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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

#81

Post by jltsiren »

arjie wrote: post #65
ComputerPerson wrote: post #2

The mathematics are above my intellectual capacities, but I still find the man and his lifestyle fascinating. Though I admit his heart attack probably wasn't a coincidence, sadly. I wonder if national, institutional, or otherwise "eccentric" sponsorships (encouraging a similar migrant-madman approach to academic cultivation) of some of the folks on HN wouldn't lead to meaningful discoveries in CS. I often see comment…

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.

Re: Why Erdős Problems Are Falling to AI

#82

Post by inigyou »

parineum wrote: post #53
eru wrote: post #29

Earlier quoted context omitted.

That's why OpenAI also published machine-checkable proofs. The process is very, very faintly similar to running a typechecker over your software sources.

They're publishing machine checkable proofs because that's the only way they, themselves, can check them. They don't understand the math either.

For 3.5 days we had a machine checkable proof the Collatz conjecture was false. There turned out to be a bug in the machine checker.

Re: Why Erdős Problems Are Falling to AI

#83

Post by kadoban »

torginus wrote: post #75
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…

There's no honor in being stupid, yet I must admit I am stupid, as there's even less honor in being stupid and pretending otherwise. A lot of people were super hyped about OpenAI's 10 discoveries, but I still don't understand what they mean, and even if I did, what are the implications. Like, what are non-sofic groups, and what follows from the conclusion that they exist? I mean in the sense that quantum mechanics mi…

> What practical things are possible or might be possible due to these results?

Not much, they're all fairly minor problems that have been solved, of ~entirely niche academic interest. It's so far more just that AI _can_ solve novel math problems, ones that humans didn't accidentally train the answer in and it just spit it back out.

Re: Why Erdős Problems Are Falling to AI

#84

Post by lacunary »

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…

> 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?

Re: Why Erdős Problems Are Falling to AI

#85

Post by inigyou »

ComputerPerson wrote: post #21
e4325f wrote: post #19

Earlier quoted context omitted.

> Though I admit his heart attack probably wasn't a coincidence, sadly. He died aged 83...

I didn't know that. A poor assumption I made that his amohetamine use shortened his life significantly.

Most drugs aren't as unhealthy as the government wants you to believe.

Re: Why Erdős Problems Are Falling to AI

#86

Post by boothby »

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 think this is where Terry Tao hit the nail on the head. OpenAI has made claims, but the mathematics community does not accept garbage. It's less that they've done a disservice to the community; more that they've done a disservice to their own reputation.

Re: Why Erdős Problems Are Falling to AI

#87

Post by sarchertech »

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.

To continue the analogy, it’s like writing answers that are so convoluted that no one can understand them.

Re: Why Erdős Problems Are Falling to AI

#88

Post by HarHarVeryFunny »

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.

It's hard to see problems in those fields falling at anywhere the same rate as math, because they are all experimental fields.

There may be some problems of type type "why does X happen?" that appear answerable in terms of known science, but even these would need verification. If you want to make advances in fundamental physics, then a promising AI-generated theory might take a decade and billions of dollars to prove or disprove.

Math is a rather unique field in being entirely theoretical, axiomatic and self-referential. It is basically the best possible case not just for AI to advance without needing experimental verification, but also specifically for today's AI technology of auto-regressive LLMs and RL training, whereby valid reasoning steps learnt in one context will also be valid in another context (i.e. there is some generalizability of learnt reasoning) as long as you have learnt the pertinent aspects of that context that the validity depends on.

Re: Why Erdős Problems Are Falling to AI

#89

Post by boothby »

glitchc wrote: post #62
tumdum_ wrote: post #9

Earlier quoted context omitted.

Latest presentation of Terence Tao on what current advancements in AI mean for math discusses (among other things) those issues: https://teorth.github.io/tao-web/slides/age-of-ai-icm-2026.p...

Thanks for sharing. Regrettably his conclusion on proof exposition increasing in importance reflects the broad trend in every field: When someone (or something) else is doing the actual work, all that's left for the original folks is to find a way to sell it. Pretty soon mathematics will be awash in marketing with slogans such as "I am a famous mathematician, I checked this proof and it looks legit. Trust me." Absolu…

No. More like "I am mathematician and I have digested this sloppy writing and understand this to be an application of Foo theory to the Bar problem with a Baz twist. The prior publication omits crucial references to..."

Re: Why Erdős Problems Are Falling to AI

#90

Post by boothby »

stymaar wrote: post #46
Levitz wrote: post #33

Earlier quoted context omitted.

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.

We are not at the end, how could you possibly be confident in this claim?
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