8 ms·

Why Erdős Problems Are Falling to AI

155 points by pseudolus 2mo ago

ComputerPerson 2mo ago
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 comments that some of the users here long to "make a computer do neat tricks all day", and I can't help but think meager sponsorship could go a long way in this area. Existing grant structures, being much more traditional, are constrained by their cost.
- qsort 2mo ago
  > 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. The problem is that Paul Erdos was eccentric but he was also Paul Erdos. I'm not saying you're implying that, but I feel it's a similar line of thinking to how popular culture often romanticizes autism and Asperger's because some very smart people are (allegedly) affected. The same group includes people who need 24/7 care. Computer science at the frontier is as specialized and hard as mathematics, you need years of study to truly understand a field well enough to make meaningful contributions. You can try sponsoring me if you really want, but I don't think you'd be spending your money wisely in expectation.
  - ComputerPerson 2mo ago
    FWIW I've worked in two different academic labs, both of which would have benefited from a runaway worker barreling down random paths, as long as they were being productive in some capacity.
    - eru 2mo ago
      > [...] as long as they were being productive in some capacity. I'm afraid you only know that after the fact. You might also like TempleOS, or at least the context and history behind it.
      - ComputerPerson 2mo ago
        I guess that's the crux of the issue
- e4325f 2mo ago
  > Though I admit his heart attack probably wasn't a coincidence, sadly. He died aged 83...
  - ComputerPerson 2mo ago
    I didn't know that. A poor assumption I made that his amohetamine use shortened his life significantly.
    - inigyou 2mo ago
      Most drugs aren't as unhealthy as the government wants you to believe.
- arjie 2mo ago
  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.
  - jltsiren 2mo ago
    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.
    - arjie 2mo ago
      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".
- AussieWog93 2mo ago
  Honestly, I think our extremely liberal disability benefits scheme achieves this to a large extent (albeit possibly with downsides for both the recipient and society). Look at all of the crazy stuff made by unemployed trans people/autists over the past decade, disproportionately from countries with strong welfare states.
12k5ha 2mo ago
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?
- cinntaile 2mo ago
  There is no such thing as the Renaissance Fund. Quanta receives funds from the Simons Foundation... I also don't know where you get it from that Renaissance Technologies is heavily invested in AI. Any sources to back up your claims because I suspect you made all that up?
  - duskdozer 2mo ago
    There is. As for their investments, I don't know: https://www.simonsfoundation.org/about/our-history/ https://www.simonsfoundation.org/about/our-history/
- mosura 2mo ago
  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.
  - HarHarVeryFunny 2mo ago
    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 ?
- RandomLensman 2mo ago
m3kw9 2mo ago
is probably a mix of other math papers that combined can solve this, the ingredients were already out there and they got trained with it.
- ewidar 2mo ago
  could be, I guess we'll see. but even that, if done correctly, is quite impressive. It sure sounds very useful given the number of papers out there.
throwatdem12311 2mo ago
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 could we ever make anything useful with them? Are we ready to just let go of our intellectual faculties and give them to a giant supercomputer nobody understands? How do we tell truth from fiction?
- tumdum_ 2mo ago
  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.pdf https://teorth.github.io/tao-web/slides/age-of-ai-icm-2026.p...
  - glitchc 2mo ago
    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." Absolutely nothing wrong with that scenario right? Except for the logically fallacious appeal to authority and the inevitable corruption over time.
    - boothby 2mo ago
      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..."
      - glitchc 2mo ago
bsaul 2mo ago
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.
- ForgotIdAgain 2mo ago
  Those long standing math problems solution may serve as building block for solving the experimental science ones.
- numbers_guy 2mo ago
  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.
  - inglor_cz 2mo ago
    Connect the AI to a fully automated lab and it may be able to generate and harvest those necessary data. It may also produce Supercovid in the process, so... We don't have to worry about new math as much as we do about new biology. Although "breaking all current encryption" would have some nasty consequences as well.
