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Why Erdős Problems Are Falling to AI

quantamagazine.org

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Re: Why Erdős Problems Are Falling to AI

#41

Post by LPisGood »

the_sleaze_ wrote: post #12
throwatdem12311 wrote: post #6

When OpenAI posted about their 10 breakthroughs, I saw lots of career research mathematicians say things mostly along the lines of “I don’t understand any of this it’s way over my head”. Are we missing the forest for the trees here? If a math problem falls in the forest but nobody is around to understand it does it make a sound? How can we possibly make use of these breakthroughs if we don’t understand them? How coul…

I don't mean to be dismissive, are these just old puzzles with no practical use whatsoever?

One of the most famous mathematicians of all time studied number theory. He wrote an apology to humanity for all of the very smart and capable people wasting time studying number theory, since these things are clearly unless old puzzles with no practical use whatsoever. Now, 100 years later, these results underpin all of modern public key cryptography.

Re: Why Erdős Problems Are Falling to AI

#42

Post by numbers_guy »

bsaul wrote: post #28

something i've just realized : today long-standing maths problems are falling. It's great intellectually but won't probably have an immediate impact on our lives. Now, what will happen once long-standing physics ( and chemistry and biology) problems will start to fall and at the same rate ? Then we're going to enter a totally different world.

Not really. These same techniques fall flat on their face when applied to most physics and chemistry problems. All of academia has already been doing ML4Science for the last 8 years. God knows how many billions have been spent.

The only two major highlights are weather modeling and folded protein backbone prediction.

Mostly everything else, either lacks enough data, or there are contraits on the size of the foundational models that render them impractical or they just fail to generalize.

Re: Why Erdős Problems Are Falling to AI

#43

Post by HarHarVeryFunny »

throwatdem12311 wrote: post #6

When OpenAI posted about their 10 breakthroughs, I saw lots of career research mathematicians say things mostly along the lines of “I don’t understand any of this it’s way over my head”. Are we missing the forest for the trees here? If a math problem falls in the forest but nobody is around to understand it does it make a sound? How can we possibly make use of these breakthroughs if we don’t understand them? How coul…

I get the impression that the value of unproven conjectures is more in the new math and techniques that may be discovered - by humans - trying to prove/disprove them, rather than much utility in any eventual result.

Take something like Fermat's last theorem - I'd be curious to hear of any use of the result itself, but there was a massive amount of new mathematics generated by those working on it, whether ultimately successful or not.

These AI math proofs are interesting testament to the power of reinforcement learning applied to math, obviously reflecting the axiomatic self-consistent nature of math itself, but it doesn't seem they have the same value as a humans working on these problems since they are using known math to solve them rather than inventing anything new.

However, it would still be interesting to analyze the LLM lines of reasoning that lead to any of these results, since there may be value there even if no new math, just as human Go players have found value in analyzing computer Go.

Still, as Demis Hassabis has himself said, the real goal with AI is discovery and creativity - you want to create the thing that could design the game of Go in the first place, not just play it. Similarly with math, while there is interest in seeing an AI "play math" using the rules of the game, what would be of much more interest is the AI that can create new math, in the same way as Andrew Wiles did while proving Fermat's last theorem.

Re: Why Erdős Problems Are Falling to AI

#44

Post by nsxwolf »

throwatdem12311 wrote: post #6

When OpenAI posted about their 10 breakthroughs, I saw lots of career research mathematicians say things mostly along the lines of “I don’t understand any of this it’s way over my head”. Are we missing the forest for the trees here? If a math problem falls in the forest but nobody is around to understand it does it make a sound? How can we possibly make use of these breakthroughs if we don’t understand them? How coul…

Maybe the "beautiful, elegant" math is really just accidentally that way, just the tiny cross section our dumb human brains can understand. The vast majority of it could be inscrutable, ugly, chaotic and seemingly meaningless.

Re: Why Erdős Problems Are Falling to AI

#45

Post by Jensson »

gglitch wrote: post #34
WarmWash wrote: post #16

Earlier quoted context omitted.

Mathematics is an unusually dense (if not the most dense...by a few large steps) field. So lots of areas of mathematics are extremely deep and narrow without any real shortcuts, even for seasoned mathematicians.

