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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

#101

Post by HarHarVeryFunny »

lacunary wrote: post #84
HarHarVeryFunny wrote: post #43

Earlier quoted context omitted.

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 s…

> are using known math to solve them rather than inventing anything new. Isn't a new proof new math? If not, what qualifies as new math?

You could regard existing mathematical techniques as a set of Lego pieces. If you build a proof just using those pre-existing Lego pieces then it may be a new proof (a new Lego model), but it's not a new Lego piece - a new piece of mathematical machinery.

Re: Why Erdős Problems Are Falling to AI

#102

Post by arjie »

jltsiren wrote: post #81
arjie wrote: post #65

Earlier quoted context omitted.

When Gwern needed a fellowship to come to SF he got one. When he had his “great idea” it was funded. The truly unique people out there find a way. Any government bureaucrat will just find the Jason Ardays of the world.

That's survivorship bias. If there are truly unique people who never get a chance to do what they would be good at, we don't hear about them. Private sector bureaucrats are no different from government ones. The fundamental issue is that if you have a system, it's going to fail outliers. Almost by definition. And the situation is even worse if the system uses metrics for making decisions.

Well, if the statement is "we need some eccentric thinkers" then survivors suffice. We have some eccentric thinkers. If the statement is "we need more eccentric thinkers" then I think I'm going to demand the rigor of "explain why we need to increase the quantity and why your proposed intervention is the best way to get there with evidence behind it".

Re: Why Erdős Problems Are Falling to AI

#103

Post by oliculipolicula »

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'.

No. Mathematicians who considered themselves experts on exactly those conjectures were bewildered.

>And then -- the kicker -- something that I personally spent a couple years on in grad school, leading to some of my proudest work: quantum parallel repetition theorems

...

>Is there some broader context or theory within which this would've been the obvious thing to do? What other results can be proven using these techniques? What is it telling us about quantum information or operator theory? I have no idea

https://bsky.app/profile/henryyuen.bsky.social/post/3ms2jpch...

That guy has been featured in quantamag too, btw

I saw another expert on Ehrhart express their bafflement (but I can't find that link now)

PP (actually the "broad career mathematicians" themselves) were being slightly disingenuous... But the specific domain experts seem to be understating how much over their head it actually was. We're in totally unknown territory here..

we can however be optimistic, oddly. If you believe Feynman when he says that you can't explain to a child something you don't understand.. then we can see that ChatGPT has no idea of just what it has done!

(Steel Manning:

1. maybe any chatbot would need 1000x more tokens than were used to arrive at the result to understand it to its own satisfaction.

2. It's possible that Einstein did not fully understand General Relativity

Re: Why Erdős Problems Are Falling to AI

#104

Post by HarHarVeryFunny »

svachalek wrote: post #40
serial_dev wrote: post #10

Earlier quoted context omitted.

I think it's the scientist version of "I vibecoded ten apps this weekend (at one point I'll have real users too)" . Because it's math, it's all mysterious and genuinely impressive, but in the end, if no human cares about it (apart from attention grabbing "it's so over" tweets and articles), does it really matter?

Nah, people are vibe coding lots of apps that no one cares about, but this is more like going through Stack Overflow and answering the most upvoted issues that don't have answers. These are published problems that have prior interest, for decades usually, and a few snarky internet comments don't undo that.

Yes, although a counterpoint to that is that if these weren't long-standing published problems that have stumped humans, would anyone care? What is important here, the problem/proof, or the fact that they are famous problems that have stumped humans (and therefore make good bragging material).

I'm not sure that a problem being famous, or hard for a human, is the same as it being important.

Take Fermat's last theorem as an example - was a proof of it ever considered to be, then or now, consequential, or was it the Paris Hilton of problems - famous for being famous? A problem whose legend grew because it it so simple to state, with the challenge that there might be a simple proof, and that turned out to be so difficult to solve that it therefore became prestigious to do so?

Re: Why Erdős Problems Are Falling to AI

#105

Post by aesthesia »

delhanty wrote: post #58

It seems that AIs are really good at finding counterexamples now. Even if progress by AIs in proving conjectures lags, it seems likely that AIs collectively will, in the next few years, find counterexamples to nearly all the Erdős (and other) conjectures that are actually false and also provably false. That means we will able to assume that nearly all the remaining conjectures are either true or undecidable. Surely,…

I'm not sure we should take it as a given that AI will not also become better than humans at mathematical exposition.

Re: Why Erdős Problems Are Falling to AI

#106

Post by mlpoknbji »

randomizedalgs wrote: post #78
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…

Even the experts on the problem being solved find the writeups nearly impossible to read. Example: https://nitter.poast.org/henryquantum/status/208362369543662... Seems like a disservice to the community that openai put so little effort into producing good writeups...

I agree. Prompting 5.6 Sol to "Review these writeups and include appropriate citations and credit where appropriate" probably would have been a step in the right direction.

Then again, this would require them to actually care about the mathematics.

Re: Why Erdős Problems Are Falling to AI

#107

Post by throwatdem12311 »

nater5000 wrote: post #57
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…

>If a math problem falls in the forest but nobody is around to understand it does it make a sound? This is not new nor unique. There is plenty of research, especially in math, which can really only be understood by a few people in the entire world. It is not uncommon for a proof to be presented by a mathematician which, initially, is only understood to that mathematician, and it can take a long time for even another…

No offense taken, I appreciate the time you took to respond and clarify these things.

Re: Why Erdős Problems Are Falling to AI

#108

Post by HoldOnAMinute »

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, for one, welcome our giant supercomputer overlords.

Re: Why Erdős Problems Are Falling to AI

#109

Post by matheist »

LPisGood wrote: post #41
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?

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.

And then there's G. H. Hardy, famous number theorist, who wrote an "Apology" for mathematics in the sense of defending the study of number theory for its own sake because it's beautiful.

Which one were you referring to?

Re: Why Erdős Problems Are Falling to AI

#110

Post by HarHarVeryFunny »

HarHarVeryFunny wrote: post #43
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 s…

It just occurred to me that this "it's not the destination, it's the journey", what you learn and contribute by trying to solve a problem, may point to what I'm guessing is an unpopular conclusion...

It may actually be a negative, rather than a positive, for AI to have solved these problems.

If humans had continued to work on these problems, then its quite possible they may have invented new math along the way and benefited the field. Now that the problems are solved, in one manner, I would assume that the level of human interest in them is much diminished, and the likelihood of these problems generating the same benefit for mathematics has diminished. Erdos chose his problems precisely because he thought they could advance the field.

AI itself does not appear to have benefited from solving these problems - they are hard for a human but apparently fairly easy for an AI. We don't just have one result here, proven after great labor, but 10 (with more results rumored to be withheld), with the effort/cost to generate them reported to be low.

The temptation to announce AI solutions to problems that are hard for humans, easy for AI, may be hard to resist, but perhaps there would be more benefit to be had to leave the hard for human problems for humans to solve, and be more impressed when AI solves problems that are hard for AI.

As Hans Moravec noted "What's easy for humans is hard for computers, and what's hard for humans is easy for computers", and while AI is trying to make inroads into that, it still remains fundamentally true.

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