- HarHarVeryFunny 2mo ago
  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.
pfdietz 2mo ago
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 (or disprove) them, with the discovered arguments then being used as training for refined models. This will require autoformalization to check the results as there will be too many for manual verification.
- CBLT 2mo ago
  Generating new conjectures is easy. It's generating novel conjectures that's hard.
  - pfdietz 2mo ago
    On the other hand, quantity has a quality all its own. Conjectures that survive the gauntlet of immediate proof/disproof could also be interesting.
    - nadermx 2mo ago
      I applaud how you articulated this.
operation_moose 2mo ago
Disappointing lack of "Why" in an article that starts with it. Are they actually doing something new and novel, or are they just absorbing that "a=b as was proven in transcendental hyper-circular group theory; and b=c was proven in universal quantum superposition"; and they're the first to find the connection that a=c? And several of the problems are counterexamples, not novel proofs of correctness? Its fascinating either way, but it'd be nice to actually understand more of what is happening.
- LPisGood 2mo ago
  One of the recent problems was actually proven with elementary results. It did use a technique not usually applied to the problem space, but it wasn’t a hyper specialized result.
vatsachak 2mo ago
Because pattern recognition and deduction is automatable and LLMs are superhuman at low depth high depth problems. Next
delhanty 2mo ago
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, that's good for folks who just want to know where the truth boundaries in mathematics lie. It's obviously causing a lot of soul-searching amongst professional mathematicians. Arguably, they should have given less weight for the last 100 years to Hardy's view in 'A Mathematician's Apology' [0]: > It is a melancholy experience for a professional mathematician to find himself writing about mathematics. The function of a mathematician is to do something, to prove new theorems, to add to mathematics, and not to talk about what he or other mathematicians have done. Rota takes a much more balanced view in 'Indiscrete Thoughts' [1]. "Problem Solvers" take Hardy's view: > ... The mathematical concepts required to state mathematical problems are tacitly assumed to be eternal and immutable. Mathematical exposition is regarded as an inferior undertaking. ... While for "theorizers": > Mathematical exposition is considered a more difficult undertaking than mathematical research. If professional mathematicians can reinvent themselves, there will be plenty of work left to do to explain the results of AIs to other humans. There probably needs to be a new career path into professional pure mathematics other than doing novel research in a PhD. [0] https://en.wikipedia.org/wiki/A_Mathematician%27s_Apology https://en.wikipedia.org/wiki/A_Mathematician%27s_Apology [1] https://ncatlab.org/nlab/show/Gian-Carlo+Rota https://ncatlab.org/nlab/show/Gian-Carlo+Rota https://ncatlab.org/nlab/show/Gian-Carlo+Rota https://ncatlab.org/nlab/show/Gian-Carlo+Rota
- aesthesia 2mo ago
  I'm not sure we should take it as a given that AI will not also become better than humans at mathematical exposition.
- RugnirViking 2mo ago
  > That means we will able to assume that nearly all the remaining conjectures are either true or undecidable I mean, we could. But it doesn't really make sense to think that AIs are perfect at finding counterexamples, just because they are good (or even better than us). My general opinion is that they are orthagonally intelligent, that is, they are intelligent in an entirely different way to the way that people are. They are undoubtably clever, but the distance between when they are better than us at their best skill (or even most skills) and when they are better than us in all aspects is going to be MASSIVE.
wasabi359 2mo ago
[flagged]
deleted 2mo ago
[deleted]
deleted 2mo ago
[deleted]
mark_something 2mo ago
> “A big problem is AI is being used a lot by people who aren’t mathematicians, who don’t have a huge mathematical background and are not capable of verifying the output,” Bloom said. “They like to move fast, ask their AI to check it, it grows and grows. We’re seeing a lot more of these 100- to 200-page papers that people are posting. ‘I solved this theorem; I got AI to generate the proof and check the proof and write the paper.’ But no human has read it, and no human is going to read it. It’s a huge challenge now.” Mathematics has the same problem as open source projects that are overwhelmed with AI slop!

