Why I do mathematical research
Of the many purposes of mathematics, my favorite is to search for the simplest explanation of a particular phenomenon. This often requires inventing a clarifying abstract language to isolate a common pattern from distracting specifics: not necessarily the maximal level of generality but the “right” one. As with mathematical aesthetics, the simplest proof, most clarifying abstraction, and right level of generality are also matters of taste — and that’s a good thing! While mathematicians tend to agree about what’s true and false, we have countless different opinions about what is intuitive or interesting. […]
Software engineers learn that all abstractions leak, so they keep the lower layers in mind in case one breaks. Mathematical abstractions don’t leak: a theorem proved abstractly holds in every instance, so you are free to keep climbing. At the right level, seemingly magical statements become “trivial”.
To give an example, when I was first learning about sets and functions, it really bothered me that the inverse image preserves both unions and intersections of subsets, while the direct image preserves only unions. The direct image seemed simpler to define, so why was its behavior more complicated? Several years later I learned that the inverse image can be regarded as a functor between powersets, and that functor admits both right and left adjoints, while the direct image only admits a right adjoint. Since left adjoints preserve colimits and right adjoints preserve limits — by a proof that strikes me as straight from the book, expressed exactly at the right level of generality — I now consider my set-theoretic confusion resolved. (Your mileage may vary.)
Mathematical maturity in action: with the right abstraction, the confusion dissolves. It also shows theory building versus problem solving—no new theorem is proved here. AI is good at the latter; much less so at the former.
My mathematical tastes tend towards this sort of “abstract nonsense” (a phrase that, like “queer,” is often used affectionately by insiders). Mathematics like this is sometimes called “theory building,” whose aim is to make it easier for more people to hold increasingly complicated mathematical thoughts in their heads.
This matters a lot when designing research software. A good library has the right level of theory built in, so there is an obvious way to extend or compose it to reach your goal. Libraries with great documentation are usually built right at this theory level: the design is sound, and the documentation teaches it to you.