TITLE: Mathematical Habit
QUESTION [18 upvotes]: Whenever I have to do proofs or understand a concept, I start with examples and generalize. As an aspiring mathematician, I have fears that it might hurt me, especially when we have so many counterintuitive things. It is more convenient to do things this way, when I am proving known facts or understanding those concepts. However, it might not be efficient when I am trying to prove/disprove unknown situations. 
Do you have any suggestion as to how to avoid generalization? Is this only my problem? If so, what would be a better approach to proofs?
Thank You.

REPLY [2 votes]: You can't reach the stars without generalization. The problem is how to prove the generalization, which it turns out always follows from the definition of what you're studying some way or another.
If you take the definition of a ring and a definition of a homomorphism, you can deduce the definition of an ideal, and then the isomorphism theorems, quite directly. From there you can prove the spectrum of a commutative ring is a topological space, again quite directly. These are clues that backing up your intuitions is mostly feasible, but that when it's infeasible you're on the way to something deep, such as how your intuition can fail, that that intuition is not a direct reflex of the language used to formulate the concept (syntactically tautological), or that you didn't realize that your intuition had a pleasant syntax (the isomorphism theorems, in application, repeatedly point this out), and hence the meaning loaded in such a theorem. Fermat's last theorem was an unproven intuition.