An example that really stuck with me is that you can draw an arrow on a blank piece of paper and measure its length and direction without needing to define an x

An example that really stuck with me is that you can draw an arrow on a blank piece of paper and measure its length and direction without needing to define an x/y coordinate system on the paper. There is something there (the manifold and the action?) that is independent from the ways we can choose to describe the space.