 Non-eliminative structuralism, in Shapiro’s and other versions, involves the thesis that all that matters about mathematical objects are their structural properties (as opposed to further intrinsic properties). In fact, such properties are taken to determine the objects’ identities. But then, objects that are indistinguishable in this respect—“structural indiscernibles”—should be identified, it seems. As several critics pointed out around the year 2000, this leads to the “identity problem” for structuralism (cf. Keränen 2001, earlier Burgess 1999).  It arises prominently for systems or structures that are non-rigid, i.e., allow for non-trivial automorphisms. In such cases, there are supposedly distinct objects that are indistinguishable in the relevant sense. A widely known case is the system of complex numbers (with the conjugate numbers i and −i); but geometry and graph theory, among others, provide further examples. The simplest example is probably an unlabeled 2-element graph with no edges, whose two vertices are structurally indiscernible. 