A note on Isomorphism and Identity

This note argues that, insofar as contemporary mathematics is concerned, there is overwhelming evidence that if mathematical objects are structures, then isomorphism should not be taken as their identity condition. This goes against a common version of structuralism in the philosophical literature. Four areas are presented in which identifying isomorphic structures or objects leads to contradiction or inadequacy. This is followed by a philosophical discussion on possible ways to approach the distinction, and a section on the possibility of proceeding intensionally, as is done in e.g.\ the Univalent Foundations program.