":::: @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.    
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