Suppose the study of some kind of mathematical object has evolved in an "organic" fashion, until it reaches the level of maturity that one wants to take an axiomatic approach to the study of those objects.
How does one go about finding a "minimal" set of axioms so that the axiomatically-constructed object has all the desired properties of the "organic" object, and no more?
To give some context to the question: in my math class, we spent the first half of the semester studying the (real and sometimes the complex) projective plane. Now we're looking at the axiomatic approach (begun by Hilbert?) to projective geometry, starting by defining the axioms for an "incidence geometry".