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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).

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  • $\begingroup$ The title suggests an even broader scope than likely you intended, from reading the body of your Question. Perhaps "justifying the study of specialized fields of mathematics" would better frame your Question. To improve it you should give specific examples of "narrow" fields that you think need such justification, or examples of where justification has been satisfactorily provided for fields (in your opinion). Details will help your Question receive considered attention and not get closed as overly broad or primarily opinion based. $\endgroup$ – hardmath Aug 19 '14 at 16:52
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    $\begingroup$ Maybe look at "The Unreasonable Effectiveness of Mathematics $\dots$" by Wigner. Googling on this title should locate other discussions. $\endgroup$ – André Nicolas Aug 19 '14 at 16:53
  • $\begingroup$ @hardmath, I thought I had done that with my examples of complex analysis and an obscure branch of finite group theory. I'll try to change the title appropriately, however. $\endgroup$ – amcalde Aug 19 '14 at 16:55
  • $\begingroup$ Perhaps Davis & Hersh's "The Mathematical Experience"? $\endgroup$ – Blue Aug 19 '14 at 16:59
  • $\begingroup$ A quotation from Lighthill: "Pure Mathematics is a very important part of Applied Mathematics." $\endgroup$ – André Nicolas Aug 19 '14 at 17:15
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Maybe you'll find the Princeton Companion to Mathematics helpful. It has some nice and accessible surveys of mathematics, both past and present.

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  • $\begingroup$ This looks promising. I'll have to check this one out. Thanks. $\endgroup$ – amcalde Sep 26 '14 at 20:18
  • $\begingroup$ So far I like this book. In section 7 of the preface they state exactly my dilemma seeking answers on the internet. $\endgroup$ – amcalde Sep 26 '14 at 20:59
  • $\begingroup$ This is a wonderfully rich book which looks at mathematics from many perspectives and, thus, gives a picture of the subject that while having a certain "random" quality is also very nuanced. As one reads more of the specialized essays one comes to see the way different aspects of the subject, from its theory to its applications, to the extraordinary people from many countries and cultures who contributed to it, make mathematics an exciting and wonderfully rich subject. $\endgroup$ – Joseph Malkevitch Sep 27 '14 at 11:35
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The founder of the Institute for Advanced Study, A. Flexner, once published a paper called "The usefulness of useless knowledge" http://library.ias.edu/files/UsefulnessHarpers.pdf. This paper justifies all theoretical sciences.

I get a kick out of reading the article.

I think about another one, "The Spirit and the Uses of the Mathematical Sciences", which is pretty inspirational and beauteously written.

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  • $\begingroup$ This is a good read, but not really what I'm looking for. $\endgroup$ – amcalde Aug 19 '14 at 20:51

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