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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

#231

Post by imawakegnxoxo »

johnsmith1840 wrote: post #155
gpm wrote: post #144

Earlier quoted context omitted.

> Unless [...] mathematicians are effectively useless? It's always been a bit bizarre that this isn't the case. Mathematicians are almost always working on problems that there is no good reason to expect to have utility in the real world... problems they selected because of their elegance or whatever... yet there is a strong historical trend of their work having huge importance after the fact. Sometimes in fields tha…

So solving and discovering math is or is not the core value a mathematician provides? If AI can perfectly replicate their work but faster and better then what? SWE have nobody crying for them as they've been massively disrupted.

Think of it like software going from programmers understanding every instruction, knowing where every byte of memory was being used and why, and using this knowledge to build optimised systems

During the process of optimising and understanding the programmer might learn something new or have some kind of "aha" moment of insight that might lead them down a new path of study where fantastic new technologies and capabilities can be realised

Fast forward to 2026

Most web pages take several seconds to load

Applications crash often for no apparent reason

A vast majority of programmers have no idea what their applications are actually really even doing anymore, so they stack bloat on top of bloat and if something breaks, well I guess that's someone elses problem cos I have no idea what's going on anymore

There's something to be said about levels of abstraction being useful, but abstracting away understanding of the task itself is not the path to generating useful knowledge or applications for humanity

We might be gaining the "what" but we are losing the "why" and the "how" and these are generally fundamentally more important

The answer is 42 but what is the question?

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

#232

Post by vikramkr »

torben-friis wrote: post #65
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…

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?

A lot. In fields where knowledge is incrementally building on previous work the reason the whole field hasn't collapsed from the replication crisis is that usually the results that are really high impact are replicated in as an initial step in new research building on it. It's almost never the focus of the paper but you'll often find a quick mention in methods/supplemental of some previous work that was verified to be valid by a replication of a key technique etc. you'll have crisis where old tools are found to be problematic and findings end up revisited etc. Plus fields like clinical research where there's an awful lot of focus on replicating findings using staged clinical trials with increasing statistical power to determine if new interventions work - that's driven by regulatory requirements grounded in good science and a lot of people make careers in just that.

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

#233

Post by kevinmchugh »

Retric wrote: post #148
derektank wrote: post #102

Earlier quoted context omitted.

>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…

From a pure statistical perspective the first scientific paper shouldn’t update your beliefs as much as the independent replication study. People don’t behave this way, but a high percentage of all papers have known flaws and that goes up even higher when you consider unknown flaws. Replication doesn’t own its own solve the underlying issue, but independent replication removes a huge range of potential issues on top…

I agree, with the caveat that it probably matters how novel the paper is, the effect size, and the confidence interval. A reputed new psychological phenomenon I'd be more skeptical of than, idk, a newly detected exoplanet.

That alleged superconductor from a few years ago - everybody kind of held their breath and waited for the reproduction.

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

#234

Post by monkpit »

blantonl wrote: post #174
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…

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?

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

#235

Post by pixl97 »

srcreigh wrote: post #100
ModernMech wrote: post #38

Earlier quoted context omitted.

If math is solved then move to an area that’s not. Why do fields need “protecting” from ai?

Math can never be fully solved by a computer, if only for lack of computational resources.

Math can never be solved, depending on exactly what set of axioms you're using. Certain problems can be though.

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

#236

Post by porridgeraisin »

meken wrote: post #213

I don’t see why it makes a meaningful difference if a human solves a math problem versus AI - it seems like the same amount of understanding will come out in the end. Either the understanding will come from humans arriving at the proof in the former case, or the understanding will come from humans understanding the proof that the AI came up with in the latter.

Because the problem has almost no value unto itself. The clay statement of navier stokes is not relevant to how CFD is done in practice.

It's about what is non verifiable versus verifiable. The same way it produces "slop" code (which, if you give it test cases, will be 100% correct), it also produces "slop" math.

Code that serves a business function, it's ok if its slop. Math that serves directly a business function also can be slop.

But most open problems are not directly for a particular usecase. People agree widely to attack it due to the perceived possibility of encountering useful mathematical objects along the way, that will then expand the world's mathematical toolset. This is not something that you can easily express in a verifier, and is thus something that is hard to force an LLM system to do.

You are right in that understanding it retrospectively is possible, but that is not going to be as useful as the desired "elegant" objects that expand and unify mathematics. You can't represent these concepts in verifiers.

Again, if you let AI rip at something like say "beat shannon capacity" and suppose it comes up with MIMO as paulraj did, great! It's useful and you can retrospectively understand it, say by expanding shannon to multiple dimensions, as foschini and telatar did. But most math problems are not in that category.

The question then is, if AI is really good at this type of math, how much of the existing mathematical community+process is necessary? I think it will still be necessary, just maybe in fewer cases. Wherever the primary purpose of the math is in a domain and that domain has a verifiable target, we can directly optimise it to that verifiable target in-domain rather than reach for the mathematical community. How well will this work? We'll see. It's not clear if it's even possible to represent most problems this way.

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

#237

Post by itissid »

vessenes wrote: post #3

That’s not untrue. But it’s also a misstatement of mathematical history. Many leading mathematicians historically have been highly competitive — Gauss comes to mind. Woe betide the lesser intellect that sent Gauss some ideas. The Newton Leibniz controversy was very serious business at the time in the UK and the continent. It was considered at the least a sin to reveal that sqrt(2) was irrational to those outside Pyth…

Yeah but a highly productive last two decades of math research from https://en.wikipedia.org/wiki/Polymath_Project has come from collaboration.

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

#238

Post by silver92bullet »

I think this highlights one of the fundamental differences between humans and our current AI systems. They can still only try to solve problems in the given well defined parameters they are given (with some exceptions). The human is able in the effort to solve problems to intuit where there may be new interesting problems adjacent to the current problem.

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

#239

Post by pixl97 »

vouaobrasil wrote: post #46
ModernMech wrote: post #38

Earlier quoted context omitted.

If math is solved then move to an area that’s not. Why do fields need “protecting” from ai?

The reason is because the entirety of society, historically, has been based upon humans using their differential skills to further it, which in turn promotes societal cohesion. If most human endeavours are solved, then we will enter a period of abundance that paradoxically will erode the glue holding society together. In short, endless abundance of solutions and ideas cannot coexist with a healthy society. Only those…

I mean the past 10 to 20 years of social media have shown a lot of societies glue already breaking.

A society can live just fine in a period of abundance. The societies that we currently have on earth do make it questionable of 'we' can right now. I mean I see people posting stuff like "I'd rather burn it all to the ground rather than see one cent more tax" kind of stuff when they have millions. That kind of person doesn't want more people uplifted and it takes away from their idea of being special.

>naive belief in a Star Trek

The naive ones don't read into ST lore to know it comes after WWIII.

The dangerous ones do.

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

#240

Post by [deleted] »

magicalist wrote: post #194
drivebyhooting wrote: post #160

Earlier quoted context omitted.

Math academia was not working well at all. Almost every single graduated from my PhD program wound up working in ads or finance. The gatekeeping in math academia is extremely unfair, or should I say objectively fair but personally unfair. I won’t cry crocodile tears.

> wound up working in ads or finance Because there's lots and lots of money in that and there's not in funding pure math. It sounds like your problem is with the people holding the purse strings.

[deleted]
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