For quite a few years of my adult life, I was in a situationship with pure mathematics. Specifically, a small lonely subfield of functional analysis. In his book on the subject, John Conway quipped that functional analysis, itself a somewhat modest corner of real analysis, is so vast that two researchers both working in it might not understand each other’s work. From my tiny specialized subfield, math looked very, very large and whatever progress I might make looked very, very inconsequential.
The situationship ended, and my ascendant career as a data-themed, quantitative substacker eventually began. However, I never lost at least high-level interest in mathematical progress and now it turns out that 2026 is mathematics’ annus mirabilis.
If you haven’t been following this, it started in May with OpenAI disproving the unit distance conjecture, which had been open since 1946. Wow! Then in July an Anthropic researcher disproved the Jacobian conjecture, open since 1939, by way of Twitter.
The professional mathematics community was under siege. The tech bro barbarians were at the gates!
The barbarians broke through the gates on September 8th when OpenAI published a long, recondite paper claiming to resolve one of the Clay Institute’s stated Millennium problems, the Navier-Stokes global regularity problem. These problems are the crown jewels of cutting edge mathematical research, though just this morning I saw a preprint claiming the formalization of this result might be sketchy. This was so shocking that some people (completely implausibly in my view) believed OpenAI had looked at the user chats from mathematicians working on the Navier-Stokes problem in coming up with their solution.
And I’d say the city was sacked in October, two days ago as of my writing this, when OpenAI published a github repository, laconically entitled “math”, with almost 800 novel mathematical results, a titanic corpus sprawled over almost all quantitative fields. This included significant advances toward resolving another Millennium problems, the Riemann Hypothesis
This siege lasted for five months. For comparison, Alaric’s first siege of Rome in 408 lasted around three months, though it would take longer for him to sack the city.
With this huge dump of varied results, we finally have the makings of a real dataset describing some dimensions of the accumulation of mathematical knowledge. In the thousands’ year history of math, something truly novel and unexpected has happened; this shocking sea change in how math will get done in the future provides an interesting natural experiment we might be able to use to study how math progresses. The rest of this post will be a descriptive study of how datasets like the OpenAI dump or the Erdős problems can be decomposed to shed light on this subject; by looking at what sorts of problems are resolved in these datasets, we might be able to understand what actually drives progress in mathematical research.
Background
Lots of mathematical progress happens in fit and bursts. On a given problem, sometimes it takes ages for a new idea or solution strategy to emerge, but once that happens, progress towards a complete solution can happen incredibly quickly. My favorite example of this is the case of Yitang Zhang.
In 2013, he was an obscure Chinese mathematician working as an untenured lecturer at UNH. It would be rude to call this a mathematical backwater, but it was not glamorous. That year, he published a major advancement in the study of twin primes. No really substantial progress had been made in the centuries since the twin prime conjecture had been formulated.
Twin primes are pairs of prime numbers that differ by 2. Euclid proved that there are infinitely many prime numbers, but this conjecture asks whether there are infinitely many such pairs of prime numbers that differ by 2. This is a much harder problem. Zhang published a novel argument that bounds the gap between pairs of prime numbers at less than 70 million infinitely often. Quite a long way off from the preferred bound of 2, but his breakthrough shocked the math world and brought him instant fame.
Almost immediately, work began to refine Zhang’s strategy and the bound got smaller. Much, much smaller. In just a year, the bound had shrunk all the way down to 246. The full solution is currently still out of reach, but it shows how unexpected progress can be and how fast it can work out.
It’s this dynamic that I’m interested in. What sorts of open problems get unexpected solutions or major breakthroughs and which ones permanently resist solution? What’s the model of causation here? Does one even exist or is the history of mathematical progress one giant path-dependent mess?
OpenAI data
The most interesting thing to me about this year’s progress in math was that the first and many of the subsequent discoveries were constructed counterexamples, not proofs. What this means is that someone guesses that some proposition is true, and then someone else (maybe an AI) comes along and makes an elaborate example showing that the proposition is false. Typically, it means some other assumption has to be made to get the proposition to be true or that the part of mathematical reality under investigation is queerer than we supposed. Some of the counterexamples in the OpenAI dump might offend whatever sense of aesthetics mathematicians have.
