I had an absolute beast of an algebra professor who impressed upon us the fact that things are defined they way they are for a reason, and we, as mathematicians

I had an absolute beast of an algebra professor who impressed upon us the fact that things are defined they way they are for a reason, and we, as mathematicians must know them. It is not enough to be able to remember the definition of a normal subgroup, for instance. We must know why we define it the way we do, and what happens if the definition is altered in the slightest, weakened, strengthened, negated etc. It is not enough to know that quotient groups are defined modulo normal subgroups only. You should know why defining a quotient group modulo a non-normal subgroup is meaningless. Even for much more complicated categories, like manifolds, one must absolutely know why all the bits and pieces are needed. Why do we require the transition functions for real n-manifolds to be C^r? What if r=0? What happens if two charts on some underlying topological space are not compatible? Why do we need the atlas to be maximal? Why on earth do we need a countable base? Hausdorff? What is this business about needing manifolds to be Hausdorff? [0]. Why is it that bijective homomorphisms are the isomorphisms in the category of groups - i.e. why is it that its inverse is not required to be a homomorphism as part of the definition? Learning definitions by rote is the surest route to mathematical mediocrity. "Don't just read it - fight it!" - P.R. Halmos. [0] Yes, I know these can be relaxed, but this is the standard presentation in Kobayashi-Nomizu/Spivak/Lee/Tu/Barden-Thomas/Do Carmo/Petersen etc etc.