Say My Name. An Objection to Ante Rem Structuralism

In this paper I raise an objection to ante rem structuralism, proposed by Stewart Shapiro: I show that it is in conflict with mathematical practice. Shapiro introduced so-called “finite cardinal structures” to illustrate features of ante rem structuralism. I establish that these structures have a well-known counterpart in mathematics, but this counterpart is incompatible with ante rem structuralism. Furthermore, there is a good reason why, according to mathematical practice, these structures do not behave as conceived by ante rem structuralism