OpenAI, the Partition Principle, and mathematics

By the time I got out of bed yesterday, I had three people ask me if I'd seen the OpenAI announcement that the Partition Principle does not imply the Axiom of Choice. The number kept increasing during the day, and I can see why. I am famously interested in this problem.

Indeed, at least two people asked if I plan on sending Sam Altman a bottle of whisky, as promised in my Problems page. The answer to that is no. And let me explain to you why, and why you should be very angry at OpenAI.

Let me clarify something first. I am not entirely against the use of AI. I understand that it is a tool, and many people find it useful, and I am happy to agree that it is merits. I think that we need to better understand how this technology changes our field and how we want to use it before we rush to throw our lot with it. This is why I generally avoid using AI for mathematics (I am happy to ask LLMs to consolidate information for me, or to generate a useful infographic, or to proof read an email, etc.), it's just the framework of using it that is currently missing in my eyes. Still, we are in the stage of discovery, people use it and we see the consequences unfold and we can decide as a community to adopt it further or restrict its use. For example, arXiv introduced rate limiting as a consequence of these tools, and almost all papers should include some AI statement at this point (even to state that no AI was used, I guess). We need to decide how we judge the contribution of the author, e.g., do we require the chat log be made public, or at least available to the reviewers and the editor? These are questions that we need to contend with and see where they take us.

But, back to the Partition Principle. I took a brief look at the preprint released by OpenAI (not at the Lean code, since I know very little of the actual usage of Lean, and that code was enormous). It sucked. It is unclear, muddled, and has a strange structure. The terminology is "a bit off", there are theorems that I would not expect to be proved, that are also stated in a very strange manner. Lemma 7.4, again, is not something you'd expect to see in a paper like this, the same goes to Lemma 8.1.

Then there are the references. One of my papers on the Bristol model is cited for some basic introduction of symmetric extensions, which is not the reference I would have chosen; at least three unpublished, unrefereed, and non-arXiv'd lecture notes are used as references (including mine, which is used to refer to one proposition that I am certain appears in some published papers of mine and of others).

Compare this to any paper in set theory that isn't dealing with the technicalities of a subfield I'm not familiar with. I can almost certainly skim the paper and have a good idea as to what is going. I might not be familiar with some of the terminology, or the arguments, but for the non-technical stuff, I should be able to understand the strategy and the construction. And for papers on my expertise, on the Axiom of Choice, I will generally need to skim the paper briefly to have a good idea as to what the authors did (or intended to do). Not to say that I am always right, or that I always get it exactly as the authors intended, but this is what I've spent over 15 working on, and by now I have a pretty good idea how it works. But with OpenAI's paper on the Partition Principle this is impossible.

No, if this was an academic paper, it should be issued a desk rejection for the quality. The onus is on the author to conform and adhere to the communications level. Much like a paper written in Swedish would likely be rejected outright from the Proceedings of the American Mathematical Society, because the onus is on the authors to make it accessible for the readers. And therein lies the rub.

OpenAI drops some hundreds of "solutions", incomprehensibly written "solutions", and we are all expected to jump on them and what? Appreciate their contributions? Why? I am trying to finish several papers, I am supervising a number of students, and I have active research of my own to do. When am I supposed to sift through a badly written paper? Why should I bother, when they don't bother?

The OpenAI press release just suggests "progress". But the media cycle, for everyone else, these are "solutions". This is a perfect way to have your cake and eat it too. But it is the equivalent of a Denial of Service, should we want to take it seriously: stop everything else and figure this one out. And some people worked for years on a problem that they might just do that, but as a whole... Well. If this was real attempt to share progress, the quality of the writing would be better, it would receive some input from actual experts on the actual topic. OpenAI is a busy company, but I am sure that if they want to be genuinely helpful, they can find the time to contact a few mathematicians for advice and input, if nothing else, then on the quality of the outputs. Because even if this is a huge leap forward in improvement, it is not nearly at the acceptable level. But hey, everyone can claim that ChatGPT solved a long-standing problem (this one, about the Partition Principle, is over a century old), so why should OpenAI care?

And what are we, the mathematicians, left with? We are left with a public, funding bodies, and policy makers that are not academics, not experts, not mathematicians, nor they are familiar with how mathematical research is done (since it is very different from research in biology, psychology, or philology), who see these "results" and think that we can replace mathematicians with bots. This is the equivalent of replacing the chef with the kitchen tools, even if the knife is automatic and can dice veggies on its own, there is still a significant distance between that and palatable food that you'd want to eat.

It seems to me, that the AI tech companies would like us to conform to their standards, rather than spend the time and energy to conform to ours. And I am afraid that this just will not do. So, no, I will not be sending Sam Altman a bottle of whisky anytime soon, nor I am planning on spending my time reading through that paper and trying to make sense of it.