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Mathematics in the age of AI

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Re: Mathematics in the age of AI

#81

Post by cma »

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?

You could have one really hard to understand proof of a theorem and then a lot of interesting human-understandable stuff that relies on that theorem. We already have lots of proofs with oracles, where you can work out consequences of what kind of structures and solutions could exist if you had some magic thing to solve a hard part, so it just seems like a variation on that. Many people learn calculus or even the real numbers without understanding the complete formalization from set theory.

Re: Mathematics in the age of AI

#82

Post by patcon »

RandomLensman wrote: post #74
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…

If the proof is formally verified but impossible to understand how would anyone be able to be sure the formal verification is correct? Complex software is bound to have bugs, no?

Sincere question, as a non-expert trying to situate your comment: are you a mathematician with experience in proofs?

Re: Mathematics in the age of AI

#83

Post by zahlman »

kriro wrote: post #35
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.

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.

Re: Mathematics in the age of AI

#84

Post by ianm218 »

GPerson wrote: post #50
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 it’s pointless to spend time trying to prove a conjecture which is ultimately false if along the way you figure out a bunch of different true variations on the conjecture, which is how mathematics actually works. This is something I’m a bit worried about with LLMs since it gets you to the end too fast.

LLMs seem to have worse intuition than experts and compensate by being able to cover a much wider surface area of ideas, so we might just need to extract the intermediate progress along the way.

Re: Mathematics in the age of AI

#85

Post by RandomLensman »

timjver wrote: post #78
RandomLensman wrote: post #74

Earlier quoted context omitted.

If the proof is formally verified but impossible to understand how would anyone be able to be sure the formal verification is correct? Complex software is bound to have bugs, no?

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.

Re: Mathematics in the age of AI

#86

Post by Jblx2 »

RandomLensman wrote: post #74
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…

If the proof is formally verified but impossible to understand how would anyone be able to be sure the formal verification is correct? Complex software is bound to have bugs, no?

Yes, you still need to be careful, especially if you have reason to think that the proof was from a malicious actor.

https://leodemoura.github.io/blog/2026-8-1-postmortem-for-ke...

(N.B. from August 2026)

Re: Mathematics in the age of AI

#87

Post by RandomLensman »

patcon wrote: post #82
RandomLensman wrote: post #74

Earlier quoted context omitted.

If the proof is formally verified but impossible to understand how would anyone be able to be sure the formal verification is correct? Complex software is bound to have bugs, no?

Sincere question, as a non-expert trying to situate your comment: are you a mathematician with experience in proofs?

My time proving things is long in the past and any systems way back when I was studying (some math among other things) certainly were different and usually quite narrow.

My point was rather more motivated by having seen so many weird ways for machines to fail/not work as expected that I wonder how to deal with that if the output were to be incomprehensible to humans.

Re: Mathematics in the age of AI

#88

Post by frozenseven »

eloisant wrote: post #70
frozenseven wrote: post #45

Earlier quoted context omitted.

Out-of-hand dismissal of Terence Tao is certainly a take. And the term "artificial intelligence (AI)" has been the name of the field for 70 years and counting. If anything, "LLM" is a misnomer that's been lingering around since 2018-19. When the term was coined, these systems were relatively small, experimental, and could only produce impractical facsimiles of the English language. This is obviously no longer the cas…

"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.

Re: Mathematics in the age of AI

#89

Post by a2ff6eeb0 »

cubefox wrote: post #58
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…

> If Amazon uses AI math to come up with better routin Most research mathematics is pure mathematics which is completely useless. No routing algorithms. It's only relevant because we (or at least mathematicians) are interested in it. So an AI producing incomprehensible proofs would be completely pointless. That's why Tao insists on the importance of human understanding.

If it's just a hobby, why would you use an LLM at all?

Re: Mathematics in the age of AI

#90

Post by fooker »

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…

Your analogy is great.

What happens when it's the cats who get to decide what's published?

Not an ideal scenario, but that's exactly the situation here. Mathematicians decide what gets reviewed and published in a a top journal.

In the long run, this can and should make journals obsolete.

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