I was recommended the book Geometry by Michele Audin by a professor when I asked about learning more about affine geometry. I like the book, but it's raised a question. To me, it seems that it would be nicer to lay down the foundation of geometry with affine geometry and then move on to geometry described by vector spaces. But this is the opposite approach taken by the book. This books builds up the concept of an affine space from the concept of a vector space.
Could we define an affine space without the concept of a vector space? Is there a textbook which does this? Thanks.