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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

#151

Post by qlte »

johnsmith1840 wrote: post #95
_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…

So what happens to this world view when AI not only clears the forest of problems we couldn't solve but also in the future discovers more forest with trees bigger than anything we've ever seen before? Not sure what the point of this argument is. Do we have mathematics for the sake of mathematicians good mental health and career or to solve and discover novel problems? Why should we care if mathematicians can understa…

Pure mathematics (defined by anything without a known application) exists not to "solve problems" in the real world, but by whatever mathematicians find interesting or lacking in current knowledge. Based on the agreed set of rules formed over time that ensure rigor.

It just so happens that even bizarrely esoteric math can later turn out to have some extremely useful and economically valuable applications. And even more useful to have mathematicians available who already understand that specific math.

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

#152

Post by david-gpu »

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 until these frontier models are used to create new interesting conjectures. I don't get Tao's line of reasoning.

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

#153

Post by jdoliner »

My model of mathematical intelligence for a little while now has been 3 levels:

1. I give you a proof, you tell me if it's correct

2. I give you a theorem, you give me a correct proof

3. I give you nothing, you give me a theorem

1. is largely solved by modern LLMs and they took a big step toward 2. today with the Navier-Stokes proof. But they're definitely not there yet. It's unclear what progress is being made toward 3. for the time being that remains the realm of humans.

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

#154

Post by johnsmith1840 »

keithnz wrote: post #140
johnsmith1840 wrote: post #95

Earlier quoted context omitted.

So what happens to this world view when AI not only clears the forest of problems we couldn't solve but also in the future discovers more forest with trees bigger than anything we've ever seen before? Not sure what the point of this argument is. Do we have mathematics for the sake of mathematicians good mental health and career or to solve and discover novel problems? Why should we care if mathematicians can understa…

there's a book I read "The Practice Effect" such that technology becomes super advanced based on using something, it gets better and better, but the people regress and become more like a medieval society as they just care that using things improves them.

Fun suggestion, thanks!

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

#155

Post by johnsmith1840 »

gpm wrote: post #144
johnsmith1840 wrote: post #126

Earlier quoted context omitted.

So...why can't an AI do the exact same thing? Make AI so it understands math better for future math to understand more math. Unless your argument is that mathematicians are effectively useless? I am assuming that's not your point though.

> 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.

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

#156

Post by johnsmith1840 »

qlte wrote: post #151
johnsmith1840 wrote: post #95

Earlier quoted context omitted.

So what happens to this world view when AI not only clears the forest of problems we couldn't solve but also in the future discovers more forest with trees bigger than anything we've ever seen before? Not sure what the point of this argument is. Do we have mathematics for the sake of mathematicians good mental health and career or to solve and discover novel problems? Why should we care if mathematicians can understa…

Pure mathematics (defined by anything without a known application) exists not to "solve problems" in the real world, but by whatever mathematicians find interesting or lacking in current knowledge. Based on the agreed set of rules formed over time that ensure rigor. It just so happens that even bizarrely esoteric math can later turn out to have some extremely useful and economically valuable applications. And even mo…

In that case isn't it actually more valuable to have an AI do this? It works faster and solves more math.

The random engineer looking at a funny problem 10 years later now has the literal author of the math to talk to about it and implement it.

I have never even spoken to a world class mathematician and now I can have them design with me?

How is this not better in almost everyway?

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

#157

Post by gpm »

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.

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 mathematicians won't be as good at this if they aren't also doing the first part.

The value to society only comes when they do both "well", and it's 2 which is really the black magic where we don't understand why they've been so useful to us.

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

#158

Post by dbmikus »

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…

An AI-generated solution always provides two pieces of info:

    1. proof that there is a solution
    2. a solution that you can work backwards from to build understanding
Maybe the solution is pretty inscrutable, but it's almost always better than nothing.

So, both of these pieces of info would be at least marginally useful for advancing human knowledge.

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

#159

Post by ppsreejith »

@Practal's comment is interesting:

> Pure mathematics is dead. Long live mathematics. I think all of interesting mathematics is applied mathematics in the end. Powerful AI means that the level at which we can do applied mathematics will be so much higher, though, and many more people will be able to be "mathematicians". The importance of pure mathematics is often argued for by citing examples of important applications that used pure mathematics invented a long time before the application became apparent. We can reverse this argument: by properly developing the mathematics our applications need, we surely will obtain all of interesting pure mathematics.

Perhaps the pace of applied mathematics would rise sharply, given cheap intelligence. And this* may end up being the forefront driving progress in mathematics.

*Or maybe a split between the human domain and the practical real world. Where the human domain might end up with a variation of a "No machine contributions" policy. Sorta like the recent gcc policy.

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

#160

Post by drivebyhooting »

gpm wrote: post #144
johnsmith1840 wrote: post #126

Earlier quoted context omitted.

So...why can't an AI do the exact same thing? Make AI so it understands math better for future math to understand more math. Unless your argument is that mathematicians are effectively useless? I am assuming that's not your point though.

> 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…

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.

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