Awesome observation. What would you consider denser?

Physics possibly.

Re: Why Erdős Problems Are Falling to AI

#46

Post by stymaar »

Levitz wrote: post #33
the_sleaze_ wrote: post #12

Earlier quoted context omitted.

I don't mean to be dismissive, are these just old puzzles with no practical use whatsoever?

It's just really hard to affirm that something has "no practical use whatsoever". Maybe it doesn't have use "now", maybe it doesn't have "direct" use but can be used for another finding that is useful, Math has a long story of finding out stuff that turns useful later on.

True. Though ~99% is still useless in the end.

Re: Why Erdős Problems Are Falling to AI

#47

Post by jnwatson »

the_sleaze_ wrote: post #12
throwatdem12311 wrote: post #6

When OpenAI posted about their 10 breakthroughs, I saw lots of career research mathematicians say things mostly along the lines of “I don’t understand any of this it’s way over my head”. Are we missing the forest for the trees here? If a math problem falls in the forest but nobody is around to understand it does it make a sound? How can we possibly make use of these breakthroughs if we don’t understand them? How coul…

I don't mean to be dismissive, are these just old puzzles with no practical use whatsoever?

Isn't most of math this way? Occasionally, we find a practical use for it, but that's usually not the point.

Re: Why Erdős Problems Are Falling to AI

#48

Post by lstodd »

eru wrote: post #27
throwatdem12311 wrote: post #6

When OpenAI posted about their 10 breakthroughs, I saw lots of career research mathematicians say things mostly along the lines of “I don’t understand any of this it’s way over my head”. Are we missing the forest for the trees here? If a math problem falls in the forest but nobody is around to understand it does it make a sound? How can we possibly make use of these breakthroughs if we don’t understand them? How coul…

> When OpenAI posted about their 10 breakthroughs, I saw lots of career research mathematicians say things mostly along the lines of “I don’t understand any of this it’s way over my head”. Math is an incredibly broad field. I mean, you don't expect a traffic engineer to understand anything about nuclear reactors, do you? Yet, they are all 'career engineers'.

I would expect anyone calling themselves an engineer of any field to understand basics of fission power generation. Come on, it's 7th (school) grade material.

Re: Why Erdős Problems Are Falling to AI

#49

Post by pfdietz »

CBLT wrote: post #37
pfdietz wrote: post #30

Non-Erdős problems are also falling. The AIs seem to have some combination of very broad familiarity with math (enabling relevant things from other subfields to be brought in to the proof) as well as patience and "sitzfleisch" (stamina in working through details even if they aren't immediately obviously promising.) An obvious area for improvement would be automated generation of new conjectures and attempts to prove…

Generating new conjectures is easy. It's generating novel conjectures that's hard.

On the other hand, quantity has a quality all its own.

Conjectures that survive the gauntlet of immediate proof/disproof could also be interesting.

Re: Why Erdős Problems Are Falling to AI

#50

Post by goodmythical »

throwatdem12311 wrote: post #6

When OpenAI posted about their 10 breakthroughs, I saw lots of career research mathematicians say things mostly along the lines of “I don’t understand any of this it’s way over my head”. Are we missing the forest for the trees here? If a math problem falls in the forest but nobody is around to understand it does it make a sound? How can we possibly make use of these breakthroughs if we don’t understand them? How coul…

The field has been dealing with this for a long time.

Since at least 2014, which was my first brush with the phenomenon when someone published a 13GB proof [0].

The consensus is that such a proof is potentially illuminating, though further work is likely required. If for instance, conjecture A is true if and only if conjectures B & C are true, and B is proven false through one of these such proofs, then we can see that A is also false given that we accept the disproof of B.

Though, the sense is that further work is likely required because it is easy to see that further work along the same direction, or in directions depending on the proof will be hard or impossible if there are not enough humans or agents that are capable of understanding and utilizing the proof. Making it 'more elegant' will increase it's utility despite not proving anything new.

This is adjacent to all of the work done to create multiple proofs using different techniques. Having the same information (that X is so) in different languages (algebraic, geometric, via harmonic analysis, etc) allows for researchers not familiar with the original technique to participate in further research.

[0] https://www.newscientist.com/article/1997488-wikipedia-size-...

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