This was a really interesting article that made me think a lot. Still, I see this as an outsider, i.e., I am not a professional mathematician, just a person who'd like to teach himself math autodidactically up to at least an undergraduate level (fighting now with Bartle and Sherbert's Introduction to Real Analysis). As a learner, I can't quite seem to agree with "The implicit purpose of math homework is to learn how to be stuck, to test ideas, to learn how to fail and, better yet, learn from your mistakes and take them to the next problem. The implicit goal of math education and research is to build intuition and to learn how to ask questions. When we give problems to models, we’re testing them". I mean, I can see this is the objective for turning a math student in the long run into the type of professional mathematician who solves problems for a living. Getting stuck sucks really bad and is extremely antipedagogical, which I suspect is one of the reasons why so many people end up hating math: they find it a continuous torture of trying and failing to solve exercises and feeling stupid in the process. And even if they solve exercise x, the next ones are just another sisyphean slog. Myself, I am trying to do all the exercises in the book and just get angry at how I get stuck for more than half an hour with most of the exercises (maybe the book is not the most pedagogical, and/or analysis is just absurdly hard). And I am a case of someone that *just wants to learn*! I am not taking an exam or anything. But I still get demotivated by the slowness of progress. In this regard, we should be considering perhaps developing AIs into excellent 1 on 1 tutors to help us and combat the inevitable frustration, instead of just considering 'well, it's too tempting and cheating to get the AI and use it in some manner while dealing with exercise sets'.
I so appreciate you weighing in! I think you are getting at one of the core tensions of AI in education which is, as you put it, "the slowness of progress." The important bit here is: progress towards what? Depends on your specific goal. Like you said, learning to be stuck when testing ideas and building intuition is definitely important for the goal of turning a math student into a profession mathematician, but is maybe less necessary for someone who is doing math recreationally with the goal of learning something new. And if your goal is just to get some exposure to some subject you find interesting, this slowness—in being frustrating and sometimes demoralizing—can actually lead to the opposite of discouraging interest in the subject you wanted to learn about. Here, AI can definitely be useful in helping work through problem-specific road blocks, especially if you don't have a professor/academic to ask to explain things to you (and if there are no solutions in the back of the book).
I think this also goes back to the question in the article about the psychology of learning (which is a bit of a rabbit hole, so I won't get too much into it here). But in short, there is a delicate balance to be struck between the amount of time/struggle you invest in a problem that is useful before it becomes harmful to your sense of curiosity and learning. Crossing that line is what leads to the narratives we tell ourselves about what we are and are not good at, which I think becomes very difficult to change once internalized.
Sorry to have rambled, hope this was helpful in some way! Thanks for reading :)
Thank you for this wonderfully written piece. As an econ student, I feel that math plays an interesting role in the quantitative social sciences. Mathematical skills have long been one of the biggest barriers to entering the field, and so I have some hope of LLMs making econ research more accessible and perhaps more diverse. At the same time, my own thinking has been so thoroughly molded by the math classes (and subsequent late-night problem set sessions) I've had the joy of struggling through; I'm now so grateful to have taken some of those before the introduction of LLMs.
I see many parallels to my field more generally and am also nervous about the flood of AI-generated results, among other things. But this gave me a lot to think about and hope for—and captures so much of what makes learning great :)
This was incredibly comprehensive. Probably the most informative look at the issue that I've seen. Thanks so much for doing the work to produce such an incredible piece of math communication.
My own articles on this stuff have not been nearly as well-researched as this one. They are just explorations of my emotional response and of the speculation these developments have forced me to engage in. The larger problem for me is preserving the sacred human experience of encountering truths.
Again, thank you for writing this. I absolutely slept on it, I wish I had read it when it dropped but I'm glad to have found it now.
Thank you for your very kind words Kirwin! That means a lot, especially because I really emotionally resonated with your piece on this topic. Thanks for sharing Simon’s article. Looking forward to reading more of your writing!
I keep coming back to read this piece, it is really wonderful.
