Internalism and the Determinacy of Mathematics.

This article critically examines the internalist view in the philosophy of mathematics, which explains mathematical content through internal categoricity results in second-order logic, as developed by Button and Walsh. While internalism successfully accounts for the uniqueness of mathematical structures—showing that second-order arithmetic picks out a unique natural number structure internally—it fails to adequately explain the determinacy of mathematical discourse. The authors argue that internalism cannot establish that every mathematical statement is determinately true or false without appealing to metalinguistic resources or stronger theories, which conflicts with naturalistic constraints and Gödelian incompleteness phenomena. Consequently, although internal categoricity results clarify uniqueness, they do not suffice to resolve the metasemantic challenge of explaining why mathematical claims have determinate truth-values.