Why Erdős Problems Are Falling to AI

Since at least 2014, which was my first brush with the phenomenon when someone published a 13GB proof [0].

The consensus is that such a proof is potentially illuminating, though further work is likely required. If for instance, conjecture A is true if and only if conjectures B & C are true, and B is proven false through one of these such proofs, then we can see that A is also false given that we accept the disproof of B.

Though, the sense is that further work is likely required because it is easy to see that further work along the same direction, or in directions depending on the proof will be hard or impossible if there are not enough humans or agents that are capable of understanding and utilizing the proof. Making it 'more elegant' will increase it's utility despite not proving anything new.

This is adjacent to all of the work done to create multiple proofs using different techniques. Having the same information (that X is so) in different languages (algebraic, geometric, via harmonic analysis, etc) allows for researchers not familiar with the original technique to participate in further research.

[0] https://www.newscientist.com/article/1997488-wikipedia-size-...

Take something like Fermat's last theorem - I'd be curious to hear of any use of the result itself, but there was a massive amount of new mathematics generated by those working on it, whether ultimately successful or not.

These AI math proofs are interesting testament to the power of reinforcement learning applied to math, obviously reflecting the axiomatic self-consistent nature of math itself, but it doesn't seem they have the same value as a humans working on these problems since they are using known math to solve them rather than inventing anything new.

However, it would still be interesting to analyze the LLM lines of reasoning that lead to any of these results, since there may be value there even if no new math, just as human Go players have found value in analyzing computer Go.

Still, as Demis Hassabis has himself said, the real goal with AI is discovery and creativity - you want to create the thing that could design the game of Go in the first place, not just play it. Similarly with math, while there is interest in seeing an AI "play math" using the rules of the game, what would be of much more interest is the AI that can create new math, in the same way as Andrew Wiles did while proving Fermat's last theorem.

Math is an incredibly broad field. I mean, you don't expect a traffic engineer to understand anything about nuclear reactors, do you? Yet, they are all 'career engineers'.

Because it's math, it's all mysterious and genuinely impressive, but in the end, if no human cares about it (apart from attention grabbing "it's so over" tweets and articles), does it really matter?

Huh?

Mathematics is an extremely wide subject, it's perfectly normal even for two professional mathematicians not to understand each other's work. Have you considered that maybe they just don't work in that area?

Are you implying OpenAI's paper (which was, by the way, edited by humans and provided Lean certificates for most of the proofs) is actually gibberish? That's flat-earth levels of conspiracy.

I can't tell if xkcd #435 is still true, or if math is just as mushy as everything else seems to be. When a math proof can only be understood by a handful of people, what does that mean about that proof? I think the LLMs are pushing a problem that existed already and pushing it further.

[0] https://xkcd.com/435/

The problem is that Paul Erdos was eccentric but he was also Paul Erdos. I'm not saying you're implying that, but I feel it's a similar line of thinking to how popular culture often romanticizes autism and Asperger's because some very smart people are (allegedly) affected. The same group includes people who need 24/7 care.

Computer science at the frontier is as specialized and hard as mathematics, you need years of study to truly understand a field well enough to make meaningful contributions. You can try sponsoring me if you really want, but I don't think you'd be spending your money wisely in expectation.

He died aged 83...

The only two major highlights are weather modeling and folded protein backbone prediction.

Mostly everything else, either lacks enough data, or there are contraits on the size of the foundational models that render them impractical or they just fail to generalize.

Except it's not? It's owned by a private foundation where the only link is both were founded by the same dude who hasn't run either in like 15 years.

That seems like a pretty big thing to just gloss over like it's just set-dressing.

It would be tempting to assume the commenter here has no idea what Renaissance is or how they made so much money.