The above image is pages 15 and 16 of Bamberg and Sternberg, A course in mathematics for students of physics.
I understand that this chapter is trying to explain the significance of affine functions but I'm struggling to understand what's being said here. This is my first introduction to affine transformations and I don't understand them in the context of physics at all.
What is meant by "you cannot write a formula for it"? I don't understand what "it" refers to. If you feel my question itself is unclear, please feel free to just explain the page because any explanation would help me at this point.
Also the last paragraph says we cannot write a formula for f because E and P are not numbers. And apparently this is solved by introducing affine coordinate functions t and x. What exactly is being solved here and how do those help?
I'm currently studying physics in college and have always taken for granted representing forces on coordinates. So I'm not quite able to understand these subtleties which are more abstract than I'm used to. I also haven't started analytic mechanics yet, so any help to make the transition to these more abstract notions would be greatly appreciated.
(Please don't hesitate to "ramble". Just go for it)
Thank you for your time.