Chow, T. Y. (2008). A beginner’s guide to forcing (arXiv:0712.1320). arXiv. https://doi.org/10.48550/arXiv.0712.1320
> “All mathematicians are familiar with the concept of an open research problem. I propose the less familiar concept of an open exposition problem. Solving an open exposition problem means explaining a mathematical subject in a way that renders it totally perspicuous. Every step should be motivated and clear; ideally, students should feel that they could have arrived at the results themselves. The proofs should be “natural” in Donald Newman’s sense [13]:
> This term . . . is introduced to mean not having any ad hoc constructions or brilliancies. A “natural” proof, then, is one which proves itself, one available to the “common mathematician in the streets.””
In 1976, the proof of the Four Color Theorem was controversial because it was done with a computer examining over 1000 cases by brute force and was essentially not comprehensible by humans. But mathematicians ended up accepting it. So mathematics has a 50-year precedent of not requiring human-scale proofs. How is the current situation different?
(Disclaimer: Apologies if this sounds dismissive or argumentative. I genuinely think that the Four Color Theorem should play a role in these discussions and suspect that many people are unaware of the controversy over it.)
As AIs become smarter and smarter, there will be no amount of clarity that will make more complex proofs understandable to humans - this is an inevitable effect of the cognitive capacity gap.
Complaining about bad style can make some sense now (I disagree anyway), but it's an argument that will be dead shortly.
Maybe no human will fully understand a future proof, but they could fully understand a little piece of it. And many humans in aggregate could understand it, each with their own little piece.
https://en.wikipedia.org/wiki/Inter-universal_Teichmüller_th... seems like a counterpoint, but IANAM. (I am likely cherrypicking the far end of the bell curve re: straightforward here)
>Mochizuki and a few other mathematicians claim that the theory indeed yields such a proof but this has so far not been accepted by the mathematical community.
Proof can't be understood, proof doesn't matter.
Someone at OpenAI, please, work on this.
In other domains I have seen first hand overwhelming evidence of how things that cause the AI to make mistakes also cause humans to make the same mistakes.
I wonder if the proofs being produced that are hard for humans to interpret are also hard for other LLMs to interpret.
In other words, I wonder if humans are still much better at compressing understanding into proofs than the best LLMs, and what it will take for LLMs to exceed them.
It kind of an explicit example of how the LLMs can be materially less intelligent than people, but still be more productive through scaling, and yet they also can't replace people because they are a categorically different kind of intelligence. It's like all the AI debates compressed into one example showing countwr-intuitive answers.
Isn't it fairly established that (generally [0]) manually written / optimized skill files perform a lot better than generated ones? Meaning that yes, this likely does hold.
[0] or to be specific, that the pecking order is: ai generated < human co/written < hyperoptimized for the specific model via some convergence process
> This looks like an AI IPO PR powerplay,
Interestingly, the post has actually also an argument for this:
> Experience has shown that, even now, there will still be people explaining in patronizing tones why none of this is real and none of it counts. If such people were capable of being impressed by anything that happens in the empirical world, of updating on anything, they would’ve already been impressed and already updated several years ago, long before things had reached the point of an actual Mathocalypse.
> So, they’ll say, maybe the alleged solutions are not solutions at all, but just “AI slop.”
There's clear benefit in a babelfish that can coordinate disparate efforts, the only problem with the current iteration is giving credit to said efforts.
Google went quite far down the road to hell, but stopped short of taking credit for websites' content since the company understood that poisoning the well only goes so far. At this point, one can safely conclude that _Chat_GPT was an intentional attempt to squeeze out more data once they mined the internet dry.
Who do we demand this from? The AI companies? Or the mathematicians who are worried they will have nothing left to do?
As the old saying: great claims require great evidence.
Also from what I can tell from the few fields I understand, the proofs aren't that long or complicated they are just terribly written.
The entire US federal budget for math research is something like $100M annually. And mathematicians in other countries are hardly making bank either. How does one reconcile how the market has historically valued mathematics with the cash-strapped frontier labs ploughing so much money into that enterprise?
Why should that make a material difference to the IPO? Because of the vibes, and investors are indeed all about the vibes.
Developers and people in CS in general seem to have gotten used to the idea that most productive SWEs don't need to exactly know how to produce assembly or trace every branch prediction or even most of the optimization the CPU (or even their compiler) is running. Mathematicians will get there.