AI is revolutionizing mathematics, and it will not go away. So somehow we have to arrange ourselves with it and find our way of doing mathematics under these changed conditions.

I am of course also fascinated by asking AI (ChatGPT in my case) to solve all my problems. And sometimes there are indeed answers which seem to be okay and which promise some real progress. But those answers are usually complicated and quite technical, and I don’t really feel like becoming a little helper and argument checker for AI.

I agree with many voices saying that asking the meaningful questions, and understanding and presenting the answers, will remain some of the main tasks for us. But this should happen in a way that we still feel good about going along with it. For me this means that I would like to take from AI mainly some ideas, but then think myself about whether they could be true, what they actually mean, how one could prove them, and what is the best way to present them and convince others.

Of course, like everybody else, I don’t have a final answer to how this should work in practice, or whether this is a route which will help mathematics to survive in anything resembling its present form.

Anyhow, I suppose we just have to try different routes. Here is one experiment I would like to make.

At the moment I am thinking about determinants of operator-valued semicircular elements. Together with Tobias Mai, we derived a few years ago a formula for the determinant of such an element S in terms of its covariance map. This was, however, for the case of vanishing mean of S, and I wondered whether there could be a nice and useful extension to the case of general mean.

Okay, so I asked ChatGPT. And it did what two years ago would have seemed totally out of range, but has now become almost ordinary: it gave me an answer, proved it, and also made some comments on possible implications for questions about the Brown measure of operator-valued circular elements.

It even wrote a manuscript about all this.

I could now just put my name on it, say that the ideas were developed in conversations with AI, and declare that I take full responsibility for the results. But this does not feel right to me. And, more importantly, it does not feel very satisfying.

So here is what I would rather like to do.

I will give some background, explain what ChatGPT proposes as the answer, and then open the problem for public discussion: Is the statement correct? Is it perhaps obvious? How is it related to things that somebody already knows? What is the right proof? And what other interesting questions or conjectures might come out of it?

My dream would be that the answers — and also the questions — are actually self-thought, and not just copies of something another AI conversation produced. Of course everybody could now ask AI about the problem and perhaps produce a paper on it. But this should not be about publishing or priority. It should be about understanding.

Maybe one could think of it as telling a mathematical story together, in a somewhat grassroots way: somebody starts with a question, somebody else recognizes a connection, another person finds an argument, somebody points out that the whole thing is wrong, or suggests the right formulation, and gradually we understand what is really going on. Of course, we can use AI in the background for information, references, or inspiration. But AI should not tell the story.

I have no idea whether this will work.

But let us try.

In my next post I will give the concrete mathematical problem and the beginning of the story.