 The first step of Poincaré’s construction of geometrical space via groups proceeds from the observable fact that a set of impressions can be modified in two distinct ways: on the one hand without our feeling muscular sensations, and on the other, by a voluntary motor action accompanied by muscular sensations. So, as in Carnap’s Aufbau, the starting point here is the definition (guided by experience) of two two-place relations: an external change \(a\) (with ‘\(x a y\)’ for ‘\(x\) changes in \(y\) without muscular sensation’) and an internal change \(S\) (with ‘\(x S y\)’ for ‘\(x\) changes in \(y\) accompanied by muscular sensations’). In the second step, he introduces a classification of external changes, some of which can be brought back to the starting point by an internal change (changes of position), while others (changes of state) cannot. One presupposes “by convention” or better by decision, that the internal compensation for external changes is exact and not approximate. In the third step, Poincaré defines the equivalence class of changes of position through an identity condition with respect to the compensation by internal changes and calls it a displacement: