3 ms·

Much of math (or science) research has the strange quality of being mostly curiosity-driven, but having giant benefits that occasionally spin out to the public.

by turzmo 4mo ago

Much of math (or science) research has the strange quality of being mostly curiosity-driven, but having giant benefits that occasionally spin out to the public. Some questions are more urgent and practical. My feeling is that the more directly practical a question is, the more likely the research community is to support AI usage in that question. The annoying thing about recent AI advances is that they target questions on the wrong end of the spectrum: Erdos problems are exactly the sort of "useless" questions that people might answer purely for the love of the game. The sort of questions that a young person might cut their teeth on and gain confidence. Solving questions like these automatically, I think, is not good for the long-term health of research. At least for the foreseeable future you still would like people to become interested and develop skills in these fields. These developments, and especially how they are presented, directly discourage that.

BigGreenJorts 4mo ago
Sounds like yet another example of how AI is kneecapping industries from the bottom by "removing the barrier to entry" but really just removing the training path by doing the work itself with no guidance for juniors.
- brador 4mo ago
  We are on tiny 1-5T parameter models with local power stations. We can reach Q models just by throwing resources at it. That’s a million times current B models.
  - bdamm 4mo ago
    Is this a known or quantifiable thing? I thought that the limit had already been determined i.e. the existing models top out and at some point it doesn't matter how much time or energy you let the model consume, it won't improve the result. And with regards to training parameters, I thought we were equally limited there, e.g. the existing models can't benefit from a larger parameter space. I was under the impression that improvements are arriving via how the models are trained and how model prompting context is constructed, rather than just by how much data or how much energy is spent searching over the model space for a particular prompt. Is there some evidence that we have not reached a pleateau with just resource consumption on existing models?
    - int_19h 4mo ago
      The existing models "top out" not because they don't get better, but because it is uneconomical. What we do know is that a model "tops out" wrt training data - that is, for a model of a given size, there's only so much training data you can squeeze into the set before you stop seeing gains. But conversely it means that if you already have a model of say 1 Ttok that is "trained to capacity", then a model of 2 TTok needs roughly twice as much training data to fully utilize all those weights. Which means that the cost of training it is not 2x but 4x (twice as many params x twice as many tokens). And then of course serving it is 2x more expensive, but even with optimal training the gains aren't 2x. So it very quickly becomes uneconomical. A good example of that kind of model is (was) GPT-4.5. The prices and the consequent lack of demand show why companies don't really do that sort of thing anymore. But no, there's no evidence of a plateau as such. I'm not sure what "evidence that we have not reached a plateau" would even look like.
  - jazzyjackson 4mo ago
    You cannot think fast enough when your wires are kilometers long. The only way up is in, and silicon transistors just cannot compete with density with biologic brains, ergo, super intelligence is a pipe dream
    - fc417fc802 4mo ago
      Baseless assertions. Fab tech continues to improve. There's no reason ML model internals have to be strictly serial - in fact we're already seeing some shifts away from that.
  - sterlind 4mo ago
    what is a B model vs. a Q model? what do these letters mean?
    - brador 4mo ago
      B Billion parameter, T trillion, Q Quadrillion.
- scottLobster 4mo ago
  Yep, and if history is any guide the only way to play it is to take part and get rich while you can, or play the super long game and be positioned for the collapse. Businesses will not adapt until they are incentivized to do so, and very few businesses have a multi-decade outlook. Even before AI, the senior 10x employee who retired and took all his domain knowledge with him because there was never any funding to train his replacement was a problem.
yieldcrv 4mo ago
That's an interesting perspective and I wholly disagree with the conclusion You are saying that tough problems with no applicability are useful because people that you happen to respect got good by their curiosity and pursuit of trying to solve these kinds of problems and failing, but branching off into other cognitive areas as mathematicians Now if I know anything about math for the sake of math, and academics, these are the same people that lament the idea of intelligent people going to the finance sector or any other trade they just happen not to respect as much The similarity being that their exact criticism of why, something they don't respect and view as having little utility, is the exact reasoning presented here now that AI can solve their pointless problems What I'm seeing is that human mathematicians have a laundry list of problems they have failed to solve for decades, centuries, which is what they are funded and employed to do. "Computer" used to a human job title too. This leads me to being excited about AI one-shotting these problems, let move on to something else.
- ccppurcell 4mo ago
