 All the variants of structuralism described up to this point are forms of “philosophical structuralism”, or more precisely, of “metaphysical structuralism”.  What that means is that these positions are intended to provide answers to questions about what mathematical structures are (even when this involves an eliminative stance), including views about their existence, abstractness, identity, dependence, etc.  Now, one can distinguish this whole variety of philosophical positions from what is sometimes called “mathematical structuralism”, or again more precisely, “methodological structuralism” (cf. Reck &amp; Price 2000, earlier Awodey 1996).  In fact, we suggest that recognizing this basic dichotomy—“metaphysical versus methodological”—to is crucial with respect to both systematic and historical discussions of structuralism.  (Among others, it allows us to connect more naturally with categorical structuralism; see, e.g., Corry 2004 and Marquis 2009.) 