Let's discover phenomena instead of solving problems
Remember the Mandelbrot set?
The Mandelbrot set is the set of complex numbers \(c\) such that if we define \(x_1 = 0\) and \(x_{n + 1} = x_n^2 + c\), then \(|x_n|\) doesn't go to infinity. It's a simple definition, but when you plot it, you get mind-blowing fractal images like the ones in the animation above. What an amazing phenomenon!
I bring it up because mathematical disciplines are currently going through a kind of existential crisis. The latest AI models have astonishing abilities to prove theorems and solve open problems. We are finally getting answers to many longstanding questions, which is good, but we are left wondering what we humans ought to do going forward.
I propose that in this new era, we focus on identifying new, interesting mathematical phenomena, almost like finding new plant or animal species in a rainforest. We can contribute novel definitions, questions, conjectures, and theorem statements. It's okay if coming up with proofs isn't really part of the job anymore.
This mindset is a big change for me, but I think the new job I'm describing is still a job I can love. We get to do more than just study and understand AI-generated proofs. We can continue to make discoveries, which is a cool, exciting, valuable thing to do! Discovering something like the Mandelbrot set would be an excellent contribution to mathematics. It wouldn't necessarily solve any important open problems, but it would introduce new ones. As far as I'm aware, Benoit Mandelbrot never proved any deep theorems about his namesake set, but he did advance plenty of conjectures about it.
A considerable amount of pre-AI research already fits into the paradigm I'm describing. Many papers, including some of my own, answer questions that nobody had even realized they should be asking. I'm proposing that we write more papers like that.
Of course, there might come a day when AI models have superhuman abilities to generate interesting mathematical definitions, questions, conjectures, and theorem statements, and then maybe mathematical research will have to evolve further. But I don't think that day is here just yet.
Acknowledgments
I thank Alicia Torres Hoza for helpful comments on a draft of this blog post.
AI Disclosure
I read AI-generated reviews of drafts of this blog post as part of my writing process.