OpenAI, the Partition Principle, and Mathematics

You might argue that these aspects of math are less important in the new AI accelerated math world, because agents will inevitably be smarter than humans, but I think clear framing and communication is even more important than before because with this technology we can choose to augment our intelligence instead of defer it

Thankfully, very few mathematicians share Hardy's opinion, just as very few share his opinion that "mathematics is a young man's game" (and indeed we now have prizes like the Abel Prize with no age limit).

In fact, many of the greatest mathematicians throughout history have taken exposition very seriously, e.g. Euclid, Euler, Lagrange, Cauchy, Dirichlet, Kolmogorov etc. all wrote textbooks. Many mathematicians today carry on that tradition of taking exposition seriously and write books and freely share their lecture notes.

So we should not take Hardy's opinion as representing the opinion of all mathematicians or even most mathematicians. In fact, Hardy's statement is somewhat self-contradictory since he himself wrote several expository books (e.g. "A Course of Pure Mathematics").

Wait, who are those people? Who is that Hardy and what did he really say? What did he really mean? Who else said or meant the same things?

Who are you criticising, exactly?

A first rate mind communes with mathematics not matter terrestrial. An Uber-Erdos entering to lecture is indifferent to the auditorium as well as it's contents. He's at the board silently communicating new mathematics. Not for himself, not for you, but for mathematics.

I wish they take a bit of more time to communicate the findings effectively.

But this is even worse, because there is no way that OpenAI "trained" on this data between September 8, 2026, the date Chenglong Ma uploaded the paper to ChatGPT; and October 6, 2026, the date that OpenAI released a paper with "identical proof strategy and specific choices of notation" (Ma). That's one month, that's not the timescale for model training.

So this implies _not_ that OpenAI is training on user input, in the conventional sense of adjusting weights; but rather that they are *straight-up channeling ideas from user input*, and with a very short lag. You would think there would be about a million controls to prevent this.

This is next-level alarming. I would be very interested in knowing whether Chenlong activated the privacy (do not train, etc) options in ChatGPT, and any other details of their setup (which plan, etc). Also, note that "do not train" might be, in a lawyerly sense, considered by OpenAI to be strictly about weights, and not covering "we hoover up your results and regurgitate them".

I don't see how it's related to quality at that point. Call your technique "the banana method" and wait to see OpenAI invent "the banana method"

People who are invested in the idea that we've invented a general intelligence, now, which includes all these companies that are literally financially invested in this claim they are making, will tend to believe that its results can already be trusted in domains like this. Some mathematicians seem to believe some of the proofs written by their models, and some, like this one, don't. I do think it's valid for an expert to push back against the claim that the best use of their time right now is to verify the poorly written work of everyone who's claimed to solve the problem

If you just strip mine the answers and Sam Altmans magic button solves 100/100 problems, what's next? Who is left to come up with a new interesting question for the magic button to solve?

Lastly, life and the present moment is all there is, if there is no enjoyment in anything we do, then what's the point of all the "living for ever" Altman et al want to achieve.

We will live forever to read boring papers generated by LLMs? Literally sounds like an eternal hell.

However, there are always smarter, hungrier people out there and this is a buffet.

Some output is going to be wrong or incomplete. I am willing to bet even those have nuggets that can be used elsewhere.

This is like "no one is forcing software engineers to use AI tooling" or "no one is forcing you to show your ID in the airport" or "no one is forcing you to own a car in your small midwestern city" - there can be no law requiring something and the practical consequences of not doing so can be so painful that you're effectively forced anyway.

> supported by a clear plurality of respondents

was referring to?

This is a misleading characterisation of the mathematicians' position.

The very first paragraph of the AGMAI recommendations explicitly states:

"we do not endorse this practice, and we ask them to stop testing advanced mathematical problems on proprietary models." You appear to have acknowledged this by saying “Mathematicians would prefer if OpenAI didn't use open math problems as an eval…”.

The mathematicians did not ask OpenAI to produce these results. They explicitly asked AI labs to stop producing them in this manner. Their subsequent recommendations concern what labs should do if they have already produced significant results, not an endorsement of the practice.

Furthermore, the recommendation was not simply to release the results, but to responsibly release already existing results. Section 2.B, Step I, explicitly recommends "...labs that have AI mathematical output that is not understood by the people who prompted the AI systems", to search the literature for relevant prior work, provide appropriate attribution, and improve the exposition of AI-generated proofs before releasing them, rather than leaving this work to mathematicians afterwards.

OpenAI published the results on GitHub while still exploring repositories that meet the committee's guidelines. So they followed some of the recommendations, but not all of them and hence, did not release the results as requested by the mathematicians.

I do not think it is a settled matter whether this was done out of goodwill. This is because releasing these results as they were can benefit OpenAI more than releasing them according to the AGMAI recommendations. AGMAI recommended in section 2.B, Step 1.5 that "Each time a solution to a problem is released, it should be clearly documented how exactly AI came to be used on that particular problem. If many results are released at once, then in addition to the results themselves a further document should be written and made public that references all of the released results and explains how many other problems of comparable difficulty the models tried and failed to solve, as well as how the problems were chosen." If the results are released, it is easy to expect that the media will discuss the capabilities of the AI used in the work, as indeed happened. If this AGMAI recommendation was followed, the media would plausibly have also discussed the number of failed attempts and then the overall attitude would not be as favourable to OpenAI as it is now when it comes to the capabilities of the AI that was used. OpenAI did release on GitHub that approximately 4,000 problems were attempted and resulted in 719 manuscripts (after 3 containing suspected errors were removed by OpenAI) across 372 families of problems, but this does not give a calculable number of problems it failed to solve. I do not claim to know OpenAI's intentions or reasoning when these results were released and am not arguing that it was done with improper intentions, only that whether it was done out of goodwill is not a settled matter.