When all of this started happening in May, there were only a handful of examples of problems settled one way or another by AI. Now we have a lot more, an actual dataset. When the counterexamples started coming out, I wondered how much of this had to do with academic prestige; when a famous person conjectures something is true, do people put in less effort to construct a counterexample given the problem’s background? Are they maybe unconsciously avoiding contradicting a prestigious peer or perhaps believe construction of a counterexample is dumb because the conjecture is probably right? Those are all things that AIs don’t care about at all which might explain all the counterexamples. I started by looking at the 722 papers OpenAI published.
In the OpenAI dataset, of the resolved problems proposed by a famous mathematician, 19% are counterexamples. For what it’s worth, that seems like a pretty fantastic predictive record to me. Of the non-illustriously begotten conjectures, only 11% are counterexamples, an odds ratio of 1.8. So far, this looks pretty good for the academic prestige theory.
However, it might be wrong. Maybe we should adjust this for how long the conjecture has been open. After all, the longer it’s been open maybe the harder it is regardless of the prestige effects we’re supposing. When you do that adjustment, the odds ratio shrinks to 1.33. This still seems substantial, but is not statistically significant. Is such an adjustment even right though? That depends on why the age of the problem and its fame are related.
Erdős problem data
Paul Erdős was the 20th century’s most prolific mathematical problem solver. His interests spanned mainly graph theory, number theory, and combinatorics. Given all the math stuff that’s happened this year, I’m currently most of the way through Hoffman’s excellent biography of Erdős entitled The Man Who Loved Only Numbers, which provides some interesting details about his life I wasn’t aware of before reading it.
Erdős would offer cash bounties for the solution of mathematical problems he proposed and couldn’t quite solve himself. The size of the bounty was proportional to how difficult he judged the problem to be. The mathematician Thomas Bloom maintains erdosproblems.com. I chatted a little with him in May when the initial Erdős problems (including the unit distance conjecture mentioned above) started getting knocked down by the frontier AI labs and felt a little silly not knowing who he was.
Fig 3. Don’t you know who I am?!?!
The Erdős problem dataset provides an interesting comparator to the OpenAI dump. Erdős was basically the Mick Jagger of the math world for large parts of the 20th century (less the womanizing, but not the drug use). All of the Erdős problems had a famous proposer by definition.
I broke the resolved Erdős problems down by whether they occurred before or after the frontier labs seriously started pursuing mathematical greatness. Before, 26% of the resolved problems were counterexamples created by humans versus a stunning 39% after with AI use. There are some convoluted explanations for this, but the most parsimonious explanation here is that AIs are simply better than humans at constructing computationally convoluted counterexamples, and I think this is borne out in the field-level data.
Fig 4. Change in conjecture refutation rates before and after AI among Erdős problems.
Number theory sits near the bottom here, perhaps less amenable to such constructions. A proposition in these fields often respects a bound or it does not; a counterexample is often a specific integer in a rigid structure or one that must be astronomically large. There are fewer computational degrees of freedom that an AI system can use to make a counterexample to a proposition, or so I think.
I also want to rubberneck here a little bit on Bloom’s remark above that Erdős’ instincts were so great. Granted, Bloom didn’t have access to the OpenAI dataset when he said that, but Erdős had substantially more overturned conjectures than those in the resolved OpenAI problems that were proposed by famous people (26% versus 19%), and clearly this gets worse with AI (39% versus 19%). We don’t know the exact mechanism by which problems appear in the OpenAI dump, but we can look at the field-level composition of the two datasets.
Fig 5. Dataset composition by mathematical subfield.
OK, this actually sort of resolves it. The fields OpenAI and Erdős worked on are very different, but the effect goes both ways. The Erdős fields like combinatorics and graph theory are construction heavy, but number theory is not. I need to think a little more about this, but I’m not prepared to let Erdős off the hook completely here.
