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This is a problem that frequently arises while I self-study. I come across a new definition in a book, and I don't understand why that definition is made.

My thought about definitions is that they should be in some way 'natural': maybe it is a useful construction for solve a specific interesting problem (e.g. the definition of homology to solve the problem of classification of spaces), or some object/property that is interesting by itself (e.g. Complex differentiability, ordinals and such), or it is a generalization of some objects that encapsulates some property that those objects share (e.g. the definitions of a group, ring, field, module, etc). I know they should be natural because it is a human(s) that made that definition, and that person must have made the definition for some good reason as in the above.

But the majority of books on mathematics seems to just throw the definition at you. And the idea is that as you study the subject further, this definition will make more sense to you and you will understand that this is a good definition. I have not seen many books that tries to motivate definitions like Artin's Algebra. And this makes me hard to proceed further into the text, since I really don't feel motivated to study further I don't feel that the concept is, well, motivated.

The most recent definition I had problems with is that of morphisms of algebraic (quasi-projective) varieties. So I saw the definition of regular functions and morphisms, and while I understand why we would want to consider polynomial maps between affine varieties, I can't really understand why we consider functions that are locally given by rational functions, specifically in the Zariski topology (I think it is the 'local in the Zariski topology' part that makes this definition hard to understand for me). What would be ideal is if I could feel that I could have made the definition myself, but since this is hard to do for many cases (it would be not easy to come up with the definition of compactness by oneself) I would like to know, and feel what the thing is that this definition is trying to encapsulate. What is this definition trying to do? Why do we define the category of algebraic varieties like this?

So, when cases like this happens, what I do first is look up on stackexchange/the internet. Sometimes there are good explanations (I found a good explanation for tensor products after browsing the internet), but sometimes I search for an entire day for stackexchange answers and other books on the subject, and look in the book after the definition for some examples, and sometimes I still don't get it. And this makes me feel bad because I feel I didn't do anything that day. So these are my questions:

  1. Should I drop my need for motivation a little bit? Should I get used to moving on first in the text? (There are times I do this, but I don't feel good) Is this a problem that many people face while self-studying?
  2. What is a good way to find the motivation of definitions or just the subject in general? The ideal way would be knowing the history and motivating questions/problems of the subject or definition, but I don't know how to learn about the history. Plus, I know that the actual history is probably very complicated, I probably need an idealized history.
Hyunbok Wi
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  • Good books always let you know the motivations in the first section of chapter or very briefly before each section of chapter irrespective of whether they are undergraduate text or graduate text. –  Oct 11 '22 at 20:22
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    I agree with @Avenger; try other books first! Also consider reading backward; find the big 'conclusion' theorem at the end of the section, and look for its dependence on the definition at hand. If it's not explicitly in there, see if you can mentally map the 'tree of dependence' that starts at your definition and ends at the theorem. – Sort of Damocles Oct 11 '22 at 20:26
  • I wouldn't drop the need for motivation but perhaps have a bit more patience. Maybe on first read, it's not entirely clear why a definition includes certain details, but allow yourself to work through this discomfort. Then circle back (perhaps multiple times) to reread definitions as you become more familiar with how they are used in particular examples, lemmas, and theorems. – Sammy Black Oct 11 '22 at 20:28
  • I suspect that you might get something out of the comments and answers to this question about the well-known John von Neumann quip, "Young man, in mathematics you don't understand things. You just get used to them." – Sammy Black Oct 11 '22 at 20:35
  • While some bad authors do hide behind formalism, there are plenty of excellent math books which do not provide motivation. It's partly a stylistic or aesthetic thing as some authors find it too informal and assume you will get this from a lecturer. My advice is not to avoid such books but be patient, work with the material, and talk to people who know the subject. – blargoner Oct 12 '22 at 00:15

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