> I’ve had the experience quite a few times now: I try to autoformalize something, and an AI will tell me “I did it; look, the proof checks out!” But, actually, in some sense it cheated: instead of formalizing what I intended, it found a (sometimes very squirrely) way to interpret what I asked so that it could successfully prove it. It’s often very hard to tell, though, that this is what happened—not least because the formalized versions of things (say as expressed in popular proof assistant systems) tend to be very low level, very verbose and very hard for us humans to understand.
This has largely been my experience in programming, too. Often when given a broad objective within an existing code base, even the frontier models seem to stand up a half dozen tests proving compliance and then add 3 new branches solving for those specific tests alone. My best guess is that our code base is quite far out of the distribution (and not for just good reasons). The agents are thus reduced to these tactics rather than extending and refactoring well known abstractions.
The reason we (as in university-funding society) need mathematicians is so that there's a steward over math that can help me, a complete layman, understand and apply mathematics. "Being a steward" will sometimes not look like much. Sometimes it's just going to be dicking around with LLM's.
But math is too important for it to be abandoned by humans and left to models. I, the tax-paying bonehead engineer, need mathematicians to exist. They can have their ego beaten up by Claude, sure, whatever, but they're not allowed to just close up shop in despair.
Much more rejections due to more submissions, and AI will probably not help you (unless you solve a major open problem).
Even frontier models still struggle at proving small conjecturers despite all the hype about major breakthroughs. It really depends a lot on the prompt, the type of problem, among other factors. But AI does not suddenly make publishing in a journal easier, although it does make it easier to produce papers.
I see two assumptions in the core argument, both of which are open to challenge.
1. AI can do proofs, but deciding which problems to solve, which math is useful, is by humans.
2. The math that's picked needs to be understandable by humans.
For #1, AI may be able to play a significant, if not a takeover, role for even figuring out what math is useful.
For #2, understandability by humans may be good for now, but could also turn out to be a significant constraint. Correctness is a goal, trust is an important requirement, human understandability may be an intermediary for that, but not necessarily the end goal.
In other words, the article may stand the current state of the art, but may not stand merely a couple years down the road.
Not sure I get that. If AI is not working for humans' benefits (i.e. it is not working towards an optimization problem set-up by humans), whom is it supposed to benefit, then? (i.e. how is that better than an entropy-producing machine)?
Same for 2, there are so many infinite ways to boil the oceans, but so few oceans to boil to begin with. Better make sure that this insane energy (both in the physical, due to natural resources scarcity, as well as intellectual) is spent towards meaningful and useful ends. We can no longer be the judges of that if we can't comprehend what we got in return.
>> not working towards an optimization problem set-up by humans
I am suggesting neither of the two.
Not a great analogy but a parent may be working for an infant's benefit without the infant yet being able to understand. Taking an arbitrarily broad example, the problem to optimize for could be "help humanity advance", and other aspects could be subgoals of the same including what mathematics is useful.
- by definition, the AI isn't human, irrespective of its computational abilities, it needs a human to tell it what being human is, and the ways "humanity can be advanced" need to be validated on this basis
- "advancing humanity" isn't a one dimensional problem/single-KPI optimization game, you might optimize certain things (e.g. expected life expectancy) at the detriment of others (e.g. freedom of movement). If you are not understanding the proposition, you are not understanding the target outcome.
- even if the target outcome is perfectly formally specified and commonly understood (which it can't), you want your AI to rationalize that the journey to get there is the most direct and efficient.
- especially so since subsequent runs of the same prompt will provide different plans
- anything less than that is begging to be conned: as a sensible person, you wouldn't give unlimited power and all your faith to a single individual trusted to "advance humanity" if they cannot be understood. On what ground would you give the machine a pass?
In all, such comments really worry me. It's like AI is triggering for some people the kinds of oppressive religious feelings whereby the individual should submit itself to the will and desires of a pretended all-powerful being. Do you really think our ancestors chose to cut off their legs in abandonment when they figured that some animals could outpace them? No, they built traps and throwing weapons to catch them and see how they taste.
I agree that AI needs alignment and validation, and this is already been worked on [*1]. I'll assume below that this is a given.
For the rest, AI itself may be able to handle better than most humans. It already has good idea about what being human is [*2], understands that this isn't a one-dimensional problem, can do balancing like humans would, find more direct/efficient paths than humans. For many things I discuss with AI, I find that it already reasons much better than most humans (not even considering that AI knows so much more than any human).
I think we over-index on "the same prompt will provide different plans". Humans would do the same too. Humans, working in isolation or with collaboration can self-correct, but the same applies to AI.
