Abstract

Blue (2026) criticizes Ash and Clarke-Doane (2025). Our thesis is that the justification of mathematical axioms relies on reflective equilibrium applied to data about which there is reasonable disagreement, as in philosophy. Blue responds with a survey of the case for Definable Determinacy and a speculative scenario involving Baire category principles, arguing that the analogy breaks down. But Blue’s response rests on a conflation that we took pains to flag -- between agreement over what follows from what and agreement over what non-logical axioms are true. Our concern is emphatically with the latter. Conditional agreement is cheap. Given agreement over logic, one can achieve it in any regimented area, from ethics to astrology. The case that Blue focuses on establishes that a wide variety of sufficiently strong theories imply Projective Determinacy. But getting from those conditional results to the conclusion that Projective Determinacy is true requires appeal to theoretical virtues (naturalness, unification, proof compression) that do not admit of proof, and whose application is reasonably disputed. In this note, I argue that Blue’s paper is, in fact, a detailed case study in the thesis that it attacks. Every element of the argument that he develops -- the crosschecks, theoretical virtues, “intended interpretations,” feedback loop between theorems and philosophical ideas, and even his classification of axiom disputes as “philosophical” or “mathematical” -- exhibits the dependence on non-deductive, non-provable, reasonably disputed considerations that Ash and I argue underwrite foundational inquiry. Blue’s mistake is to think that because these considerations are good --responsive to deep mathematics -- they cease to be philosophical.