I apologize: I am going to be obtusely thick-headed and focus on the mathematical example you used to introduce your essay rather than the point you used it to make.
I'll start with Michael Harris's thoughtful post "About that Erdős problem" (https://siliconreckoner.substack.com/p/about-that-erdos-problem). I won't try to summarize this post, but I want to draw on one point it makes explicitly and a second by implication. The former is that we don't know how many proofs were wrongly claimed and discarded before finding one that worked. The latter is that the Erdős unit distance conjecture was *disproved by finding a counterexample* (which can be extended to a schema for finding counterexamples.) In other words, *as it turns out*, this problem is well-suited to computer strengths and human weaknesses, in that it can be addressed by generating candidate disproofs and testing them; nobody is going to be surprised that a computer can do this faster than a human. The distinctively "AI" contribution is that the disproof (apparently spontaneously) drew on published human work to generate its construction in a way that its authors had not anticipated. It is not at all obvious that this approach would have worked had the Erdős conjecture been correct.
That is novel and valuable, compared to previous computer proofs where the computer had to be told explicitly what to do, but now try reading this Quanta summary (https://www.quantamagazine.org/two-researchers-are-rebuilding-mathematics-from-the-ground-up-20260520/) of work by Peter Scholze and Dustin Clausen on "condensed sets", an idea they have for replacing topology in a way that would make it more compatible with other branches of mathematics. "Scholze prefers coming up with new definitions rather than coming up with new proofs", writes the author, and another mathematician is quoted saying that "they are solving a problem we didn’t know we had.” Do you believe that the current development direction of LLMs will be capable of this sort of work in the foreseeable future? Well I do not. It is of a different order entirely than finding a counterexample.
ah, makes sense that disproving a conjecture by counterexample is the easiest kind of math for AI. try thousands and check each one and keep what works? i guess brute forcing w/ verification is where computers have been good. But to your point, to be fair, we don't see humans failed attempts either.
Please excuse me; I ought to have explained that point more fully. Here is the full quotation of Melanie Matchett Wood reproduced by Harris:
"This result does not show us all the times AI has claimed to have a proof of something and been wrong. Without that context (which many of us have just from personal experience), it is also easy to draw incorrect conclusions about the current state of AI and research mathematics."
Mathematicians have wrong ideas all the time, but I do not agree that they often assert that their wrong ideas are correct. The only live example I can think of is Mochizuki's claimed proof of the ABC conjecture. The mathematical community does not accept it, although they are reluctant to assert that he is wrong.
yes. had to look up the Mochizuki case but that sounds right - human mathematicians don't usually assert wrong results as correct bc they have internal quality control and AI does not, or at least not in the same way
Great post. I do also wonder whether AI can used in interesting simulator ways for when we do use social science to make reality à la deterrence theory or McKenzie engine not a camera world (https://mitpress.mit.edu/9780262633673/an-engine-not-a-camera/). Like how does the world change when people do start behaving in these very specific ways that theory says they ought to
Yes very much what I was trying to get at. In the McKenzie example they literally take futures pricing theory and make sure that the rules of the market start adhering to its assumptions
so the model shows what the world does as agents start behaving the way a theory says they ought to? that sounds interesting. at first glance I want to say no, since the sim "confirms" the theory because the agents learned the theory. That's the "trapped in training data" point. But then thinking about it some more though, if in the real world, theories work through people internalizing it, then LLM agents internalizing the theory is the same mechanism. I don't know enough about this, but sounds interesting
"So naturally that got me thinking: do we have anything resembling Erdos Problems in social science?"
No. We don't even have anybody resembling an Erdos.
Which is a joke but also not a joke. It's not surprising that the same LLMs that were good at Go and chess would be good at Erdos problems. These are are all closed systems containing discrete problems, and since AI models are similar to game engines they are great at simulating game outcomes. Inferring from those to non-discrete problems in open systems is how you end up blowing up a girls' school in Tehran on the road to WW3, while telling everyone how committed to "progress" you are.
I'm sure 'AI' could solve this-or-that higher-order game theory problem better than many/most human researchers, because game theory is a sandbox (which is to that it is abstract math, highly stylized, like an Erdos problem). In fact, we've been using computer programs to solve game theory problems for close to 50 years now.
