 There is also the problem of applicability, which Frege thought an insuperable one for formalists. What can the meaning be of applied mathematical notions, such as «the number of \(\phi\)s», where mathematical and non-mathematical discourse is mixed together? Unless the formalist wishes to go down the Dummettian anti-realist route and generalise the notion of proof to a notion of verification appropriate for empirical language, she will have to find a way of combining, without too much ad hocness, a proof-theoretic semantics for pure mathematics with a different, perhaps a realist, truth-conditional semantics, for empirical language. 