(Part 2 after months) Odd question once again, I asked my discomfort about coordinate systems before but I just can't seem to be able to do this, in any form of continuous mathematics (especially differential geometry). Whenever I think of some sort of curve, shape, manifold I end up thinking of it as a physical object, and the arbitrary choice of coordinates make it annoying for me to work with them. Like for instance, when we are thinking of R² I sometimes obsess over what is the basis vectors we are working with -- we can assume to be 1 for each one of the pair of R, then I ask '1 of what'? Or whenever we consider maps say f:M->R, I just feel some discomfort thinking what is the scale in R we are mapping in?…