>> On what ground would you give the machine a pass?
The benchmark I hold is humans themselves. Humans currently have the pass, and have had it since history. Yet, there are many irrational decisions everywhere around.
I hear complaints that AI hallucinates. Yes, it does. Humans do too, and more often than they are willing to admit. The concept of 'god' may entirely be a hallucination (i.e., something not supported by facts). There is no good scientific evidence of prayers working, yet many humans believe in the same.
The real issue is that AI seems to learn and hallucinate in a different way than humans -- it sometimes makes some very silly mistakes. No disagreement, we need to make it better. The pace at which AI can become better however could easily surpass the speed at which humans learn or change. I am not suggesting we give AI the pass till it becomes good enough, there's enough progress on alignment, etc.
>> Do you really think our ancestors chose to cut off their legs in abandonment when they figured that some animals could outpace them?
Great! Here lies an important point. Between humans and animals, we have nature's evolutionary processes, survival of the fittest, ... Depending on how one sees it (and this is a real debate in my mind), we could use AI to enhance our survival and progress faster, or we could see AI as an enemy (like it is another species) and compete.
For some people, the goal is exactly advancements of humans. I, so far, see it as evolutionary progress, whatever form it takes.
[*1] Whether we are doing enough for that or not, is a valid debate.
[*2] AI cannot experience it, it cannot 'know' it in the human sense of knowing if that means something different, but it could emulate well enough.
It can still benefit humans, but there are plenty of situations where humans prioritize guarantees of formal correctness over the ease of navigating granular details.
> AI can do proofs, but deciding which problems to solve, which math is useful, is by humans.
This is within a very narrow view before the emergence of always running “minds” within any given domain. The only reason they don’t exist now is because they’re expensive.
Pretty soon we’re going to have always running minds that are constantly thinking about every domain imaginable and coming up with their own proofs and improvements and everything else imaginable within those domains.
Isn't this basically the hitchhikers guide to the galaxy passage on the computer that spends millions of years to determine the ultimate answer to be 42 and then when people get upset at how meaningless that answer is offers to build another even bigger computer to figure out what the question was.
Love this reference but the Minds are self building intelligences and (iirc) run their internal logic in hyperspace to get around the limits of the speed of light among other fantastical things. If we had genuine super intelligence like that then the discussion on how to use it would be moot as we wouldn’t be in that discussion.
I think the big question is whether these recent gains continue. It’s entirely plausible that the ai firms are going to run out of human trainers capable of further refining the model. It’s also possible we are on the cusp of real superintelligence.
Wolfram gives a very important and sober take on the present state and future of pure math. I came away with the following key takeaways:
1. An essential goal of mathematics is human understanding. The computation of proof terms doesn't necessarily enrich human understanding. The proof of the four color theorem result is a good example, and formal verification/SAT solving gives many more: these are results that can be trusted up to our trust in the system used to produce them, and they can be used in practice, but they don't necessarily enrich our understanding. Imagine a computer with near infinite proof search powers set loose with the current human definitions, theorems, and understanding of mathematics. Suppose it constructs a proof for a new theorem at our mathematical frontier. The shortest such proof in terms of currently understood definitions and concepts could be so long and mechanical that the entire lineage of humans until the end of the universe could not finish reading it. So though it overlaps with the activity of mathematicians, this type of computational proof search is not mathematics as such. This is an important distinction that many people do not seem to grasp and some dismiss as cope.
2. The human activity of theory building, rendering otherwise monstrous proofs like the one I discussed above into light conceptual arguments a person can understand, appears at this time out of reach of models. Maybe they will do this in the future, but it is not yet the case. Human theory building drastically compresses the spaces of theorems and their proofs: this is why great theory builders like Groethendieck are so important to the field; grinding has its value too, but runs up against computational limits in both humans and computers. These limits are collapsed by the conceptual shortcuts created by theory builders.
Gowers has a nice and arguably better-grounded article on the mathematical capabilities of recent LLMs that I think is enlightening to read alongside Wolfram's bird's eye view of the implications of those capabilities: https://gowers.wordpress.com/2026/08/12/what-sort-of-maths-a...
I've heard the human understanding line a whole bunch.
But it begs the question. What is the value to the rest of humanity that a small group of people possesses something that can be called human understanding? Particularly when that group of people is historically terrible at communication (as is routinely demonstrated in Calculus classes), and most humans are not capable of learning that understanding (though more are capable than think they are capable - that is another story).
