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Ten advances in mathematics and theoretical computer science

openai.com

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Re: Ten advances in mathematics and theoretical computer science

#251
post #6

My main gripe here is the lack of transparency around the total experiment and construction. I doubt that they simply pointed their model at these ten specific problems alone and gave the model one shot; therefore the $2000 number could be completely misleading, similar to P-value hacking by not disclosing the total experimental setup. I want to know: 1. How many total problems were given to the model, and what perce…

It seems like they threw it a decently large battery of open math problems and probably limited it to something like $200-500 per problem:

https://x.com/polynoamial/status/2083478171975082334

As a complete guess, it seems like they tested hundreds to thousands of problems with a relatively low per-problem budget

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The linked tweet from Noam Brown at OpenAI reads:

> And yes we did try other major problems without success. Sadly no Millennium Prize problems (yet).

> But also, we didn’t spend a lot on each problem. It’s possible to push test-time compute much further.

Re: Ten advances in mathematics and theoretical computer science

#252
post #238
post #232

Earlier quoted context omitted.

I can see this being important if you only care about the results as evaluations of AI progress, but if what you care about is the math itself why should you care about the prompt or anything other than the proof?

Because the math isn't solely about the proof being correct. You don't need to take my word for it, here's one of the most famous living mathematicians' take on it: https://teorth.github.io/tao-web/slides/age-of-ai-icm-2026.p...

I don’t see Tao suggesting what you have suggested there. Instead he suggests that humans responsibly disclose AI use, and that mathematicians develop a set of norms to deal with an overabundance of AI generated results. For example, he suggests that authors should be able to discuss their results in detail to demonstrate understanding before publication.

Re: Ten advances in mathematics and theoretical computer science

#253

one of the early premises of how ai takeoff would go was that a system that could solve open problems in advanced mathematics would also discover novel advances in math and computer science that directly unlock drastically better software performance. we are seeing frontier level math breakthroughs (ie performance that would put it in the top 100 or 1000 mathematicians in the world if it were a human, meaning top .00…

> we are also seeing incredible advances in software performance Incredible? > open ai announced like 15% improvement by fixing gpu kernel issue That is... ordinary software optimization.

a 15% improvement at a trillion dollar scale company is massive

Re: Ten advances in mathematics and theoretical computer science

#254
post #79

Earlier quoted context omitted.

Sorry, OpenAI's take is correct here. If you're not convinced, here is how they prompted LLM: https://cdn.openai.com/pdf/04d1d1e4-bc75-476a-97cf-49055cd98... [0] A slightly smarter highschooler could write these. I could write these. It's clear as day that the LLM, not the human, did the heavy lift. It'd be ridiculous to give full credit to whoever wrote the prompt. [0]: Not one of the proofs in the linked article, b…

But why can’t we prompt the LLM “just do math research”? This is what I don’t understand.

100% agree. If the models are so capable that they're advancing math, it doesn't seem like a stretch to expect they should be able to determine with "doing math research" entails and the best way to use their capabilities towards that end. Why do we need to hand hold the models by telling them to do parallel research, keep threads independent, etc.

Re: Ten advances in mathematics and theoretical computer science

#255

In a way the most remarkable thing about this is that it isn't even at the top of the HN homepage. Even if this is a step up from what we've seen before, we're no longer astonished by the idea that AI can make significant advances in mathematics and computer science.

This is not at the top as it is actively flagged by people that can't psychologically cope with the advances of AI. Hacker News is no longer a web site of an elite.

> Hacker News is no longer a web site of an elite.

It was always mainly a website for employees of an elite.

Re: Ten advances in mathematics and theoretical computer science

#256

At what point can we say AGI has been achieved? What is the test? AI is solving mathematical problems that humans have not been able to solve for decades. Is that not enough? Sam Altman has said "If superintelligence can't discover novel physics, I don't think it's a superintelligence." Is that the test? How far away are we from AI discovering novel physics? It seems within reach.

It’s artificial, it’s general, and it’s intelligence. The people who believe “AGI” is an important and unattained goal need to start coining and defining their terms better.

Re: Ten advances in mathematics and theoretical computer science

#257

At what point can we say AGI has been achieved? What is the test? AI is solving mathematical problems that humans have not been able to solve for decades. Is that not enough? Sam Altman has said "If superintelligence can't discover novel physics, I don't think it's a superintelligence." Is that the test? How far away are we from AI discovering novel physics? It seems within reach.

It’s artificial, it’s general, and it’s intelligence. The people who believe “AGI” is an important and unattained goal need to start coining and defining their terms better.

A true AGI will continuously improve itself without periodic retraining from scratch. Just like humans.

