As a fellow former mathematician, I share your sense that “mathematics as poetry” seems like the most likely outcome, at least for pure maths and barring more fundamental AI-driven changes to the human condition (e.g., extinction, or humans being replaced by very different kinds of minds).
Rather, as a more explicit (and leading) form of the question, I prefer
How do mathematicians advance human understanding of mathematics?
This question brings to the fore something that is fundamental and pervasive: that what we are doing is finding ways for people to understand and think about mathematics.
The rapid advance of computers has helped dramatize this point, because computers and people are very different. For instance, when Appel and Haken completed a proof of the 4-color map theorem using a massive automatic computation, it evoked much controversy. I interpret the controversy as having little to do with doubt people had as to the veracity of the theorem or the correctness of the proof. Rather, it reflected a continuing desire for human understanding of a proof, in addition to knowledge that the theorem is true. On a more everyday level, it is common for people first starting to grapple with computers to make large-scale computations of things they might have done on a smaller scale by hand. They might print out a table of the first 10,000 primes, only to find that their printout isn’t something they really wanted after all. They discover by this kind of experience that what they really want is usually not some collection of “answers”—what they want is understanding.
The product of mathematics is clarity and understanding. Not theorems, by themselves. [...]
I think of mathematics as having a large component of psychology, because of its strong dependence on human minds. Dehumanized mathematics would be more like computer code, which is very different. [...]
In short, mathematics only exists in a living community of mathematicians that spreads understanding and breaths life into ideas both old and new. The real satisfaction from mathematics is in learning from others and sharing with others. All of us have clear understanding of a few things and murky concepts of many more. There is no way to run out of ideas in need of clarification. The question of who is the first person to ever set foot on some square meter of land is really secondary. Revolutionary change does matter, but revolutions are few, and they are not self-sustaining—they depend very heavily on the community of mathematicians.
To be clear, I do think that AI comes with bigger challenges for, and will have bigger impacts on, the maths community than the kind of ‘automatic computation’ or ‘computer code’ Thurston mentions in these quotes because LLMs can do a much wider range of maths tasks, including communicating about maths in many of the ways human mathematicians do, and so their use likely won’t be limited to narrow use cases outside an exclusively human-driven core activity of generating and transmitting ‘understanding’ to each other. And so in particular I agree with the broad picture you outline where AI use is a pervasive feature of a “mathematics as poetry” world.
As a fellow former mathematician, I share your sense that “mathematics as poetry” seems like the most likely outcome, at least for pure maths and barring more fundamental AI-driven changes to the human condition (e.g., extinction, or humans being replaced by very different kinds of minds).
Some of your discussion reminded me of Field Medalist Bill Thurston’s 1994 note “On Proof and Progress in Mathematics”:
Similarly, Thurston wrote on MathOverflow:
To be clear, I do think that AI comes with bigger challenges for, and will have bigger impacts on, the maths community than the kind of ‘automatic computation’ or ‘computer code’ Thurston mentions in these quotes because LLMs can do a much wider range of maths tasks, including communicating about maths in many of the ways human mathematicians do, and so their use likely won’t be limited to narrow use cases outside an exclusively human-driven core activity of generating and transmitting ‘understanding’ to each other. And so in particular I agree with the broad picture you outline where AI use is a pervasive feature of a “mathematics as poetry” world.