I am speaking as someone who nearly finished a PhD in mathematics. I understand why mathematicians would wish to continue in the age of AI. But, barring something like universal basic income, it isn't obvious why the rest of humanity would support them in this endeavor.
The idea that the rest of humanity needs to support them is absurd. A couple billion in private trust funds could fund mathematics researchers in perpetuity.
The idea of being supported by the rest of humanity, is not a requirement that the rest of humanity support it equally.
The idea that "someone rich will take care of it", reminds me of a passage from https://en.wikipedia.org/wiki/The_Logic_of_Collective_Action. It talks about "the exploitation of the large, by the small". Where a public good (in this case mathematics) is provisioned by a large entity that finds it worthwhile for their own reasons, and the remaining players who value it, feel no need to contribute anything themselves.
What does it mean for a mathematics to be "provisioned"? A well-funded math foundation run like the Linux Foundation could cover the lifestyles of many elite mathematicians. I don't understand why you imagine it would be centralized behind one "large entity" or that this large entity, so long as it is not a government, can't be trusted to fund good things.
People unaffiliated with the Linux foundation contribute to Linux. Regularly.
I think the central problem with that line of reasoning is that it supposes we are already living with AGI. Even the latest PR from Anthropic admits that we are not [0] there yet. If we don't have AGI capable of integrating and employing the mathematical results produced by an artificially superintelligent theorem prover so that they become useful to the rest of science and engineering, we will need humans to do at least some of the work in mathematics. Moreover if we freeze current AI capabilities today, "digestion" probably doesn't scale, so mathematical research will probably end up AI-assisted in ways not far off from what we're already seeing in the sciences.
I am stating a counter-argument to a common argument.
You are simply giving a different argument than the one I'm giving a counter to. Yes, of course, if including human understanding results in strictly better results than AI, then there is an argument for the value supplied by human mathematicians.
But that's an argument that human understanding is on the path to better outcomes. It's not an argument that the intrinsic worth of human understanding is a reason to support mathematicians.
I kind of feel motte-and-bailey'd here. You began by questioning the value of human understanding in mathematics and I've explained (exactly as you've said) why it is currently on the critical path of outcomes in science and engineering with value to humanity almost nobody would dispute. This new question, where human understanding of mathematics isn't on the critical path of anything, is something I can't argue, in the same way that I can't argue for the intrinsic value or meaning of essentially any human activity without resorting to what amount to basically religious axioms.
Okay, now, when AI and robots can do surgeries and advise people better than modern doctors do (the bar is not that high, ask anyone with an autoimmune disease or gut issues)...
People in the humanities regularly write things that many people find interesting and entertaining. One need not be a historian to understand https://acoup.blog/2026/01/30/collections-the-late-bronze-ag... and find it interesting. The value here is not that they possess human understanding. It is that they can share human understanding, in a way that many humans can be interested in.
But, for example, take my first paper: https://dspace.library.uvic.ca/server/api/core/bitstreams/66... The title was, "Derivations whose iterates are zero or invertible on a left ideal." In order to understand the title, you need to learn what a ring is, what a derivation on a ring is, what an invertible element of a ring is, what an ideal of a ring is, and why these are concepts that anyone would have invented. In order to read the theorem, you have to further understand what division ring is, a matrix ring is, the characteristic of a ring, and a polynomial over a ring. The proof is even worse.
Good luck interesting anyone who wasn't a mathematician. (And good luck interesting most mathematicians!)
The desperately needed TL;DR is that perhaps the actual mathematics itself (i.e. proofs, calculations, etc.) can be done by AI, but why we do it and deciding which problems to solve can only be done by AI. Therefore mathematician do maths.
I'm not a mathematician, but this seems like a weak and slightly bizarre argument.
I'm wondering how we are going to maintain a critical mass of people who understand frontier mathematics if in another five or ten years the only "mathematicians" truly working at the frontier anymore are AIs. Or maybe "understanding" at depth will become a thing of the past, superseded by broad-strokes grasp of results plus machine verification.
For one thing, every topic can be explained with motion graphics offering another 1 or 2 dimensions (time and for the 3d stuff that is hard to see structure of in static 2d, plus things like vr will allow more direct depth perception and different views) than a typical blackboard lecture.
On Proof and Progress in Mathematics talks about how much gets lost of the geometric understanding when translated to a paper. The kinds of things many mathematicians visualize in their head will be much easier to transmit.
I don't know if that is enough to offset the other affects, but learning and transmitting the understanding should be able to get much easier for a lot of people in principle.
If training data are purely historical, then how does the AI look forward?
And if human mathematicians are drummed out of producing future training data, then can AI end up proving itself so much "eating the seed corn", only at scale?
