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Not to denigrate the moment (AI ingress into theory which this is a part of) or the result here, but these headlines are perhaps overstating the importance - so

Not to denigrate the moment (AI ingress into theory which this is a part of) or the result here, but these headlines are perhaps overstating the importance - some of the theories and conjectures are available for AI-assisted exploration because they are quite niche and not very important. Maxwell's name being invoked here for instance implies a hundred year old foundational problem like Fermat, but it's just a recent conjecture that was inspired by reflections from the great man on his work.

deleted 3mo ago
[deleted]
ashleyn 3mo ago
They might be low-hanging fruit but two things immediately come to mind: * As more of the small stuff is just proven for free, the more they can be used as a basis for other proofs. If you know something is true or false for certain, that can be a significant tailwind for the much harder, much more important problems. Fermat's last theorem looks deceptively simple and invited many failed amateur attempts at solving it, but Wiles' proof drew on a diversity of seemingly-distant subfields within mathematics that were better understood. * What are aspiring math Phd's supposed to do, now that the bar is much higher these days? The net effect of this appears to be that we'll see far fewer, but far more elite math Phd's, potentially discouraging many young people from the field.
- throwaway0123_5 3mo ago
  > The net effect of this appears to be that we'll see far fewer, but far more elite math Phd's, potentially discouraging many young people from the field. It seems plausible that the value of education will go down for the vast majority of fields and as a result less people will be getting degrees of all types. Not a good outcome I think for humanity to be less educated, even if people are provided for when they can't get jobs... things like mathematical and scientific literacy, as well as history knowledge (which even STEM majors often receive via undergraduate degree breadth requirements), etc. I would expect strongly result in more informed and harder to deceive citizens.
  - CWuestefeld 3mo ago
    A large proportion of degrees awarded today are not useful for any practical application. But having a degree of some kind still serves as a signaling mechanism, demonstrating lots of things including the ability to "play the game". to follow instructions, and so forth.
    - zarzavat 3mo ago
      If I were 18 I'd be sceptical about the value of degrees compared to the value of trades. Yes degrees are a prerequisite for white collar jobs but if there is competition for ever fewer white collar jobs that's a life gamble perhaps not worth taking unless you are supremely confident in your abilities in a field. If we're being honest, the value of a degree has been in decline for a long time. It was only being held up by STEM and even that is on shaky ground now. The days when an e.g. English degree would open doors are long over.
      - ConceptJunkie 2mo ago
        There's going to be a huge realignment in white collar work in the next decade or two, and then there's the fact that college degrees themselves (except for maybe STEM), are becoming less and less valuable because higher education is so politicized and taken over by nonsense. My college education (Computer Science, 1987) served me very well over my career, but I don't know what I'd be doing if I were 18 now.
    - throwaway0123_5 3mo ago
      I agree somewhat, but for the most part the signal is that you can succeed in some kind of knowledge work. If AI replaces ~all knowledge work, that signal isn't actually going to be valuable I think. While I don't find it certain that AI will do that, it seems increasingly plausible with every model release. In that world, I don't think very many people are pursuing degrees.
    - cedws 3mo ago
      I never understood why being able to “play the game” is considered a positive signal. Getting a degree is the safe path, it doesn’t require any risk taking, and not much initiative. It’s about having the money (or going into debt for most people) and sitting your ass in a chair until you’re told you’re worthy. That idea is repulsive to me.
- julianeon 3mo ago
  Is the bar really higher? Those math PhD's can use GPT too; they benefit equally from AI assistance.
  - fn-mote 3mo ago
    This approach totally changes the game, in ways we are still discussing. The current consensus is that domain experts get the most out of using AI on problems. How will domain expertise develop when AI is doing the work? I’m not saying that there won’t be another approach that builds on the strengths of AI, but we have to look for that and develop it. There’s a lot to talk about.
    - hgoel 3mo ago
      A decent chunk of the work a student does during a PhD is stuff that their advisor would have done in much less time, but are intentionally delegating to the student to allow them to learn. So, I don't think AI changes the way expertise is earned. It might change the cost tradeoffs depending on where AI costs eventually settle.
- chermi 3mo ago
  Maybe masters will become more popular. I have no problem with bars being raised on PhDs, but it's crazy history that math may be the first one to have it raised (or brought back to old levels).
  - StableAlkyne 3mo ago
    IMO, I don't think this will raise the bar for PhDs. In every field of science, PhD student research is largely incremental. Very rarely is a thesis groundbreaking. The point of it is all is to function as an apprenticeship for that PhD student to become a scientist. Sometimes a particularly gifted or lucky one hits an important result, but that's rare and isn't the purpose.
    - chermi 3mo ago
      I agree, but there's degrees (no pun intended!). Many PhDs are just an advisor telling a student exactly what to do. The training is still valuable, but it's not really what a PhD is "supposed" to mean. The example I give because of familiarity is many PhDs in chemistry/physics are just clicking run on simulations + running predefined analysis with results already predicted by the PI. Or in chemistry specifically OChem wet lab, it's often basically a minimum wage factory job.
- hgoel 3mo ago
  I don't believe this is raising the bar for minting fresh PhDs. We need to maintain perspective here, a PhD is essentially work done by a researcher at the least experienced, least skilled point of their career. Their primary goal is to demonstrate that they are capable of contributing to research. They are not competing against AI to publish a counterexample to a known conjecture.
- ndriscoll 3mo ago
  The bar for math PhDs already ruled out like 99.9% of the population, so I don't see it having much effect on discouraging people. The gap between even a bright student who takes AP calculus or whatever and someone studying e.g. spectral sequences is already incomprehensibly large. Like you literally could not even convey to a smart young person how far away they are from the boundary of today's understanding. I don't think I even have a reasonable sense with a math bachelor's! People who get into math do it because they can't not do it.
