 For Russell, one is free to take a space and assign a ‘distance’ to each pair of points (subject to some simple conditions which we omit). Relative to this sense of distance, one can say if, as a region is moved around, points in it remain the same distance apart or not. We have done this for our sense of distance on the surface of the Earth, and we can do this whether or not we also have some rigid body motions. In mathematical terms, Russell would be happy with what is called a metric space. The point is not that one could impose a metric on the surface of the Earth in which a particular pair of points, say in Cambridge, were a metre apart and London and Paris were only half a metre apart—one could—but that one can talk about distance without presupposing the action of a group. Some metric spaces admit the action of groups that preserve distance, others do not, but distance can be defined without talking about a group. Poincaré was never confronted with exactly this argument—metric spaces are an invention of the 20th century—but we know what he would have said. He would have said that it was valid mathematics but entirely formal and could not be considered as genuine knowledge because it lacked a psychophysical dimension. We know this because it was his criticism of the axiomatic geometries constructed by Hilbert (see below).