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Tao: Open math problems being non-renewably mined by AI

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Re: Tao: Open math problems being non-renewably mined by AI

#281

Post by augment_me »

akoboldfrying wrote: post #262
augment_me wrote: post #204

Earlier quoted context omitted.

Well maybe its time to pivot from mathematics, and science as whole from personal attribution to being about progress of the field? Maybe your contribution to humanity as a mathematician is to find the right meaningful question to ask, and not to stamp your name on some fact?

Yes. But this is hard for mathematicians to stomach, because like everyone else, deep down in a place where they don't like to talk about at parties, they have egos and a sense of purpose based in part on demonstrating mastery of a technically difficult field, as well as social connections based on their participation in it, and taking all that away from them probably feels like a kind of death. The situation is not…

I totally understand and agree, I just feel like with the progress that we have in automating informational work, you are going to have an exponential amount of these "deaths" as new fields where humans can provide any kind of value get more and more short-lived.

At some point in my suggestion the machine will ask better questions than you, and that will be pointless as well, and you keep doing what you like doing, or you move on to something new. But if you keep tying your value to outcome and recognition instead of process you are going to have some incredibly depressing years ahead, and every time will just be as hard to stomach because of your ego.

Re: Tao: Open math problems being non-renewably mined by AI

#282

Post by marsten »

augment_me wrote: post #204
_alternator_ wrote: post #55

Earlier quoted context omitted.

Yes, and they will. But what's happening here is that the system that cultivates mathematics (and mathematicians) is recieving likely the biggest shock of its history. How do you reward merit and identify talen when people can't absorb the number of proofs being generated, much less understand them? Perleman's proof of the Poincare conjecture took several years for the mathematical community to digest; the proof of N…

Well maybe its time to pivot from mathematics, and science as whole from personal attribution to being about progress of the field? Maybe your contribution to humanity as a mathematician is to find the right meaningful question to ask, and not to stamp your name on some fact?

This is the direction of experimental particle physics and observational astronomy, where the budgetary scale at which progress occurs means we now fund these efforts at a societal level. These fields have graduated beyond "tabletop science".

For 3000 years mathematics has only been a "tabletop science". Even big programs like the classification of finite simple groups have been comprised of small teams chipping away at different (publishable) parts of an overall program.

This latest Navier-Stokes advance cost something like $22m in tokens, already well beyond what a mathematician's research grant can fund. As the easy open problems get mined, the cost of frontier progress will continue to climb. Some part of mathematics as a field will need to transition from tabletop science to big science: Coordinated top-down programs addressing high-priority objectives.

TBD is what the role of individual mathematicians will look like in a "big science" paradigm, but we could look to experimental high energy physics for ideas. For all practical purposes, AI converts math from a theoretical field into an experimental/observational one.

Re: Tao: Open math problems being non-renewably mined by AI

#283

Post by esteban0x »

david-gpu wrote: post #152

Aren't we in a similar position to what chess went through in the 2000s when Deep Fritz came out, and a desktop PC was able to defeat a reigning World Chess Champion? Did chess players just give up and stop playing? No, they didn't. They used these new chess engines to become better players. Computer programmers and mathematicians will probably go through something analogous. Presumably it is only a matter of time un…

Perhaps his takes are evidence of just usual human fear to new things. I see he relies quite much on the "community" or "social" aspects of the discussion.

I probably have delusional expectation of what a mathematician of his level should be talking about, but I expected from him a pure objective analysis on what to do with this new AI thing , what are its limitations, how it can improve the field and the creation of human knowledge, etc.

Re: Tao: Open math problems being non-renewably mined by AI

#285

Post by inkysigma »

monkpit wrote: post #234
blantonl wrote: post #174

Earlier quoted context omitted.

Why was there a prize attached to this problem then? What does humanity get out of this being proved?

This is my question too. If we are all just going “well that sucks” after AI solves this problem, why did anyone care about the problem being solved in the first place? Is the bummer that we got a solution we didn’t want - that navier-stokes is not always applicable or something, but we hoped it was?

I think the Navier Stokes problem kind of illustrates what he’s highlighting. I think most people even before AI expected that this would resolve in the negative and that you could get finite time blow up. There wasn’t really ever going to be a situation where the resolution to this question, or really any of the other Millenium Prize problems as far as I know, gives some kind of immediate massive practical feedback.

The hope with many of these problems in math is that in trying to prove that, we get some additional insight into why it blew up that could be applied elsewhere to more general PDEs that cannot be easily controlled.

I think the observation from Tao and many others is that when humans solved these problems, the additional insights into intuition and theory building came for free since humans can give expository on what they found hard or what was their own intuition. This is much more difficult or tedious to extract from an AI model. Even when people did have access to the chain of thought, it wasn’t always very helpful to figure out what was the exact thing that made it all click. This is even more difficult how that the CoT are hidden but I would think the sort of difficulty of extracting the key ideas for a human might be worse now with more advanced models.

There’s a long term aspect to this too where we have historically used these problems as markers for the other parts of mathematics but if AI can solve it all, then suddenly this signal is not very meaningful.

Maybe to bring it closer to home. If an oracle just gave you P \neq NP, then this would be generally uninteresting since this was already expected. There’s a deeper question of why that needs to be answered. However, one would hope that creating such a separation would give us tools that allow us to create lower bounds on a lot more problems we do care about and perhaps some bigger insight onto what makes a problem intrinsically hard or easy. These long term considerations are helpful but are definitely more vague. The remarkable part is that AI is separating the part about proving theorems and the “free” insight you get.