  - ian-g 3mo ago
    I think it'll become increasingly important to have folks thinking about how to explain new math in a way that makes sense to your math bachelors students. And to your bright AP Calc students. If we can point ChatGPT at these problems and get eventual answers, that's awesome, but it'll still be important to figure out how to tell people why this matters in ways they understand.
  - ConceptJunkie 2mo ago
    I'll look at math papers on Arxiv, etc., and browsing the titles will find maybe one in a hundred where I even know what it's talking about. I've been a math enthusiast since I was a kid, but the field is so immense I can't even wrap my head around how big it is, leave alone the actual concepts. Out of all the topics mentioned in the recent AI advances at finding counterexamples to conjectures, etc., about the only one I understood was the one where they've made advances in a bounds for kissing numbers in higher dimension, a topic I find fascinating even if I don't really know much about it.
- jauntywundrkind 3mo ago
  Notably we have had very few conjectures proven true, so it doesn't feel like there is that widening base of established map to draw yet more proofs upon.
- altairprime 3mo ago
  As a potential elite math Ph.D, though sadly I can't afford to complete studies in that direction yet, both before and after A.I. I have been interested in the field for synthesis of different fields of mathematics into novel outcomes. A.I. is, as with protein and antenna folding, excellent at constraint solving and optimization — but human creativity continues to excel at creating new frontiers that are not found within the 'local' maxima explored by algorithms. Consider that we have brute-force calculation tests that can be applied to an entire floating-point 2^32 space and I guarantee a lot of A.I. effort invested in finding more efficient storage methods, but there's still room for human invention of floating-point storage simply by a person making a leap between Field One and Field Two that hasn't been proven yet: > We would like to stress that the key idea behind ALP is to design for vectorized execution; it led us to analyze and uncover unexploited opportunities from a vector perspective in a variety of datasets. — doi.org/10.1145/3626717, via ALP: Adaptive lossless floating-point compression (6 days ago, 4 comments) https://news.ycombinator.com/item?id=49051355 https://news.ycombinator.com/item?id=49051355 Perhaps future mathematicians use A.I. to prove it, but that's still a human invention — and even if we brute force the entire space of mathematics, it's functionally useless without an ontology to help humans narrow an A.I. 'problem-solving' response from 'all possible solution spaces' to 'the interesting solution spaces', and even if that ontology is A.I. created, a human is still going to be evaluating 'interesting' through their own cognitive experiences, biases, and dissonances in order to identify novel connections for other humans to build with. Just because we can `echo 0..2^64-1 >> file.txt` doesn't mean that we're deriving value from it, even if it's indexed by number of digits or nearest power of two or whatever. OEIS exists, and cannot be easily replaced by an A.I., because it's not just a list of numbers indexed by sequence (which no doubt computerized proofs plus generative algorithms will eventually automate the generation of), it's a list of interesting to humans sequences, and that's something A.I. can't be substituted for. > Director Cary Fukunaga mentioned a complex narrative structure in his 2018 big pharma miniseries Maniac being nixed because of the audience loss predicted by the data. — via Bland, for fans of everything (11 months ago, 7 comments) https://news.ycombinator.com/item?id=45049412 https://news.ycombinator.com/item?id=45049412
moomin 3mo ago
Yeah, the Jacobian conjecture counter-example was big news. In particular, it would have been news even if an AI hadn't done it. That's where the bar is now. Settling Erdős conjecture 7529 or whatever no longer qualifies as AI news.
- tsunamifury 3mo ago
  Real talk. AI solving these makes me feel like mathematicians put far more importance on their work than was actually there. Many solutions seems to be tautological games, and games of logic where conjecture puzzles that few work on or care can be solved by AI which doesn’t care what it works on. It seems to always be some form of this: Mathematician: “Propose conjecture a and conjecture b can’t be true simultaneously” AI: “they can” Everyone: “ok…” I know this might be unfair or out of ignorance but it genuinely is how this field feels today. Games of games with self importance added in. Edit: the point I should have made is, should we be using AI to figure out what proofs MATTER now vs games of proofs?
  - efficax 3mo ago
    "tautological games". All proofs are a form of tautology, you have to end up back at the point your theorem proposed. Math is games of logic. That's what it is.
  - dwaltrip 3mo ago
    Look up how pure mathematics connects back to reality in countless unexpected and useful ways, time and time again.
  - deeznuttynutz 3mo ago
    Don't do that...
  - chermi 3mo ago
    Very useful games, self-importance or no. See pattern, conjecture generalization, test generalization. It's almost like empirical math. I like it and I also like mathematicians doing it the old way.
astro1234 3mo ago
I find them useful bellweathers of genuinely out of domain performance and capability, regardless of their theoretical importance. What I see is that performance trends are remarkably stable both upstream (miraculous scaling laws of pretraining on validation loss) and downstream performance (epoch capability index). We get the equivalent of a GPT4->GPT5 performance leap every ~16-18 months, and we are not hitting ceilings nor do we see any deceleration. Today we can solve nontrivial open problems. What will we be able to do next year or the year after? 18months ago no one was using a coding agent seriously. Now for a large segment of the population you cannot do your job without them.
gjm11 3mo ago
I'm not sure that's right. The conjecture is (claimed to be, by the people who explicitly made it) simply a reformulation of a claim made by Maxwell in his "Treatise on Electricity and Magnetism" of 1873.
spullara 3mo ago
It is much harder to prove these conjectures true than false. On some of these people had spent years of their life trying to prove them true. By showing they are definitely false that can be avoided.
paulpauper 3mo ago
some of the theories and conjectures are available for AI-assisted exploration because they are quite niche and not very important. The fact this has gone viral, shows otherwise. It is important to apparently enough people that the story went viral.
- subsistence234 3mo ago
  "AI solved another conjecture" is exciting for a lot of people who have otherwise no interest in math. So the fact that it goes viral may have very little connection to the mathematical importance of the result.

