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A misalignment of AI in mathematics

mathandai.org

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Re: A misalignment of AI in mathematics

#401

Post by matco11 »

tmhn2 wrote: post #200

As a mathematician maybe I am a little more optimistic than this declaration. I am thinking of Mochizuki's abc conjecture: He worked in relative isolation, and dumped a huge incomprehensible proof on the community (to oversimplify a bit). That's not totally unlike what might happen if AI generates a huge, incomprehensible proof of let's say RH. Well, what is the result? In the Mochizuki case, it was a lot of skeptici…

I get excited at the idea of a world in which advanced mathematical problems (and their solutions) become much more accessible to a much greater number of people. As a result, making mathematics much more loved at a societal level.

Imagine a world where these most complex mathematical problems are not accessible to a few hundred people, but a few hundred thousands people. ...Those original few hundred gifted mathematicians would have an even more prominent role, and their names and achievements would be known by orders of magnitude more people that they are now.

Re: A misalignment of AI in mathematics

#402

Post by Sharlin »

soerxpso wrote: post #390
omnicognate wrote: post #201

Earlier quoted context omitted.

The statement is not about AI but about the behaviour of AI companies. OpenAI have put vast resource into solving open maths problems: many millions of dollars of compute just on the Navier-Stokes result, plus whatever they spent on the broader Millenium Prize problems initiative and the other results they have published. Anthropic are doing the same. The statement is asking them to stop doing this. AI companies are…

Are they claiming that the only value in solving these problems was for their field's personal development process? I thought Navier-Stokes (and some of the other millenium prize problems) actually had implications for useful technology. It would be insane to demand that people avoid making progress on technology that can save lives or improve general quality of life, just to protect the sanctity of your karate belt…

No, there's most likely zero practical benefit of having found a pathological edge case in which the Navier-Stokes equations do not work. We're most assuredly not talking about "saving lives" here. Unless advanced aliens show up and tell us they'll destroy Earth unless a counterexample to the N-S equations is provided within 24 hours.

Re: A misalignment of AI in mathematics

#403

Post by ggm »

To me, the "meaning" of proof is twofold. First is the understanding which is completely absent from a proof which depends on exhaustive iterations of instances or is asserted by fiat from myriad individually contestable paths. That's what I think makes AI proofs risky: they absence of understanding.

The second is the utility. We get to do navigation because the maths about angles and spheres checks out. The social utility downstream of AI proofs may be huge.

I don't like this disjunction.

Re: A misalignment of AI in mathematics

#404

Post by soerxpso »

yzydserd wrote: post #6

> solving problems is only a tool and proxy for achieving the primary goal of conceptual understanding and insight. This is the effect of AI on most intellectual disciplines, and it’s a real worry.

The existence of solutions to problems does not prevent you from solving the problems yourself anyway, if your goal truly is personal development. You're free to go solve Navier-Stokes yourself right now. You're free to manually do all of the AI work in any field, actually. You won't get grant money or prestige (which are not a part of conceptual understanding and insight) but you will get all of the conceptual understanding and insight you're after. You have not been deprived of it.

Re: A misalignment of AI in mathematics

#405

Post by andai »

So what's the actual point here? It's too fast and we can't keep up?

Isn't that just a function of the technology itself, and the same problem being faced by every other field? And going to get exponentially "worse" every year!

Or is the issue that they're bad at explaining things, in a way that produces actual learning? (e.g. AI is amazing for learning but the net effect on students so far appears to be negative.)

Re: A misalignment of AI in mathematics

#406

Post by newprint »

The level of things happening in the last several months is just SciFi level, especially last two weeks. 1. Overlords of AI threatening to destroy someone if their demands aren't met over the Millennia problem. Proving the point that given the chance, owners of the AI companies will immediately use their power to squash or control other people. (See Butlerian Jihad) 2. At least for me, the Exponential progress didn't sound true until this week. If AI indeed solved all those problems and proofs and solutions are correct, we went from "Write me English 101 college essays" to "Help me solve the Navier–Stokes existence and smoothness problem" in a just 3-4 years. 3.Looks like someone's work might have been stolen, by AI company and there are strong evidence. See point 1. 4. War between AI & AI companies. 5. It produces a machine verifiable LEAN proofs for the hardest problems know to us !!! Proving that point that best use of AI so far is to create verifiable systems. I.e the absolutely chaotic AI creates verifiable (orderly) proofs. It is Maxwell Demon: chaos to order.

