Neo-Fregeanism: An Embarrassment of Riches

Abstract

In the last decade or two there has been a revival of interest in logicism recast not as the doctrine that mathematics is logic but rather as the claim mathematical truths have something like the status assigned to them by the logicists. The ‘neo-logicist’ contention is that mathematical truths are known, where they are, neither by some mysterious form of direct intuition nor by empirical confirmation, even of an indirect and holistic fashion via the scientific theories they contribute to. Rather mathematical knowledge arises on the basis solely of the understanding of the basic mathematical and logical concepts which anyone who grasps the mathematical truths has. This view might be interpreted as saying that mathematical truths are analytic, are true by virtue of meaning, similarly that fundamental mathematical inference rules are meaningconstitutive. Since the notion of analyticity is still under a cloud, in some quarters, a more broadly acceptable goal for the neo-logicist might be to try to establish that mathematical axioms are implicit definitions since, prima facie, anyway, this does not commit one to the notion of analyticity; this, indeed, is the direction which recent work has taken (see Hale and Wright [11]).