In that time we've learned much more about how analytical reductionism can generate over-confidence that produces calamity. So the social science story of the past 25 years is that of minimizing reliance on highly-abstracted frames.
AI systems are being used to target kill and people. That is the "problem" that will be "solved" by them, the most analogous thing to an Erdos problem: "we want to kill bin Laden, but we can't find him, so let's see what the bots can do".
But the only enduring lesson of IR, really, is that we believe killing people will *not* solve more problems, at least generally. What solves problems is removing the ability for people to do harm. AI is not reducing the ability to cause harm, it is massively MASSIVELY increasing it at all scales.
So the main implication from IR for AI is simple: we should shut it down until it is strictly regulated at global level, in more or less the same way as nuclear technology. Hopefully we don't have to have experience some Hiroshimas before that happens, but I suspect we will.
partly a joke. someone else said to find the Erdos of social science we must first identify the top itinerant amphetamine addict of social science
the reductionism-breeds-overconfidence point: I agree but it almost proves too much because it becomes an argument against formalization generally. if the problem is that abstraction generates false precision then forget AI, most of modern quant social science is guilty too
"But the only enduring lesson of IR, really, is that we believe killing people will *not* solve more problems, at least generally." - yep, nothing to add or disagree with here
"we must first identify the top itinerant amphetamine addict of social science"
It says a lot about the current sociology of social science that Erdos could not have a career in our era. Certainly not at a top institution, which are still debating "challenges to the liberal order" last time I checked.
"it becomes an argument against formalization generally"
Agreed, or at least it recasts the project. Lots of expansive thinkers have used formalization at various points, but not for purposes of "testing" narrow hypotheses.
"most of modern quant social science is guilty too"
Not only do I also agree with that, it is the central argument of my research career (which is largely why I don't have one anymore).
It's not even an argument anymore: social science is by-and-large silent while the world it thought it was explaining disintegrates in ways it completely failed to anticipate, at all scales. Top researchers at elite institutions hope bots will figure all this out for them? They've been wrong about almost everything else, why would they be right about this?
Modern quant social science is premised on a set of observably false assumptions It wants a frictionless world, like the world of mathematics (elites always want to pretend things are less structurally-determined than they are). But as Elinor Ostrom noted in her Nobel Lecture, in 2009 (following a decade of events that social scientists also failed to anticipate or even satisfactorily explain):
"To explain the world of interactions and outcomes occurring at multiple levels, we also have to be willing to deal with complexity instead of rejecting it. Some mathematical models are very useful for explaining outcomes in particular settings. We should continue to use simple models where they capture enough of the core underlying structure and incentives that they usefully predict outcomes. When the world we are trying to explain and improve, however, is not well described by a simple model, we must continue to improve our frameworks and theories so as to be able to understand complexity and not simply reject it."
Social science mostly ignored this for the past 17 years, instead doing survey experiments on irrelevant questions and calling that "the gold standard of science". Which is an anti-scientific claim in and of itself.
To the extent the bots are trained on anything they are trained on these crap models. Which is why they end up blowing up girls' schools in Tehran.
The epistemic stance at the end resonated with me — as much progress can come from jettisoning old frameworks as adopting new ones. Which raises a question implied in your AI argument: if we over-rely on AI and end up removing humans from the process of thinking and discovery, do we risk even more path dependency than we have already, institutionalizing the map at the expense of exploring more territory?
IMO this type of question is why Gadamer is the most important philosopher for AI - his entire career was spent on trying to ascertain the limits of "Method" (logical/mathematical thinking of the kid AI excels at) for the humanities and social sciences, mostly because method is constrained by not considering interpretation and intellectual history as important parts of understanding
damn I have not heard that name in a while. I know very little of gadamer but the basic idea that no method gets you to understanding without interpretation seems very correct
This is a great post! One thing though is the Erdos problems are just problems that the mathematician Paul Erdos - the most prolific of the twentieth century - deemed noteworthy and offered a cash prize to anyone who solved one of varying amounts according to difficulty and importance. They are not necessarily the most important problems in all of mathematics. For example, what is often seen and argued as the most important unsolved problem in mathematics - the Riemann hypothesis - is not an Erdos problem. If that ends up being solved by an AI that is essentially the point that math researchers would be under-laborers or midwives to AI, if the profession would survive at all.
