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Mathematics in the age of AI

arxiv.org

121–130 of 292 posts

Re: Mathematics in the age of AI

#121
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.

Math isn't "cooperative". Math is about truths. The length of circumference. The area of a triangle. The formulas for these are true in an objective sense irrespective of whether you understand them. That said, without understanding, Math can't evolve. Comprehension of a proof is very important, but not what Math is fundamentally about. Computer programs are Math. You can use them without understanding how they work.

this is primitive understanding of math. what does "truth" mean here? usually arguing over definitions is something i hate, but that's the whole point of mathematics.

it starts as a tool for humans, then evolves into a set of interesting properties of those tools, then grows into an art form, a set of "games" where cooperation is half of the point. the other half is discovering beauty in this weird parallel world of our reasoning and imagination. once tools become autonomous and start making up their own games we can't even play then mathematics loses it's meaning as a discipline. the only retort you can come up with is that "it's going to be useful". how would you know? because your autonomous tool that's too smart for you told you so? they could be as useful as morning orange juice to Claude Shannon was in terms of inventing information theory. I.e. you drinking it won't make you any closer to inventing anything of the sort anytime in your lifetime.

do triangles exist IRL? is the world discrete or continuous? can you prove it? If you have an answer to all of those I know you're wrong.

also in your computer program example just shows you don't understand it at all. those programs ARE NOT understood by you, but someone else who built them did. someone who bothered to read and architect it did. The whole Google codebase might be incomprehensible in its totality if you go bottom up but it is comprehensible by construction by us. Same with math. You don't understand all the bits of it, but someone built every brick and so you know it is "true". once the bricks become black boxes you're screwed.

Re: Mathematics in the age of AI

#122
post #100

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…

>This is obviously no longer the case today. Advancement in capability does not mean the mechanism is the different. The LLM name denotes a very specific mechanism..

>The LLM name denotes a very specific mechanism..

No, not really. This is just the term that stuck around. The "large" is now up to five orders of magnitude larger and "language model" has gone far beyond any simple notion of modeling a singular natural language. And anything you'd cite about transformers, or tokens, or autoregression, etc., is more of a factoid about what works best and happens to be the most convenient in the here and now. I see all of this as an unbroken continuation of work that's been going on since the 1940s.

Instead of trying to play word games, why can't you just read Tao's article?

Re: Mathematics in the age of AI

#123
post #102

Earlier quoted context omitted.

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!

We can meaningfully talk about (1) the existence of a field and (2) the said field hitting its stride. The title of Tao's article denotes the latter.

And oh, what a stride it is: https://vibemathed.com/stats

Re: Mathematics in the age of AI

#124
post #97
post #32

Earlier quoted context omitted.

For an exhaustive search, if you can explain to me: - how to exhaustively list the cases that need to be checked, and why that method is exhaustive - how to check each case, and why that works and then conclude with "we've had a computer do this exhaustive search, and the result came up as X", for me that satisfies completely understanding the proof.

But the "computer" is magic, to you. I could prove anything by claiming I completed a trivial-to-explain exhaustive search. The only support or refutation would be someone doing their own search. It's a very weak foundation. We already had the ABC conjecture crisis: A theorem with a human-written proof so complex that no one besides the author can understand it. Some people claim to have refuted it. Most mathematicia…

If you prove that the theorem prover’s true and false determinations are correct—in the cases in which it can make them—then Bob’s your uncle.

Re: Mathematics in the age of AI

#125

Earlier quoted context omitted.

I think it's impossible to be half in. AI will eventually be better at things than people, and people will simply be rocks in the gears of progress. The only thing to do is to be all in, or get run over.

If what you’re saying is true, what’s the use (or even meaning) of being “all in”? You’re a rock either way.

It's not there yet, and every move we're currently making is towards an extremely inequitable future. Unless things change, not everyone is going to benefit from AI. What are you doing today to end up on the team that wins?

Once AI starts under its own direction, human brains won't be competitive. It feels like we're a breakthrough or two away, and with trillions of funding, we'll get there. Every dollar we spend on Claude subscriptions gets us closer.

Re: Mathematics in the age of AI

#126
post #78

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.

I don't think Lean is as rigorous as you implied here.

https://en.wikipedia.org/wiki/Collatz_conjecture#In_proofs_o...

> In July 2026, a disproof of the Collatz conjecture was verified not only by Lean, but another formal verification system Nanoda. However, investigation quickly revealed that the proof exploited bug(s) in these verifiers.

Re: Mathematics in the age of AI

#127
post #83
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 is also useful. If someone showed that p = np tomorrow in a formally verified proof I don't care if no one can understand it.

Re: Mathematics in the age of AI

#128
post #107

Earlier quoted context omitted.

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…

It sounds like the idea is to turn on a math generator and keep running it until it generates something interesting. And it might be fun to try it. But if it’s too much output to read and we don’t understand the output either, how does anyone recognize when it’s done something that’s practically interesting? The output might make a cool screen saver as-is, but we probably need a way to evaluate it somehow.

This is relevant, but only after AI has solved all the open problems including Millenium problems. Until then, as AI keeps solving harder open problems, people will pay attention and be interested.

Re: Mathematics in the age of AI

#129
post #110
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.

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, the…

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Re: Mathematics in the age of AI

#130

Terence Tao's quote about AI's math proofs is relatable outside of pure math: "the writing very often dwells at length on trivialities while passing briefly through — or even actively obscuring — the most interesting and novel portions of the argument."

That's also true for regular math proofs.

No one talks about why the proof works, but they will happily spend thousands of pages explaining how it works.

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