How do we know that these "internal models" are not just half computer and half a giant team of mathematicians?
How do we know that OpenAI actually came up with these solutions and didn't steal them from outside researchers?
It would be extremely unlikely human mathematicians able to solve these kinds of problems would accept not getting credit that would set them for life professionally.
Also lean proofs are notoriously tedious and slow to write so this level of output is very likely to be from LLMs. The number of people able to understand this level of math and prove it using Lean is a few hundred at most.
Hm that's the question, innit. If you were a mathematician, and you have a chance at either being known for solving a particular problem, and the accolades that come with that, and the potential for money, or $X,000,000, for large values of X, in OpenAI stock, what value of X would it take to accept the OpenAI stock? Being known for solving an obscure math problem might get you money, but as academia is unreliable and full of politics, the OpenAI stock is also not a given to set you up with generational wealth, so it's also a gamble.
We'd have to accept that there are hundreds of such mathematicians willing to take the OpenAI stock options for that conspiracy to be true, and that no one leaked being offered that at all, so personally I don't think that's possible. Which leads me to conclude that OpenAI's AI did indeed create the proofs, with some level of help from humans to guide it.
That's all true but there is a reason why these problems have remained unsolved for decades and are now all falling at the same time. The reason is not mathematicians being more incentivized.
It is an interesting game theory scenario. But I think the kind of person who gets a math PhD would value the professional recognition for solving a famous hard problem more than the money.
And that is part of the reason why, if you are a great mathematician and you have very good understanding of programming, you want to be local and not do anything on someone else’s server, and that doesn’t just apply to mathematics, but to many other high-level fields of learning. Can you afford to have an original idea? be appropriated by the data center crew?
We all know it’s only a matter of time before lawsuits in this area become widespread particularly in lawyer happy America.
This would that the proofs would be almost done anyway, which statistically could not be the case, or that these mathematicians were already progressing thanks to ChatGPT. Still a provenance question, but the impact of AI is unquestionable with regards to outcome.
The proofs would only need to be well on their way enough that the only thing needed to finish it were thousands of terawatthours of energy. Something which the mathematician having their work stolen does not have.
(also answered similarly to another comment; this is a common question)
There are suddenly many new solutions to problems that have resisted sustained attacks (e.g. the Uniform Games Conjecture as detailed in TFA at some length). Where do you think they are coming from? Why is there suddenly a bunch of results to be stolen?