deleted 3mo ago

[deleted]

ashleyn 3mo ago

They might be low-hanging fruit but two things immediately come to mind: * As more of the small stuff is just proven for free, the more they can be used as a basis for other proofs. If you know something is true or false for certain, that can be a significant tailwind for the much harder, much more important problems. Fermat's last theorem looks deceptively simple and invited many failed amateur attempts at solving it, but Wiles' proof drew on a diversity of seemingly-distant subfields within mathematics that were better understood. * What are aspiring math Phd's supposed to do, now that the bar is much higher these days? The net effect of this appears to be that we'll see far fewer, but far more elite math Phd's, potentially discouraging many young people from the field.

throwaway0123_5 3mo ago
> The net effect of this appears to be that we'll see far fewer, but far more elite math Phd's, potentially discouraging many young people from the field. It seems plausible that the value of education will go down for the vast majority of fields and as a result less people will be getting degrees of all types. Not a good outcome I think for humanity to be less educated, even if people are provided for when they can't get jobs... things like mathematical and scientific literacy, as well as history knowledge (which even STEM majors often receive via undergraduate degree breadth requirements), etc. I would expect strongly result in more informed and harder to deceive citizens.
- CWuestefeld 3mo ago
  A large proportion of degrees awarded today are not useful for any practical application. But having a degree of some kind still serves as a signaling mechanism, demonstrating lots of things including the ability to "play the game". to follow instructions, and so forth.
  - zarzavat 3mo ago
    If I were 18 I'd be sceptical about the value of degrees compared to the value of trades. Yes degrees are a prerequisite for white collar jobs but if there is competition for ever fewer white collar jobs that's a life gamble perhaps not worth taking unless you are supremely confident in your abilities in a field. If we're being honest, the value of a degree has been in decline for a long time. It was only being held up by STEM and even that is on shaky ground now. The days when an e.g. English degree would open doors are long over.
    - ConceptJunkie 2mo ago
      There's going to be a huge realignment in white collar work in the next decade or two, and then there's the fact that college degrees themselves (except for maybe STEM), are becoming less and less valuable because higher education is so politicized and taken over by nonsense. My college education (Computer Science, 1987) served me very well over my career, but I don't know what I'd be doing if I were 18 now.
  - throwaway0123_5 3mo ago
    I agree somewhat, but for the most part the signal is that you can succeed in some kind of knowledge work. If AI replaces ~all knowledge work, that signal isn't actually going to be valuable I think. While I don't find it certain that AI will do that, it seems increasingly plausible with every model release. In that world, I don't think very many people are pursuing degrees.
  - cedws 3mo ago
    I never understood why being able to “play the game” is considered a positive signal. Getting a degree is the safe path, it doesn’t require any risk taking, and not much initiative. It’s about having the money (or going into debt for most people) and sitting your ass in a chair until you’re told you’re worthy. That idea is repulsive to me.
julianeon 3mo ago
Is the bar really higher? Those math PhD's can use GPT too; they benefit equally from AI assistance.
- fn-mote 3mo ago
  This approach totally changes the game, in ways we are still discussing. The current consensus is that domain experts get the most out of using AI on problems. How will domain expertise develop when AI is doing the work? I’m not saying that there won’t be another approach that builds on the strengths of AI, but we have to look for that and develop it. There’s a lot to talk about.
  - hgoel 3mo ago
    A decent chunk of the work a student does during a PhD is stuff that their advisor would have done in much less time, but are intentionally delegating to the student to allow them to learn. So, I don't think AI changes the way expertise is earned. It might change the cost tradeoffs depending on where AI costs eventually settle.
chermi 3mo ago
Maybe masters will become more popular. I have no problem with bars being raised on PhDs, but it's crazy history that math may be the first one to have it raised (or brought back to old levels).
- StableAlkyne 3mo ago
  IMO, I don't think this will raise the bar for PhDs. In every field of science, PhD student research is largely incremental. Very rarely is a thesis groundbreaking. The point of it is all is to function as an apprenticeship for that PhD student to become a scientist. Sometimes a particularly gifted or lucky one hits an important result, but that's rare and isn't the purpose.
  - chermi 3mo ago
