Research in Mathematics after AI

Mathematicians can spend years on a problem and still find the time well spent, even if they never solve it. I enjoy doing mathematics, though I can see how returning to the same stubborn question day after day might look odd from the outside. We persist even on days when the work is frustrating and progress is hard to see. For many of us, there is meaning in that effort: we are cultivating how we think.

The style of an argument, the pictures we learn to draw, and the colleagues we exchange ideas with are central to that process. As we hone our craft, we become able to understand and use concepts that once seemed beyond us. We take pride in having earned that understanding. Mathematicians have many reasons for pursuing the subject, including its applications. But the solution to a problem is only part of what we gain through the work and can share with others.

One thing that unsettles me about AI doing mathematical research is that the work through which we learned to think may no longer be necessary to produce new mathematics. What becomes of that work, and what should we be aiming for as mathematicians?

When proofs outpace understanding

Let me indulge a mathematical habit and state some assumptions. Imagine AI systems that, for a substantial range of research problems, can:

1. produce proofs of statements we propose;
2. express those proofs in a form a computer can check against a precise statement;
3. explore possible arguments much faster than a person can.

These are working assumptions. OpenAI's initial release of 722 mathematical manuscripts [1] and its proposed Navier–Stokes proof [2] give us reason to consider this situation now. Formal verification covers only part of this collection, and the results remain under review.

Mathematicians have emphasized that having a proof and developing the thinking needed to find one are different achievements [3, 4]. The declaration A Severe Misalignment of AI in Mathematics reflects this concern, warning that using problems as AI benchmarks can undermine the work of shaping ideas and deepening understanding [5].

At the research frontier, AI could produce theories that remain beyond our individual understanding, even after considerable study. We might be able to check an argument step by step without seeing enough of the surrounding ideas to work with them. A wall of correct but alien mathematics would leave us with an uncomfortable question: how do we participate in a subject we can no longer fully grasp?

The other product of research

Research changes what a mathematician can do. Alongside a proof, we develop methods, examples, and a sense of where an argument might work. How we arrive at a result deserves a place in what we share, so that others can question our choices and put the ideas to use.

One way to pursue this with AI is to build a research-trained AI: a system that has learned through a particular investigation to use and extend the mathematical ideas and methods developed there. It could distill that experience into ways of working that others could learn to use. Such a system would itself be a result of the research.

There is already work in this direction. In biology, Paper2Agent makes a paper's computational methods available through interactive agents [6].

Suppose I receive a proof of a conjecture I care about. It is correct, but I cannot tell which idea made the breakthrough. With an expert beside me, I could ask:

Which step required a new idea or a change in perspective?

Can you show me a simple case to illustrate the main idea?

Could this method help with the problem I am working on?

I would put these questions to a research-trained AI much as I would to an expert at the blackboard. I would work through its answers, drawing another picture or trying a simpler example, until I could begin to use the idea myself.

Now, suppose an AI develops a mathematical theory of quantum gravity and spits out a 5,000-page verifiably correct write-up. It might contain methods relevant to my own work, yet I would have little sense of where to look or how to recognize them. But the AI system itself, supported by a structured record of how it developed the theory, could help me find and use those methods without first mastering the theory as a whole.

I would want such a system to have some judgment about where to begin. It should choose examples that reveal the mechanism of an argument and recognize when a different formulation makes a method easier to use. It should be able to move between a calculation and the structure it suggests, lingering over a special case when there is something to learn from it. Such command of an idea is part of what we recognize in a mathematical expert.

Responsiveness to someone else's line of thought matters, too. If I offer a picture or an unfinished argument, the system could help develop that idea far enough to see what it can do. It should be willing to reconsider an approach when I raise an objection, or follow a promising digression. Sometimes an exchange might leave us with a better question.

An AI-generated proof is only one product of research; a research-trained AI would preserve the craft developed in finding the proof.

What we keep cultivating

We may eventually need AI as an intermediary even within a subject we call our own. I find that prospect uncomfortable. The concern raised in A Severe Misalignment of AI in Mathematics applies here, too: such a system could displace the human conversations through which understanding grows. That concern should guide how we use a research-trained AI. I would want it to help us bring difficult ideas into our seminars, collaborations, and teaching. The time spent discussing, simplifying, and questioning those ideas would remain part of the research.

We also have to judge what the system tells us, even when its proof has been verified. It can give a fluent but mistaken account of why an argument works. Its explanations should point to supporting steps and distinguish established consequences from suggestions. After an exchange, I would want to be able to explain the idea, recognize when it applies, and adapt it to another problem.

Our own learning need not depend on staying ahead of a machine. We can still take up difficult problems, develop approaches, and follow them far enough to discover what works. What we try and why—including the attempts that fail—is worth sharing as the investigation unfolds. Other mathematicians could build on that work, and AI systems could learn from it, acquiring methods they did not have when we began.

What might we recognize as a mathematical contribution in that setting? A solved problem gives us something clear to point to; the ability developed along the way is harder to see. A method that others can adapt, an example that exposes an obstruction, or an account of why an approach failed can also advance the subject. This process of solving difficult problems remains part of what draws me to mathematics. We are still cultivating thinking, just not solely our own anymore.