  I think you've slightly straw manned the lamentation there. Not that I agree with the lamentation, but using your talent to make the rich richer (which is what quants do, they are paid a fixed amount to provide a larger value up the chain), as opposed to advancing human knowledge, is the reason for the lament, not some sort of respectability issue.
  - le-mark 4mo ago
    Quants benefit from substantial bonus structure as part of their compensation.
    - ccppurcell 4mo ago
      Well exactly. One way to make money is to create a great product that solves a problem people have and market it effectively. That's a sort of idealised situation, an aspiration, but it's what I would call socially valuable. Another way is to help people who are already rich move their money around so that they become richer, in return for a fraction of the increase in wealth. I personally have no problem with that, everybody has to make a living, but that is not socially valuable. It is debatable how socially valuable pure mathematics research is. But you take a fixed fee and all your best work is public domain.
math_dandy 4mo ago
To me, the most interesting feature of the OpenAI solution of the Unit Distance (Erdös) Problem is that the solution - using deep algebraic number theory as a source of extremal combinatorial/geometric constructions - is much more interesting than the problem’s elementary statement might lead one to expect. Writing off Erdös’s problems as random, useless, or meaningless dismisses his mathematical intuition, second-to-none, and strikes me as somewhat uncharitable. Finally, I agree that AI threatens mathematical training by rendering an entire class of acolyte-level research problems solvable by prompt. But the Unit Distance Problem is not of this class.
- pfdietz 4mo ago
  > much more interesting than the problem’s elementary statement might lead one to expect This is reinforced by the immediate (human) use of the idea to resolve in the negative another significant problem, the sum-product conjecture on reals. Explanation of what was involved: https://www.erdosproblems.com/forum/thread/blog:6 https://www.erdosproblems.com/forum/thread/blog:6
- danbruc 4mo ago
  I am not a mathematician and did not read the unit distance solution too carefully, but my impression was that it used a variation of a known technique to solve the problem. And that makes perfect sense to me, there are a lot of techniques and lot of less relevant problems, I am not surprised that one can solve some of them with known techniques that just nobody has tried [hard enough] before. I am much more sceptical when it come to the important unsolved problems where every known technique has probably been tried several times over. In those instances it will probably take a true leap in understanding to solve them and I am sceptical that large language models are well suited for that because of the way they work.
  - math_dandy 4mo ago
    We're very fortunate to have had some very eminent mathematicians backfill the OpenAI proof with history, context, and a literature review [1]. Ideas behind the proof seem to have been "in the air". Indeed, looked at certain point of view, the OpenAI construction can be viewed as a high-dimensional generalization of a known low-dimensional one. In this vein see the remarks of Gowers, Sawin and Tsimerman in [1]. Are LLMs capable of "true leap[s] in understanding"? I have absolutely no idea. But LLMs keep surprising me. [1] https://arxiv.org/html/2605.20695v1 https://arxiv.org/html/2605.20695v1
azeirah 4mo ago
Do you not think that solutions to erdos problems might end up stepping stones to other important problems? Either by introducing new tools, or by proving things that were previously unproven that end up helping in unexpected ways? That's often how math goes, isn't it?
- math_dandy 4mo ago
  This is, indeed, how math often goes.
lovemenot 4mo ago
>> At least for the foreseeable future you still would like people to become interested and develop skills in these fields. These developments, and especially how they are presented, directly discourage that. This assumption may well turn out to be correct, but it is not self-evident. Nearly everyone who has ever got interested in mathematics got discouraged at some point and they left the field. Mathematics is very hard. Those very few that remained certainly have talent, but they also have characteristics that are necessary for success in a competitive field, which are perhaps less valuable per se. Such characteristics as may be over-represented in males for instance. This is not a point about gender differences, but about the intrinsic merit of different success factors. It seems equally possible that the above assumption will turn out to be diametrically incorrect. People that would have been discouraged before LLMs will now retain their curiosity longer. Democratisation is surely a possible outcome. Arguably, chess has never been as popular and accessible. And that discipline fell to AI three decades ago.
lo_zamoyski 4mo ago
