Automorphism
A symmetry of a structure: a way of swapping its parts around that leaves every relation in it exactly as it was. A structure with a non-trivial automorphism contains positions that nothing inside the structure can tell apart.
Greek autos, “self”, and morphē, “form”.
Take a structure and relabel its parts. If every relation that held before still holds after, the relabelling is an automorphism. The trivial one — leave everything where it is — always exists; what matters is whether there is another.
If there is, the structure contains positions that nothing internal to it distinguishes. An unlabelled two-vertex graph with no edge between the vertices is the simplest case: from inside the structure, the two vertices are interchangeable.
That is why the concept turns up in a theological argument. Eustratius of Nicaea's triangle is isosceles on purpose — the Son and the Spirit have to be perfectly co-ordinate, or the Latin addition to the creed sneaks back in. But an isosceles triangle admits the reflection that swaps its base corners. The symmetry he needs is the symmetry that makes his two derived positions indiscernible.
Where this comes up
Sources
- SEP, “Structuralism in the Philosophy of Mathematics”, on the identity problem