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Mathematics in the age of AI

arxiv.org

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Re: Mathematics in the age of AI

#101

Post by rowanG077 »

sonicrocketman wrote: post #8

Tao's Rule of Thumb (which applies very well to software): > My own suggested rule of thumb: if the authors cannot convincingly demonstrate that they are able to give a clear, expert-level talk on their results, one that is correct and properly attributed, then the result should not be published. A proof that no human can properly explain should be viewed as incomplete, even if it has been formally verified.

I just don't see that to be true. If tommorow someone pulls a proof that n = np out of their ass but is not able to explain it, it will still have immense value.

Re: Mathematics in the age of AI

#102

Post by qsera »

frozenseven wrote: post #88
eloisant wrote: post #70

Earlier quoted context omitted.

"artificial intelligence" is a vague and moving target. I'm not sure I would call that "a field". It's been used to talk about computers playing chess, then machine learning, and now LLM-based systems.

Of course it's been "used" to talk about those things, because all of those things are examples of AI. Always have been.

Then it has always been the age of "AI"...So the article title is inaccurate!

Re: Mathematics in the age of AI

#103

Post by mlmonkey »

sonicrocketman wrote: post #8

Tao's Rule of Thumb (which applies very well to software): > My own suggested rule of thumb: if the authors cannot convincingly demonstrate that they are able to give a clear, expert-level talk on their results, one that is correct and properly attributed, then the result should not be published. A proof that no human can properly explain should be viewed as incomplete, even if it has been formally verified.

What if the result is a counter-example? A fact that disproves the conjecture? Is that not publication-worthy?

Re: Mathematics in the age of AI

#104

Post by rowanG077 »

RandomLensman wrote: post #85
timjver wrote: post #78

Earlier quoted context omitted.

The whole point of Lean is that you don't need to understand the entire proof to be sure that it's correct. You only need to understand the definition of the theorem being proven, and you need to trust that the relatively small core of Lean is correct.

Doesn't Lean also have libraries? Anyway, there could also be hardware errors, I suppose.

Lean does have libraries, but since they are also in lean they are subject to the same rules. It's basically a super strong type checker. If it compiles the proof is valid. Unless there is a bug in the type checker.

Re: Mathematics in the age of AI

#105

Post by rowanG077 »

Jblx2 wrote: post #65
lacker wrote: post #59

Earlier quoted context omitted.

I don't think the mathematicians are going to be able to make that work, because journals are already struggling to keep up with their review load, and AI seems like it will make that harder. So a solution that involves "journals will do a lot more effort to review each paper" doesn't seem practical. It would work better as a bar for hiring, rather than as a bar for publishing.

It will be interesting to see the evolution of journals in the next ten years for sure. Have they outlived their usefulness? Maybe everyone will just upload papers to arXiv, along with a copy of the formal proof.

Just package the proof as a library and put it in some source code repository like github.

Re: Mathematics in the age of AI

#106

Post by drdeca »

thechao wrote: post #95
zahlman wrote: post #83

Earlier quoted context omitted.

I don't think this is a valid counterpoint at all. Math is cooperative, and comprehension is the point : the proof has value exactly because (and only to that extent) it empowers humans to understand an abstract truth. Chess is competitive: the memorized line has value because it makes you incrementally more likely to defeat your opponent.

I hope I'm remembering this right: a mathematician claims to have a proof for the ABC conjecture, but can't conceive any other mathematician it's right — it's "too weird", so the proof is rejected?

Not quite? It is more that 1) someone has gone through it, identified a step he thinks isn’t a valid step, and the author hasn’t been willing to work with that person 2) most consider the proof, due to its length combined with those doubts as to its validity, not worth their time and effort to work through and understand (because it would take a lot of time, and they have jobs to do, doing research and teaching, etc.)

Re: Mathematics in the age of AI

#107

Post by nilkn »

skybrian wrote: post #49
nilkn wrote: post #48

Earlier quoted context omitted.

I believe this rule of thumb will come to fail. The combination of superhuman mathematical reasoning and synthesis in upcoming AI models plus the rapid build-out of scalable formal verification infrastructure means this exponential in math is going to take off quite explosively, and we've barely seen anything yet. Mathematics is going to decisively move beyond human ability fairly soon (within our lifetimes, if not m…

Maybe that will be true when it's math with practical applications, but most theoretical math isn't like that. If it's not practical and it's not for mathematians to understand, what good is it?

We have thousands of years of precedent that suggests that breakthroughs in mathematics tend to accumulate into broader technology breakthroughs in other domains.

Why does this tend to be the case, even when some of the smartest people in the world have historically predicted incorrectly that certain branches of math would forever be useless (e.g., number theory)? I can only offer my own theory on that, but my guess is that mathematics is simply a predictive framework based on pattern compression. A more powerful pattern compression framework accelerates every single field that relies on pattern recognition or prediction of the unknown based on patterns.

Re: Mathematics in the age of AI

#109

Post by Calazon »

TSiege wrote: post #37
a2ff6eeb0 wrote: post #25

I don't know why anyone should care about understanding the results if the AI is better at math than us. It'd be like demanding that human mathematicians are banned from publishing until their cats understand the theorems. If Amazon uses AI math to come up with better routing, the cats can benefit from cheaper delivery fees just as much as humans can. No understanding needed. The human brain is being obsoleted, soon…

I’m not anti AI but thinking the human brain is obsolete and using it will become a hobby is a dystopian view of the future where no one has any agency anymore. By your logic since our brains provide no value why not just shoot ourselves in the head while we’re at?

I think they meant obsolete relative to economic, scientific, and engineering objectives.

That doesn't mean they don't provide any value of any kind to anyone.

Re: Mathematics in the age of AI

#110

Post by nilkn »

zahlman wrote: post #83
kriro wrote: post #35

Earlier quoted context omitted.

The counterpoint to this comes from chess. High level engines "prove" certain lines correct (not in the mathematical sense) but those "engine lines" are really hard to explain to humans, even by GMs. They can sort of explain that something is a good line but not why. Engines crush GMs and are considered ground truth even if noone really understands what is happening. Would it be a nightmare if math was the same, not…

I don't think this is a valid counterpoint at all. Math is cooperative, and comprehension is the point : the proof has value exactly because (and only to that extent) it empowers humans to understand an abstract truth. Chess is competitive: the memorized line has value because it makes you incrementally more likely to defeat your opponent.

Math that humans don't understand but nonetheless allows AI systems to develop breakthroughs in various fields of science, technology, physics, engineering, medicine, etc., would have great value to humanity even if it doesn't help humans understand abstract truth at all.

Imagine if humans couldn't understand multivariable calculus, but we had access to an AI system that developed it, it initially seemed useless, then another AI system found a predictive model of electromagnetism using it.

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