A Carnapian Solution to the Puzzle of Extrinsic Justifications

Set theorists commonly give extrinsic justifications forExtrinsic justification axioms, i.e., they present the fruitfulness of axioms with respect to certain mathematical goalsMathematical goals as evidence for their truth. The puzzle of extrinsic justificationsExtrinsic justification is the question of how the fruitfulness of an axiom can be evidence for its truth. In this paper, I first discuss some problems for Penelope Maddy’s solution to the puzzle of extrinsic justificationsExtrinsic justification. I then propose an alternative, Carnapian solution according to which the fruitfulness of an axiom can be evidence for its truth because there are analytic truths to the effect that sets are fruitful with respect to certain mathematical goalsMathematical goals. I argue that the Carnapian solution is independently plausible, that it avoids the problems faced by Maddy’s solution, and that it neatly aligns with Maddy’s epistemological and metaphysical views concerning set theory, as well as with her methodological commitments.