However, this proposal faces two difficulties. The first concerns the ontological status of the structural equivalence between the (applied) mathematical and the non-mathematical domains. On what grounds can we claim that the structures under consideration are mathematically the same if some of them concern ‘material’ objects? Of course, given that the structural equivalence is established by an isomorphism the material objects are already formulated in structural terms—this means that some mathematics has already been applied to the domain in question. In other words, in order to be able to represent the applicability of mathematics, Hellman assumes that some mathematics has already been applied. This means that a purely mathematical characterization of the applicability of mathematics (via structure preservation) is inherently incomplete. The first step in the application, namely the mathematical modeling of the material domain, is not, and cannot, be accommodated, since no isomorphism is involved there. Indeed, given that by hypothesis the domain is not articulated in mathematical terms, no isomorphism is defined there.