What are mathematical definitions? When(at what stage) and how do mathematicians come up with the basic definitions of the new mathematical concepts they have found?

I BELIEVE in math I BELIEVE it's super useful, however, I see sometimes hundreds of definitions in a math book before the author starts explaining the resulting relations, and to make things worse, then I can not even remember how they made use of all those many definitions(if they did) they provided in the earlier chapters. I feel I need to know what mechanism is behind this convention of providing many definitions and lemmas and etc. in beginning sections so that I can go through them more easily and get less tired(please don't get me wrong I'm neither saying math is inherently boring nor I'm after "applications" or "usages" in math, I know mathematics is about finding relations that can be proved to be right and it's cool and enough, although then also we(or someone else in hundred years or later) may find out that their observation of some behavior of a natural phenomena conforms our relations and formulas and boom!).

I really enjoy math lessons but this question has always been in my head and I thought I may need to eventually start sharpening my view on it. Specially when I encounter some of the definitions talking about boundaries and special cases of the concept being taught(example: 0!=1)(of course that's a very simple example from high school math but you get the idea, I couldn't think of a more publicly understandable one right away; these kinds of special case definitions are really present in ALL textbooks of various fields of math) I again ask this from myself. Maybe if I really get to understand what's happening in the "expert side" of the math world who prepares math lessons for me and if I learn the answer to this question I can be more sticked to my math books and finish them faster and maybe even motivate other people to go for their mathematical textbooks more often too!!!

My guess is that the process of structuralizing and formulating mathematical concepts in lessons and textbooks is actually "reverse" of what happens in reality. Maybe they first see the big picture of the underlying relations and then try to find the best building blocks that fit the covering layers(which is finally the big picture) and then start putting them in this bottom to top order since it'll be inevitably the only way the big picture could be conveyed to those who are unfamiliar with it, although those blocks may sound boring in the first encounter. I'm not a math student or a math guru or something like that at all I'm just an engineering student who is obviously always engaged with so many math lessons and thus I'm making these "guesses".

(Thanks in advance for the answers. English isn't my mother tongue so sorry for probable typos and mistakes.)


closed as off-topic by Jasper, Joel Reyes Noche, Adam, Ben Crowell, Opal E Nov 16 '18 at 17:24

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    $\begingroup$ Your "reverse" guess is right. Also this question has little to do with mathematics education. $\endgroup$ – Jasper Nov 16 '18 at 11:42
  • $\begingroup$ @Jasper Sorry. Thanks. $\endgroup$ – M-J Nov 16 '18 at 12:27