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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!

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    $\begingroup$ I think this is all likely to be very subjective. Fermat, for instance, is just one of a (seemingly endless) list of hard Diophantine problems. I don't know that it's especially interesting because of its nature. Of course it became interesting in the same way that scaling Mt. Everest became interesting....it was obviously very hard and brilliant mathematicians had fallen to their deaths trying (metaphorically, at least). $\endgroup$
    – lulu
    Aug 24, 2022 at 21:38
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    $\begingroup$ In case it's of interest, this question overlaps a little with this one, insofar as being interesting hinges on being in the pure sweet spot between applied & recreational. $\endgroup$
    – J.G.
    Aug 24, 2022 at 21:45
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    $\begingroup$ One interesting path might be to look at questions which are serious but somehow deemed "not interesting". Classification of Finite Simple Groups was obviously "interesting" but Classification of Finite Perfect Groups isn't, though I'd have said that was the inevitable next step. Why? Well, I suppose because the Simple Group part was so incredibly brutal that nobody has the stomach for round $2$. $\endgroup$
    – lulu
    Aug 24, 2022 at 21:54
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    $\begingroup$ @FShrike: I find this question interesting --- As I began reading I expected something in which the issues were a bit trite (e.g. "beauty" is subjective), but as I continued reading I changed my mind, especially with the identification of "clusters", where I'm sure some will disagree with the examples chosen but my experience has been that there is a fair amount of truth with the examples chosen. nonagon -- I assume you know of books such as The Mathematical Experience by Davis/Hersh, Loving + Hating Mathematics by Hersh/John-Steiner, How Mathematicians Think by Byers, etc.? $\endgroup$ Aug 24, 2022 at 21:57
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    $\begingroup$ This post may be subjective, so an opinion $\endgroup$ Sep 24, 2022 at 17:24

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