Affine space for Minkowski space time 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?
 A: The elements of A (the events in Minkowski space) exist indepently of your choice of coordinate system on A. 
Similarly, one can define translations (the action by a vector) in a purely abstract fashion, without referring to any set of coordinates at all.
Hence the abstract notion of an affine space (or vector space, or manifold) is more fundamental than the structure provided by a coordinate system. 
Of course, it is often convenient to think about affine spaces like Minkowski space entirely in terms of coordinates, but keep in mind that there is a more abstract structure underneath.
Furthermore, Minkowski space is a rather harmless kind of space, but in more complicated settings, not being careful enough in making the distinction between a space and its coordinates can lead to unwanted side-effects, which is why mathematicians often prefer coordinate-free notation in the first place.
Think of the real world around you. 
The universe is a kind of space as well (which we are supposed to describe as a Minkowski space), but it does not come with any canonical coordinate system at all: you are free to choose any set of points and give them the coordinates you like, as long as your coordinate system makes sense.