I think nuance gets lost in these conversations. Your distinction between the practical and the theoretical is important. Practicality is important - everything we do is a matter of practicality of means or method, even how we pursue theoretical ends - but two points. First, there is more to life than the practical. Some truths are known for their own sake, even if they also tell us about still more profound truths (also known for their own sake) or may have incidental practical relevance and consequences in some other context. Second, while the theoretical terminus is the truth for its own sake, the practical terminus is always something other than itself. Well, what is that "something else"? You can't have an infinite regress of practicality. The meaning of a proximate, practical end is always other than itself. The practical requires an end beyond itself to justify it. I agree that most people don't seem to inquire much about such ultimate ends. Their thoughts are confined to the proximate. Of course, how have they determined what the proximate should be? Something for people to contemplate. Where science is concerned, it depends. On the one hand, there are fields that are certainly more theoretically oriented. It's not "the game" that motivates theory - that would make it mere recreation, with the truth taking a backseat - but the truth. (For this reason, I hesitate to call Erdos theoretically motivated. AFAICT, he was motivated by the challenge of problem solving and not the truth, insight, and understanding to be gained which would have been merely incidental and instrumental for him.) However, I would also say a good chunk of science is motivated by a background motivation of technology production and the mastery of nature. Think Francis Bacon who viewed science as an instrument of power and showed a preference for the "how" over the "what" (τόδε τι) or the "why" (τὸ διότι). This set the tone for a great deal of modern science. A great deal does less explaining and more predictive modeling, because predictive modeling can be sufficient for control. Indeed, a truly theoretical causal account and understanding of a thing's nature can be less useful as a practical instrument than a merely predictive model. Now, AI is a practical tool. I think they can be enormously useful as research aids, even in theoretical contexts, provided that one 1. understands their nature; 2. understands the purpose of the theoretical activity undertaken. What is their nature? Well, they're statistical models that can unearth interesting and useful correlations and patterns. But they are not reasoning and knowing things. Their results are generated mechanically and mindlessly. Knowing this means taking their results with a healthy skepticism and a critical eye. What about the purpose of theory? By analogy, think of a student in school who uses AI to complete all his assignments. Has he satisfied the purpose of those assignments? No, because the purpose of the assignments isn't to produce the effect - the solutions - per se, but to learn something. Theoretical work is like that; it's purpose is to understand and to grasp some truth. An AI can be used to assist this process, just as a calculator or a search engine can, but if you use it in a manner that circumvents that purpose instead of supporting it, then you're not achieve that purpose and wasting your time. What's the point?
empath75 4mo ago
I've been spending 3 weeks, as a non mathemetician, chasing down a particular, very simply-stated, but secretly quite complex problem, and AI has been _so incredibly helpful_, not just in making progress on it, and doing obvious stuff like formalizing in lean, doing literature searches, reading through 10 or 15 papers and summarizing the results for me and how they apply to what I'm doing, giving me enough of an introduction to _entire fields_, that I can talk intelligently about it (I've had email correspondence with a couple of professional mathematicians in a few different fields about it, who agreed that it's an interesting, simple, but difficult problem). I've gone from "this should be easy", to "okay, I've almost got a proof", to "this is impossible", to literally just nailing down a few remaining sub-cases out of an infinite family. I don't want to call anyone out, but I emailed one fairly famous mathemetician, and he literally said: "This is very interesting, I thought about it for a while, couldn't figure it out, but I thought ChatGPT had an interesting response..." and he linked me to his chatgpt transcript... (which, was actually helpful, because he asked it a better question than I was asking). I have a suspicion that math will quite soon be exactly like programming and fall to the same machinery that coding is. One thing that I noticed is that a common workflow I had was isolating hard subquestions in a self contained way and then "surveying" multiple different LLMs in a totally clean context. They would often say: "Oh, this is a obvious example of such-and-such" and immediately clear the barrier.
- losvedir 4mo ago
  I'd be very cautious about "AI psychosis" here, or at the very least becoming a "crank". I've read too many stories of people convincing themselves they're on the verge of some great discovery to not hear "3 weeks to become conversational in mathematical fields" and not see all kinds of red flags. I studied math at MIT and have several friends who are professors now and they deal with cranks all the time and since they're very kind and conflict averse people they tend to respond with perfunctory emails when they get inbounds like that. So just be wary. Your external validation may not be as strong as you think it is, though kudos to you for at least trying to step out of the AI vortex to attempt to ground yourself.
  - empath75 4mo ago
fasterik 4mo ago
It seems to me that when you have a tool that automates part of the work, it doesn't make the curiosity go away, it changes the landscape of what problems humans find interesting. Maybe Erdos problems are no longer a good entry-level benchmark for a researcher, but that's going to drive young researchers to explore other areas that might have been out of reach before AI-human collaboation.

