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The cultural divide between mathematics and AI

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Re: The cultural divide between mathematics and AI

#181

Post by pathsjs »

coffeeaddict1 wrote: post #78
m0llusk wrote: post #18

> The last mathematicians considered to have a comprehensive view of the field were Hilbert and Poincaré, over a century ago. Henri Cartan of the Bourbaki had not only a more comprehensive view, but a greater scope of the potential of mathematical modeling and description

I would also add Grothendieck to that list.

Grothendieck did not have a comprehensive view of mathematics, nor he ever claimed to. There are vast swathes of mathematics (e.g. PDE or probability) that never fell under Grothendieck's radar

Re: The cultural divide between mathematics and AI

#182

Post by nicf »

oersted wrote: post #179
xg15 wrote: post #171

Earlier quoted context omitted.

Ah, thanks for the clarification. Then the whole thing makes a lot more sense - though I'd say the outlook also becomes more optimistic. I thought the rhetoric sounded somewhat like the AGI/accelerationist folks who postulate some sort of eventual "godlike" AI whose thought processes are somehow fundamentally inaccessible to humans. So if you had a proof that was only understandable to this sort if AIs, then mathemat…

There's an argument to be made that incomprehensible proofs are still very useful. I understand that mathematicians are all about understanding why a theorem is true, not just checking that it is, and then using that understanding to push forward other research. Same as most scientists care more about understanding how the world works rather than developing methods to control and manipulate it. But proven theorems ar…

Incomprehensible proofs are indeed still useful to some extent, and I don't think you'll find many mathematicians who would reject them as an answer to the binary question of whether the result is true.

But when you talk about "getting a lot more done," I want to ask, get a lot more done to what end? Despite what mathematicians sometimes write in their grant applications, resolving most of the big open problems in the field probably won't lead to new technologies or anything. To use the Riemann Hypothesis example again, most number theorists already think it's probably true, and there are a lot of papers being published already which prove things like "if the Generalized Riemann Hypothesis is true, then [my new result]".

No one is really waiting around just for the literal, one-bit answer to the question of whether RH is true; if we got that information and nothing else, I'm sure number theorists would be happy to know, but not a whole lot about the work being done in the field would change. It's not just being "satisfying to the curious"; virtually the entire reason we want a proof is to use the new ideas it would presumably contain to do more mathematics. This is exactly what's happened with the proof of the Poincare Conjecture, the only one of the Millennium Problems that's been resolved so far.

This is what I was lamenting in my comment earlier: the thing you're describing, where we set proof-finding models to work and they spit out verifiable but totally opaque proofs of big open problems in math, very well might happen someday, but it wouldn't actually be all that useful for anything, and it would also mean the end of the only thing about the whole enterprise that the people working in it actually care about.

Re: The cultural divide between mathematics and AI

#183

Post by nicf »

zmgsabst wrote: post #173
nicf wrote: post #119

Earlier quoted context omitted.

This is actually a metaphor I've used myself. I do think the woodworking community is both smaller and less professionalized than it would be in a world where industrial furniture production didn't exist. (This is a bizarre counterfactual, because it's basically impossible for me to imagine a world where industrial furniture production doesn't exist but YouTube does, but like pretend with me here for a moment.) I don…

That’s no different than high-end artisanal woodworking, which I would argue is comparable to self-directed, cutting-edge research: - apprenticing - journeyman phase - only finally achieving mastery CNC never replaced those people, rather, it scaled the whole field — by creating much higher demand for furniture. People who never made that full journey instead work at factories where their output is scaled. What was d…

Woodworking is very far from my world, so I don't really have any grounds to judge how comparable the two things actually are. I'll say two things instead.

First, right now presumably the reason a few people still become master woodworkers is that their work is actually better than the mass-produced furniture that you can get for much less money. Imagine a world where instead it was possible to cheaply and automatically produce furniture that is literally indistinguishable from, or maybe even noticeably superior to, anything a human woodworker could ever make. Do you really think the same number of people would still spend years and years developing those skills?

Second, you've talked about business logic and "math experts at the company" a few times now, which makes me wonder if we're just referring to different things with the word "mathematics". I'm talking about a specific subset, what's sometimes called "pure math," the kind of research that mostly only exists within academia and is focused on proving theorems with the goal of improving human understanding of mathematical patterns with no particular eye on solving any practical problems. It sounds like you're focused on the sort of mathematical work that gets done in industry, where you're using mathematical tools, but the goal is to solve a practical problem for a business.

These are actually quite different activities --- the same individuals who are good at one stand a decent chance of being good at the other, but that's most of what they have in common, and even there I know many people who are much more skilled at one than the other. I'm not really asking anyone who doesn't care about pure math to start caring about it, but when I'm talking about the effect of AI on the future of the field, I'm referring specifically to pure math research.

