The main point underlying these considerations has been stressed often enough (although in a different context): isomorphism does not seem to be an appropriate condition for capturing the relation between mathematical structures and the world (see, e.g., da Costa and French 2003). There is, of course, a correct intuition underlying the use of isomorphism at this level, and this relates to the idea of justifying the application of mathematics: the isomorphism does guarantee that applied mathematical structures S and the structures M which represent the material situation are mathematically the same. The problem is that isomorphism-based characterizations tend to be unrealistically strong. They require that some mathematics has already been applied to the material situation, and that we have knowledge of the structural equivalence between S and M. What is needed is a framework in which the relation between the relevant structures is weaker than isomorphism, but which still supports the applicability, albeit in a less demanding way (e.g., Bueno, French and Ladyman 2002).