The Incommensurable Group ⟨T⟩ Mathematically Formalizes "the Aesthetic"

As a criterion of selection in nature, "the aesthetic" appears to be dominant, yet it is not generally regarded as something to be objectively formalized. Here, let Ω be the "set of operators whose eigenvalues include an irrational number" (Otsuka 2023, Chapter 1), and define the group ⟨T⟩ = {Tⁿ : n ∈ ℤ} generated by any invertible element T of Ω as the "incommensurable group." This group ⟨T⟩ mathematically formalizes a truly objective criterion of "the aesthetic," or "the convergence of evolution," as follows. In a system whose time evolution is given by the transformation T, the degree of evolution can be measured by how closely it has asymptotically approached an irrational ratio, that is, a ratio that is infinitely divided. Then, since there exist infinitely many steps until convergence, infinite evolution is guaranteed.