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.
> 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.
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.
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!)
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.
> 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?
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.
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?
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.
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 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.
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.
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.
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.
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...
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 "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.
If you're going to ask that then you need to ask the same thing about essentially every non-STEM department.
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!)
However, such experiences are not absolute truths and cannot be equated with the current situation.
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.
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?
I'm not a mathematician, but this seems like a weak and slightly bizarre argument.
Sorry, was there a typo here? Both sides of the comparison are AI, and in the affirmative?
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?
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.
Well, through the metaphysical lens that has both powered innovation and stumped the Really Smart Types since antiquity.
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.
Because it being validated as correct resolves the main issue with incestuous training, which is compounding error.
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.