I think the one take-away from this is what should have been obvious for decades: we need generalists. Specialization to a degree is important, but we need people who are passionately interested in many different things as well as engagement with others of seemingly completely different fields. This is the main way that true innovation can flourish.
I am also very critical of democratic peace theory. It is such a small sample size that we are looking at, its defenders engage in a 'no true Scotsman' whenever people provide a counterexample while ignoring that each democracy has a different system itself, and there are how many other variables involved such as most democracies have a large established middle class. It reminds me of the man who wrote about the 'great illusion' that, with the expansion of free markets war could never be in anyone's interest, and so the Concert of Europe had done away with the scourge of war - published in the decade preceding WWI.
Thanks! you're right erdos problems aren't the summit of mathematics, I mean it as a jumping off point. But even solving something like the Riemann would still mean solving a problem with a clear answer, which is different from the kind of conceptual reframing that I think is the harder question for AI.
DPT - man, I don't know what to say. It's the phlogiston of political science theory. I'm not a fan. But as an example of an empirical finding preceding theorization, it cannot be beat
AI is undoubtedly gonna see a lot of (mis)use in defense analysis, where you already have a lot of discussion about simulation, what makes it good or useful, its limits, its hazards (and plenty of practice ignoring all that discussion).
I'm not sure the main problem is reflexivity--though that certainly is a problem! But the core problem is more like self-containment. Pure math is completely self-contained; if you know the axioms of your mathematical system, in some sense you know everything there is to know.
Anything touching the physical world is not self-contained, there will always be problems that are out of sample, unless your sample literally recapitulates the entire physical world at every possible level of analysis. This problem gets worse as you go from physics to social science, but as a continuum rather than a sharp break.
interesting but if self-containment was the core problem shouldn't AI be crushing conceptual innovation in math. that's as fully-contained as it gets. I mean you're right that self-containment is necessary but it's not sufficient. The world generates out-of-sample novelty that isn't in the training data, but it doesn't explain the hardest cases even within math. also I think reflexivity is a specific kind of non-self-containment that's qualitatively different
"The kind of theoretical breakthroughs that reshape social science involve a specific cognitive move: they take what everyone already sees and reorganize it in a way that makes new things visible." - yes, I agree. A kind of synthesis, not bound by one field, or identity attachment, but across multiple layers, and uniting them. This is the outcome of an article I published, my framework on human suffering that came from my own experiences.. and in trying to understand it, I built it. I don't really know where else to share, other than saying, I resonated with your quote.
It is here in case you ever want to look at it. Thank you.
Consider me as a tool of our simulation. There are at least two possible, if not necessairy changes for a much better economical approach. Our past tendency to mount all kinds of societal costs and duties in the price of a settlement can be altered by seperating this part and have it borrowed by the (level of)state for a reasonable cheap rate as states can. Banks only need to lend for the stones so to speak. The state can choose which areas, communities onto persons to target in time. etc.etc. Second is investment in youth Lots of people even older are fully occupied with keeping sheltered without enough escape perspectives. They are to be mobilised into cheaper living, learning more and guided saving all in one. Government should promote organised camping in all possible ways, like what is happening already especially in the USA. In my country, the Netherlands, I figured a difference of $ 1000, since the cheapest campings can functions from off $200. And a family house grosses at least $1200. The result is freeing a lot of housing. you're igniting lots of connecting and learning for minimum engagements of like ten years before accessing the full savements made, in at least a steady form of house ownership. That is for economics as a start. Mind you, this is after my introduction of reorganising all the first principles of physics. For which I am waiting to have results. You can contact me on vosforr@gmail.com
I apologize: I am going to be obtusely thick-headed and focus on the mathematical example you used to introduce your essay rather than the point you used it to make.
