@sepdek's

The thing starts with invariance, conservation and symmetry… Invariant is something that is unchanged by a coordinate transformation. Conserved is something that is unchanged by a process within a given coordinate system. Symmetry could be considered a broader property of some ‘universe’ in which some entity of interest remains the same under a specific operation. […]

In order to be a measure (distance) d must: $latex \displaystyle d:X \times X \rightarrow R \; \text{(it should map to the set of real numbers)} &s=1$ $latex \displaystyle \exists d_0 \in R: -\infty \textless d_0 \leq d(x,y) \textless +\infty , \; \forall x,y \in X \; \text{(a minimum should exist)} &s=1$ $latex \displaystyle d(x,x)=d_0, […]