How do a mathematician "come up with" a good definition? How long does it take? Is there any strategy to shorten the time it take?
I'm not very good at math, but from my understanding the most important part in math is definition. It took me some time to appreciate its importance after trying to prove something really hard( or at least trying to understand others' proofs of proving it), for me.
And finally, would you consider it annoying, when you found resources, maybe about other subjects not just math, using vague wording after being a mathematician?
(Although I'm not very good at it, and the issue is quite annoying for me, mathematics is still my favorite subject.)
My initial idea is that I'm thinking about how those mathematicians who are at the frontier of mathematics create new idea. I cannot create one, but I guess that must be hard but exciting, and once the "define" process is done, and with some luck, it can attack those unsolved problems, which is the biggest challenge in mathematics...
To give an example why definition is important: I once read about the C++ draft and some definitions are very hard to understand, but it does "push" the frontier of it to include more powerful features. I think math works the same way if new idea are condensed into definition.