    I agree, but there's degrees (no pun intended!). Many PhDs are just an advisor telling a student exactly what to do. The training is still valuable, but it's not really what a PhD is "supposed" to mean. The example I give because of familiarity is many PhDs in chemistry/physics are just clicking run on simulations + running predefined analysis with results already predicted by the PI. Or in chemistry specifically OChem wet lab, it's often basically a minimum wage factory job.
hgoel 3mo ago
I don't believe this is raising the bar for minting fresh PhDs. We need to maintain perspective here, a PhD is essentially work done by a researcher at the least experienced, least skilled point of their career. Their primary goal is to demonstrate that they are capable of contributing to research. They are not competing against AI to publish a counterexample to a known conjecture.
ndriscoll 3mo ago
The bar for math PhDs already ruled out like 99.9% of the population, so I don't see it having much effect on discouraging people. The gap between even a bright student who takes AP calculus or whatever and someone studying e.g. spectral sequences is already incomprehensibly large. Like you literally could not even convey to a smart young person how far away they are from the boundary of today's understanding. I don't think I even have a reasonable sense with a math bachelor's! People who get into math do it because they can't not do it.
- ian-g 3mo ago
  I think it'll become increasingly important to have folks thinking about how to explain new math in a way that makes sense to your math bachelors students. And to your bright AP Calc students. If we can point ChatGPT at these problems and get eventual answers, that's awesome, but it'll still be important to figure out how to tell people why this matters in ways they understand.
- ConceptJunkie 2mo ago
  I'll look at math papers on Arxiv, etc., and browsing the titles will find maybe one in a hundred where I even know what it's talking about. I've been a math enthusiast since I was a kid, but the field is so immense I can't even wrap my head around how big it is, leave alone the actual concepts. Out of all the topics mentioned in the recent AI advances at finding counterexamples to conjectures, etc., about the only one I understood was the one where they've made advances in a bounds for kissing numbers in higher dimension, a topic I find fascinating even if I don't really know much about it.
jauntywundrkind 3mo ago
Notably we have had very few conjectures proven true, so it doesn't feel like there is that widening base of established map to draw yet more proofs upon.
altairprime 3mo ago
As a potential elite math Ph.D, though sadly I can't afford to complete studies in that direction yet, both before and after A.I. I have been interested in the field for synthesis of different fields of mathematics into novel outcomes. A.I. is, as with protein and antenna folding, excellent at constraint solving and optimization — but human creativity continues to excel at creating new frontiers that are not found within the 'local' maxima explored by algorithms. Consider that we have brute-force calculation tests that can be applied to an entire floating-point 2^32 space and I guarantee a lot of A.I. effort invested in finding more efficient storage methods, but there's still room for human invention of floating-point storage simply by a person making a leap between Field One and Field Two that hasn't been proven yet: > We would like to stress that the key idea behind ALP is to design for vectorized execution; it led us to analyze and uncover unexploited opportunities from a vector perspective in a variety of datasets. — doi.org/10.1145/3626717, via ALP: Adaptive lossless floating-point compression (6 days ago, 4 comments) https://news.ycombinator.com/item?id=49051355 https://news.ycombinator.com/item?id=49051355 Perhaps future mathematicians use A.I. to prove it, but that's still a human invention — and even if we brute force the entire space of mathematics, it's functionally useless without an ontology to help humans narrow an A.I. 'problem-solving' response from 'all possible solution spaces' to 'the interesting solution spaces', and even if that ontology is A.I. created, a human is still going to be evaluating 'interesting' through their own cognitive experiences, biases, and dissonances in order to identify novel connections for other humans to build with. Just because we can `echo 0..2^64-1 >> file.txt` doesn't mean that we're deriving value from it, even if it's indexed by number of digits or nearest power of two or whatever. OEIS exists, and cannot be easily replaced by an A.I., because it's not just a list of numbers indexed by sequence (which no doubt computerized proofs plus generative algorithms will eventually automate the generation of), it's a list of interesting to humans sequences, and that's something A.I. can't be substituted for. > Director Cary Fukunaga mentioned a complex narrative structure in his 2018 big pharma miniseries Maniac being nixed because of the audience loss predicted by the data. — via Bland, for fans of everything (11 months ago, 7 comments) https://news.ycombinator.com/item?id=45049412 https://news.ycombinator.com/item?id=45049412