AGMAI's October 6 statement explicitly clarified that its advisory role should not be interpreted as an endorsement of OpenAI's process, and that it was up to the mathematical community to assess how successfully its recommendations had been followed.

Recommending how to responsibly handle the outcomes of something you oppose is not the same as asking for it to happen.

As the author of the post points out, there is no way this is “the best they could do”. It’s a write up that didn’t involve someone with the math + communication skills required to clearly explain the result.

Maybe this is just a matter of what model developers choose to invest training resources in, but I don’t think it’s inevitable unless clarity is made a higher priority

What do you do if those results suck like in the article?

The alternative you are proposing implicitly is even crazier. OpenAI should not release a proof that is most likely correct so that it doesn’t burden others. What? It’s not about that guy dude. It’s about the society TM. One can’t delay progress because a guy may be burdened.

“Guys plz don’t release this thing that is absolutely correct but I’m kinda busy with other things ok?”

If it verifiably works and solves important outstanding issues, then quite possibly yes.

Those who really understand a topic are usually also able (and great at) explaining it in very clear and "simple" terms. This may be part of my personal bias; I see theory builders as those who advance the field the most, and these are usually also amazing at explaining it. On the other hand, those who mostly "grind" through problems (approach them as complicated puzzles) with effort/time were often bad at explaining.

I observed the same for programming: the "architects" usually explain very well, the "debuggers" often don't. LLMs very much remind of the grind/puzzle approach. It does makes sense that RLVR, which in my understanding enables a lot of these results, would lead to a more mechanical approach.

Of course, I can't make any predictions on whether it will stay that way. But I strongly suspect that we need different ways of training for LLMs to write better text and explain better (I suspect the vagueness of LLM language is the result of RLHF as vague expression is less often incorrect).

Were you satisfied with the paper?

Having read the paper, do you understand "what you missed" in those 3000 hours?

https://news.ycombinator.com/item?id=50016860

??? That's literally what everyone does. You are creating a hypothetical that is nonsensical. Here's your hypothetical:

1. Agent gives you 5000 lines of slop

2. You reject it and just do it yourself

This is reality

1. Agent gives you 5000 lines of slop

2. you realise that it has done a lot of research and is mostly in the correct direction and you ask it nicely to refine it

3. verify that you understood it and push it to prod

Are mathematicians babies that they need a completely different approach?

I can't speak for the rest of the world, but I think it's amazing that multiplication and 3SUM are sub-quadratic despite that being "obviously" impossible.

I wonder what this means long term.

Long term, stronger models than what OpenAI used will be available to everyone. Some mathematicians will take advantage of those tools and do great things; others will continue their whiny gatekeeping and become irrelevant.

Today, you are training the machine to accept the poorly specified inputs of the CEO's dumb nephew tomorrow.

They didn't just "start" it though - they released slop papers.

That's finishing it - not starting it as far as scientific publishing is concerned.

> Or are you saying the community has no incentive to do that because there is no reward/recognition for doing that?

Not exactly - but that is part of it.

I think what would have went over better is:

1. Immediately announce a solution has been found.

2. Do not publish the solution.

3. Put out an open request for anyone with experience in the area who wants to get involved to help collaborate on a construction and human-comprehensible paper. Accept anyone who can demonstrate potentially useful work/experience in the field/problem. Share the solution with them after they sign some kind of NDA that they won't independently publish or share the solution/work.

4. Work with people until a paper is ready (I mean actually ready - not the kind of slop that they released).

5. Publish. Include names of everyone who made meaningful contributions to the paper (not just the proof).

EDIT: Notice the incentive with my proposed second path is that it gives OpenAI an incentive to improve the interpretability of its proofs. This is a good thing! The maths community would be thrilled to actually gain understanding from such releases, and OpenAI would be happy because they could more quickly and independently publish their results. At the moment the "value" of their mathematics research "product" is low because of the lack of this interpretability, and this current approach is simultaneously destroying the opportunity value of the community as well as the incentive for OpenAI to ever improve on what's missing.

To me, science is not a database of facts and data, and the scientific process is not just the process of adding to this database.

It is more than that.

Science is building the database of facts and data, but it is also sharing/disseminating that database so that others can write to it too.

We both agree that "openness" is critical for this, but to me, the scientific process includes things like the actual words that a scientist uses to communicate their ideas/data, the paper they wrote, the collaborations they had, the work that they built upon. Science is not like a database only - its more like a distributed system, and the scientific community is like a living, breathing thing. It is the health of that system which has been damaged in my opinion.

So I have no problem that OpenAI updated the database.

They didn't just make the information available though.

They updated the knowledge database, then released low quality slop papers.

They made it clear that they don't give a damn about sharing/disseminating information.

They are trying to fundamentally say "we can do science without prioritizing sharing.

We don't think its important. We'll release the proof because it makes our stock go up, but we're gonna keep going and building and not spend any time on communication or comprehension because we think that it doesn't matter, and your contributions don't matter, and AI is gonna do all of this independently anyway. We spent $20m on the proof because it made our stock go up $20b, but we'll spend 20 min on the paper. Priorities clear.

Does any of that make sense?