..

Some thoughts on AI (and math)

It appears to me that the current trajectory of mathematics is not looking good.

I am writing this late May 2026. Since I am lazy, I will not provide links to pages where in "normal" posts one would do so. A disclaimer is that this post was written entirely by me and AI was not used for any purpose during the creation of this post. Another page worth reading is here. Another disclaimer is that I stopped following closely this subject because it was causing me unnecessary anxiety, so I might have not noticed some recent developments.

I was prompted to write this after witnessing the recent surge in results proved by/with assistance of AI (most notably the unit distance problem), reading the comments and answers to several mathoverflow threads, and online and real-life discussions with various other mathematics people.

I believe mathematics is not about proving the largest amount of theorems by any means possible. I think this is not controversial and is stated also by the proponents of AI-assisted mathematics. It however seems that this is in conflict with their actions.

Mathematics is (at least for me) about having fun and pushing the boundaries of our mind. I believe AI-assisted mathematics is neither fun nor pushes the boundaries of our mind.

Consider the prompt of OpenAI given to their internal model which solved the unit distance problem: "[...] Resolve Erdős’s planar unit-distance problem completely [...]" (source: their paper, page 3). Now, I understand that the post-AI mathematician imagined by AI proponents is not purely a prompt engineer who just asks the AI to solve any problem they want and then the job of the mathematician is just to read and try to understand the AI proof. If the future were to be like this, I think it is almost immediately clear that this would not be good for mathematicians in general: I don't think anyone pursues mathematics with hopes of being a prompt engineer and/or debugger for AI. Doing mathematics oneself is significantly more fun than checking someone else's math, and even that would be, because of the "social aspect", significantly more fun than checking AI's math. It should also be clear that this does not push the boundaries of one's mind.

I believe now I have disposed of the case where the job of the post-AI mathematician is to check AI's math (in the sense that I "established" that this would be a bad outcome for mathematicians).

Therefore consider now a more likely option which seems to be actually believed by some people: The future post-AI mathematician has some sort of back-and-forth with an AI model and together they produce some mathematics. I can dispose of this option also as follows: One of the biggest appeal of mathematics is that it is more egalitarian than other fields of science in the sense that one can do meaningful mathematical research without any expensive equipment unlike more experimental fields (which require labs). In this case mathematicians with access to better AI models will inevitably be "more productive" in producing mathematical research. Also consider that mathematicians in less developed countries will lack access to paid models and will fall behind in their output. These seem to be common points made in the discussions I witnessed. I have seen some people arguing that AI makes mathematics more egalitarian in the sense that one no longer needs a long extensive training in mathematics to conduct research. However this is only true for people who have the privilege to afford the powerful models.

I believe the AI-assisted mathematicians of the future can result in a "the rich get richer" phenomenon where the people initially chosen with some criteria ("being in a select group at some t=0" ?) are given access to better models that are not available to others at t=1, who produce more/better mathematics using these models at t=2, who are then given access to even better models at t=3, et cetera. This essentially closes the door behind people who were not in this select group at t=0. "Obviously" this is not good for anyone other than the select group.

Another bad thing about this situation is that I don't think it is a wise move to tie the practice of mathematics fundamentally to the products of a few companies with ethically questionable practices. Note that these companies do not necessarily have the best interests of mathematics (or society at large, but probably this has been talked about so much that I don't have much to add to that discussion) when operating. For instance one does not know if one day the "good models" become very expensive and we have to buy them because we can no longer conduct research without AI. Obviously this would also be bad. We already have one leech on academia in the form of publishing companies. We don't need another one in the form of AI companies.

I now talk about some related things.

There seems to be an opinion held by some people that AI allows/will allow mathematicians to focus on the "deep aspects" of mathematics and leave the boring technical aspects to AI. I don't think this is really possible unless one is extremely talented and can see the deep phenomena immediately through the technical details. I don't think there are many such people; I am certainly not one of them. I need to work through many examples, go through technical details, et cetera to get a feel for the objects. And when I don't do this myself, I will not have a good understanding. It would be similar to reading a math book like a novel. I believe this is the case for a lot of people, especially early-career.

