This post was written by Michael Freedman (Harvard CMSA & Logical Intelligence), and typeset by Milo Moses.
Frontier models exhibit astonishing mathematical knowledge and power despite being in their infancy. They will surely strengthen further through routine maturation of the technology, and quite likely through future paradigm shifts in architecture and training. Where does this leave the human mathematician?
Already, it has made us study harder than we have since school. The AIs speak to us in a broad language, equally at home in all fields, and we scurry to keep up. So those of us (not me) who were great students, are back in their natural environment, learning, interpreting, organizing knowledge as it flows in. The challenge of absorbing this knowledge will make good teachers into great ones. But what about those whose lives revolved around quiet hours struggling with an open problem, where do we go? My advice: lean in, let the AI’s teach you everything they can about the problem and its context; push them until they are stuck.
One of the great feelings in research is the (rare) sense that you have at your command all the information relevant to the problem at hand, and all you need to do is to come up with your own idiosyncratic lens to see through it. Conversing with the AIs is often eye-opening on how much existing mathematics is out there and might be brought to bear. Let the AIs put you in this state of ideal preparation. If they flat out solve the problem on their own it means it was not hard enough. We (humans) should focus on the hardest problems and be creative in perturbing them into useful variants, stepping stones. I used to tell graduate students that if they worked on the central problem of the field, any progress they made would have to be absorbed - their work would have to be read. It makes little sense, now, to clean up loose ends - the AIs excel at this- we should be lost in meditation on the largest issues.
There is a risk. Adi Shamir once told me, “I stopped working on P vs. NP when I realized there was no test I could apply to distinguish working on that problem from doing nothing at all.” In short, work needs to be documentable, both for one’s own sanity and one’s career. Here is a thought on documentation (of both kinds). Some mathematicians may choose to keep the AIs at arm’s length, but for those who work with them let’s create a shared work space. I envision the frontier AIs sharing a math-workspace, where their human collaborators agree to have all their inputs and queries used as training data to strengthen the AIs (in both research and exposition) and create a library of “human contributed research ideas”. This library could be drawn on in real time by any other mathematician; we accept being scooped as part of our collective endeavor. But the AIs can watch what's going on and tell us (or our department chair) how useful our recent ideas have been world-wide. It would be a new, parallel, publication process.
We would not have to write a full paper, just introduce a workable idea - if it is fruitful, the AIs will pass it around to our peers (with a pleasant acknowledgement). Basically, blog to the AIs about your great new idea and let them pass it on when someone else’s conversation could benefit from it. This could be turned into a score, if needed, to buttress a tenure decision, job application, or a grant proposal (assuming the relevant institutions pass through the phase transition we are approaching).
So I see human researchers thinking about, or at least around, the most intractable problems: the Andrews-Curtis conjecture, the smooth 4D Poincare conjecture, the Riemann Hypothesis, even P vs. NP. The art will be to avoid “doing nothing”: Find the special case or context in which progress can be made, documented, and can contribute to the work of others, with the AIs acting as the internet did before it - to accelerate communication by another factor of ten (recall conversing via snail mail.)
I see mathematics quite differently post AI. I can imagine new fields like “the statistical physics of truth (or provability)”. “Phase transitions“ in k-sat have long been studied. This is an area where myriad similar statements can profitably be considered at once. I expect that AI’s will permit this kind of “bulk math” to be extended first to areas like diophantine equations and later to many other domains (knot theory and 3-manifold topology come to mind).