Re: Ten advances in mathematics and theoretical computer science

#258
post #226
post #211

Earlier quoted context omitted.

I believe we're seeing a new kind of mathematics that will require completely new formats for publication, a bit similar to those used in experimental sciences. AI-powered mathematics should be fully reproducible, so it's the authors' responsibility to disclose the exact model type, inference settings/seeds and the full prompt history leading to the result. Of course that would ideally require open weights models. It…

I think this is a bit optimistic compared to my view (wrt portability). There's a large stack of software that is involved in training and probably less so in inference. I'm not saying it's impossible but there are definitely different levels of reproducibility and the academic incentive structure doesn't really prioritize reproducibility in my experience. I'm sure it varies quite a bit, I'd be curious to know how th…

I know it sounds unrealistic and not aligned with academic incentive structures. But those are the exact structures that gave us a lot of headaches in the experimental sciences. I think it would be a good north star to aim for something that resembles how those are trying to address the reproducibility crisis. Better than to embrace the most black-box version of math that AI systems can produce (million-line proofs without context). Even if a reproducibility crisis is seemingly impossible (although agents so far have also been pretty good at finding compiler bugs).

Re: Ten advances in mathematics and theoretical computer science

#259
post #221

Earlier quoted context omitted.

You shouldn't be getting downvoted for something that a majority of pure math and art enthusiasts believe to be true. The truth is many of these entrepreneurs and VCs are obsessed with AI not for money or human progress, but because it makes them feel closer to being a "god" rather than a mere mortal. Much of it (especially AI art) is out of spite for human creativity, which is done by mortals with limitations.

Taking the stance of moral superiority is kind of funny. And pure math and art enthusiasts don't think they are closer to being a "god" from understanding/"discovering" math? Stop coping and deluding yourself mate. To begin with, whether AI is the one doing the discovering or not makes no difference. Any "pure math" person would aim to understand regardless - and would be quite glad that they have a longer paved path…

Lmao what a ridiculous response. Yes, some mathematicians and artists are in it to feel smart. But the vast majority also just enjoy the process. Having a computer do all the work for you and just typing prompts in ruins that completely. As Ronny Chieng said in his Harvard speech, the journey is the point.

>Any mathematician in academic or industry is more than likely not a "pure math" person (tainted by capitalism)

Ignoring that I meant pure as in non applied math, let's just make it clear: you agree that mathematicians who are against capitalism encroaching on this process should be allowed to dislike it without criticism of being pretentious?

Re: Ten advances in mathematics and theoretical computer science

#260
post #193

Earlier quoted context omitted.

I wasn't trying to give a model. The point was that I don't think it's necessary to give one. You didn't address any of what I wrote, let alone provide any counterarguments. Which part of what I wrote do you think was wrong?

Making definitive claims about whether LLMs do or do not have specific properties absolutely does require precise definitions of those properties that can be used to evaluate those questions. Merely hand waving that LLMs didn't undergo the same evolutionary process is not a definitive argument. For example, the Turing machines and the lambda calculus don't look anything alike, but they are fundamentally interconverti…

I just don't think linguistic (or other symbolic) representations alone can contain the information, in any sense of the word, of what e.g. human subjective experiences actually are like. The concepts we express with language get their meaning from our physical reality, even if quite indirectly in case of some abstract concepts.

Hunger as a concept doesn't mean anything without the physical need. Politeness or bluntness, even in writing, don't mean anything without social dynamics. And we have social dynamics (and neural structures that directly process social cues and associated feelings) because we've evolved into social animals for whose survival that was important.

I see no reason to believe that a model trained only with symbolic representations, with no connection to the physical world phenomena that those symbols represent, could contain the subjective experience itself.

Neural network models may be able to derive novel (or at least novel-looking) output rather than just an obvious rehash of their input, but I don't think any set of bytes can fundamentally contain information that was never entered into it. (Even if e.g. a model produces previously unknown mathematical results, those results can in principle be derived from the information that they were trained with.)

I'm not saying that artificial neural networks couldn't, in principle, be aware. ANNs and biological neural nets may be equivalent in the sense that any information and processing structures represented by a biological one could in principle be represented by an artificial one. If that's the case, and awareness is purely a product of our neural systems as materialism would imply, it should be possible for an ANN to be aware, too.

But when the model has been trained with only language, and IMO the subjective experience can't be derived from the symbolic representation alone, I can't see how the model could include the actual subjective human experience.

An AI model could of course have an awareness and subjective experiences that are totally different than our human experience. But then the fact that it happens to produce output resembling what humans find meaningful shouldn't be considered indicative of such awareness.

This is of course more of a philosophical argument than a technical one, and I'm happy to hear counterarguments, but not on the level of off-hand dismissal.

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