This is also how it works for humans too.
What portion of human art produced in 2025 or 1925 do you think was fresh and novel, compared to derived and repetitive?
The same is true for science and Engineering.
What matters is if you have a method of sorting through the repetitive trash
If training data are purely historical, then how does the human look forward?
Seems to me not impossible that given current knowledge, AI generate one nugget more of knowledge (eg a proof of Navier Stokes), and given current knowledge + the nugget, generate yet some more new knowledge.
> back in 1988, when we first introduced Mathematica, there was also some of the same kind of talk about math being taken over, and made pointless. Of course that’s not how it worked out at all.
Makes sense.
> For me, its greatest use in mathematical pursuits has been its ability in effect to thematically mine the knowledgebase of human mathematics. ... Modern AI is, first and foremost, a way of leveraging the existing corpus of human knowledge.
AI is more a database of knowledge (stolen knowledge but let's leave that discussion asside) than a thinking machine. You can query a compressed version of millions of books.
That is very useful. (Are we already StarTrek-communists?
> generating useful mathematics is a much more exacting activity than generating language.
This is something that most people forget. Generative AI is mostly LLMs, and they are chatbots not mathbots.
> It’s a frustrating feature of modern times that someone like me gets sent many AI-generated documents every day that have the “statistical texture” of math papers, but that one at least expects have a very low probability of being meaningfully correct
And here is the trick. A million monkeys with a million typewriters may write a Shakespeare masterpiece. But they would not be able to differentiate it from garbage text.
> So, yes, there’s every reason to expect a bright future—now with some additional help from AI—for that most rarefied of human pursuits: research in pure mathematics.
> AI is more a database of knowledge than a thinking machine. You can query a compressed version of millions of books.
No…? Isn’t the entire reason we are having this discussion because LLMs are coming up with results that are not already represented in the training data?
AIs have representations of the training data, but they also do search. Search creates new knowledge. For example, a chess program doesn't have a representation of all chess games, but can search and in doing so create new games. This isn't (just) random generation, but generation constrained by an evaluation function.
>AI is more a database of knowledge (stolen knowledge but let's leave that discussion asside) than a thinking machine. You can query a compressed version of millions of books.
Outdated view. Reasoning models do a lot of "thinking", and can solve novel problems - even quite difficult problems.
The whole reason we're even having this conversation is because AI solved a millennium prize problem that no human knew an answer to.
> The whole reason we're even having this conversation is because AI solved a millennium prize problem that no human knew an answer to.
Well, in this case, some mathematicians were on the finish line, and the AI combined (presumably, since nothing is published) existing work to obtain the new result, barely beating the original authors.
I like Wolfram and always enjoy reading his posts. An irreconcilable thing here is Wolfram clearly wants the Mathematica language to be central to the development of math as a field, but it's proprietary and so there is no guarantee it will survive the dissolution of the company if/when that happens. With Lean & others you can fairly safely assume that a particular version will be archived somewhere, and so a proof formalized for that version can be checked at any time. Not so with Mathematica. It's unfortunate, Mathematica is a cool language, but that's just the structure of incentives at this time. This is without getting into what a proof formalized in Mathematica would even mean, the relative maturity of the kernel and the possibility of there being multiple kernel implementations for cross-checking, and so on.
> I’ve had the experience quite a few times now: I try to autoformalize something, and an AI will tell me “I did it; look, the proof checks out!” But, actually, in some sense it cheated: instead of formalizing what I intended, it found a (sometimes very squirrely) way to interpret what I asked so that it could successfully prove it. It’s often very hard to tell, though, that this is what happened—not least because the formalized versions of things (say as expressed in popular proof assistant systems) tend to be very low level, very verbose and very hard for us humans to understand.
This has largely been my experience in programming, too. Often when given a broad objective within an existing code base, even the frontier models seem to stand up a half dozen tests proving compliance and then add 3 new branches solving for those specific tests alone. My best guess is that our code base is quite far out of the distribution (and not for just good reasons). The agents are thus reduced to these tactics rather than extending and refactoring well known abstractions.
The reason we (as in university-funding society) need mathematicians is so that there's a steward over math that can help me, a complete layman, understand and apply mathematics. "Being a steward" will sometimes not look like much. Sometimes it's just going to be dicking around with LLM's.
But math is too important for it to be abandoned by humans and left to models. I, the tax-paying bonehead engineer, need mathematicians to exist. They can have their ego beaten up by Claude, sure, whatever, but they're not allowed to just close up shop in despair.
Much more rejections due to more submissions, and AI will probably not help you (unless you solve a major open problem).