155 points by pseudolus 2mo ago

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 comments that some of the users here long to "make a computer do neat tricks all day", and I can't help but think meager sponsorship could go a long way in this area. Existing grant structures, being much more traditional, are constrained by their cost.

qsort 2mo ago
> 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. The problem is that Paul Erdos was eccentric but he was also Paul Erdos. I'm not saying you're implying that, but I feel it's a similar line of thinking to how popular culture often romanticizes autism and Asperger's because some very smart people are (allegedly) affected. The same group includes people who need 24/7 care. Computer science at the frontier is as specialized and hard as mathematics, you need years of study to truly understand a field well enough to make meaningful contributions. You can try sponsoring me if you really want, but I don't think you'd be spending your money wisely in expectation.
- ComputerPerson 2mo ago
  FWIW I've worked in two different academic labs, both of which would have benefited from a runaway worker barreling down random paths, as long as they were being productive in some capacity.
  - eru 2mo ago
    > [...] as long as they were being productive in some capacity. I'm afraid you only know that after the fact. You might also like TempleOS, or at least the context and history behind it.
    - ComputerPerson 2mo ago
      I guess that's the crux of the issue
e4325f 2mo ago
> Though I admit his heart attack probably wasn't a coincidence, sadly. He died aged 83...
- ComputerPerson 2mo ago
  I didn't know that. A poor assumption I made that his amohetamine use shortened his life significantly.
  - inigyou 2mo ago
    Most drugs aren't as unhealthy as the government wants you to believe.
arjie 2mo ago
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.
- jltsiren 2mo ago
  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.
  - arjie 2mo ago
    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".
AussieWog93 2mo ago
Honestly, I think our extremely liberal disability benefits scheme achieves this to a large extent (albeit possibly with downsides for both the recipient and society). Look at all of the crazy stuff made by unemployed trans people/autists over the past decade, disproportionately from countries with strong welfare states.

12k5ha 2mo ago

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?

cinntaile 2mo ago
There is no such thing as the Renaissance Fund. Quanta receives funds from the Simons Foundation... I also don't know where you get it from that Renaissance Technologies is heavily invested in AI. Any sources to back up your claims because I suspect you made all that up?
- duskdozer 2mo ago
  There is. As for their investments, I don't know: https://www.simonsfoundation.org/about/our-history/ https://www.simonsfoundation.org/about/our-history/
mosura 2mo ago
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.
- HarHarVeryFunny 2mo ago
  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 ?
RandomLensman 2mo ago

m3kw9 2mo ago

is probably a mix of other math papers that combined can solve this, the ingredients were already out there and they got trained with it.

ewidar 2mo ago
could be, I guess we'll see. but even that, if done correctly, is quite impressive. It sure sounds very useful given the number of papers out there.

throwatdem12311 2mo ago

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 could we ever make anything useful with them? Are we ready to just let go of our intellectual faculties and give them to a giant supercomputer nobody understands? How do we tell truth from fiction?

tumdum_ 2mo ago
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.pdf https://teorth.github.io/tao-web/slides/age-of-ai-icm-2026.p...
- glitchc 2mo ago
  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." Absolutely nothing wrong with that scenario right? Except for the logically fallacious appeal to authority and the inevitable corruption over time.
  - boothby 2mo ago
    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..."
    - glitchc 2mo ago

bsaul 2mo ago

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.

ForgotIdAgain 2mo ago
Those long standing math problems solution may serve as building block for solving the experimental science ones.
numbers_guy 2mo ago
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.
- inglor_cz 2mo ago
  Connect the AI to a fully automated lab and it may be able to generate and harvest those necessary data. It may also produce Supercovid in the process, so... We don't have to worry about new math as much as we do about new biology. Although "breaking all current encryption" would have some nasty consequences as well.
HarHarVeryFunny 2mo ago
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.

pfdietz 2mo ago

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 (or disprove) them, with the discovered arguments then being used as training for refined models. This will require autoformalization to check the results as there will be too many for manual verification.

