 For Shapiro, the structures at issue thus have a dual nature: they are “universals”, in the sense that the natural number structure, say, can be instantiated by various relational systems (the system consisting of the finite von Neumann ordinals, or of the Zermelo ordinals, etc.); but they are also “particulars”, to be named by singular terms and treated as objects themselves. To defend the latter further, Shapiro develops a general structure theory, i.e., an axiomatic theory that specifies which structures exist. While modeled on set theory, this theory is justified independently (more on how below). As such, it is meant to underwrite “ante rem structuralism”.  Shapiro’s terminology of “ante rem versus in re” involves an explicit reference to Medieval discussions of universals. The crucial point in our context is that the structures specified in his theory are meant to be ontologically independent of, indeed prior to, any instantiations of them. In other words, the structures do not just exist  in their instantiations, but separate to and  before them. 