Daniel Litt@littmath
This is a characteristically thoughtful and coherent account of mathematics from my colleague Jacob, and I agree with much of what he writes. But I want to push back on some aspects, which don't accord with my experience of or motivation for doing mathematics.
Problem-solving
I fully agree with Jacob that, as currently practiced, problem-solving is a fundamental aspect of doing mathematics; like Jacob, I identify as a "problem-solver" more than a "theory-builder." (A related axis: I identify more as a "frog" than a "bird.")
Why do we solve problems? For some of us, it's more or less about enjoyment. That is NOT why I solve problems. I enjoy parts of that process: getting the solution, some little moments of understanding along the way. But my primary emotional experience of problem-solving is not fun: it's frustration. I try to understand something and get confused and I HATE that feeling, and need to resolve it. For a while my bio on here read "forever confused" -- that's not an exaggeration.
I think the main reason I (and many other mathematicians) solve problems is that it's the only way we know how to ground ourselves in mathematical truth. Without solving problems and working out examples, our work inevitably devolves into bullshit.
The activity of mathematics
So is 80%+ of mathematics about problem-solving? I think this is a coherent account of mathematics but it's not my experience. Like Jacob and many other mathematicians my work is indeed guided by some big problems: for me, the Grothendieck-Katz p-curvature conjecture, some questions about mapping class groups, some questions about fundamental groups of algebraic varieties. Many of these problems have occupied me for a decade+ now.
My experience of thinking about these problems is, perhaps paradoxically, not about "problem-solving." Rather, these problems benchmark our failure to understand certain fundamental phenomena: differential equations, surfaces, polynomials. It's useful to have rigorously stated problems like this to guide the field, but I think they have relatively little influence on my day-to-day work. That looks more like: trying to identify the most basic situation in which our understanding fails, and develop it in that basic situation.
In this model, problem-solving is secondary: my typical experience is that I think I understand something new, often non-rigorously, and then try to operationalize it to solve some problems both to test the correctness of this understanding, and to measure its effectiveness. It's not uncommon in this model for a problem and its solution to appear at the exact same time. In fact, for me, it's somewhat unusual to write down a rigorous statement of a lemma that I do not already know how to prove, though this does of course happen.
Oracles
Jacob proposes the a thought experiment, where one has access to an AI oracle that can solve rigorously-stated problems better than humans but has less capability in other areas of the mathematical process. Like him, I do not expect this to be the long-term situation--eventually I expect AI mathematics to exceed humans in every mathematical capability--but let's run with it for a second. What would mathematical activity look like with such an oracle?
Jacob writes: "Well, you make a definition, and want to know if it’s the right one. You immediately ask your oracle a thousand questions. From “are these basic properties true” to “ooh, so is this deep conjecture true?” and start getting back answers, and amending your definitions. You could invent and resolve entire research directions in days. But the confusion you would have had to push through to flesh out your theory would largely (probably not entirely) be instantly resolved and the whole process sped up tremendously by your oracle. A big part of the process would be gone."
I think this is where I most strongly disagree with what he writes. I think you start getting back answers, and then to continue, you have to UNDERSTAND them. And the dirty little secret of mathematics is that it's impossible to understand what anyone else is saying. Conveying one's mathematical intuition is incredibly hard: at least for me, the experience of acquiring understanding from someone else's work is nearly identical to that of discovering it on my own.
Of course, what the mathematics of the future will look like depends (like all AI prognostication) on the precise shape of future AI capabilities; I do not think the picture of an uncreative oracle is realistic. I expect future AI mathematicians to be creative, and also, not to be oracles. I think a lot of the questions we view as fundamental will remain open for some time. Basic mathematical questions can be arbitrarily hard! And we will still want to understand them.
Doing math
Most of what I love about the practice of mathematics is: talking to colleagues about math, learning and understanding new things, developing intuition and resolving confusion, etc. My sense is that these parts of math survive with arbitrarily capable AI tools.
I also like a lot of other aspects of the job: I get paid and can afford to eat, I have a lot of intellectual freedom, I have great colleagues (like Jacob), I don't have a boss and can work sprawled out on a couch. Absent a real attempt for the profession to adapt to the coming changes, it's possible that the shape of the profession changes in a way that makes it much less enjoyable, even as most of what I like about doing math survives. There are questions as to why society should support human mathematicians if and when machines have absolute advantage over us in all aspects of mathematics. I think we'll have advantage in some aspects of mathematics for some time, but it's worth thinking about this endpoing for the profession, as it is for all other professions.
That said, I think there's a future here where we continue to ask basic questions about fundamental mathematical phenomema. Sometimes we get an answer from a machine, and sometimes the machine gets stuck, and so do we. And when we get stuck, we get frustrated--we get an itch--and we don't give up.
