Abstract:  The existence of non-standard models of important mathematical theories, such as first-order Peano-Arithmetic, threatens to undermine the claim of the moderate mathematical realist that non-mysterious access to mathematical structures is possible on the basis of our best mathematical theories. The move to frameworks stronger than FOL to articulate ‘better’ versions of these theories is denied to the moderate realist on the grounds that it merely shifts the indeterminacy ‘one level up’ into the meta-theory by — illegitimately, — assuming determinacy of the notions needed to formulate such logics.

In this talk I want to outline the beginnings of a response to the determinacy challenge facing the moderate mathematical realist. I argue that the unique determinability of notions that enable categorical characterizations of important mathematical structures provides grounds for claiming naturalistically acceptable access to these structures, sufficient to resolve the determinacy challenge. I will illustrate the idea by showing how the mathematical realist may achieve arithmetical determinacy, and discuss ways to extend this approach to richer theories and structures.