Unpacking "for reasons of symmetry": Two categories of symmetry arguments

Hermann Weyl succeeded in presenting a consistent overarching analysis that accounts for symmetry in (1) material artifacts, (2) natural phenomena, and (3) physical theories. Weyl showed that group theory is the underlying mathematical structure for symmetry in all three domains. But in this study Weyl did not include appeals to symmetry arguments which, for example, Einstein expressed as "for reasons of symmetry" (wegen der Symmetrie; aus Symmetriegründen). An argument typically takes the form of a set of premises and rules of inference that lead to a conclusion. Symmetry may enter an argument both in the premises and the rules of inference, and the resulting conclusion may also exhibit symmetrical properties. Taking our cue from Pierre Curie, we distinguish two categories of symmetry arguments, axiomatic and heuristic; they will be defined and then illustrated by historical cases.