If we have a configuration of, let's say, five point particles moving around in an otherwise empty space (i.e., the relative distances/angles between the particles change) and no other information about the system, how could we, theoretically, determine the geometry of the space the configuration moves in? How could we, for example, decide whether four particles lie on a plane or not?
Put more generally: How can we describe the kinematics of a configuration without starting by presupposing a complete framework like Newtonian or Riemannian space, but instead determining it from within the system (i.e., without taking some kind of external ruler or clock)?
