26 - Symmetry and equivalence

Summary

As other contributions to this volume also testify, the notions of symmetry and equivalence are closely connected. This paper is devoted to exploring this connection and its relevance to the symmetry issue, starting from its historical roots. In fact, it emerges as an essential and constant feature in the evolution of the modern notion of symmetry: at the beginning, as a specific relation between symmetry and equality; in the end, as a general link between the notions of symmetry, equivalence class, and transformation group.

Symmetry and equality

Weyl's 1952 classic text on symmetry starts with the following distinction between two common notions of symmetry:

If I am not mistaken the word symmetry is used in our everyday language in two meanings. In the one sense symmetric means something like well-proportioned, well-balanced, and symmetry denotes that sort of concordance of several parts by which they integrate into a whole.… The image of the balance provides a natural link to the second sense in which the word symmetry is used in modern times: bilateral symmetry, the symmetry of left and right …

Bilateral symmetry is in fact a particular case of the scientific notion of symmetry, the symmetry being defined as invariance with respect to a transformation group (in the case of bilateral symmetry, the group of spatial reflections).