I'll start with Michael Harris's thoughtful post "About that Erdős problem" (https://siliconreckoner.substack.com/p/about-that-erdos-problem). I won't try to summarize this post, but I want to draw on one point it makes explicitly and a second by implication. The former is that we don't know how many proofs were wrongly claimed and discarded before finding one that worked. The latter is that the Erdős unit distance conjecture was *disproved by finding a counterexample* (which can be extended to a schema for finding counterexamples.) In other words, *as it turns out*, this problem is well-suited to computer strengths and human weaknesses, in that it can be addressed by generating candidate disproofs and testing them; nobody is going to be surprised that a computer can do this faster than a human. The distinctively "AI" contribution is that the disproof (apparently spontaneously) drew on published human work to generate its construction in a way that its authors had not anticipated. It is not at all obvious that this approach would have worked had the Erdős conjecture been correct.
That is novel and valuable, compared to previous computer proofs where the computer had to be told explicitly what to do, but now try reading this Quanta summary (https://www.quantamagazine.org/two-researchers-are-rebuilding-mathematics-from-the-ground-up-20260520/) of work by Peter Scholze and Dustin Clausen on "condensed sets", an idea they have for replacing topology in a way that would make it more compatible with other branches of mathematics. "Scholze prefers coming up with new definitions rather than coming up with new proofs", writes the author, and another mathematician is quoted saying that "they are solving a problem we didn’t know we had.” Do you believe that the current development direction of LLMs will be capable of this sort of work in the foreseeable future? Well I do not. It is of a different order entirely than finding a counterexample.
ah, makes sense that disproving a conjecture by counterexample is the easiest kind of math for AI. try thousands and check each one and keep what works? i guess brute forcing w/ verification is where computers have been good. But to your point, to be fair, we don't see humans failed attempts either.
Please excuse me; I ought to have explained that point more fully. Here is the full quotation of Melanie Matchett Wood reproduced by Harris:
"This result does not show us all the times AI has claimed to have a proof of something and been wrong. Without that context (which many of us have just from personal experience), it is also easy to draw incorrect conclusions about the current state of AI and research mathematics."
Mathematicians have wrong ideas all the time, but I do not agree that they often assert that their wrong ideas are correct. The only live example I can think of is Mochizuki's claimed proof of the ABC conjecture. The mathematical community does not accept it, although they are reluctant to assert that he is wrong.
yes. had to look up the Mochizuki case but that sounds right - human mathematicians don't usually assert wrong results as correct bc they have internal quality control and AI does not, or at least not in the same way
Great post. I do also wonder whether AI can used in interesting simulator ways for when we do use social science to make reality à la deterrence theory or McKenzie engine not a camera world (https://mitpress.mit.edu/9780262633673/an-engine-not-a-camera/). Like how does the world change when people do start behaving in these very specific ways that theory says they ought to
Yes very much what I was trying to get at. In the McKenzie example they literally take futures pricing theory and make sure that the rules of the market start adhering to its assumptions
so the model shows what the world does as agents start behaving the way a theory says they ought to? that sounds interesting. at first glance I want to say no, since the sim "confirms" the theory because the agents learned the theory. That's the "trapped in training data" point. But then thinking about it some more though, if in the real world, theories work through people internalizing it, then LLM agents internalizing the theory is the same mechanism. I don't know enough about this, but sounds interesting
"So naturally that got me thinking: do we have anything resembling Erdos Problems in social science?"
No. We don't even have anybody resembling an Erdos.
Which is a joke but also not a joke. It's not surprising that the same LLMs that were good at Go and chess would be good at Erdos problems. These are are all closed systems containing discrete problems, and since AI models are similar to game engines they are great at simulating game outcomes. Inferring from those to non-discrete problems in open systems is how you end up blowing up a girls' school in Tehran on the road to WW3, while telling everyone how committed to "progress" you are.
I'm sure 'AI' could solve this-or-that higher-order game theory problem better than many/most human researchers, because game theory is a sandbox (which is to that it is abstract math, highly stylized, like an Erdos problem). In fact, we've been using computer programs to solve game theory problems for close to 50 years now.
In that time we've learned much more about how analytical reductionism can generate over-confidence that produces calamity. So the social science story of the past 25 years is that of minimizing reliance on highly-abstracted frames.
