Marina Imocrante

I submit that in Pincock’s structural account [SA] the request of a priori justifiability of mathematical beliefs [AP] follows from the adoption of semantic realism for mathematical statements [SR] combined with a form of internalism about mathematical concepts [INTmc]. The resulting framework seems to clash with Pincock’s proposal of an “extension-based” epistemology for pure mathematics [EBE], in that the endorsement of [EBE] seems to ask for a form of conceptual externalism [EXTmc] that would not provide us with the a priori justifications for mathematical beliefs requested. I claim that Pincock’s overall account of pure and applied mathematics would be made more stable if the assumption of [INTmc] was replaced by [EXTmc]. In eliminating [INTmc], [SA] would not entail any necessary commitment to forms of a priori justification for mathematical beliefs anymore, preventing the tension with [EBE]. Someone could object that the combination of [SR] and [EXTmc] would lead to ontological realism for mathematical objects [OR]. I answer by arguing that the kind of [EXTmc] that can be endorsed within Pincock’s framework takes the content of mathematical concepts to be determined by contingent facts in the historical development of mathematical practice, so that no commitment to the existence of mathematical objects is required.

Penelope Maddy (Defending the axioms: on the philosophical foundation of set theory. Oxford University Press, New York, 2011) suggests a new naturalistic account of the objectivity of mathematics, grounding it on the fruitfulness of mathematics both internally, in mathematics itself, and externally, in the applications of mathematical concepts to empirical sciences. In light of the notion of mathematical depth, I will try to defend Maddy’s epistemological project from Jeffrey Roland’s criticisms concerning the reliability of mathematical beliefs in her account and Maddy’s ontological agnosticism between Thin Realism and Arealism. Although Maddy’s presentation of mathematical depth could profit from certain clarifications, I suggest that it could nevertheless ground an account of the reliability of mathematical beliefs, in response to the first objection. With regard to the second, I will argue that Roland’s criticism is misplaced.