What Do Privileged Coordinates Tell Us about Structure?

Abstract We examine whether the “privileged coordinates” of a geometric space encode its “amount of structure.” In doing so, we compare this coordinate approach to comparing amounts of structure to the more familiar automorphism approach. We first show that on a natural understanding of the former, it faces one of the same well-known problems as the latter. We then capture a precise sense in which the two approaches are closely related to one another, and we conclude by discussing whether they might still prove useful in cases of philosophical interest, despite their shortcomings.