How do we as mathematicians justify the study of math to students? Or, indeed, how do we justify it to the general public? How do you justify your particular field? I'm particularly interested in justifications of fields of mathematics that are more off the beaten path. My question is this: Is there a catalogic reference for this? I'm looking for a good article, paper, blog, or book about this. (Specifically justifying various fields of higher math.)
For example: If I were asked by a young scientist or engineer why she should have to study complex analysis, I could point to any number of results that may literally be termed art (like pretty pictures of fractals), not to mention the calculation of integrals if that tickles their fancy, or quantum mechanics or whatever else might actually apply to their field.
This seems a reasonable question to me, for I often find myself struggling to understand the point of someone else's research. For example I recently read an article about a project that classified some particular groups. I laughed out loud when I read the author's justification for his research stating that classifying these groups has allowed for the classification of other groups! Lockhart's Lament discusses this deeply: (https://www.maa.org/external_archive/devlin/LockhartsLament.pdf)
All that said, I often see such beauty in math that compels me personally to ever study and learn more of those beautiful theorems and ideas that mathematicians have uncovered. As in many fields, I suppose, you can claim that the mathematics is simply beautiful. But I'd like a reference, a book that speaks to some of the beauty in these fields: WHY one studies groups and their classifications; WHY one studies Differential Geometry; WHY one studies algebraic topology; etc. Each "WHY" above could be answered by a few unusual results, or descriptions of connections to other math or science.
I've read the articles and books suggested so far and I'm not satisfied so I'm posting a bounty. A catalogic reference justifying many fields of mathematics. A book like this should exist (or a blog, etc).