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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

#251

Post by onetimeusename »

dvt wrote: post #64
BeetleB wrote: post #60

Earlier quoted context omitted.

Mathematicians will be less likely to work on a problem if there is a solution - even an incomprehensible one.

> Mathematicians will be less likely to work on a problem if there is a solution Yes, that is Tao's premise, I'm just not sure I buy it. Suppose an oracle existed which could answer any question truthfully. Let's ignore the mechanics of this for now, but it could say things like "the Riemann hypothesis is False" or whatever and we would take it as gospel. Does this mean that we wouldn't have mathematicians or physici…

> Does this mean that we wouldn't have mathematicians or physicists or computer scientists or biologists anymore?

In the case of mathematicians, I think not as researchers. What would a research mathematician do? I don't think there would be any reason to try to gain insight from proofs that AI made for the sake of understanding. I don't see what that would achieve besides just retaining extremely niche knowledge (which AI or the oracle already does). The whole point of having that knowledge was to build toward novel work which the AI/oracle does. Also, the time spent and difficulty understanding them could be very high but with no payoff besides just understanding them because the AI/oracle would be used to solve all the problems anyway.

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

#252

Post by Alien1Being »

Tao's central point seems to be:

"In short, the indiscriminate use of powerful solution-extraction tools can achieve the immediate short-term goal of solving problems at hand, but at the cost of sustaining the ecosystem for the next wave of progress, or in understanding the progress already obtained. "

I am no mathematician, may have misunderstood his point and would be delighted to receive any corrections.

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

#253

Post by Jblx2 »

pvillano wrote: post #179

The way to tame a profit-maximizer is to make the most profitable choice the one that creates the most societal good. I would like to see the Clay Institute give zero recognition for formalizations without human-readable proofs. That would incentivize OpenAI to scram or create something that's actually useful.

OpenAI has already said they aren't going to claim the $1,000,000. If this proof claim holds up, then the Millennium prizes will be 2 for 2 for rejections of the prize money for valid solutions. Maybe that will be the precedent for other AI labs as well.

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

#254

Post by akoboldfrying »

silver92bullet wrote: post #238

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.

I don't think we can confidently say, yet, that LLMs can't discover those connections. We just haven't explored them yet, because 99.99% of the prestige is locked up in proving hard results, which also happen to be easier to assess objectively (thanks to automated proof checkers), and so that's where all the effort has thus far been exerted.

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

#255

Post by mrbungie »

meken wrote: post #248
mrbungie wrote: post #226

Earlier quoted context omitted.

For sure, but this was supposedly ~18 million dollars of compute, afaik 100 pages paper / lean proof and only god knows how many bytes of chat interactions + thought traces. Scale matters.

I bet it can be decomposed quite nicely though. At the top level, there are probably only like five steps. Dig as deep as you want into any of those steps (i.e. engineering).

Hopefully. We'll need to wait until mathematicians confirm how easy it is to digest whatever GPT did.

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

#256

Post by vessenes »

magicalist wrote: post #130
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…

Surprised to see someone on HN arguing against open science. Seems like the opposite of the lessons we should learn from Newton and Gauss, actually, hoarding results for decades at the expense of progress. (the Pythagorean thing isn't really competition either, is ahistorical, and from what we actually do know it's again people hoarding results instead of sharing them). FWIW, your post comes off as a middlebrow dismi…

Not arguing against open science - it's super valuable. I'm saying that pearl clutching by people reading Tao isn't useful, because it misses some long history which tells us that this kind of science has been seen as fundamentally competitive for millennia.

Should it be competitive? Is it more useful to be collaborative? How collaborative can it be when it's fundamentally competitive? Is it only fundamentally competitive because of some common 'quirks' of math types, or are there deeper forces pressuring it to be competitive?

These are all questions that I think are worth discussing, as is the note that the pendulum seems to be swinging away from cooperation in the face of competing for $trillion+ valuations (and a real enthusiasm for proving cool math stuff). The alternative, tweeting complaints on twitter without some context, is mostly a waste of space. I mentioned the history in hopes we could get informed complaints on twitter.

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

#257

Post by senshan »

aethelraed wrote: post #249
bityard wrote: post #208

Earlier quoted context omitted.

I don't see why that should be a problem, as we already know the answer is 42 in any case.

There is as yet insufficient Data for a meaningful answer. [1]: https://www.imdb.com/title/tt0708807

??? http://www.thelastquestion.net/

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

#258

Post by cobbal »

dbmikus wrote: post #158
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.

This is definitely true in an information theory sense: having more knowledge is always better than less knowledge. However, it may not be true in math as a social human endeavor, and having answers without interesting paths to get there may not expand human mathematics in the same way.

If Fermat had a book with larger margins, would Weil have devoted so much time to proving the Taniyama-Shimura conjecture? No one can say.

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

#259

Post by esteban0x »

porcoda wrote: post #90
jfengel wrote: post #75

I didn't realize that open math problems were a finite resource. I recall a story about some famous mathematician (Gauss?) dismissing interest in Fermat's Last Theorem claiming that he could crank out problems of equivalent interest. Clearly Tao knows a hell of a lot more than I do about this, but I'm surprised that math that close to completion.

They aren't, but the problem is that open problems tend to emerge when people are working on other problems. If fewer people are spending time deeply thinking about current problems since a handful of labs are solving them with AI without an eye towards understanding and only on verification, the pool of open problems won't be continuously growing. There is a fear that there will be a chilling effect on the community…

That explains a lot on why his arguments always focus on the "social part"

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

#260

Post by ramraj07 »

Alien1Being wrote: post #252

Tao's central point seems to be : "In short, the indiscriminate use of powerful solution-extraction tools can achieve the immediate short-term goal of solving problems at hand, but at the cost of sustaining the ecosystem for the next wave of progress, or in understanding the progress already obtained. " I am no mathematician, may have misunderstood his point and would be delighted to receive any corrections.

Seems to be it, though Im not particularly concerned about this problem personally.

The fact that we all readily accept that modern AI systems can likely solve any math problem that no living genius can, tells me that no task is beyond this system we just need the right harness around it. The exhaustion of meaningful math problems to motivate mathematicians minds seems to be the least of my worries at that point.

Inb4 someone suggests that this is not proof that these AIs generalize, I agree thats a popular opinion, but both sides are merely that, with no possible way to prove. I will wallow in my existential dread while you do whatever it is that gives you comfort.

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