10 Logic ficulties in relation to the axioms of whatever foundational theory they favour instead (cf. Shapiro 1991). Perhaps it is for this reason that some mathem- aticians (e.g. Mayberry 1994) have tried simply to deny that mathematics has a foundation. But plainly more needs to be said if this is to be anything more substantial than an indefinite refusal to address the question. Another response to these difficulties, more popular among mathematicians than among philosophers, has been to espouse a stricter formalism, a version, that is to say, of the view that the primitive terms of an axiomatic theory refer to nothing outside of the theory itself. The crudest version of this doctrine, pure formalism, asserts that mathematics is no more than a game played with symbols. Frege’s demolition of this view (1893–1903, II, §§86–137) is treated by most philosophers as definitive. Indeed it has become popular to doubt whether any of the mathematicians Frege quotes actually held a view so stu- pid. However, there are undoubtedly some mathematicians who claim, when pressed, to believe it, and many others whose stated views entail it. Less extreme is postulationism — which I have elsewhere (Potter 2000) called axiomatic formalism. This does not regard the sentences of an axiomatic theory as meaningless positions in a game but treats the primitive terms as deriving their meaning from the role they play in the axioms, which may now be thought of as an implicit definition of them, to be contrasted with the explicit definitions of the non-primitive terms. ‘The objects of the theory are defined ipso facto by the system of axioms, which in some way generate the material to which the true propositions will be applicable.’ (Cartan 1943, p. 9) This view is plainly not as daft as pure formalism, but if we are to espouse it, we presumably need some criterion to determine whether a system of axioms does confer meaning on its constituent terms. Those who advance this view agree that no such meaning can be conferred by an inconsistent system, and many from Hilbert on have thought that bare consistency is sufficient to confer meaning, but few have provided any argument for this, and without such an argument the position remains suspect.