by turzmo 4mo ago

BigGreenJorts 4mo ago

Sounds like yet another example of how AI is kneecapping industries from the bottom by "removing the barrier to entry" but really just removing the training path by doing the work itself with no guidance for juniors.

brador 4mo ago
We are on tiny 1-5T parameter models with local power stations. We can reach Q models just by throwing resources at it. That’s a million times current B models.
- bdamm 4mo ago
  Is this a known or quantifiable thing? I thought that the limit had already been determined i.e. the existing models top out and at some point it doesn't matter how much time or energy you let the model consume, it won't improve the result. And with regards to training parameters, I thought we were equally limited there, e.g. the existing models can't benefit from a larger parameter space. I was under the impression that improvements are arriving via how the models are trained and how model prompting context is constructed, rather than just by how much data or how much energy is spent searching over the model space for a particular prompt. Is there some evidence that we have not reached a pleateau with just resource consumption on existing models?
  - int_19h 4mo ago
    The existing models "top out" not because they don't get better, but because it is uneconomical. What we do know is that a model "tops out" wrt training data - that is, for a model of a given size, there's only so much training data you can squeeze into the set before you stop seeing gains. But conversely it means that if you already have a model of say 1 Ttok that is "trained to capacity", then a model of 2 TTok needs roughly twice as much training data to fully utilize all those weights. Which means that the cost of training it is not 2x but 4x (twice as many params x twice as many tokens). And then of course serving it is 2x more expensive, but even with optimal training the gains aren't 2x. So it very quickly becomes uneconomical. A good example of that kind of model is (was) GPT-4.5. The prices and the consequent lack of demand show why companies don't really do that sort of thing anymore. But no, there's no evidence of a plateau as such. I'm not sure what "evidence that we have not reached a plateau" would even look like.
- jazzyjackson 4mo ago
  You cannot think fast enough when your wires are kilometers long. The only way up is in, and silicon transistors just cannot compete with density with biologic brains, ergo, super intelligence is a pipe dream
  - fc417fc802 4mo ago
    Baseless assertions. Fab tech continues to improve. There's no reason ML model internals have to be strictly serial - in fact we're already seeing some shifts away from that.
- sterlind 4mo ago
  what is a B model vs. a Q model? what do these letters mean?
  - brador 4mo ago
    B Billion parameter, T trillion, Q Quadrillion.
scottLobster 4mo ago
Yep, and if history is any guide the only way to play it is to take part and get rich while you can, or play the super long game and be positioned for the collapse. Businesses will not adapt until they are incentivized to do so, and very few businesses have a multi-decade outlook. Even before AI, the senior 10x employee who retired and took all his domain knowledge with him because there was never any funding to train his replacement was a problem.

yieldcrv 4mo ago

That's an interesting perspective and I wholly disagree with the conclusion You are saying that tough problems with no applicability are useful because people that you happen to respect got good by their curiosity and pursuit of trying to solve these kinds of problems and failing, but branching off into other cognitive areas as mathematicians Now if I know anything about math for the sake of math, and academics, these are the same people that lament the idea of intelligent people going to the finance sector or any other trade they just happen not to respect as much The similarity being that their exact criticism of why, something they don't respect and view as having little utility, is the exact reasoning presented here now that AI can solve their pointless problems What I'm seeing is that human mathematicians have a laundry list of problems they have failed to solve for decades, centuries, which is what they are funded and employed to do. "Computer" used to a human job title too. This leads me to being excited about AI one-shotting these problems, let move on to something else.

