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Mathematics in the age of AI

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Re: Mathematics in the age of AI

#191

Post by glimshe »

Jensson wrote: post #185
glimshe wrote: post #166

Earlier quoted context omitted.

I think Hardy would be very much on the same page with me, as well as Godel and many others. Mathematical truths exist independently from our feelings and processes to discover them. People do and should argue about which truths are interesting to pursue and refine, but all of them are out there to be discovered... or not. Whatever philosophy you prefer, math is about establishing objectively valid logical results, c…

Axioms don't exist independently from our feelings and processes, we pick axioms we feel are good, and axioms defines mathematics. Mathematicians even argue which axioms we should have, it isn't objective in the slightest, mathematics is therefore very closely linked to our feelings and intuition. Remove that and you just have formal logic, a very different field.

Is formal (aka "mathematical") logic part of mathematics? Now that's a philosophical question.

From my perspective, I feel you restated what I said with the opposite conclusion. You say that axioms "defines" mathematics. If I were Claude, I'd say that the word "define" is doing a lot of work, is load bearing or something like that.

"Define" is where we turn these axioms into consequences - what I call "truth". As opposed to all the other stuff people could say that don't follow from these axioms. These are nonsense and, most certainly, un-mathematical.

Re: Mathematics in the age of AI

#192

Post by ziiinq »

GPerson wrote: post #137
raincole wrote: post #118

Earlier quoted context omitted.

Math is not about truths, at least not by the meaning of "truth" as a word in daily use. Math has been almost purely arbitrary since ~ late 19th/early 20th century. There are uncountably many correct mathematical theorems. Almost all of them can't even be written in symbols. Even if you have a tape with infinite length (which is already longer than the whole physical universe!) filled with theorems, they are still on…

The tastes and interests, and cultural pretext, of human beings are not arbitrary just because they’re not easily described.

Yes and therefore?

You seem to be vaguely waving in the general direction of a point, without making any concrete claims or bothering to engage with the GP’s argument.

The tastes and interests of humans are absolutely not arbitrary. They are dictated by fate, the sun and the moon gods. Or maybe by the unitary evolution of the universe’s quantum state. Or by the probability distribution of finding ourselves in a particular branch of the universe.

What bearing does this have on whether math is a collaborative endeavor?

And what is unique to math, that your argument wouldn’t apply equally to physics, sociology or financial markets? All, “truth seeking” disciplines.

Re: Mathematics in the age of AI

#193

Post by a2ff6eeb0 »

Jensson wrote: post #183
a2ff6eeb0 wrote: post #178

Earlier quoted context omitted.

sure, agreed, but once you have a proof, you can probably get AI to reduce other problems to that problem in P.

Why wouldn't you be able to do that without a proof? I don't see the value of the proof here, just ask the AI to solve the problem you want and the proof isn't needed.

From my understanding, the two are equivalent; if you can reduce an NP problem to a P problem, you've proven P=NP. The rest is application.

Re: Mathematics in the age of AI

#194

Post by czgov »

Jensson wrote: post #190
czgov wrote: post #187

Earlier quoted context omitted.

You have an unfortunate view of the value of knowledge. The frontiers of science would be static if everyone believed as you do.

The whole point why anyone cares about these proofs is that the things we learn as we make the proof might add value, proving p = np itself isn't interesting, that knowledge has no application and therefore no value in itself. I have published mathematics so I do value knowledge, but for most of mathematics the value of the knowledge isn't the thing you try to prove it is all the things you learn as you try to prove…

You are wrong as far as most mathematicians believe. The fact is important. The proof of the fundamental theorem of algebra is interesting and important but the theorem itself is also important.

For what? Which product becomes better if it is correct?

This sentiment is anti-thetical to the whole point of pure math and theoretical science. No product became better when Euler proved the fundamental theorem of algebra.

Re: Mathematics in the age of AI

#195

Post by brador »

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.

This artificially limits mathematics to the limit of human ability.

It should be ignored and refused.

Re: Mathematics in the age of AI

#196

Post by butwaitTheres4 »

unified101 wrote: post #165
butwaitTheres4 wrote: post #160

Earlier quoted context omitted.

Tao has yet to produce work that outshines those whose work he studied and memorized. Not worth the reverence merely being a VHS copy of history. He's a typical person otherwise, politically aware of how he barters for food; until proven otherwise this can be seen as little more than social moat defense. To paraphrase a quote attributed to Upton Sinclair; hard to get a worker to understand something when their payche…

wtf?! Tao is the only mathematician I can name, and widely considered the foremost living one.

Says more about your own effort to learn math than Tao's ability.

Re: Mathematics in the age of AI

#197

Post by customguy »

czgov wrote: post #187
Jensson wrote: post #176

Earlier quoted context omitted.

> It would be very useful to have an oracle tells us whether or not RH is correct. For what? Which product becomes better if it is correct?

You have an unfortunate view of the value of knowledge. The frontiers of science would be static if everyone believed as you do.

Someone claimed it would be "useful", without saying what for. Hence the questions "what for?". To try to shame people for that question in the name of science of all things is wild.

Re: Mathematics in the age of AI

#198

Post by black_knight »

a2ff6eeb0 wrote: post #89
cubefox wrote: post #58

Earlier quoted context omitted.

> 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?

It is not a hobby when you are paid to do it! But I take it you mean “Done for the art of it”. Which I guess is a concept foreign to many.

A few different reasons why use an LLM when mathematics is done for its own sake:

Formally verifying my proofs catches any mistakes I make, but verifying is also hard work. LLMs shaves off a lot of time when formally verifying a proof.

I can still read through an LLM generated proof and understand it. This is a way for me to understand the result I am working on (usually in order to know what to prove next, results are not proven in a vacuum).

My experience thus far is that, while correct, an LLM generated proof is often unnecessarily complicated or inelegant. I take pleasure in elegant proofs and will spend time iterating on the first proof until I find it conveys the idea in the most elegant way. Having the initial LLM proof to start with is really useful, but is thus far rarely the final product.

Re: Mathematics in the age of AI

#199

Post by unified101 »

butwaitTheres4 wrote: post #196
unified101 wrote: post #165

Earlier quoted context omitted.

wtf?! Tao is the only mathematician I can name, and widely considered the foremost living one.

Says more about your own effort to learn math than Tao's ability.

You should definitely inform the wikipedia editors as well https://en.wikipedia.org/wiki/Terence_Tao#Recognition .

(I don't know why you're so butthurt BTW - neigher of your ad-hominem comments actually outline your concern)

Re: Mathematics in the age of AI

#200

Post by js8 »

HWR_14 wrote: post #164
noosphr wrote: post #127

Earlier quoted context omitted.

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.

If a magic oracle tells you p=np, that's useless. How would that change anything?

Oh it would change a lot. It would be an enormous psychological boost for everyone to find a practical algorithm.

In any case, I think it's better to read PP as somebody would find a practical, albeit incomprehensible, algorithm for solving NP complete problems.

Although I probably disagree with PP, because even a candidate algorithm that mysteriously works without proof would have practical value, so this case is not predicated on proving.

I think a better example of genuinely practical but rather uninteresting (YMMV) mathematical proofs are proofs of convergence of numerical methods, FEM for example. (I have been through it in school, it was a torture.)

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