On a philosophical level, the diversity of proofs (even of the same theorem) within the Oct. 6th OpenAI release, have shifted my thinking on what can be proved. From Scott Aaronson I learned the phrase “true, but for no good reason”, a category that might apply to many number theoretic statements that have been checked into the billions and where, in a sense, the “probability” of finding a counterexample decays rapidly with size. The Goldbach conjecture (every even number is the sum of two primes) might be of this nature. Why should there be a proof from the axioms just because it’s true? (The joke regarding Goldbach is that it is difficult because you are not supposed to add primes.) A metaphor is that you are standing under a willow tree looking up at the sky. Each point of the sky is a true theorem. The tree is what gets you there - the proof structures. All points of the sky occluded by the tree and its leaves have proofs, but if you see blue, no proof has reached that point. Now, post AI, I think of the proofs as a vastly overlapping jungle, so a typical point of the sky is blocked (proved) in myriad ways. I always “knew” there were many proofs of the same fact, but the Oct. 6th release brought the point home. Years ago, I was visiting MSRI when Kaplansky was director and Bill Thurston was giving a series of lectures. Someone asked Kaplansky how Bill proves such and such. The answer, “Any way he wants.” I feel the AIs have a similar command, you can tie one hand behind their back and still have them prove it. I now expect to see less sky. Of course Gödel taught us that there are true statements without proofs, but I now suspect mathematics gets to most places unless there is a good reason it can’t go there.
So here is my advice. Don’t be discouraged (yet). Fight back, or if you prefer, bounce back. There is no evidence one way or the other on whether the sun is setting on the era of human insight into mathematical structure. We have an immensely powerful tool for seeing connections and carrying out complex developments and calculations. But we do not know exactly how humans jump or whether LLMs can or cannot jump[1]. In spring 2026 I thought the AIs could form the “closure” of human mathematical knowledge, by the summer I was thinking “convex hull”, after Oct. 6th, it feels like more. But how can we even tell if a mathematical idea is in the convex hull of the millions of known facts or is a totally fresh idea. Only a philosopher could attempt to understand what a fresh idea means - or if truly “there is nothing new under the sun”. Personally, I feel like I have seen, and perhaps had, “fresh ideas”, but introspection is difficult. Maybe the new science of mathematics will illuminate creativity and tell us if we and/or the machines have much of it. My guess is that creativity is a real thing and that we and our machines have occasional access. A newspaper reporter once asked Einstein if he kept a pad of paper by his bed in case he had an idea in the middle of the night. Einstein replied, “Oh, I think you misunderstand. A person is lucky if they have one or two ideas their entire life.”
My hope for human mathematicians is that they will study hard - learn what is being shown them, lie back and allow creativity to strike, and then openly share their inventions with the AIs and through that medium communicate inspiration to the rest of us. My ask of the large AI companies is that they join the math community in building this monument to their intellectual creation: A shared, attributive, workspace bridging human and agentic mathematics.
This post was written by Michael Freedman (Harvard CMSA & Logical Intelligence), and typeset by Milo Moses.
Frontier models exhibit astonishing mathematical knowledge and power despite being in their infancy. They will surely strengthen further through routine maturation of the technology, and quite likely through future paradigm shifts in architecture and training. Where does this leave the human mathematician?
Already, it has made us study harder than we have since school. The AIs speak to us in a broad language, equally at home in all fields, and we scurry to keep up. So those of us (not me) who were great students, are back in their natural environment, learning, interpreting, organizing knowledge as it flows in. The challenge of absorbing this knowledge will make good teachers into great ones. But what about those whose lives revolved around quiet hours struggling with an open problem, where do we go? My advice: lean in, let the AI’s teach you everything they can about the problem and its context; push them until they are stuck.
One of the great feelings in research is the (rare) sense that you have at your command all the information relevant to the problem at hand, and all you need to do is to come up with your own idiosyncratic lens to see through it. Conversing with the AIs is often eye-opening on how much existing mathematics is out there and might be brought to bear. Let the AIs put you in this state of ideal preparation. If they flat out solve the problem on their own it means it was not hard enough. We (humans) should focus on the hardest problems and be creative in perturbing them into useful variants, stepping stones. I used to tell graduate students that if they worked on the central problem of the field, any progress they made would have to be absorbed - their work would have to be read. It makes little sense, now, to clean up loose ends - the AIs excel at this- we should be lost in meditation on the largest issues.