Despite being not the best in my math classes and never winning much at math competitions, I really fell in love with it during my undergrad because of the "implicit goals" of math. I loved attending lectures in college because it was one of the first times math was portrayed as a story. I loved that professors had opinions and notes about the math, it made the subject feel less rigid to me and it definitely made me understand why so many people compare math to art. Doing math feels like that old parable of blind men trying to figure out what an elephant is, and sometimes someone's insight or perspective on a subject enlightens us to see the whole elephant.
If we keep using AI to answer questions in math, we might lose some of the good kinds of failure that have been pillars of the field, and that is scary. But I loved how you framed it here. I am hopeful that AI will develop a new consciousness in math that prioritizes digestion and communication over answers. This is the part of the field that I fell in love with, and I am excited to see where it goes.
thank you jess!! i totally agree, I love the opinions/stories that are told in math classrooms that usually don't make it into the textbooks — i love the elephant analogy. :)
Such a great article. Especially coming from a non math background and the other side of the US. I'm a sophomore at Harvard and it is striking how different the administration's response to AI and learning is compared to Stanford. The Stanford professors seem so open to discussing what to do about learning, AI, and PSETs, while at Harvard it feels like decisions about grading and AI is kept under close. We're all feeling the same things as our attention spans decrease, as it gets easier to just one shot anything, and I feel that too. It goes back to, I think, first order and second order desires about what do I actualy want to want versus what do I want.
This was such an interesting read! I loved doing math in school but eventually drifted away from it. I’ve found LLMs surprisingly helpful when trying to work through the math in ML papers now, especially since so much is often left to the reader as a “fun exercise.”
I had a conversation with Bessis about this but it lead to something different. I think it’s worth posting again:
I see the “new consciousness of mathematics” differently.
The issue is not simply that math is human, beautiful, difficult, communal, or slow. All of that is true, but it does not quite reach the core.
For me, mathematics lives in a three-layer movement:
Intuition explores.
Formalism constrains.
Invariance confirms.
Intuition matters because it lets us see possibilities before we can prove them. But intuition alone is not enough. Intuition can be wrong.
So what exposes mistaken intuition?
Formal constraint.
Proof is not a bureaucratic afterthought. It is the discipline that tests whether an intuition can survive contact with structure.
And when formal constraint does its work, what remains?
Invariance.
That is where mathematics becomes more than cleverness. A result matters when something survives change of perspective, survives reformulation, survives constraint, and still remains coherent.
So I do not think the consciousness of mathematics changes simply because AI can generate more proofs. Proof abundance does not make understanding obsolete. It makes the deeper question more important:
What survives constraint?
That is the question I keep returning to.
If intuition were sufficient, we would not need proof.
If formalism were sufficient, we would not need insight.
If invariance were not necessary, we would not talk about structure.
A lot of mathematicians seem to ground mathematical reality in persistence across perspectives. I prefer to ground it in structural coherence under constraint.
Those are close, but they diverge when intuition persists despite incoherence. If several intuitions converge on something that later violates formal constraint, we do not call it real. We call it mistaken.
That way of looking at math has served me well. It keeps me from becoming attached to a single idea. Instead, I keep asking:
What survives?
And when you frame mathematics that way, patterns start showing up across domains. The same structural ideas reappear in mathematics, physics, logic, and even philosophy.
At that point, you are not just solving problems anymore.
You are beginning to see the architecture underneath them.
That, to me, is the consciousness of mathematics.
And it does not change because machines can produce more proofs. It becomes more necessary.
I unfortunately had to stop reading after the lauding of all such great figures, including Tao and Buzzard, both of which I have interacted with directly, worked with on Lean I suppose, and both of which disappoint greatly. If interested in the nuances of formalisation and how precarious this all is, I would recommend looking at my piece on the matter. Unfortunately, this social aspect in my view, and the inherent qualities of academia, is paralysing any counter-action. Whereas the “AI side” has a much more unified resolve, so I fear the more and more this is deliberated without action, the more doomed it becomes.
