Abstract:

In this talk, the tenability of extending arithmetical knowledge by way of confirmation is examined, where the relevant notion of confirmation is understood probabilistically in the manner familiar from Bayesianism. The motivation here is to see what can be said for a pre-Fregean view to the effect that mathematical induction-- one of the Peano axioms-- is akin to enumerative induction in certain of its epistemic features. I will focus on two apparent problems with this view. First, there is the problem that if a certain probabilistic omega-rule is adopted, then the arithmetically probable will end up alining with the arithmetically true. Second, there is an obvious tension between this view and the admittedly intuitive thought that genuine mathematical justification for a universal hypothesis should be resistant to improvement through the examination of particular cases.