What's the future for pure math research in the age of AI?
writings.stephenwolfram.com
writings.stephenwolfram.com
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.
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.
>> 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.
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.
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.
However, such experiences are not absolute truths and cannot be equated with the current situation.
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.
AI creates novel discovery> incorporates this information > makes new discovery
This is how it works for humans too.
"AI slop", for example, appears a regression toward some "mean".
The same is true for science and Engineering.
What matters is if you have a method of sorting through the repetitive trash
Rather, that AI is unlikely to produce a fresh Marcin Patrzalek[1].
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.
[0] https://www.anthropic.com/research/claude-shaped-science
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.
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 unaffiliated with the Linux foundation contribute to Linux. Regularly.
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!)
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?
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.
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.