Thoroughgoing implicationism — the view that mathematics has no sub- ject matter whatever and consists solely of the logical derivation of con- sequences from axioms — is thus a very harsh discipline: many mathem- aticians profess to believe it, but few hold unswervingly to what it entails. The implicationist is never entitled, for instance, to assert unconditionally that no proof of a certain proposition exists, since that is a generalization about proofs and must therefore be interpreted as a conditional depending on the axioms of proof theory. And conversely, the claim that a proposition is provable is to be interpreted only as saying that according to proof theory it is: a further inference is required if we are to deduce from this that there is indeed a proof. One response to this difficulty with taking an implicationist view of set the- ory is to observe that it arises only on the premise that set theory is intended as a foundation for mathematics. Deny the premise and the objection evap- orates. Recently some mathematicians have been tempted by the idea that other theories — topos theory or category theory, for example — might be better suited to play this foundational role