 While Resnik’s and Shapiro’s structuralist positions are sometimes identified, this is somewhat misleading given the differences already mentioned. Nevertheless, there is significant overlap. Both recognize mathematical structures as certain patterns with positions in them etc. (whether these patterns are then treated as full-fledged objects, in a realist sense, or not). And for both Resnik and Shapiro, the notion of isomorphism is crucial (or some related, more general notion of equivalence; see Resnik 1997 and Shapiro 1997). That is to say, for both a relevant structure/pattern can be instantiated by any of a relevant class of isomorphic relational systems. This corresponds to the fact that the axiomatic systems at issue, for the natural numbers, the real numbers, and similar cases, are all categorical (or quasi-categorical in the case of set theory). Not every mathematical axiom system has that feature, of course, e.g., the axiom systems for group or ring theory allow for non-isomorphic instances or models. For Resnik as well as Shapiro, such “algebraic” theories are to be treated in a different, more derivative way; their structuralist point of view is meant to apply primarily to “non-algebraic” theories, paradigmatically arithmetic.