I'm studying Minkowski space time (M4) and they say it's a 4 dimensions real affine space. M4 is an affine space so there is a non-empty set A, a 4 dimension real vector space V, and there is a function f: AxA-->V (with the proper property). The elements of A are the events.
The doubt: Have I to think that I’m in certain inertial frame of reference with a certain coordinate system so that every element of A is identify with 4 numbers? Or have I to think of A as a very abstract set of points and I set the coordinates of A after setting the basis for V?
I'm asking this because I have read that the affine coordinate system comes after the definition of the affine space, but in this case, I don't understand how it's possible to identify an event before setting a coordinate system. Are coordinate system and affine coordinate system two different things?