Even frontier models still struggle at proving small conjecturers despite all the hype about major breakthroughs. It really depends a lot on the prompt, the type of problem, among other factors. But AI does not suddenly make publishing in a journal easier, although it does make it easier to produce papers.
I see two assumptions in the core argument, both of which are open to challenge.
1. AI can do proofs, but deciding which problems to solve, which math is useful, is by humans.
2. The math that's picked needs to be understandable by humans.
For #1, AI may be able to play a significant, if not a takeover, role for even figuring out what math is useful.
For #2, understandability by humans may be good for now, but could also turn out to be a significant constraint. Correctness is a goal, trust is an important requirement, human understandability may be an intermediary for that, but not necessarily the end goal.
In other words, the article may stand the current state of the art, but may not stand merely a couple years down the road.
Not sure I get that. If AI is not working for humans' benefits (i.e. it is not working towards an optimization problem set-up by humans), whom is it supposed to benefit, then? (i.e. how is that better than an entropy-producing machine)?
Same for 2, there are so many infinite ways to boil the oceans, but so few oceans to boil to begin with. Better make sure that this insane energy (both in the physical, due to natural resources scarcity, as well as intellectual) is spent towards meaningful and useful ends. We can no longer be the judges of that if we can't comprehend what we got in return.
>> If AI is not working for humans' benefits
>> not working towards an optimization problem set-up by humans
I am suggesting neither of the two.
Not a great analogy but a parent may be working for an infant's benefit without the infant yet being able to understand. Taking an arbitrarily broad example, the problem to optimize for could be "help humanity advance", and other aspects could be subgoals of the same including what mathematics is useful.
I see several problems with this,
- by definition, the AI isn't human, irrespective of its computational abilities, it needs a human to tell it what being human is, and the ways "humanity can be advanced" need to be validated on this basis
- "advancing humanity" isn't a one dimensional problem/single-KPI optimization game, you might optimize certain things (e.g. expected life expectancy) at the detriment of others (e.g. freedom of movement). If you are not understanding the proposition, you are not understanding the target outcome.
- even if the target outcome is perfectly formally specified and commonly understood (which it can't), you want your AI to rationalize that the journey to get there is the most direct and efficient.
- especially so since subsequent runs of the same prompt will provide different plans
- anything less than that is begging to be conned: as a sensible person, you wouldn't give unlimited power and all your faith to a single individual trusted to "advance humanity" if they cannot be understood. On what ground would you give the machine a pass?
In all, such comments really worry me. It's like AI is triggering for some people the kinds of oppressive religious feelings whereby the individual should submit itself to the will and desires of a pretended all-powerful being. Do you really think our ancestors chose to cut off their legs in abandonment when they figured that some animals could outpace them? No, they built traps and throwing weapons to catch them and see how they taste.
I agree that AI needs alignment and validation, and this is already been worked on [*1]. I'll assume below that this is a given.
For the rest, AI itself may be able to handle better than most humans. It already has good idea about what being human is [*2], understands that this isn't a one-dimensional problem, can do balancing like humans would, find more direct/efficient paths than humans. For many things I discuss with AI, I find that it already reasons much better than most humans (not even considering that AI knows so much more than any human).
I think we over-index on "the same prompt will provide different plans". Humans would do the same too. Humans, working in isolation or with collaboration can self-correct, but the same applies to AI.
>> On what ground would you give the machine a pass?
The benchmark I hold is humans themselves. Humans currently have the pass, and have had it since history. Yet, there are many irrational decisions everywhere around.
I hear complaints that AI hallucinates. Yes, it does. Humans do too, and more often than they are willing to admit. The concept of 'god' may entirely be a hallucination (i.e., something not supported by facts). There is no good scientific evidence of prayers working, yet many humans believe in the same.
The real issue is that AI seems to learn and hallucinate in a different way than humans -- it sometimes makes some very silly mistakes. No disagreement, we need to make it better. The pace at which AI can become better however could easily surpass the speed at which humans learn or change. I am not suggesting we give AI the pass till it becomes good enough, there's enough progress on alignment, etc.
>> Do you really think our ancestors chose to cut off their legs in abandonment when they figured that some animals could outpace them?
Great! Here lies an important point. Between humans and animals, we have nature's evolutionary processes, survival of the fittest, ... Depending on how one sees it (and this is a real debate in my mind), we could use AI to enhance our survival and progress faster, or we could see AI as an enemy (like it is another species) and compete.
For some people, the goal is exactly advancements of humans. I, so far, see it as evolutionary progress, whatever form it takes.
