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This is more of a personal philosophical question, but bear with me. So, I have been trying to study K-theory, but the study has been more focused on studying category theory. It has been two weeks of consistent reading, lecture videos and engaging in practice problems. At this point, I'm seeing good improvement on my side, I understand all the formal things, etc. But I'm still hit with the fundamental question: To what end?

This same question hit me with module theory. As I was reading a course syllabus about modules, the aim of the class is to build modules from other modules, and, analogously, now I figure that my aim is to build categories from other categories--as these theories seem to be interested in building such and such from other such and such. With that, I am at least pacified with the aim.

Still though, even reflecting on passing the most basic math course on the first year calculus series, I still do not know what to do with an integral. Don't get me wrong, I know a decent deal of integration--I know a good deal of identities and could solve a challenging one from some math competitions--but I do not know my aim with it--not like I was intending to study analysis or anything.

In sum, I guess what this post is asking: Do any of you have an aim when you start to study a subject?

By the way, I'm not stopping with my study, but I want to know if this "malaise" happens to any of you.

Masacroso
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Hoptopcop
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    Maybe try to step back and ask yourself what brings you pleasure. What kind of book makes you excited just by looking at the preface and table of contents. If you are not in touch with yourself this could take a while - you could even come up empty-handed. But forcing yourself to study things that you don't enjoy gets pretty miserable after a while. – Novice Mar 28 '22 at 06:25
  • D. Mumfort writes in the preface of his small book Curves and their Jacobians " “[Algebraic geometry] seems to have acquired the reputation of being esoteric, exclusive, and very abstract, with adherents who are secretly plotting to take over all the rest of mathematics. In one respect this last point is accurate.” This sentence is hovewer not Mumfort's final verdict against algebraic geometry. Quite the contrary. In that book Mumfort enthusiastically lays out an "amazing synthesis" which "should really not be taken for granted" between Algebra, Geometry and Analysis. – Kurt G. Mar 28 '22 at 06:56
  • Part of the problem might be jumping into something more abstract and advanced than you're ready for. For example, in the case of module theory, you should be familiar with linear algebra at least at the level of Hoffman/Kunze (e.g. "level 2" in this answer) and some group theory (especially abelian groups) and be at the maturity level where you've worked with (or are ready to work with) commutative diagrams and exact sequences and such. Also, it might be helpful to read over some of the introductory parts (continued) – Dave L. Renfro Mar 28 '22 at 07:16
  • where modules are introduced in some of the standard graduate level algebra texts (Lang, Hungerford, Jacobson, etc.) to see how they motivate the introduction of modules and the prototypical examples they give, and maybe also (this all being quickly do'able pulling down books from the shelves of a hopefully-nearby university library) look at some books specifically devoted to modules (such as given in comments here) and read over their prefaces and introductory motivational parts. For me doing something like this (continued) – Dave L. Renfro Mar 28 '22 at 07:28
  • before taking a course on something (or deciding on a topic for self-study) always seemed to be an obvious thing to do. Even if the course was something required for my degree (or strongly suggested), I was always curious as to what it was all about, so I would scan over books about it in the library and look at the text in the university bookstore (back when they used to have all such books, even from past semesters), sometimes this being to see whether I was mathematically ready for the course and not just curiosity, and talking to students around the dept. who had taken the course before. – Dave L. Renfro Mar 28 '22 at 07:37
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    This is in general an important question! See https://mathoverflow.net/questions/376084/reference-request-examples-of-research-on-a-set-with-interesting-properties-whi/376119, for example, about a case where someone expended a lot of effort proving properties of something that cannot actually exist. If you have a reason to study a thing, you have a better chance of at least showing that you're not expending effort on something useless. – Patrick Stevens Mar 28 '22 at 08:37
  • Such apparenty opinion-based questions should be discussed in chat-rooms, not in the question-answer section.Ten people will have twelve different opinions. – Peter Mar 28 '22 at 09:35
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    @Patrick Stevens: That mathoverflow question is about research, not learning some existing subject for school or personal purposes, the latter being what the OP is concerned about. Nonetheless, that mathoverflow question has a lot of answers that people seeing your comment might find interesting. Naturally, I like this particular answer :) – Dave L. Renfro Mar 28 '22 at 10:11

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My aim to generally study something follow one of these three reasons:

  1. I'm curious about something
  2. I need some background in some area to understand another area about I'm curious
  3. I need some knowledge for a job or to get some academic titulation.

This is all. So I never study something not related to these three reasons. This can be applied to anything not just mathematics. As mathematics is a hobby for me then I just apply the first two points above.

Masacroso
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