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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

#91

Post by HarHarVeryFunny »

mosura wrote: post #5
12k5ha wrote: post #3

Quantamagazine is owned by the heavily AI invested Renaissance Fund. The whole article is an ad that covertly or overtly inserts how websites are built with ChatGPT, how humans say that AI is better than them etc. This is incidentally the future of chatbots. I could not have written this comment without Illy Espresso. Would you like to find a cafe near you?

We have now passed the point where the AI bots were annoying. Now it is the reactionary “everything even possibly slightly remotely tangentially connected to AI must be deplatformed” rants everywhere that are even worse. It would be tempting to assume the commenter here has no idea what Renaissance is or how they made so much money.

The "commentator"'s account also has a karma of negative 3 ! I've never even seen a negative karma before !

There is a pattern where the most divisive comments, that one might suspect of being from bots, do tend to come from accounts with single or double digit karma. Maybe HN tries to identify and remove these, which is why they tend to be new ?

Re: Why Erdős Problems Are Falling to AI

#92

Post by davidivadavid »

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?

My intuitive understanding is that deriving new theorem from existing concepts is "new math" insofar as it conclusively proves whether existing conjectures are in fact true or not by deriving proofs within an existing formal "system" (loosely understood), but it's not "new math" as it doesn't introduce new concepts to the system. It's the usual problem solver vs. theory builder dichotomy.

An interesting thought experiment would be: assuming AI can solve any given problem (or prove it's undecidable), and thus that the "proving" activity becomes trivialized, what's the interesting part that remains? Can we work on "refactoring" mathematics to make it more intuitive? More "powerful" in some sense? What are other refactorings that are worth exploring?

Re: Why Erdős Problems Are Falling to AI

#93

Post by glitchc »

boothby wrote: post #89
glitchc wrote: post #62

Earlier quoted context omitted.

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..."

The writing's already pretty good, even Terrence Tao calls it flawless, if you read the slides. That's not where the LLM requires help.

Re: Why Erdős Problems Are Falling to AI

#94

Post by WarmWash »

Jensson wrote: post #45
gglitch wrote: post #34

Earlier quoted context omitted.

Awesome observation. What would you consider denser?

Physics possibly.

Physics has bounds of concern, where as mathematics would encompass all physics, as well as all possible alternative physics, in all possible forms. Maybe not unbounded, but close to it.

Re: Why Erdős Problems Are Falling to AI

#95

Post by twotwotwo »

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 of mathematics is smart about this and knows the difference between a pile of Lean code and understanding, and mathematicians try to get from the unintuitive explanations to something that makes more sense, e.g. https://terrytao.wordpress.com/2026/07/21/a-digestion-of-the... (where incidentally Tao used a chatbot to help take apart the problem, but with a lot of interaction and work from his side).

Making things understandable is mathematics, and more generally a kind of intelligence, and is crucial to continued progress. You couldn't use algebraic geometry to disprove a conjecture if people hadn't organized (what could have been just) a pile of random observations into something called algebraic geometry.

Historically LLMs have done best where it's possible to train using an objectively verifiable reward function. Computer programs are pretty good on this front and so are Lean proofs. (Of course, they don't only do things you can RLVR heavily, but those have progressed fastest.) Not sure where 'making mathematical knowledge more understandable' falls on that spectrum.

Understandability isn't only important for advanced math. Keeping computer programs from becoming a mess is a challenge in high-level organization too, and the chat with the user is an explanation task. If you look online at what people say about large LLM-built codebases (SlopCodeBench is a neat effort to make make it concrete, but common wisdom seems to mostly agree on the general problem) and chatbot prose, I don't think everyone considers those solved problems!

