I have a big problem:
When I read any mathematical text I'm very skeptical. I feel the need to check every detail of proofs and I ask myself very dumb questions like the following: "is the map well defined?", "is the definition independent from the choice of representatives" etc... Even if the author of the paper/book says that something is a easy to check, I have this impulse to verify by myself.
I think that this approach is philosophically a good thing, but it leads to severe drawbacks:
I waste a lot of time in reading few lines of mathematics, and at the end of the day I look at what I've done and I realize that I managed to go through a few theorems without learning enough. Remember that when one is a (post)graduate student (s)he has plenty of things to learn, so the time is almost never enough.
This kind of learning could be affordable for undergraduate texts, but very often is almost impossible to read a paper with a skeptics point of view. At a certain point things become very complicated and the only way out is to accept results on faith.
And finally the real object of my question:
3. Despite the big effort I've employed in reading very carefully something, after few weeks or months I obviously forget the details. So, for example if I try to read again a proof after a while, maybe I would remember the big picture but probably I would check again the details as though I'd never done it yet.
Therefore, even if the common rules for a mathematician say that "learning" should ideally be done skeptically, I've finally realized that maybe this is not very healthy. Now, could you recommend a sort of royal road for reading mathematics? It should be a middle way between accepting every result as true and going through every detail. I'd like to know what to do in practice.