For quite a few years of my adult life, I was in a situationship with pure mathematics. Specifically, a small lonely subfield of functional analysis. In his book on the subject, John Conway quipped that functional analysis, itself a somewhat modest corner of real analysis, is so vast that two researchers both working in it might not understand each other’s work. From my tiny specialized subfield, math looked very, very large and whatever progress I might make looked very, very inconsequential.
The situationship ended, and my ascendant career as a data-themed, quantitative substacker eventually began. However, I never lost at least high-level interest in mathematical progress and now it turns out that 2026 is mathematics’ annus mirabilis.
If you haven’t been following this, it started in May with OpenAI disproving the unit distance conjecture, which had been open since 1946. Wow! Then in July an Anthropic researcher disproved the Jacobian conjecture, open since 1939, by way of Twitter.
https://x.com/__alpoge__/status/2079028340955197566?s=20
The professional mathematics community was under siege. The tech bro barbarians were at the gates!
The barbarians broke through the gates on September 8th when OpenAI published a long, recondite paper claiming to resolve one of the Clay Institute’s stated Millennium problems, the Navier-Stokes global regularity problem. These problems are the crown jewels of cutting edge mathematical research, though just this morning I saw a preprint claiming the formalization of this result might be sketchy. This was so shocking that some people (completely implausibly in my view) believed OpenAI had looked at the user chats from mathematicians working on the Navier-Stokes problem in coming up with their solution.
And I’d say the city was sacked in October, two days ago as of my writing this, when OpenAI published a github repository, laconically entitled “math”, with almost 800 novel mathematical results, a titanic corpus sprawled over almost all quantitative fields. This included significant advances toward resolving another Millennium problems, the Riemann Hypothesis
This siege lasted for five months. For comparison, Alaric’s first siege of Rome in 408 lasted around three months, though it would take longer for him to sack the city.
With this huge dump of varied results, we finally have the makings of a real dataset describing some dimensions of the accumulation of mathematical knowledge. In the thousands’ year history of math, something truly novel and unexpected has happened; this shocking sea change in how math will get done in the future provides an interesting natural experiment we might be able to use to study how math progresses. The rest of this post will be a descriptive study of how datasets like the OpenAI dump or the Erdős problems can be decomposed to shed light on this subject; by looking at what sorts of problems are resolved in these datasets, we might be able to understand what actually drives progress in mathematical research.
Background
Lots of mathematical progress happens in fit and bursts. On a given problem, sometimes it takes ages for a new idea or solution strategy to emerge, but once that happens, progress towards a complete solution can happen incredibly quickly. My favorite example of this is the case of Yitang Zhang.
In 2013, he was an obscure Chinese mathematician working as an untenured lecturer at UNH. It would be rude to call this a mathematical backwater, but it was not glamorous. That year, he published a major advancement in the study of twin primes. No really substantial progress had been made in the centuries since the twin prime conjecture had been formulated.
Twin primes are pairs of prime numbers that differ by 2. Euclid proved that there are infinitely many prime numbers, but this conjecture asks whether there are infinitely many such pairs of prime numbers that differ by 2. This is a much harder problem. Zhang published a novel argument that bounds the gap between pairs of prime numbers at less than 70 million infinitely often. Quite a long way off from the preferred bound of 2, but his breakthrough shocked the math world and brought him instant fame.
Almost immediately, work began to refine Zhang’s strategy and the bound got smaller. Much, much smaller. In just a year, the bound had shrunk all the way down to 246. The full solution is currently still out of reach, but it shows how unexpected progress can be and how fast it can work out.
It’s this dynamic that I’m interested in. What sorts of open problems get unexpected solutions or major breakthroughs and which ones permanently resist solution? What’s the model of causation here? Does one even exist or is the history of mathematical progress one giant path-dependent mess?
OpenAI data
The most interesting thing to me about this year’s progress in math was that the first and many of the subsequent discoveries were constructed counterexamples, not proofs. What this means is that someone guesses that some proposition is true, and then someone else (maybe an AI) comes along and makes an elaborate example showing that the proposition is false. Typically, it means some other assumption has to be made to get the proposition to be true or that the part of mathematical reality under investigation is queerer than we supposed. Some of the counterexamples in the OpenAI dump might offend whatever sense of aesthetics mathematicians have.
