  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 preference. 