Thank you for the great post. Though I believe mathematicians instead of adapting a defeatist tone and engage in metaphysical soul searching should demand clarity and rigor before uncritically accepting the grandiose claims coming out of of AI-companies regarding the 'autonomous' mathematical capacities of their 'internal' models that are trained, fined tuned, supervised and guided by large teams of researchers including very capable mathematicians. The fact that we are told nothing about the nature of these models or how these 'autonomous' results are being extracted, including the level of human involvement, beyond the 'trust me bro our internal general purpose model one shotted it' we are getting from the likes of Sebastien Bubeck, would be a giant red flag in any other scientific discipline, especially given the scale of the financial incentives at play. In the case of OpenAI's unit distance conjecture we not even get a list of authors on the paper!
The fact that a group of highly skilled mathematicians tweaking an AI model into producing a result is nowadays considered to be bigger breakthrough than mathematicians proving the thing themselves should be reason enough to question the actual mathematical capacities of these models.
....Read a lot. Try different things a lot. Fail a lot. Question a lot. How did anyone ever come up with this? Eventually, solve the question or some variant of it.... SURRENDER THE FRUITS OF KNOWLEDGE/WORK.... GRATITUDE ❣️....BE AT PEACE....DO IT ALL OVER AGAIN....Repeat again and again and again....until maybe one day.... SOMETHING EMERGES.... GRATITUDE ❣️....
Apoorva, before I invest time in reading the entirety of your piece (I’m only three paragraphs in), can you tell me if you used AI at all in stitching together your essay? I can see that it’s a hearty read (even heartier if the reader takes time to read it thoughtfully), but as it’s supposed to be a heartfelt take on this moment in time and the outlook for the profession and meaning of what it means to do math, I would hope that every word, every sentence, every word sequence and every segue came from your natural “intelligence” (whatever that word means; I’ve got my own beef with it).
Thank you, Apoorva, and at the very least I applaud your decision to shine a light on something that people everywhere, regardless of their field of work, are thinking about this “come to Jesus” moment.
This was a really interesting article that made me think a lot. Still, I see this as an outsider, i.e., I am not a professional mathematician, just a person who'd like to teach himself math autodidactically up to at least an undergraduate level (fighting now with Bartle and Sherbert's Introduction to Real Analysis). As a learner, I can't quite seem to agree with "The implicit purpose of math homework is to learn how to be stuck, to test ideas, to learn how to fail and, better yet, learn from your mistakes and take them to the next problem. The implicit goal of math education and research is to build intuition and to learn how to ask questions. When we give problems to models, we’re testing them". I mean, I can see this is the objective for turning a math student in the long run into the type of professional mathematician who solves problems for a living. Getting stuck sucks really bad and is extremely antipedagogical, which I suspect is one of the reasons why so many people end up hating math: they find it a continuous torture of trying and failing to solve exercises and feeling stupid in the process. And even if they solve exercise x, the next ones are just another sisyphean slog. Myself, I am trying to do all the exercises in the book and just get angry at how I get stuck for more than half an hour with most of the exercises (maybe the book is not the most pedagogical, and/or analysis is just absurdly hard). And I am a case of someone that *just wants to learn*! I am not taking an exam or anything. But I still get demotivated by the slowness of progress. In this regard, we should be considering perhaps developing AIs into excellent 1 on 1 tutors to help us and combat the inevitable frustration, instead of just considering 'well, it's too tempting and cheating to get the AI and use it in some manner while dealing with exercise sets'.
I so appreciate you weighing in! I think you are getting at one of the core tensions of AI in education which is, as you put it, "the slowness of progress." The important bit here is: progress towards what? Depends on your specific goal. Like you said, learning to be stuck when testing ideas and building intuition is definitely important for the goal of turning a math student into a profession mathematician, but is maybe less necessary for someone who is doing math recreationally with the goal of learning something new. And if your goal is just to get some exposure to some subject you find interesting, this slowness—in being frustrating and sometimes demoralizing—can actually lead to the opposite of discouraging interest in the subject you wanted to learn about. Here, AI can definitely be useful in helping work through problem-specific road blocks, especially if you don't have a professor/academic to ask to explain things to you (and if there are no solutions in the back of the book).
