The future of human mathematics
Cross-posted from Twitter; written for mathematicians.
About a year ago, I wrote that superhuman math AI will plausibly arrive in the late 2020s/early 2030s. It now looks like if anything, I was underestimating the pace of progress. In this post, I will try to say something about the sociological implications for mathematics.
I don't want to focus too much on the predicament of the mathematical community today; instead, I'll try to take a longer-term view, and say something about where I think human mathematics might eventually end up. For the sake of argument, I'll assume that within the next decade or so, AI systems are developed that are vastly more capable than humans at every mathematical task (including proving theorems, building theories, and explaining mathematics to humans), and that there nevertheless remains a community of human mathematicians, alive and well, and with funding to pursue mathematics as they please. What would such a community be doing?
Of course, in this hypothetical, mathematicians are not pursuing mathematics for any productive purpose; that is handled by the machines. They are pursuing mathematics for its own sake. There is nothing at all new about this; it is the same purpose that Hardy famously defended in 1940 in A Mathematician's Apology, declaring, "Judged by all practical standards, the value of my mathematical life is nil." As a former set theorist, I can certainly say the same of my own work in that area, as I am sure would many others.
What, then, does the pursuit of mathematics for its own sake look like, in the AI era? There are many different activities that mathematicians might engage in, so this is largely a question about which of those I feel best embodies the spirit of mathematics as it is practiced today. I will go through a few different options in turn.
Mathematics as chess
One possibility is that humans will solve mathematical problems with known solutions. They might do this