There is no a priori reason to prefer one equivalence class of homogeneous functions over another without argument

To every possible equivalence class [\(F\)] of homogeneous functions, there corresponds a type of metrical space. The Pythagorean-Riemannian space, for which \(F^{2}_{p} = (dx^{1})^{2} + \cdots + (dx^{n})^{2}\), is one among several types of possible metrical spaces. The problem, therefore, is to single out the equivalence class \([F]\), where \(F\) corresponds to \(F^{2}_{p} = (dx^{1})^{2} + \cdots + (dx^{n})^{2}\), from the other possibilities, and to provide arguments for this preferen