Re: The cultural divide between mathematics and AI

#184

Post by xg15 »

nicf wrote: post #182
oersted wrote: post #179

Earlier quoted context omitted.

There's an argument to be made that incomprehensible proofs are still very useful. I understand that mathematicians are all about understanding why a theorem is true, not just checking that it is, and then using that understanding to push forward other research. Same as most scientists care more about understanding how the world works rather than developing methods to control and manipulate it. But proven theorems ar…

Incomprehensible proofs are indeed still useful to some extent, and I don't think you'll find many mathematicians who would reject them as an answer to the binary question of whether the result is true. But when you talk about "getting a lot more done," I want to ask, get a lot more done to what end? Despite what mathematicians sometimes write in their grant applications, resolving most of the big open problems in th…

Yeah, I imagine in those situations, an AI proof that the conjecture is false would probably be more interesting and useful than a proof that it is true.

A proof of the conjecture would essentially just move the situation from "we think there could be counterexamples, but so far we haven't found any" to "there really are no counterexamples anywhere, you can stop looking". The interesting thing here would be the explanation why there can be no counterexamples, which is exactly the thing that the proof wouldn't give you.

On the other hand, a counterproof would either directly present the community with a counterexample - and maybe reveal some interesting class of objects that was overlooked so far - or at least establish that there have to be counterexamples somewhere, which would probably give more motivation to efforts of finding one.

(Generally speaking here. I'm not a mathematician and I can't really say anything about the hypotheses in question)

Re: The cultural divide between mathematics and AI

#185

Post by nicf »

xg15 wrote: post #184
nicf wrote: post #182

Earlier quoted context omitted.

Incomprehensible proofs are indeed still useful to some extent, and I don't think you'll find many mathematicians who would reject them as an answer to the binary question of whether the result is true. But when you talk about "getting a lot more done," I want to ask, get a lot more done to what end? Despite what mathematicians sometimes write in their grant applications, resolving most of the big open problems in th…

Yeah, I imagine in those situations, an AI proof that the conjecture is false would probably be more interesting and useful than a proof that it is true. A proof of the conjecture would essentially just move the situation from "we think there could be counterexamples, but so far we haven't found any" to "there really are no counterexamples anywhere, you can stop looking". The interesting thing here would be the expla…

Yeah, that's definitely right --- an explicit counterexample to the Riemann Hypothesis would be very surprising and interesting, and I think that would be equally true no matter whether it was found by a person or a computer! The situation that would be mostly unhelpful is a certificate that the result is true that communicates nothing about why.

Re: The cultural divide between mathematics and AI

#186

Post by SwtCyber »

sigmoid10 wrote: post #132
SwtCyber wrote: post #130

If AI-generated proofs become incomprehensible to humans, do they still count as -math- in the traditional sense?

We already have proofs by exhaustion that could only ever be verified using computers. Some people would argue they are not "elegant" but I don't think anyone would argue they are not math.

But I wonder if there's a distinction between a proof that is merely computationally intensive and one that is conceptually inaccessible

Re: The cultural divide between mathematics and AI

#187

Post by jebarker »

zmgsabst wrote: post #175
jebarker wrote: post #153

Earlier quoted context omitted.

> why would they be replicating it if they didn’t want it? Who is "they"? Most AI for math work is being done by AI researchers that are not themselves academic mathematicians (obviously there exceptions). Similarly, most AI for music and AI for visual art is being done by AI researchers that themselves are not professional musicians or artists (again, there are exceptions). This model can work fine if the AI researc…

> This model can work fine if the AI researchers collaborate with mathematicians or artists to understand that the use of AI is actually useful in the workflow of those fields, but often that doesn't happen and there is a savior-like arrogance where AI researchers think they'll just automate those fields. In my experience, the vast majority is people who are hobbyists or amateurs in those fields, who are looking to i…

I think we're actually in agreement with respect to the needs of AI in music and graphics. When I say "AI researchers" I'm talking about people in industrial and academic labs. I consider the hobbyists you describe to be the users (who obviously can also be researchers). It's the desires/needs of the latter that should drive the research agenda of the former.

I actually don't understand your position here or what you think I'm arguing for. My point is that the real musicians, artists and mathematicians (whether they're hobbyists, academics or professionals in industry) are not well served by detached AI researchers just trying to automate their work for them. They need AI researchers to understand their workflows and build tools that elevate them, i.e. bicycles for the mind (or hands?).

Again, I do recognize there may be new fully automated workflows that can come out of AI research too but I maintain that the actual artists, musicians and mathematicians today have a valuable role in guiding that development too.

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