CBLT 2mo ago
Generating new conjectures is easy. It's generating novel conjectures that's hard.
- pfdietz 2mo ago
  On the other hand, quantity has a quality all its own. Conjectures that survive the gauntlet of immediate proof/disproof could also be interesting.
  - nadermx 2mo ago
    I applaud how you articulated this.

operation_moose 2mo ago

Disappointing lack of "Why" in an article that starts with it. Are they actually doing something new and novel, or are they just absorbing that "a=b as was proven in transcendental hyper-circular group theory; and b=c was proven in universal quantum superposition"; and they're the first to find the connection that a=c? And several of the problems are counterexamples, not novel proofs of correctness? Its fascinating either way, but it'd be nice to actually understand more of what is happening.

LPisGood 2mo ago
One of the recent problems was actually proven with elementary results. It did use a technique not usually applied to the problem space, but it wasn’t a hyper specialized result.

vatsachak 2mo ago

Because pattern recognition and deduction is automatable and LLMs are superhuman at low depth high depth problems. Next

delhanty 2mo ago

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, that's good for folks who just want to know where the truth boundaries in mathematics lie. It's obviously causing a lot of soul-searching amongst professional mathematicians. Arguably, they should have given less weight for the last 100 years to Hardy's view in 'A Mathematician's Apology' [0]: > It is a melancholy experience for a professional mathematician to find himself writing about mathematics. The function of a mathematician is to do something, to prove new theorems, to add to mathematics, and not to talk about what he or other mathematicians have done. Rota takes a much more balanced view in 'Indiscrete Thoughts' [1]. "Problem Solvers" take Hardy's view: > ... The mathematical concepts required to state mathematical problems are tacitly assumed to be eternal and immutable. Mathematical exposition is regarded as an inferior undertaking. ... While for "theorizers": > Mathematical exposition is considered a more difficult undertaking than mathematical research. If professional mathematicians can reinvent themselves, there will be plenty of work left to do to explain the results of AIs to other humans. There probably needs to be a new career path into professional pure mathematics other than doing novel research in a PhD. [0] https://en.wikipedia.org/wiki/A_Mathematician%27s_Apology https://en.wikipedia.org/wiki/A_Mathematician%27s_Apology [1] https://ncatlab.org/nlab/show/Gian-Carlo+Rota https://ncatlab.org/nlab/show/Gian-Carlo+Rota https://ncatlab.org/nlab/show/Gian-Carlo+Rota https://ncatlab.org/nlab/show/Gian-Carlo+Rota

aesthesia 2mo ago
I'm not sure we should take it as a given that AI will not also become better than humans at mathematical exposition.
RugnirViking 2mo ago
> That means we will able to assume that nearly all the remaining conjectures are either true or undecidable I mean, we could. But it doesn't really make sense to think that AIs are perfect at finding counterexamples, just because they are good (or even better than us). My general opinion is that they are orthagonally intelligent, that is, they are intelligent in an entirely different way to the way that people are. They are undoubtably clever, but the distance between when they are better than us at their best skill (or even most skills) and when they are better than us in all aspects is going to be MASSIVE.

wasabi359 2mo ago

[flagged]

deleted 2mo ago

[deleted]

deleted 2mo ago

[deleted]

mark_something 2mo ago

> “A big problem is AI is being used a lot by people who aren’t mathematicians, who don’t have a huge mathematical background and are not capable of verifying the output,” Bloom said. “They like to move fast, ask their AI to check it, it grows and grows. We’re seeing a lot more of these 100- to 200-page papers that people are posting. ‘I solved this theorem; I got AI to generate the proof and check the proof and write the paper.’ But no human has read it, and no human is going to read it. It’s a huge challenge now.” Mathematics has the same problem as open source projects that are overwhelmed with AI slop!