Disclaimer: this question is more about philosophy of mathematics than technical mathematics.
Mathematicians always need to choose what to focus their work on. Many pure mathematicians like to say that they're not motivated by a problem's real-life applications, but rather by its beauty/interestingness/etc. I'm trying to build an understanding of how this "beauty" is determined. Let's assume we're talking about research level mathematics and grad students/professional mathematicians.
Clearly some of this "beauty" is subjective since mathematicians have different aesthetic preferences. However, there seems to be at least some objectivity:
- An overwhelming majority of mathematicians would find the following theorem uninteresting:
<a fixed-and-otherwise-unremarkable boolean circuit> is unsatisfiable. - Most mathematicians find Fermat's Last Theorem at least somewhat interesting.
Plus, people's interests in math subjects seem (anecdotally to me) to form clusters:
- Combinatorics and computer science are "close", in a sense that mathematicians who like combinatorics are more likely to like computer science
- Analysis and algebra are "distant", in that mathematicians who like the algebra-clustered subjects often dislike the analysis-clustered subjects
- Areas like combinatorics, logic and (higher) category theory seem to be quite polarizing -- more mathematicians have strong feelings about them, both negative and positive, compared to areas like stochastic calculus or algebraic topology.
I'm looking for philosophy / sociology references that investigate this question, both
- Empirically: surveying mathematicians to discover patterns in their "interestingness" rankings
- Philosophically: what makes a mathematical problem interesting? In the flavor of something like:
- Problems for which we have complete algorithms are uninteresting, e.g. elementary plane geometry
- Problems which connect two previously separate clusters of mathematics are interesting, e.g. Langlands program
- Theorems about ad-hoc specific cases are less interesting than theorems about a whole class of objects
- Theorems that derive complex structure from simple axioms are interesting
Thanks!