I think this also goes back to the question in the article about the psychology of learning (which is a bit of a rabbit hole, so I won't get too much into it here). But in short, there is a delicate balance to be struck between the amount of time/struggle you invest in a problem that is useful before it becomes harmful to your sense of curiosity and learning. Crossing that line is what leads to the narratives we tell ourselves about what we are and are not good at, which I think becomes very difficult to change once internalized.
Sorry to have rambled, hope this was helpful in some way! Thanks for reading :)
Thank you for this wonderfully written piece. As an econ student, I feel that math plays an interesting role in the quantitative social sciences. Mathematical skills have long been one of the biggest barriers to entering the field, and so I have some hope of LLMs making econ research more accessible and perhaps more diverse. At the same time, my own thinking has been so thoroughly molded by the math classes (and subsequent late-night problem set sessions) I've had the joy of struggling through; I'm now so grateful to have taken some of those before the introduction of LLMs.
I see many parallels to my field more generally and am also nervous about the flood of AI-generated results, among other things. But this gave me a lot to think about and hope for—and captures so much of what makes learning great :)
This was incredibly comprehensive. Probably the most informative look at the issue that I've seen. Thanks so much for doing the work to produce such an incredible piece of math communication.
A great recent addition to the conversation around "proof-abundance" is the article of Simon H: https://substack.com/home/post/p-210565116
My own articles on this stuff have not been nearly as well-researched as this one. They are just explorations of my emotional response and of the speculation these developments have forced me to engage in. The larger problem for me is preserving the sacred human experience of encountering truths.
Again, thank you for writing this. I absolutely slept on it, I wish I had read it when it dropped but I'm glad to have found it now.
Thank you for your very kind words Kirwin! That means a lot, especially because I really emotionally resonated with your piece on this topic. Thanks for sharing Simon’s article. Looking forward to reading more of your writing!
I keep coming back to read this piece, it is really wonderful.
Despite being not the best in my math classes and never winning much at math competitions, I really fell in love with it during my undergrad because of the "implicit goals" of math. I loved attending lectures in college because it was one of the first times math was portrayed as a story. I loved that professors had opinions and notes about the math, it made the subject feel less rigid to me and it definitely made me understand why so many people compare math to art. Doing math feels like that old parable of blind men trying to figure out what an elephant is, and sometimes someone's insight or perspective on a subject enlightens us to see the whole elephant.
If we keep using AI to answer questions in math, we might lose some of the good kinds of failure that have been pillars of the field, and that is scary. But I loved how you framed it here. I am hopeful that AI will develop a new consciousness in math that prioritizes digestion and communication over answers. This is the part of the field that I fell in love with, and I am excited to see where it goes.
thank you jess!! i totally agree, I love the opinions/stories that are told in math classrooms that usually don't make it into the textbooks — i love the elephant analogy. :)
Such a great article. Especially coming from a non math background and the other side of the US. I'm a sophomore at Harvard and it is striking how different the administration's response to AI and learning is compared to Stanford. The Stanford professors seem so open to discussing what to do about learning, AI, and PSETs, while at Harvard it feels like decisions about grading and AI is kept under close. We're all feeling the same things as our attention spans decrease, as it gets easier to just one shot anything, and I feel that too. It goes back to, I think, first order and second order desires about what do I actualy want to want versus what do I want.
This was such an interesting read! I loved doing math in school but eventually drifted away from it. I’ve found LLMs surprisingly helpful when trying to work through the math in ML papers now, especially since so much is often left to the reader as a “fun exercise.”
Thank you for sharing this article :)
Thank you for writing this piece. I am excited to see mathematics from my generation is starting to discuss how things could be!
Thanks for writing this up. I also loved the event.
I had a conversation with Bessis about this but it lead to something different. I think it’s worth posting again:
I see the “new consciousness of mathematics” differently.
The issue is not simply that math is human, beautiful, difficult, communal, or slow. All of that is true, but it does not quite reach the core.
For me, mathematics lives in a three-layer movement:
Intuition explores.
Formalism constrains.
Invariance confirms.
Intuition matters because it lets us see possibilities before we can prove them. But intuition alone is not enough. Intuition can be wrong.
