Tom Colclough

This paper ties together three threads of discussion about the following question: in accepting a system of axioms S, what else are we thereby warranted in accepting, on the basis of accepting S? First, certain foundational positions in the philosophy of mathematics are said to be epistemically stable, in that there exists a coherent rationale for accepting a corresponding system of axioms of arithmetic, which does not entail or otherwise rationally oblige the foundationalist to accept statements beyond the logical consequences of those axioms. Second, epistemic stability is said to be incompatible with the implicit commitment thesis, according to which accepting a system of axioms implicitly commits the foundationalist to accept additional statements not immediately available in that theory. Third, epistemic stability stands in tension with the idea that in accepting a system of axioms S, one thereby also accepts soundness principles for S. We offer a framework for analysis of sets of implicit commitment which reconciles epistemic stability with the latter two notions, and argue that all three ideas are in fact compatible.