.

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

#459

Post by gowld »

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?

It depends on if AI can invent cold fusion before running our of all the energy on Earth.

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

#460

Post by ykl »

tacitusarc wrote: post #362
ykl wrote: post #334

Earlier 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…

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 change…

I was a little muddy in my original post on distinguishing between what I think LLMs might be able to do and what AI broadly might be able to do. I'm skeptical LLMs can expand the hull or add dimensions to the space; but I also don't think the reasons for that skepticism necessarily apply to all AI system generally.
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