Re: A misalignment of AI in mathematics

#407

Post by SJMG »

globular-toast wrote: post #301
oytis wrote: post #184

Earlier quoted context omitted.

It has been best at chess for quite a while. Yet everyone knows who is Magnus Carlsen, even though at no point of his career he was stronger than the machine

Magnus Carlsen plays fellow humans at the game because people still care about human competitions. What motivation would a mathematician have for solving already solved problems by hand? Can you imagine someone spending years working on a proof for an already-proved theorem just in case it leads to new insight?

Of course not. I'm so tired of the chess analogy. It was always a game and it was always performative.

Research is fundamentally different. We are seeing the erosion of specific needs for thinking at depth. AI is the automobile for the mind. There will 100% be undesirable consequences and selective atrophy of cognitive abilities once prized. This is a loss. There's no getting the cat back in the bag at this point so long as the objective dimension of work, as in object opposed to subject, is held as the most important.

SWEs felt this same crisis late last year. Now it's the mathematicians. They won't be the last.

Re: A misalignment of AI in mathematics

#408

Post by grufkork »

bpodgursky wrote: post #186
grufkork wrote: post #47

Earlier quoted context omitted.

Going to the gym and using a forklift

There is an implicit agreement that mathematics is funded, for the most part by the public, as a way to advance the state of knowledge and propagate (even if very indirectly) what has learned for the public good. It was never about helping individual mathematicians demonstrate that they are individually good at math. It happened to work out that way, but it wasn't the goal.

Agree, my comment is rather about the fact that most work you do in school and (lower) university is not itself of interest, rather the purpose is to work out your brain. The more of your exercise you leave to the forklift, the steeper the cliff will be when you find a weight that doesn’t fit on forks. (That is, unless/until machines get good enough to completely replace people, at which point we have another few problems on hand, such as day-to-day purpose and justification of existence)

Re: A misalignment of AI in mathematics

#409

Post by HarHarVeryFunny »

k2xl wrote: post #149

I’m sympathetic to the concern, but I’m still unclear on what the concrete ask is. If the worry is that AI companies are turning open problems into benchmarks and potentially “using up” fertile mathematical problems before humans can develop the ideas around them, what exactly should the companies do differently? Also why does discovering the answers preclude humans developing ideas from them? I don't get why solving…

> If the worry is that AI companies are turning open problems into benchmarks and potentially “using up” fertile mathematical problems before humans can develop the ideas around them, what exactly should the companies do differently?

Just stop doing that. Don't treat unsolved math problems as some cheap benchmark to beat.

Leave the math for mathematicians, and let them use AI in a way that helps the field, not in a way that harms it.

Re: A misalignment of AI in mathematics

#410

Post by GPerson »

alkyon wrote: post #366
tzumaoli wrote: post #234

Earlier quoted context omitted.

Is this the scenario described in Ted Chiang's short story https://en.wikipedia.org/wiki/The_Evolution_of_Human_Science where scientists are "catching crumbs from the table" trying to decipher the results generated by superhuman intelligence?

It's still an optimistic scenario. Artificial superintelligence may develop hypermathematics of a kind that never will be accesible to human mind, enhanced or not. One can't teach geometry to ants even if you put them on a Moebius strip. It would be more like Lem's novel where it completely disappears from the human horizon: https://en.wikipedia.org/wiki/Golem_XIV

What’s optimistic or non-optimistic specifically about the machine having a system of mathematics beyond our comprehension within it? Why should we care about that in itself?
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