Empiricism, Probability, and Knowledge of Arithmetic

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.