Why are tangent spaces on a general manifold associated to single points on the manifold? I've heard that it has to do with not being able to subtract/ add one point from/to another on a manifold (ignoring the concept of a connection at the moment), but I'm not sure I fully understand this - is it simply because, even if two points lie in the same coordinate patch the coordinate map will not be Cartesian (i.e. the identity map) in general, and so subtracting/adding their coordinate values will not correspond to subtracting/adding one point from/to another on the manifold? Also, why is it that we can compare vectors at different points, and also add/subtract points in Euclidean space? For points, is it simple because the coordinate map is the identity map and so adding/subtracting the coordinates of points is equivalent to adding/subtracting points in Euclidean space. And for vectors, is it because Euclidean space is affine and so the tangent spaces at two different points in Euclidean space are naturally isomorphic (with the isomorphism given by parallel translation).