[*1] Whether we are doing enough for that or not, is a valid debate.
[*2] AI cannot experience it, it cannot 'know' it in the human sense of knowing if that means something different, but it could emulate well enough.
It's not clear current what the human benefit of current mathematics is. Can people figure out what's important?
It can still benefit humans, but there are plenty of situations where humans prioritize guarantees of formal correctness over the ease of navigating granular details.
> AI can do proofs, but deciding which problems to solve, which math is useful, is by humans.
This is within a very narrow view before the emergence of always running “minds” within any given domain. The only reason they don’t exist now is because they’re expensive.
Pretty soon we’re going to have always running minds that are constantly thinking about every domain imaginable and coming up with their own proofs and improvements and everything else imaginable within those domains.
Isn't this basically the hitchhikers guide to the galaxy passage on the computer that spends millions of years to determine the ultimate answer to be 42 and then when people get upset at how meaningless that answer is offers to build another even bigger computer to figure out what the question was.
It’s more akin to the minds from the culture series (though not really).
Love this reference but the Minds are self building intelligences and (iirc) run their internal logic in hyperspace to get around the limits of the speed of light among other fantastical things. If we had genuine super intelligence like that then the discussion on how to use it would be moot as we wouldn’t be in that discussion.
Some arguments are based on "past experience..."
However, such experiences are not absolute truths and cannot be equated with the current situation.
I think the big question is whether these recent gains continue. It’s entirely plausible that the ai firms are going to run out of human trainers capable of further refining the model. It’s also possible we are on the cusp of real superintelligence.
Read by the author https://www.youtube.com/live/gPrWX8i1htM (with multiple mentions of the Wolfram language and such)
What is a good exit strategy?
We live not to have a good time, but to make the times good.
Wolfram gives a very important and sober take on the present state and future of pure math. I came away with the following key takeaways:
1. An essential goal of mathematics is human understanding. The computation of proof terms doesn't necessarily enrich human understanding. The proof of the four color theorem result is a good example, and formal verification/SAT solving gives many more: these are results that can be trusted up to our trust in the system used to produce them, and they can be used in practice, but they don't necessarily enrich our understanding. Imagine a computer with near infinite proof search powers set loose with the current human definitions, theorems, and understanding of mathematics. Suppose it constructs a proof for a new theorem at our mathematical frontier. The shortest such proof in terms of currently understood definitions and concepts could be so long and mechanical that the entire lineage of humans until the end of the universe could not finish reading it. So though it overlaps with the activity of mathematicians, this type of computational proof search is not mathematics as such. This is an important distinction that many people do not seem to grasp and some dismiss as cope.
2. The human activity of theory building, rendering otherwise monstrous proofs like the one I discussed above into light conceptual arguments a person can understand, appears at this time out of reach of models. Maybe they will do this in the future, but it is not yet the case. Human theory building drastically compresses the spaces of theorems and their proofs: this is why great theory builders like Groethendieck are so important to the field; grinding has its value too, but runs up against computational limits in both humans and computers. These limits are collapsed by the conceptual shortcuts created by theory builders.
Gowers has a nice and arguably better-grounded article on the mathematical capabilities of recent LLMs that I think is enlightening to read alongside Wolfram's bird's eye view of the implications of those capabilities: https://gowers.wordpress.com/2026/08/12/what-sort-of-maths-a...
I've heard the human understanding line a whole bunch.
But it begs the question. What is the value to the rest of humanity that a small group of people possesses something that can be called human understanding? Particularly when that group of people is historically terrible at communication (as is routinely demonstrated in Calculus classes), and most humans are not capable of learning that understanding (though more are capable than think they are capable - that is another story).
I am speaking as someone who nearly finished a PhD in mathematics. I understand why mathematicians would wish to continue in the age of AI. But, barring something like universal basic income, it isn't obvious why the rest of humanity would support them in this endeavor.
The idea that the rest of humanity needs to support them is absurd. A couple billion in private trust funds could fund mathematics researchers in perpetuity.
The idea of being supported by the rest of humanity, is not a requirement that the rest of humanity support it equally.
The idea that "someone rich will take care of it", reminds me of a passage from https://en.wikipedia.org/wiki/The_Logic_of_Collective_Action. It talks about "the exploitation of the large, by the small". Where a public good (in this case mathematics) is provisioned by a large entity that finds it worthwhile for their own reasons, and the remaining players who value it, feel no need to contribute anything themselves.