moomin 3mo ago

Yeah, the Jacobian conjecture counter-example was big news. In particular, it would have been news even if an AI hadn't done it. That's where the bar is now. Settling Erdős conjecture 7529 or whatever no longer qualifies as AI news.

tsunamifury 3mo ago
Real talk. AI solving these makes me feel like mathematicians put far more importance on their work than was actually there. Many solutions seems to be tautological games, and games of logic where conjecture puzzles that few work on or care can be solved by AI which doesn’t care what it works on. It seems to always be some form of this: Mathematician: “Propose conjecture a and conjecture b can’t be true simultaneously” AI: “they can” Everyone: “ok…” I know this might be unfair or out of ignorance but it genuinely is how this field feels today. Games of games with self importance added in. Edit: the point I should have made is, should we be using AI to figure out what proofs MATTER now vs games of proofs?
- efficax 3mo ago
  "tautological games". All proofs are a form of tautology, you have to end up back at the point your theorem proposed. Math is games of logic. That's what it is.
- dwaltrip 3mo ago
  Look up how pure mathematics connects back to reality in countless unexpected and useful ways, time and time again.
- deeznuttynutz 3mo ago
  Don't do that...
- chermi 3mo ago
  Very useful games, self-importance or no. See pattern, conjecture generalization, test generalization. It's almost like empirical math. I like it and I also like mathematicians doing it the old way.

astro1234 3mo ago

I find them useful bellweathers of genuinely out of domain performance and capability, regardless of their theoretical importance. What I see is that performance trends are remarkably stable both upstream (miraculous scaling laws of pretraining on validation loss) and downstream performance (epoch capability index). We get the equivalent of a GPT4->GPT5 performance leap every ~16-18 months, and we are not hitting ceilings nor do we see any deceleration. Today we can solve nontrivial open problems. What will we be able to do next year or the year after? 18months ago no one was using a coding agent seriously. Now for a large segment of the population you cannot do your job without them.

gjm11 3mo ago

I'm not sure that's right. The conjecture is (claimed to be, by the people who explicitly made it) simply a reformulation of a claim made by Maxwell in his "Treatise on Electricity and Magnetism" of 1873.

spullara 3mo ago

It is much harder to prove these conjectures true than false. On some of these people had spent years of their life trying to prove them true. By showing they are definitely false that can be avoided.

paulpauper 3mo ago

some of the theories and conjectures are available for AI-assisted exploration because they are quite niche and not very important. The fact this has gone viral, shows otherwise. It is important to apparently enough people that the story went viral.

subsistence234 3mo ago
"AI solved another conjecture" is exciting for a lot of people who have otherwise no interest in math. So the fact that it goes viral may have very little connection to the mathematical importance of the result.