It's hard to tell how thoroughly the labs grasp and care about this at an organization-wide level. I'm sure at least some maybe-results exist inside labs but haven't been published because the humans couldn't verify them and didn't want to be embarrassed with a false result. (Maybe also why counterexamples are a lot of the first results published: often simple to verify, even if hard to obtain.) A good sign would be if results in a few months come out more like what mathematicians consider well-written papers explaining results in a more intuitive way, fewer shocking announcements of bare counterexamples in tweets. It's probably a slow climb to get there.

Re: Why Erdős Problems Are Falling to AI

#96

Post by vouaobrasil »

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?

Most modern pure math has no practical use whatsoever. It's amazing how few people understand this. Mathematics is a service department and maybe one in a hundred thousand results have any practical use. (Speaking as someone with a PhD and over a decade in pure math with several published papers...)

Re: Why Erdős Problems Are Falling to AI

#97

Post by lstodd »

ben_w wrote: post #66
lstodd wrote: post #59

Earlier quoted context omitted.

I can explain what the void coefficient is, why it is a dangerous simplification, why neutron moderation is needed, how it is typically controlled, why BWR reactors are inherently less stable than PWR. etc. it's simple really. All the while I have never had anything to do with nuclear power professionally or studied specifically it. I might have overstated a bit, but by 9th grade (15 year old) this is what was taught…

> why neutron moderation is needed What you learned, was it more like: 1. "You need to slow down neutrons so they can react" or 2. "Here's the graphs of how the neutron absorption and scattering cross sections vary with neutron temperature for H-1, H-2, H-3, Be-9, C-12, O-16, Fe-54, Fe-56, Fe-57, U-233, U-235, U-238, Pu-239, …" If it was the former, you didn't learn "nuclear engineering".

ofc, you can't even start talking about moderation and not mention cross-sections

(which is a simplification in itself, but that's best left until 2nd-3rd year in uni)

But for general understanding, .. there is stuff that slows neutrons. some is more effective, some less. There is also activation. It is why tanks and ifvs were lined with polyethylene or similar on the inside back in cold war - it had lots of hydrogen. But for controlling a power plant that is not enough - why?

and then we answer why.

Re: Why Erdős Problems Are Falling to AI

#98

Post by vouaobrasil »

jnwatson wrote: post #47
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?

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

Yes, that's correct. I've done research in pure math, but left after the postdoc level. I enjoyed it for its artistic sake but it's not something that gives anything practical. Most pure mathematicians don't ever work on practical problems either. They just teach and in return get to do their hyperspecialized research. It's done for its own sake. (Some pure math eventually ends up having applications but it's rarer and rarer and it wasn't very common to begin with. Like applications to crypto or whatever, but that was always a tiny slice to begin with.)

Re: Why Erdős Problems Are Falling to AI

#99

Post by insanitybit »

glitchc wrote: post #64
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…

My perception is different. I start from the axiom that AI is a massive compressed corpus of knowledge. That it can find solutions suggests to me that the solutions were already known, but simply lacked publicity. This is less about discovery and more about pattern-matching.

That axiom is clearly false. AI can interact with external systems, which means it is not just compressed knowledge - it has the ability to access new information.

Re: Why Erdős Problems Are Falling to AI

#100

Post by epiccoleman »

lstodd wrote: post #59
neutrinobro wrote: post #55

Earlier quoted context omitted.

Basics yes, details no, and if you are deep in a mathematical proof the details matter. For example, I seriously doubt you will find an average 7th grader (or even a professional engineer outside of the nuclear power field) able to give you a good explanation or even definition of the void coefficient, and how it may interact with the fuel temperature coefficient of reactivity for a particular reactor design. Do you?

I can explain what the void coefficient is, why it is a dangerous simplification, why neutron moderation is needed, how it is typically controlled, why BWR reactors are inherently less stable than PWR. etc. it's simple really. All the while I have never had anything to do with nuclear power professionally or studied specifically it. I might have overstated a bit, but by 9th grade (15 year old) this is what was taught…

Man you went to a way cooler high school than me. I definitely was not taught anything that would help me understand how a nuclear reactor work.
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