I have unfortunately seen at least one person taking the occasion of the unit distance problem being solved by AI to take a jab at combinatorics or the types of mathematics they feel is "not deep", as "even AI can do it". I believe this is harmful at many levels. I will focus on only one aspect. It is not guaranteed that when AI becomes more powerful it will not solve problems (where I count "develop a suitable theory of X" as also solving a problem) in areas one would consider "deeper". In general recall that AI companies have profit as their goal rather than the advancement of mathematics (or society at large). I believe (this should not be a controversial belief) that they use mathematics as a benchmark for the power of their models to attract investment. I now appropriate a famous quote to make my point, "First they came for the combinatorialists. I stayed silent because I was not a combinatorialist. [...]"

This is not a time to be divided. This is a time for mathematicians to stay in solidarity and push back against trillion-dollar companies with not necessarily ethical practices essentially taking advantage of our field, using it as a training ground, for more investment. What do we get in exchange? We get to use their products in our research and make them more money. This does not seem fair.

Consider the impact of AI on early-career researchers. What is the point of a PhD in mathematics? A few years ago it would be to prepare the student and train them in conducting mathematical research. It seems that the goalposts are moving. I definitely did not sign up to be pushed implicitly or explicitly to become a prompt engineer. Currently there seems to be some sort of "adopt AI or perish" mindset.

Consider the following quote from an interview of Terry Tao: "The job description is changing a lot. A graduate student who refuses to touch AI systems and just wants to prove things the way we’ve done in the past might find they have fewer opportunities, unfortunately. Those who understand maths traditionally but are also adept at using new tools can flourish.". I interpret this as follows: if one avoids the use of these tools for any reason, you will be denied jobs, grants, et cetera.

Consider now a graduate student who is working on a problem (where again problem is taken in the wide sense as in one of the previous paragraphs). Normally one would think hard and long about this problem, fail many times, and build an understanding of the objects in question. Even if in the end it turns out to have a short or easy solution, this would be a good training in mathematical research for the student. In the AI age, this would be solved in short order by AI and will not result in the increase in understanding for the student. However also if the student decides to gain understanding instead of letting AI help them, they risk someone else with powerful AI solving the problem before them, so there is an incentive to use AI, thus an incentive to have reduced understanding.

This can have the following detrimental result: once the mathematicians trained classically are all dead or retired, their understanding will be lost and a similar understanding will not have been gained by newcomers, which results in a reduced overall understanding of mathematics which is not good.

One more thing we should keep in mind is the health of fields. Consider the story of foliations mentioned in Thurston's essay (pages 13-14). Essentially, Thurston proved many things and people stopped working on the area because they thought whatever they proved, Thurston already has proven and it is sitting in a drawer. I feel that AI generated mathematics might have a similar effect. When, say, OpenAI announces a solution of a major problem, this might discourage newcomers to the field because they might think whatever they are trying to do, OpenAI is also trying to do with a crazy amount of resources.

There is also the related problem of insider knowledge. It is not an outlandish idea to think that when an internal model proves a result, this is communicated to some mathematicians before it is announced publicly. This gives these people an advantage; even if the company does not try to prove further results in the area, the people with the knowledge have the possibility to push the ideas further. I believe this can result in a similar effect with Thurston and foliations.

What do *I* do? I don't like the big AI companies. I refuse to pay them any amount of money. This means for the most part I only use the free models. I believe I have Gemini Pro for free due to being a student. I am not sure if this will ever run out or not. I however (in my research) tend to use the "free" version sparingly (usually for things like sanity checks), and the "pro" version essentially never. I know that this is putting me behind many people in terms of research productivity.

I hope/want to believe that in the future there will be at least some demand for this "human-made" mathematics that is done by little/no AI involvement. Some analogies would be: human-made art, "organic/natural" food products, natural bodybuilding/other sports where doping is frowned upon in general, playing video games without paying for in-game items that give one an advantage, et cetera. Another one: there are cars that take us from point A to point B quickly. People still walk or run, because it is fun.

Have a nice day, et cetera. Feel free to contact me to discuss this or other matters. If I receive valuable comments I might add them to this page later.