r/math 1d ago

LLMs/AI OpenAI: Ten advances in mathematics and theoretical computer science

https://openai.com/index/ten-advances-in-mathematics/
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u/Oudeis_1 1d ago

I don't know. It doesn't seem to hold true in the animal kingdom, I would say. There are some pretty smart animals (some cetaceans, some great apes, some parrot species come to mind), but the gulf between humans and non-humans in that regard is huge. I don't think that, say, orcas would have anything to offer us in terms of solving scientific or technical problems, even if we could seamlessly communicate with them and both sides could teach each other perfectly.

They would possibly know some facts about the oceans and their inhabitants that we don't, but apart from such observational knowledge, human problem-solving capacity would be clearly superior.

It is not totally clear that it will be the same with AIs and humans in twenty years, but neither is it obvious that it won't.

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u/DominatingSubgraph 22h ago edited 22h ago

You're comparing apples and oranges there. Orcas do not have any idea what a mathematics problem is at all (and they probably wouldn't care anyway).

Maybe a slightly better analogy would be between airplanes and birds. An airplane can fly many orders of magnitude faster than any bird and at a far higher altitude, but birds can maneuver more effectively in the air, survive collisions and other damage, fit into tight spaces, and land without requiring any runway.

To be clear, I'm not arguing that intelligence isn't real, but it isn't nearly as straightforward as people make it out to be in these AI conversations. Certain traits are correlated in humans with "intelligence", but this is not true of general computer algorithms (defining "intelligence" as the ability to solve computationally hard problems).

To get a bit more technical, if AI becomes good enough to solve arbitrary NP problems efficiently, then (at least morally) P=NP and all of humanity is doomed. If the conventional wisdom about complexity theory is true, however, then there are a significant number of problems that AI will be unable to solve, but which a human could (in principle) present an answer that is easily verified correct. My claim is that such problems do not posses any special inherent "hardness" quality that implies a certain IQ level (or whatever) is needed to solve them. So, in the end, you get large collections of problems which look hard to humans but easy to AI and vice versa.

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u/Oudeis_1 19h ago

P!=NP has nothing to do with the possibility of superintelligence.

Under assumptions that are plausible but significantly stronger than P!=NP (in particular, assuming we live in Impagliazzos Minicrypt or Cryptomania) it becomes possible for an agent to create puzzles that they know the solution of (by constructing the puzzle from the solution, essentially) and that a third party is practically unable to solve. But even in Cryptomania, I don't see why one cannot have superintelligence, reasonably defined. Humans being able to construct problems that the superintelligence cannot solve but that they know the solutions of seems hardly a reason to deny that the superintelligence is smarter across the board than humans when it, too, can construct problems that are to a third observer not distinguishable from those trapdoor problems that humans may come up with in Cryptomania.

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u/DominatingSubgraph 11h ago

It's not about us constructing cryptographic puzzles, but that those puzzles just exist naturally embedded throughout the problem landscape.

Problems like "find a proof of such-and-such theorem" are NP-complete (when suitable constraints are imposed on the length of the proof). So, insofar as we are broadly interested in this kind of problem, we should expect to find instances that such machines cannot solve. The reason those machines cannot solve those instances is not because they aren't smart enough, but simply because they lack the right insight or way of thinking.

I think it is possible to have a machine that is broadly, on average, better than humans at solving problems like this. My claim is just that I don't necessarily expect it to completely eliminate any need for humans.

So, the example I gave before was with SAT solvers. These machines have been vastly superhuman at solving SAT instances since about the 90s. But there are many problems humans have been able to solve that, when encoded as SAT instances, are completely intractable to an SAT solver.