What does it mean for a mathematics to be "provisioned"? A well-funded math foundation run like the Linux Foundation could cover the lifestyles of many elite mathematicians. I don't understand why you imagine it would be centralized behind one "large entity" or that this large entity, so long as it is not a government, can't be trusted to fund good things.
People unaffiliated with the Linux foundation contribute to Linux. Regularly.
For a public good to be provisioned means that the necessary resources and organization for that good, need to be supplied.
Supporting a group of elite mathematicians is not enough. There needs to be a path to becoming an elite mathematician.
Which either means that it becomes an unpaid hobby. Or there is some wider source of support for the profession.
I think the central problem with that line of reasoning is that it supposes we are already living with AGI. Even the latest PR from Anthropic admits that we are not [0] there yet. If we don't have AGI capable of integrating and employing the mathematical results produced by an artificially superintelligent theorem prover so that they become useful to the rest of science and engineering, we will need humans to do at least some of the work in mathematics. Moreover if we freeze current AI capabilities today, "digestion" probably doesn't scale, so mathematical research will probably end up AI-assisted in ways not far off from what we're already seeing in the sciences.
[0] https://www.anthropic.com/research/claude-shaped-science
I am stating a counter-argument to a common argument.
You are simply giving a different argument than the one I'm giving a counter to. Yes, of course, if including human understanding results in strictly better results than AI, then there is an argument for the value supplied by human mathematicians.
But that's an argument that human understanding is on the path to better outcomes. It's not an argument that the intrinsic worth of human understanding is a reason to support mathematicians.
I kind of feel motte-and-bailey'd here. You began by questioning the value of human understanding in mathematics and I've explained (exactly as you've said) why it is currently on the critical path of outcomes in science and engineering with value to humanity almost nobody would dispute. This new question, where human understanding of mathematics isn't on the critical path of anything, is something I can't argue, in the same way that I can't argue for the intrinsic value or meaning of essentially any human activity without resorting to what amount to basically religious axioms.
> What is the value to the rest of humanity that a small group of people possesses something that can be called human understanding?
Okay, now, take the field of medicine and ask the same question.
Okay, now, when AI and robots can do surgeries and advise people better than modern doctors do (the bar is not that high, ask anyone with an autoimmune disease or gut issues)...
>What is the value to the rest of humanity that a small group of people possesses something that can be called human understanding?
If you're going to ask that then you need to ask the same thing about essentially every non-STEM department.
People in the humanities regularly write things that many people find interesting and entertaining. One need not be a historian to understand https://acoup.blog/2026/01/30/collections-the-late-bronze-ag... and find it interesting. The value here is not that they possess human understanding. It is that they can share human understanding, in a way that many humans can be interested in.
But, for example, take my first paper: https://dspace.library.uvic.ca/server/api/core/bitstreams/66... The title was, "Derivations whose iterates are zero or invertible on a left ideal." In order to understand the title, you need to learn what a ring is, what a derivation on a ring is, what an invertible element of a ring is, what an ideal of a ring is, and why these are concepts that anyone would have invented. In order to read the theorem, you have to further understand what division ring is, a matrix ring is, the characteristic of a ring, and a polynomial over a ring. The proof is even worse.
Good luck interesting anyone who wasn't a mathematician. (And good luck interesting most mathematicians!)
The desperately needed TL;DR is that perhaps the actual mathematics itself (i.e. proofs, calculations, etc.) can be done by AI, but why we do it and deciding which problems to solve can only be done by AI. Therefore mathematician do maths.
I'm not a mathematician, but this seems like a weak and slightly bizarre argument.
> but why we do it and deciding which problems to solve can only be done by AI
Sorry, was there a typo here? Both sides of the comparison are AI, and in the affirmative?
Oops, there was, sorry. And now I can't correct it.
The right side (how to use the maths, what maths to do, etc.) should have said it should be done by people.
I'm wondering how we are going to maintain a critical mass of people who understand frontier mathematics if in another five or ten years the only "mathematicians" truly working at the frontier anymore are AIs. Or maybe "understanding" at depth will become a thing of the past, superseded by broad-strokes grasp of results plus machine verification.
For one thing, every topic can be explained with motion graphics offering another 1 or 2 dimensions (time and for the 3d stuff that is hard to see structure of in static 2d, plus things like vr will allow more direct depth perception and different views) than a typical blackboard lecture.
On Proof and Progress in Mathematics talks about how much gets lost of the geometric understanding when translated to a paper. The kinds of things many mathematicians visualize in their head will be much easier to transmit.
I don't know if that is enough to offset the other affects, but learning and transmitting the understanding should be able to get much easier for a lot of people in principle.
If training data are purely historical, then how does the AI look forward?
