RQG A point you re still missing is that there is more than one actual mathematical definition Different characterizations of a concept can be equivalent to each other and which one is taken to be the definition may be a matter of convenience varying with the context As I pointed out above one of the characterizations as A V superficially makes it appear that an affine space has more structure than a vector space since the vector space is only V rather than A V But I explained why that is a superficial appearance and that particular way of defining the concept of affine space is not the only way and is not sacred