When all of this started happening in May, there were only a handful of examples of problems settled one way or another by AI. Now we have a lot more, an actual dataset. When the counterexamples started coming out, I wondered how much of this had to do with academic prestige; when a famous person conjectures something is true, do people put in less effort to construct a counterexample given the problem’s background? Are they maybe unconsciously avoiding contradicting a prestigious peer or perhaps believe construction of a counterexample is dumb because the conjecture is probably right? Those are all things that AIs don’t care about at all which might explain all the counterexamples. I started by looking at the 722 papers OpenAI published.
In the OpenAI dataset, of the resolved problems proposed by a famous mathematician, 19% are counterexamples. For what it’s worth, that seems like a pretty fantastic predictive record to me. Of the non-illustriously begotten conjectures, only 11% are counterexamples, an odds ratio of 1.8. So far, this looks pretty good for the academic prestige theory.
However, it might be wrong. Maybe we should adjust this for how long the conjecture has been open. After all, the longer it’s been open maybe the harder it is regardless of the prestige effects we’re supposing. When you do that adjustment, the odds ratio shrinks to 1.33. This still seems substantial, but is not statistically significant. Is such an adjustment even right though? That depends on why the age of the problem and its fame are related.
Erdős problem data
Paul Erdős was the 20th century’s most prolific mathematical problem solver. His interests spanned mainly graph theory, number theory, and combinatorics. Given all the math stuff that’s happened this year, I’m currently most of the way through Hoffman’s excellent biography of Erdős entitled The Man Who Loved Only Numbers, which provides some interesting details about his life I wasn’t aware of before reading it.
Erdős would offer cash bounties for the solution of mathematical problems he proposed and couldn’t quite solve himself. The size of the bounty was proportional to how difficult he judged the problem to be. The mathematician Thomas Bloom maintains erdosproblems.com. I chatted a little with him in May when the initial Erdős problems (including the unit distance conjecture mentioned above) started getting knocked down by the frontier AI labs and felt a little silly not knowing who he was.
Fig 3. Don’t you know who I am?!?!
The Erdős problem dataset provides an interesting comparator to the OpenAI dump. Erdős was basically the Mick Jagger of the math world for large parts of the 20th century (less the womanizing, but not the drug use). All of the Erdős problems had a famous proposer by definition.
I broke the resolved Erdős problems down by whether they occurred before or after the frontier labs seriously started pursuing mathematical greatness. Before, 26% of the resolved problems were counterexamples created by humans versus a stunning 39% after with AI use. There are some convoluted explanations for this, but the most parsimonious explanation here is that AIs are simply better than humans at constructing computationally convoluted counterexamples, and I think this is borne out in the field-level data.
Fig 4. Change in conjecture refutation rates before and after AI among Erdős problems.
Number theory sits near the bottom here, perhaps less amenable to such constructions. A proposition in these fields often respects a bound or it does not; a counterexample is often a specific integer in a rigid structure or one that must be astronomically large. There are fewer computational degrees of freedom that an AI system can use to make a counterexample to a proposition, or so I think.
I also want to rubberneck here a little bit on Bloom’s remark above that Erdős’ instincts were so great. Granted, Bloom didn’t have access to the OpenAI dataset when he said that, but Erdős had substantially more overturned conjectures than those in the resolved OpenAI problems that were proposed by famous people (26% versus 19%), and clearly this gets worse with AI (39% versus 19%). We don’t know the exact mechanism by which problems appear in the OpenAI dump, but we can look at the field-level composition of the two datasets.
Fig 5. Dataset composition by mathematical subfield.
OK, this actually sort of resolves it. The fields OpenAI and Erdős worked on are very different, but the effect goes both ways. The Erdős fields like combinatorics and graph theory are construction heavy, but number theory is not. I need to think a little more about this, but I’m not prepared to let Erdős off the hook completely here.
Coda
The mathematical community writ large has taken the frontier labs’ conquest of math extremely badly, and Scott Aaronson has called the October 6th dump a “mathocalypse”. I don’t see why we can’t use the mathocalypse to study a higher- order theory of mathematical progress.