And if human mathematicians are drummed out of producing future training data, then can AI end up proving itself so much "eating the seed corn", only at scale?
This applies to all professions, not just mathematicians.
The most common answer I’ve heard so far is “well, AI will train on its own output… maybe”.
I don’t think that’s even possible.
AI training on its own output is (probably) fine if it’s validated, like a lean-verified proof to Navier-Stokes.
Because it being validated as correct resolves the main issue with incestuous training, which is compounding error.
But from whence come the fresh insights?
The AI provides fresh insights as the output.
AI creates novel discovery> incorporates this information > makes new discovery
This is how it works for humans too.
One wonders at the fresh/derived breakdown.
"AI slop", for example, appears a regression toward some "mean".
This is also how it works for humans too. What portion of human art produced in 2025 or 1925 do you think was fresh and novel, compared to derived and repetitive?
The same is true for science and Engineering.
What matters is if you have a method of sorting through the repetitive trash
It's not my contention that AI & people do not both conform to Sturgeon's Law[0].
Rather, that AI is unlikely to produce a fresh Marcin Patrzalek[1].
[0] https://en.wikipedia.org/wiki/Sturgeon%27s_law
[1] https://youtu.be/zbfKFa-reBE?is=pHxTCFcfyU0evbvi
If training data are purely historical, then how does the human look forward?
Seems to me not impossible that given current knowledge, AI generate one nugget more of knowledge (eg a proof of Navier Stokes), and given current knowledge + the nugget, generate yet some more new knowledge.
Not a given, but not obviously impossible either.
> how does the human look forward
Well, through the metaphysical lens that has both powered innovation and stumped the Really Smart Types since antiquity.
> back in 1988, when we first introduced Mathematica, there was also some of the same kind of talk about math being taken over, and made pointless. Of course that’s not how it worked out at all.
Makes sense.
> For me, its greatest use in mathematical pursuits has been its ability in effect to thematically mine the knowledgebase of human mathematics. ... Modern AI is, first and foremost, a way of leveraging the existing corpus of human knowledge.
AI is more a database of knowledge (stolen knowledge but let's leave that discussion asside) than a thinking machine. You can query a compressed version of millions of books.
That is very useful. (Are we already StarTrek-communists?
> generating useful mathematics is a much more exacting activity than generating language.
This is something that most people forget. Generative AI is mostly LLMs, and they are chatbots not mathbots.
> It’s a frustrating feature of modern times that someone like me gets sent many AI-generated documents every day that have the “statistical texture” of math papers, but that one at least expects have a very low probability of being meaningfully correct
And here is the trick. A million monkeys with a million typewriters may write a Shakespeare masterpiece. But they would not be able to differentiate it from garbage text.
> So, yes, there’s every reason to expect a bright future—now with some additional help from AI—for that most rarefied of human pursuits: research in pure mathematics.
Happy to hear that.
> AI is more a database of knowledge than a thinking machine. You can query a compressed version of millions of books.
No…? Isn’t the entire reason we are having this discussion because LLMs are coming up with results that are not already represented in the training data?
That's not necessarily true.
The "new" it comes up with very often is a combination of existing tools.
AIs have representations of the training data, but they also do search. Search creates new knowledge. For example, a chess program doesn't have a representation of all chess games, but can search and in doing so create new games. This isn't (just) random generation, but generation constrained by an evaluation function.
>AI is more a database of knowledge (stolen knowledge but let's leave that discussion asside) than a thinking machine. You can query a compressed version of millions of books.
Outdated view. Reasoning models do a lot of "thinking", and can solve novel problems - even quite difficult problems.
The whole reason we're even having this conversation is because AI solved a millennium prize problem that no human knew an answer to.
> The whole reason we're even having this conversation is because AI solved a millennium prize problem that no human knew an answer to.
Well, in this case, some mathematicians were on the finish line, and the AI combined (presumably, since nothing is published) existing work to obtain the new result, barely beating the original authors.
I like Wolfram and always enjoy reading his posts. An irreconcilable thing here is Wolfram clearly wants the Mathematica language to be central to the development of math as a field, but it's proprietary and so there is no guarantee it will survive the dissolution of the company if/when that happens. With Lean & others you can fairly safely assume that a particular version will be archived somewhere, and so a proof formalized for that version can be checked at any time. Not so with Mathematica. It's unfortunate, Mathematica is a cool language, but that's just the structure of incentives at this time. This is without getting into what a proof formalized in Mathematica would even mean, the relative maturity of the kernel and the possibility of there being multiple kernel implementations for cross-checking, and so on.