Re: Tao: Open math problems being non-renewably mined by AI

#286

Post by lanstin »

derektank wrote: post #102
torben-friis wrote: post #65

Earlier quoted context omitted.

Humanity is very biased for the culmination of work, considering everything that comes before and after busywork for the lower masses. Replicating a paper is just as valuable scientifically as publishing it, but how many careers advance through replication? If we move the goal from "find the solution" to "clear up the LLMs work" that doesn't bode well neither for the attractiveness of the problem nor for the career o…

>Replicating a paper is just as valuable scientifically as publishing it, but how many careers advance through replication? I don’t think this is true, especially for novel or unexpected results. I suppose it depends on what you mean by scientifically, and there is a debate in the philosophy of science about what the value of research even is, but a successful replication does not result in substantial updates to one…

It is true for mathematics certainly. I would guess it is less true for science per se.

Re: Tao: Open math problems being non-renewably mined by AI

#287

Post by dgellow »

gowld wrote: post #111
pitchlatte wrote: post #84

Earlier quoted context omitted.

his whole point is that specifically problems that have been held as important by consensus in the field are a finite resource. obvious example being the Clay millennium prize problems. seems like they function to shape the direction of future research into useful directions. which is to say, the process of developing a solution itself generates more useful problems. of course thrrr are tons of problems once you remo…

> the Clay millennium prize problems augmented Hilbert's problems of 1900. Surely mathematicians are creative enough to ask new questions? If not, then the next set of challenges will be to find questions to ask!

Did you read Tao‘s tweets? That’s what he addresses

Re: Tao: Open math problems being non-renewably mined by AI

#288

Post by nullbio »

There's nothing that AI won't be able to mine and accomplish (aside from being literally human), it's only a matter of hardware and scale at this point. Generalized problem solving is a factor of search efficiency over the problem space. The actual software part is all figured out, the only open questions are how to do things efficiently and what the trade-offs are from a hardware perspective, but if hardware paradigms are unlocked then efficiency becomes a secondary factor for the problems we care about. Why bother making an LLM twice as fast if you can make a chip that can process 100mil TPS, for example. You're already in a ballpark where it can do anything you want, with plenty left to spare.

The awkward part about all of this is that we're about to enter an age of extreme enslavement at the hands of the major tech companies if we do not focus on distribution of hardware and research, so that everyone can participate in the abundance and automate their daily lives. If we're beholden to frontier labs because they have hoarded all of the cutting edge hardware and we're left with overpriced scraps, we're collectively screwed. They will ensure a false economy is maintained so they can clutch onto a permanent class hierarchy of haves and have-nots and remain the key global decision makers. Automating hardware manufacturing is irrelevant if the hardware is not being distributed fairly, and is weighted to real scarcity instead of artifical scarcity.

Take Louis Vuitton for example. They can mass-produce their products for pennies, but they're artificially scarce and incredibly expensive. Imagine if ALL clothing was the price of LV. Now imagine this applies to every single thing you can purchase (or rather, rent - if some of these "elite" get their way), because they've cooked the economy and swallowed all industry. That's where we are headed if distribution and decentralization is not a priority for the world and we let labs like Anthropic pull off their regulatory capture stunts.

Re: Tao: Open math problems being non-renewably mined by AI

#289

Post by vkou »

dvt wrote: post #52

I'm with @nilesh on this one, and not exactly sure how merely the existence of a solution precludes the advancement of human knowledge. If a problem is "solved" (say, symbolically verified) without any insights gained, it doesn't seem very interesting to the profession. Navier-Stokes is a bit different (because there's a prize attached, so "scooping" matters), but almost all interesting problems don't have any prizes…

> and not exactly sure how merely the existence of a solution precludes the advancement of human knowledge.

You'd be more sure if you read the tweets.

Tao's point is very simple.

1. Working on problems that AI solvers can solve is a waste of human time.

2. We have no idea which problems can be solved by AI solvers...

3. ...Because the AI labs are keeping their negative results secret, and don't tell us which problems they've tried and failed to solve, and why they've failed to solve them (or succeeded at solving others).

There are additional points surrounding it, but that is the thrust of his argument. His issue is not the existence of AI, but the anti-scientific secrecy in how it is used to solve problems. All the incentives around its current use result in closed, uncollaborative work - which while very attractive to a vulture capitalist, is anathema to scientists.

---

He also posits that having a solution to a problem is a small part of the value of solving a problem. What the AI labs are doing is the equivalent of a student turning in their homework, which has 100% of the right answers, but with none of the 'show your work' steps. Those steps are a critical artifact for doing mathematics, because the process of solving a difficult problem teaches us things about other problems.

Re: Tao: Open math problems being non-renewably mined by AI

#290

Post by [deleted] »

johnsmith1840 wrote: post #164
gpm wrote: post #157

Earlier quoted context omitted.

Mathematicians provide two complementary services bundled together. 1. Proving theorems - what AI can apparently replicate faster and better. 2. Creating definitions and new theorems from those definitions to prove, selecting which of the possible statements to work on. I.e. developing the "language" of mathematics. So far there is no evidence that LLM can do this at all well. And there's some reason to think that ma…

Fair take. So I totally agree if AI also cannot do the second part better than a person. Honestly though, I wouldn't want to take that bet. I never thought that the first thing AI would become super human AGI like is math. You ask me 10years ago and I'd think the opposite. I think we all would have said we'd have super human HR employees before a super human mathematician. But here we are.

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