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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

#111

Post by gowld »

pitchlatte wrote: post #84
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.

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!

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

#113

Post by roywiggins »

thymine_dimer wrote: post #80

Doesn't this just suggest that the next frontier for powerful AI models is to ask challenging questions, not simply solve them? Terry even says this: "In fact, it is now the identification of a promising problem which is the scarce and precious resource." The creativity and insight needed to ask a question that Terry gets excited about is the next step. Perhaps OpenAI should create a set of challenging questions and…

If AI can generate questions and then answer them, what are the people for?

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

#114

Post by gowld »

monktastic1 wrote: post #93
dvt wrote: post #64

Earlier quoted context omitted.

> 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…

But this oracle doesn't just say true / false. It also gives a proof. That makes it much less exciting (not to mention beneficial for your career) to find another one (or even worse, the same one).

The "proof" is merely an appeal (unreadable program) submitted to a different oracle (Lean).

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

#115

Post by gowld »

roywiggins wrote: post #113
thymine_dimer wrote: post #80

Doesn't this just suggest that the next frontier for powerful AI models is to ask challenging questions, not simply solve them? Terry even says this: "In fact, it is now the identification of a promising problem which is the scarce and precious resource." The creativity and insight needed to ask a question that Terry gets excited about is the next step. Perhaps OpenAI should create a set of challenging questions and…

If AI can generate questions and then answer them, what are the people for?

If humans can shovel dirt, then what are the ants for?

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

#116

Post by SpicyLemonZest »

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…

> 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 understand proofs if they are correct?

Most modern mathematical problems are sufficiently abstract that their proofs or disproofs have no direct application. There's no problem you can fix or invention you can build based solely on OpenAI's construction, because analytic solutions to the Navier-Stokes equations are not used for practical purposes in fluid dynamics. The problems and their proofs are only interesting to the degree that they help us better understand how the math works.

IIUC the Navier-Stokes proof is understandable by human beings, but if it weren't it would be no more useful than a proof that 3 dimensional florg-complete entry seams have no durdle-nodes.

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

#117

Post by qlte »

thymine_dimer wrote: post #80

Doesn't this just suggest that the next frontier for powerful AI models is to ask challenging questions, not simply solve them? Terry even says this: "In fact, it is now the identification of a promising problem which is the scarce and precious resource." The creativity and insight needed to ask a question that Terry gets excited about is the next step. Perhaps OpenAI should create a set of challenging questions and…

The incentives are massively skewed towards the AI labs investing their massive amounts of compute into being the first to solve an outstanding problem.

It's a marketing game for them, any societal benefits are secondary. Winning a prize is going to get headlines and feed into the "AGI soon, machine replaces another career" narrative they crave unlike coming up with some (possibly) interesting problems.

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

#118

Post by cool_dude85 »

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.

Relevant, interesting problems that we have some immediate hope of making genuine work on might be, if not finite, quite difficult to produce. And it's also plausible that AI will not do as good a job of producing these as it does at solving them.

The other problem that Tao identifies is that math has typically been an unusually open subject in many respects. This openness may not work if big AI labs can afford to throw $X million at a problem to scoop you if the rumor gets around that you think you have something promising. Hence, less collaboration, and less chance of identifying these exciting new problems, infinite though they may be.

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

#119

Post by roywiggins »

ReflectedImage wrote: post #56
olalonde wrote: post #49

Can't mathematicians still gain novel insights by reverse-engineering AI-generated proofs? Just like chess players learn new concepts by studying what engines play.

Well no because it works by joining together existing novel insights.

Today, maybe. Where's the law of nature that says it won't be generating novel insights in two years? Five? Ten?

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

#120

Post by ThrowawayR2 »

gradus_ad wrote: post #16

>"While it may be technically infeasible to completely prohibit the use of automated tools to perform indiscriminate solution extraction, I believe that we can still designate many classes of problems as being desirous of a careful analysis that not only solves the problem, but identifies insights from the solution process, and learn more about the difficulty landscape for nearby problems, and for which raw solutions…

- If you have only a fuzzy idea of how to get to your travel destination, wrong turns and alternate routes may reveal sights and places you'd never have encountered without that wandering.

- If your GPS directs you straight to your travel destination, you are now where you wanted to be but missed out on the exploration. This is the sort of consequences the AI math proofs have.

STEM research thrives on that side exploration and unearthing unexpected things along the way. James Burke's famous documentary Connections spends the middle episodes talking about the unexpected directions that exploration has taken science. It's very hard to credibly make the case that this sort of meandering exploration is not valuable.

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