AI systems are being used to target kill and people. That is the "problem" that will be "solved" by them, the most analogous thing to an Erdos problem: "we want to kill bin Laden, but we can't find him, so let's see what the bots can do".
But the only enduring lesson of IR, really, is that we believe killing people will *not* solve more problems, at least generally. What solves problems is removing the ability for people to do harm. AI is not reducing the ability to cause harm, it is massively MASSIVELY increasing it at all scales.
So the main implication from IR for AI is simple: we should shut it down until it is strictly regulated at global level, in more or less the same way as nuclear technology. Hopefully we don't have to have experience some Hiroshimas before that happens, but I suspect we will.
partly a joke. someone else said to find the Erdos of social science we must first identify the top itinerant amphetamine addict of social science
the reductionism-breeds-overconfidence point: I agree but it almost proves too much because it becomes an argument against formalization generally. if the problem is that abstraction generates false precision then forget AI, most of modern quant social science is guilty too
"But the only enduring lesson of IR, really, is that we believe killing people will *not* solve more problems, at least generally." - yep, nothing to add or disagree with here
"we must first identify the top itinerant amphetamine addict of social science"
It says a lot about the current sociology of social science that Erdos could not have a career in our era. Certainly not at a top institution, which are still debating "challenges to the liberal order" last time I checked.
"it becomes an argument against formalization generally"
Agreed, or at least it recasts the project. Lots of expansive thinkers have used formalization at various points, but not for purposes of "testing" narrow hypotheses.
"most of modern quant social science is guilty too"
Not only do I also agree with that, it is the central argument of my research career (which is largely why I don't have one anymore).
It's not even an argument anymore: social science is by-and-large silent while the world it thought it was explaining disintegrates in ways it completely failed to anticipate, at all scales. Top researchers at elite institutions hope bots will figure all this out for them? They've been wrong about almost everything else, why would they be right about this?
Modern quant social science is premised on a set of observably false assumptions It wants a frictionless world, like the world of mathematics (elites always want to pretend things are less structurally-determined than they are). But as Elinor Ostrom noted in her Nobel Lecture, in 2009 (following a decade of events that social scientists also failed to anticipate or even satisfactorily explain):
"To explain the world of interactions and outcomes occurring at multiple levels, we also have to be willing to deal with complexity instead of rejecting it. Some mathematical models are very useful for explaining outcomes in particular settings. We should continue to use simple models where they capture enough of the core underlying structure and incentives that they usefully predict outcomes. When the world we are trying to explain and improve, however, is not well described by a simple model, we must continue to improve our frameworks and theories so as to be able to understand complexity and not simply reject it."
Social science mostly ignored this for the past 17 years, instead doing survey experiments on irrelevant questions and calling that "the gold standard of science". Which is an anti-scientific claim in and of itself.
To the extent the bots are trained on anything they are trained on these crap models. Which is why they end up blowing up girls' schools in Tehran.
Ostrom’s Nobel lecture is worth reading now more than ever btw: https://www.nobelprize.org/uploads/2018/06/ostrom_lecture.pdf
The epistemic stance at the end resonated with me — as much progress can come from jettisoning old frameworks as adopting new ones. Which raises a question implied in your AI argument: if we over-rely on AI and end up removing humans from the process of thinking and discovery, do we risk even more path dependency than we have already, institutionalizing the map at the expense of exploring more territory?
IMO this type of question is why Gadamer is the most important philosopher for AI - his entire career was spent on trying to ascertain the limits of "Method" (logical/mathematical thinking of the kid AI excels at) for the humanities and social sciences, mostly because method is constrained by not considering interpretation and intellectual history as important parts of understanding
damn I have not heard that name in a while. I know very little of gadamer but the basic idea that no method gets you to understanding without interpretation seems very correct
This is a great post! One thing though is the Erdos problems are just problems that the mathematician Paul Erdos - the most prolific of the twentieth century - deemed noteworthy and offered a cash prize to anyone who solved one of varying amounts according to difficulty and importance. They are not necessarily the most important problems in all of mathematics. For example, what is often seen and argued as the most important unsolved problem in mathematics - the Riemann hypothesis - is not an Erdos problem. If that ends up being solved by an AI that is essentially the point that math researchers would be under-laborers or midwives to AI, if the profession would survive at all.