So what exposes mistaken intuition?
Formal constraint.
Proof is not a bureaucratic afterthought. It is the discipline that tests whether an intuition can survive contact with structure.
And when formal constraint does its work, what remains?
Invariance.
That is where mathematics becomes more than cleverness. A result matters when something survives change of perspective, survives reformulation, survives constraint, and still remains coherent.
So I do not think the consciousness of mathematics changes simply because AI can generate more proofs. Proof abundance does not make understanding obsolete. It makes the deeper question more important:
What survives constraint?
That is the question I keep returning to.
If intuition were sufficient, we would not need proof.
If formalism were sufficient, we would not need insight.
If invariance were not necessary, we would not talk about structure.
A lot of mathematicians seem to ground mathematical reality in persistence across perspectives. I prefer to ground it in structural coherence under constraint.
Those are close, but they diverge when intuition persists despite incoherence. If several intuitions converge on something that later violates formal constraint, we do not call it real. We call it mistaken.
That way of looking at math has served me well. It keeps me from becoming attached to a single idea. Instead, I keep asking:
What survives?
And when you frame mathematics that way, patterns start showing up across domains. The same structural ideas reappear in mathematics, physics, logic, and even philosophy.
At that point, you are not just solving problems anymore.
You are beginning to see the architecture underneath them.
That, to me, is the consciousness of mathematics.
And it does not change because machines can produce more proofs. It becomes more necessary.
I unfortunately had to stop reading after the lauding of all such great figures, including Tao and Buzzard, both of which I have interacted with directly, worked with on Lean I suppose, and both of which disappoint greatly. If interested in the nuances of formalisation and how precarious this all is, I would recommend looking at my piece on the matter. Unfortunately, this social aspect in my view, and the inherent qualities of academia, is paralysing any counter-action. Whereas the “AI side” has a much more unified resolve, so I fear the more and more this is deliberated without action, the more doomed it becomes.
Thank you for the great post. Though I believe mathematicians instead of adapting a defeatist tone and engage in metaphysical soul searching should demand clarity and rigor before uncritically accepting the grandiose claims coming out of of AI-companies regarding the 'autonomous' mathematical capacities of their 'internal' models that are trained, fined tuned, supervised and guided by large teams of researchers including very capable mathematicians. The fact that we are told nothing about the nature of these models or how these 'autonomous' results are being extracted, including the level of human involvement, beyond the 'trust me bro our internal general purpose model one shotted it' we are getting from the likes of Sebastien Bubeck, would be a giant red flag in any other scientific discipline, especially given the scale of the financial incentives at play. In the case of OpenAI's unit distance conjecture we not even get a list of authors on the paper!
The fact that a group of highly skilled mathematicians tweaking an AI model into producing a result is nowadays considered to be bigger breakthrough than mathematicians proving the thing themselves should be reason enough to question the actual mathematical capacities of these models.
Thanks for the enlightening post. Tao's lecture (https://www.youtube.com/watch?v=Uc2zt198U_U) is private; would you consider making it public?
....Read a lot. Try different things a lot. Fail a lot. Question a lot. How did anyone ever come up with this? Eventually, solve the question or some variant of it.... SURRENDER THE FRUITS OF KNOWLEDGE/WORK.... GRATITUDE ❣️....BE AT PEACE....DO IT ALL OVER AGAIN....Repeat again and again and again....until maybe one day.... SOMETHING EMERGES.... GRATITUDE ❣️....
Apoorva, before I invest time in reading the entirety of your piece (I’m only three paragraphs in), can you tell me if you used AI at all in stitching together your essay? I can see that it’s a hearty read (even heartier if the reader takes time to read it thoughtfully), but as it’s supposed to be a heartfelt take on this moment in time and the outlook for the profession and meaning of what it means to do math, I would hope that every word, every sentence, every word sequence and every segue came from your natural “intelligence” (whatever that word means; I’ve got my own beef with it).
Thank you, Apoorva, and at the very least I applaud your decision to shine a light on something that people everywhere, regardless of their field of work, are thinking about this “come to Jesus” moment.