There is a risk. Adi Shamir once told me, “I stopped working on P vs. NP when I realized there was no test I could apply to distinguish working on that problem from doing nothing at all.” In short, work needs to be documentable, both for one’s own sanity and one’s career. Here is a thought on documentation (of both kinds). Some mathematicians may choose to keep the AIs at arm’s length, but for those who work with them let’s create a shared work space. I envision the frontier AIs sharing a math-workspace, where their human collaborators agree to have all their inputs and queries used as training data to strengthen the AIs (in both research and exposition) and create a library of “human contributed research ideas”. This library could be drawn on in real time by any other mathematician; we accept being scooped as part of our collective endeavor. But the AIs can watch what's going on and tell us (or our department chair) how useful our recent ideas have been world-wide. It would be a new, parallel, publication process.
We would not have to write a full paper, just introduce a workable idea - if it is fruitful, the AIs will pass it around to our peers (with a pleasant acknowledgement). Basically, blog to the AIs about your great new idea and let them pass it on when someone else’s conversation could benefit from it. This could be turned into a score, if needed, to buttress a tenure decision, job application, or a grant proposal (assuming the relevant institutions pass through the phase transition we are approaching).
So I see human researchers thinking about, or at least around, the most intractable problems: the Andrews-Curtis conjecture, the smooth 4D Poincare conjecture, the Riemann Hypothesis, even P vs. NP. The art will be to avoid “doing nothing”: Find the special case or context in which progress can be made, documented, and can contribute to the work of others, with the AIs acting as the internet did before it - to accelerate communication by another factor of ten (recall conversing via snail mail.)
I see mathematics quite differently post AI. I can imagine new fields like “the statistical physics of truth (or provability)”. “Phase transitions“ in k-sat have long been studied. This is an area where myriad similar statements can profitably be considered at once. I expect that AI’s will permit this kind of “bulk math” to be extended first to areas like diophantine equations and later to many other domains (knot theory and 3-manifold topology come to mind).
On a philosophical level, the diversity of proofs (even of the same theorem) within the Oct. 6th OpenAI release, have shifted my thinking on what can be proved. From Scott Aaronson I learned the phrase “true, but for no good reason”, a category that might apply to many number theoretic statements that have been checked into the billions and where, in a sense, the “probability” of finding a counterexample decays rapidly with size. The Goldbach conjecture (every even number
So here is my advice. Don’t be discouraged (yet). Fight back, or if you prefer, bounce back. There is no evidence one way or the other on whether the sun is setting on the era of human insight into mathematical structure. We have an immensely powerful tool for seeing connections and carrying out complex developments and calculations. But we do not know exactly how humans jump or whether LLMs can or cannot jump[1]. In spring 2026 I thought the AIs could form the “closure” of human mathematical knowledge, by the summer I was thinking “convex hull”, after Oct. 6th, it feels like more. But how can we even tell if a mathematical idea is in the convex hull of the millions of known facts or is a totally fresh idea. Only a philosopher could attempt to understand what a fresh idea means - or if truly “there is nothing new under the sun”. Personally, I feel like I have seen, and perhaps had, “fresh ideas”, but introspection is difficult. Maybe the new science of mathematics will illuminate creativity and tell us if we and/or the machines have much of it. My guess is that creativity is a real thing and that we and our machines have occasional access. A newspaper reporter once asked Einstein if he kept a pad of paper by his bed in case he had an idea in the middle of the night. Einstein replied, “Oh, I think you misunderstand. A person is lucky if they have one or two ideas their entire life.”
My hope for human mathematicians is that they will study hard - learn what is being shown them, lie back and allow creativity to strike, and then openly share their inventions with the AIs and through that medium communicate inspiration to the rest of us. My ask of the large AI companies is that they join the math community in building this monument to their intellectual creation: A shared, attributive, workspace bridging human and agentic mathematics.
See Tom Zahavy (2026), "LLMs can't jump" (original and discussion)