What Is the Future of Mathematics?

Academic Math

In a near future where advanced AIs can prove or disprove almost any conjecture, I wonder what the future of math higher education will be. As a PhD student with a thesis proposal, you will be strongly tempted to just go press the red “prove me” button sitting right there at your desktop. But instead you plan to spend the next three years of your life going down that rabbit hole, investigating all avenues, reading papers, presenting intermediate work at conferences, and eventually getting that PhD, proving you’re ready to enter the field — and along the way, finding that proof. Of course some people will cheat and press that button, but most will understand that the journey is the point, and will spend the time to actually learn things, establish connections, and argue and write cogently and clearly. We’ll all know we won’t ever perform at the same level as machines, but like climbing a mountain, it’s a particularly human challenge.

I see this as quite analogous to what happened in the game of Go, when AlphaGo beat Lee Sedol with the infamous move 37. It was thought that fewer humans would be interested in playing once the ceiling was raised so high, but since then interest in Go has continued to be strong, often using AIs as teachers. I wonder if we’ll see similar behavior in pure math PhDs — will people turn away because they know the machines are better, or will they embrace using them to reach new heights?

Universities and Funding

But after a few years, I wonder about how the priorities of universities themselves will evolve. Will they continue to support math grad students, knowing they could just press that red button and get the result (a proof, a conference paper, solving an important open problem — and by the way, who decides what problems are important anyway)? Why exactly are they funding the development of mathematicians who can perhaps perform almost as well as AIs? What is the end goal?

I think what will happen is math departments will eventually be treated more like, and funded more like, the humanities. Nobody cares how well an AI can write or paint or play music²; these are fundamentally human endeavors. Arts are created by humans, for humans, to express human feelings, aspirations, fears and desires. We write poetry because it helps us be more human. We study human history to understand humanity from our human perspective. Our creativity is uniquely our own, and no machine can replicate that. So I think that pure mathematics in academia will become an expression of what problems and research areas and ways to express mathematical ideas are interesting to humans — the ones that are particularly fascinating to us, or that reveal symmetries that help us see the beauty of pure mathematics, or even just the ones that require incredible feats of human effort and creativity to overcome tremendous obstacles.

Unfortunately, humanities funding has been in trouble³ for many years now, and especially in the current cultural/political climate the problems are dire. That, coupled with anti-science policies, may mean that all pure science studies end up with the same funding crises we now see in the arts and humanities.

Where do we go from here?

I wonder what fully unleashed machine-created pure math research would look like: what would be interesting to an AI? Today they are trained on human culture, but they may self-evolve into their own forms with different goals and interests, as in the prescient film Her. I’ve done some early experiments in AI creativity⁴ but the results are very inconclusive; recursive self-improvement isn’t close enough yet to allow the AIs to evolve themselves fast enough to find new tastes, interests and biases — which is likely a good thing; see my previous article on risks.

I think eventually we will learn to work together as distributed cyborgs, as we already are in other domains like software and medicine. Machines may do the heavy lifting, and even much of the work of creative discovery. In fact, it’s possible that amateur scholars in concert with advanced AIs could participate in mathematical discovery without traditional university academic support, perhaps returning in some way to the Renaissance patronage model or Enlightenment natural history scholars⁵. But in any case, I believe human creativity is irreplaceable; a true dialog between humans and machines working to discover new ideas could open up entirely new fields. My hope is that humans will still be a part of that journey, helping make discoveries that matter to us.

Hero image: Domenico Fetti, Archimedes Thoughtful (c. 1620), Gemäldegalerie Alte Meister, Dresden. Public domain.