ccppurcell 4mo ago
I think you've slightly straw manned the lamentation there. Not that I agree with the lamentation, but using your talent to make the rich richer (which is what quants do, they are paid a fixed amount to provide a larger value up the chain), as opposed to advancing human knowledge, is the reason for the lament, not some sort of respectability issue.
- le-mark 4mo ago
  Quants benefit from substantial bonus structure as part of their compensation.
  - ccppurcell 4mo ago
    Well exactly. One way to make money is to create a great product that solves a problem people have and market it effectively. That's a sort of idealised situation, an aspiration, but it's what I would call socially valuable. Another way is to help people who are already rich move their money around so that they become richer, in return for a fraction of the increase in wealth. I personally have no problem with that, everybody has to make a living, but that is not socially valuable. It is debatable how socially valuable pure mathematics research is. But you take a fixed fee and all your best work is public domain.

math_dandy 4mo ago

To me, the most interesting feature of the OpenAI solution of the Unit Distance (Erdös) Problem is that the solution - using deep algebraic number theory as a source of extremal combinatorial/geometric constructions - is much more interesting than the problem’s elementary statement might lead one to expect. Writing off Erdös’s problems as random, useless, or meaningless dismisses his mathematical intuition, second-to-none, and strikes me as somewhat uncharitable. Finally, I agree that AI threatens mathematical training by rendering an entire class of acolyte-level research problems solvable by prompt. But the Unit Distance Problem is not of this class.

pfdietz 4mo ago
> much more interesting than the problem’s elementary statement might lead one to expect This is reinforced by the immediate (human) use of the idea to resolve in the negative another significant problem, the sum-product conjecture on reals. Explanation of what was involved: https://www.erdosproblems.com/forum/thread/blog:6 https://www.erdosproblems.com/forum/thread/blog:6
danbruc 4mo ago
I am not a mathematician and did not read the unit distance solution too carefully, but my impression was that it used a variation of a known technique to solve the problem. And that makes perfect sense to me, there are a lot of techniques and lot of less relevant problems, I am not surprised that one can solve some of them with known techniques that just nobody has tried [hard enough] before. I am much more sceptical when it come to the important unsolved problems where every known technique has probably been tried several times over. In those instances it will probably take a true leap in understanding to solve them and I am sceptical that large language models are well suited for that because of the way they work.
- math_dandy 4mo ago
  We're very fortunate to have had some very eminent mathematicians backfill the OpenAI proof with history, context, and a literature review [1]. Ideas behind the proof seem to have been "in the air". Indeed, looked at certain point of view, the OpenAI construction can be viewed as a high-dimensional generalization of a known low-dimensional one. In this vein see the remarks of Gowers, Sawin and Tsimerman in [1]. Are LLMs capable of "true leap[s] in understanding"? I have absolutely no idea. But LLMs keep surprising me. [1] https://arxiv.org/html/2605.20695v1 https://arxiv.org/html/2605.20695v1

azeirah 4mo ago

Do you not think that solutions to erdos problems might end up stepping stones to other important problems? Either by introducing new tools, or by proving things that were previously unproven that end up helping in unexpected ways? That's often how math goes, isn't it?

math_dandy 4mo ago
This is, indeed, how math often goes.

lovemenot 4mo ago

>> At least for the foreseeable future you still would like people to become interested and develop skills in these fields. These developments, and especially how they are presented, directly discourage that. This assumption may well turn out to be correct, but it is not self-evident. Nearly everyone who has ever got interested in mathematics got discouraged at some point and they left the field. Mathematics is very hard. Those very few that remained certainly have talent, but they also have characteristics that are necessary for success in a competitive field, which are perhaps less valuable per se. Such characteristics as may be over-represented in males for instance. This is not a point about gender differences, but about the intrinsic merit of different success factors. It seems equally possible that the above assumption will turn out to be diametrically incorrect. People that would have been discouraged before LLMs will now retain their curiosity longer. Democratisation is surely a possible outcome. Arguably, chess has never been as popular and accessible. And that discipline fell to AI three decades ago.