I think the one take-away from this is what should have been obvious for decades: we need generalists. Specialization to a degree is important, but we need people who are passionately interested in many different things as well as engagement with others of seemingly completely different fields. This is the main way that true innovation can flourish.
I am also very critical of democratic peace theory. It is such a small sample size that we are looking at, its defenders engage in a 'no true Scotsman' whenever people provide a counterexample while ignoring that each democracy has a different system itself, and there are how many other variables involved such as most democracies have a large established middle class. It reminds me of the man who wrote about the 'great illusion' that, with the expansion of free markets war could never be in anyone's interest, and so the Concert of Europe had done away with the scourge of war - published in the decade preceding WWI.
Thanks! you're right erdos problems aren't the summit of mathematics, I mean it as a jumping off point. But even solving something like the Riemann would still mean solving a problem with a clear answer, which is different from the kind of conceptual reframing that I think is the harder question for AI.
DPT - man, I don't know what to say. It's the phlogiston of political science theory. I'm not a fan. But as an example of an empirical finding preceding theorization, it cannot be beat
Thanks for this very interesting post. A small question stemming from it: where does Anthony Giddens refer to the “double hermeneutic”? Thanks.
he talked about it forever but I think his social theory book from the 70s is where it starts.
The Constitution of Society, that's it. I had to read small parts in grad school
Thanks.
AI is undoubtedly gonna see a lot of (mis)use in defense analysis, where you already have a lot of discussion about simulation, what makes it good or useful, its limits, its hazards (and plenty of practice ignoring all that discussion).
there are going to be so many people confirming their priors with AI simulations where the parameters just happen to suit their assumptions
I'm not sure the main problem is reflexivity--though that certainly is a problem! But the core problem is more like self-containment. Pure math is completely self-contained; if you know the axioms of your mathematical system, in some sense you know everything there is to know.
Anything touching the physical world is not self-contained, there will always be problems that are out of sample, unless your sample literally recapitulates the entire physical world at every possible level of analysis. This problem gets worse as you go from physics to social science, but as a continuum rather than a sharp break.
interesting but if self-containment was the core problem shouldn't AI be crushing conceptual innovation in math. that's as fully-contained as it gets. I mean you're right that self-containment is necessary but it's not sufficient. The world generates out-of-sample novelty that isn't in the training data, but it doesn't explain the hardest cases even within math. also I think reflexivity is a specific kind of non-self-containment that's qualitatively different
"The kind of theoretical breakthroughs that reshape social science involve a specific cognitive move: they take what everyone already sees and reorganize it in a way that makes new things visible." - yes, I agree. A kind of synthesis, not bound by one field, or identity attachment, but across multiple layers, and uniting them. This is the outcome of an article I published, my framework on human suffering that came from my own experiences.. and in trying to understand it, I built it. I don't really know where else to share, other than saying, I resonated with your quote.
It is here in case you ever want to look at it. Thank you.
https://johnsplace.substack.com/p/the-prometheus-theory?r=88drbw&utm_campaign=post&utm_medium=web
Consider me as a tool of our simulation. There are at least two possible, if not necessairy changes for a much better economical approach. Our past tendency to mount all kinds of societal costs and duties in the price of a settlement can be altered by seperating this part and have it borrowed by the (level of)state for a reasonable cheap rate as states can. Banks only need to lend for the stones so to speak. The state can choose which areas, communities onto persons to target in time. etc.etc. Second is investment in youth Lots of people even older are fully occupied with keeping sheltered without enough escape perspectives. They are to be mobilised into cheaper living, learning more and guided saving all in one. Government should promote organised camping in all possible ways, like what is happening already especially in the USA. In my country, the Netherlands, I figured a difference of $ 1000, since the cheapest campings can functions from off $200. And a family house grosses at least $1200. The result is freeing a lot of housing. you're igniting lots of connecting and learning for minimum engagements of like ten years before accessing the full savements made, in at least a steady form of house ownership. That is for economics as a start. Mind you, this is after my introduction of reorganising all the first principles of physics. For which I am waiting to have results. You can contact me on vosforr@gmail.com