// HACKER NEWS — CYBERSECURITY
A Beginning for Mathematics
Three years ago, AI systems could not reliably add two numbers. A year ago, internal models at OpenAI and DeepMind received the equivalent of a gold-medal score on the IMO. Now, these systems are autonomously resolving major open questions. It’s hard to imagine this trend continuing for another year, but I expect it will. It is clear that this will require a radical rethinking of our profession.
A few weeks ago, I gave a talk titled The End of Mathematics. If you only read the title1, you might guess that this talk was about how, soon, AI will “solve” math. That’s not what it was about. The talk instead laid out a gloomy vision of the future, in which, despite the possibility of AI systems that are robustly superhuman at mathematics, the design of our institutions causes human understanding of mathematics, and possibly even mathematical progress in the abstract, to stall. I think we will avoid this future, but I also think it is plausibly the default if academic mathematics does not adapt. Despite my relative enthusiasm for the use of AI to do mathematics, I share this view with many of its detractors.
Here I want to lay out, instead, a positive vision of the future of mathematics, and the human practice of mathematics. I claim we can deepen human understanding even as the production of interesting mathematics becomes less dependent on it.
This essay will take as a premise that AI systems that are robustly superhuman at most or all aspects of mathematics will be here soon. But the concrete changes to our institutions I propose only require accepting the weaker premise that the production of mathematical text is becoming increasingly disconnected from mathematical understanding.
I think it has now become clear that there is no consensus in the mathematical community as to what our goals are. Some of us want to solve problems; some of us think of mathematics as play or as poetry. For some: “Wir müssen wissen – wir werden wissen.”2 Some of us think we are penetrating the mysteries of the platonic realm. Some of us think the goal is to embody love of and understanding of mathematics,3 and to transmit that love and understanding to the next generation.
We’re trying to produce and understand high quality mathematics.
We’re trying to produce high quality mathematicians.
These goals should be construed broadly. What high quality mathematics consists of has changed quite dramatically over time; we come to its definition as a community. We are not just training PhD students to do research in mathematics. A substantial part of our job, though perhaps an underemphasized one, is to educate the general public about high quality mathematics and mathematical thinking.4
Whatever our goals are, we’ve operationalized them primarily through proving theorems. Almost all papers or PhD theses have a main theorem, and ostensibly a proof of it. But it should be clear that the goal of mathematics is not to prove theorems; if it was, it would be trivial to automate. A computer or monkey could easily start at the axioms of ZFC and iteratively apply deduction rules to them, with no attention whatsoever paid to their meaning. It has had particular significance when a theorem resolves an open problem, especially one that has resisted substantial effort. Again this is easily automated; our computer or monkey can simply conjecture all mathematical propositions in alphabetical order.
The general attitude of our community towards a technology that can prove theorems and solve open problems suggests that these operationalizations of our values are at best incomplete.