lo_zamoyski 4mo ago

I think nuance gets lost in these conversations. Your distinction between the practical and the theoretical is important. Practicality is important - everything we do is a matter of practicality of means or method, even how we pursue theoretical ends - but two points. First, there is more to life than the practical. Some truths are known for their own sake, even if they also tell us about still more profound truths (also known for their own sake) or may have incidental practical relevance and consequences in some other context. Second, while the theoretical terminus is the truth for its own sake, the practical terminus is always something other than itself. Well, what is that "something else"? You can't have an infinite regress of practicality. The meaning of a proximate, practical end is always other than itself. The practical requires an end beyond itself to justify it. I agree that most people don't seem to inquire much about such ultimate ends. Their thoughts are confined to the proximate. Of course, how have they determined what the proximate should be? Something for people to contemplate. Where science is concerned, it depends. On the one hand, there are fields that are certainly more theoretically oriented. It's not "the game" that motivates theory - that would make it mere recreation, with the truth taking a backseat - but the truth. (For this reason, I hesitate to call Erdos theoretically motivated. AFAICT, he was motivated by the challenge of problem solving and not the truth, insight, and understanding to be gained which would have been merely incidental and instrumental for him.) However, I would also say a good chunk of science is motivated by a background motivation of technology production and the mastery of nature. Think Francis Bacon who viewed science as an instrument of power and showed a preference for the "how" over the "what" (τόδε τι) or the "why" (τὸ διότι). This set the tone for a great deal of modern science. A great deal does less explaining and more predictive modeling, because predictive modeling can be sufficient for control. Indeed, a truly theoretical causal account and understanding of a thing's nature can be less useful as a practical instrument than a merely predictive model. Now, AI is a practical tool. I think they can be enormously useful as research aids, even in theoretical contexts, provided that one 1. understands their nature; 2. understands the purpose of the theoretical activity undertaken. What is their nature? Well, they're statistical models that can unearth interesting and useful correlations and patterns. But they are not reasoning and knowing things. Their results are generated mechanically and mindlessly. Knowing this means taking their results with a healthy skepticism and a critical eye. What about the purpose of theory? By analogy, think of a student in school who uses AI to complete all his assignments. Has he satisfied the purpose of those assignments? No, because the purpose of the assignments isn't to produce the effect - the solutions - per se, but to learn something. Theoretical work is like that; it's purpose is to understand and to grasp some truth. An AI can be used to assist this process, just as a calculator or a search engine can, but if you use it in a manner that circumvents that purpose instead of supporting it, then you're not achieve that purpose and wasting your time. What's the point?

empath75 4mo ago

I've been spending 3 weeks, as a non mathemetician, chasing down a particular, very simply-stated, but secretly quite complex problem, and AI has been _so incredibly helpful_, not just in making progress on it, and doing obvious stuff like formalizing in lean, doing literature searches, reading through 10 or 15 papers and summarizing the results for me and how they apply to what I'm doing, giving me enough of an introduction to _entire fields_, that I can talk intelligently about it (I've had email correspondence with a couple of professional mathematicians in a few different fields about it, who agreed that it's an interesting, simple, but difficult problem). I've gone from "this should be easy", to "okay, I've almost got a proof", to "this is impossible", to literally just nailing down a few remaining sub-cases out of an infinite family. I don't want to call anyone out, but I emailed one fairly famous mathemetician, and he literally said: "This is very interesting, I thought about it for a while, couldn't figure it out, but I thought ChatGPT had an interesting response..." and he linked me to his chatgpt transcript... (which, was actually helpful, because he asked it a better question than I was asking). I have a suspicion that math will quite soon be exactly like programming and fall to the same machinery that coding is. One thing that I noticed is that a common workflow I had was isolating hard subquestions in a self contained way and then "surveying" multiple different LLMs in a totally clean context. They would often say: "Oh, this is a obvious example of such-and-such" and immediately clear the barrier.

losvedir 4mo ago
I'd be very cautious about "AI psychosis" here, or at the very least becoming a "crank". I've read too many stories of people convincing themselves they're on the verge of some great discovery to not hear "3 weeks to become conversational in mathematical fields" and not see all kinds of red flags. I studied math at MIT and have several friends who are professors now and they deal with cranks all the time and since they're very kind and conflict averse people they tend to respond with perfunctory emails when they get inbounds like that. So just be wary. Your external validation may not be as strong as you think it is, though kudos to you for at least trying to step out of the AI vortex to attempt to ground yourself.
- empath75 4mo ago

fasterik 4mo ago

It seems to me that when you have a tool that automates part of the work, it doesn't make the curiosity go away, it changes the landscape of what problems humans find interesting. Maybe Erdos problems are no longer a good entry-level benchmark for a researcher, but that's going to drive young researchers to explore other areas that might have been out of reach before AI-human collaboation.