•  111
    Relative categoricity and abstraction principles
    with Sean Walsh
    Review of Symbolic Logic 8 (3): 572-606. 2015.
    Many recent writers in the philosophy of mathematics have put great weight on the relative categoricity of the traditional axiomatizations of our foundational theories of arithmetic and set theory. Another great enterprise in contemporary philosophy of mathematics has been Wright's and Hale's project of founding mathematics on abstraction principles. In earlier work, it was noted that one traditional abstraction principle, namely Hume's Principle, had a certain relative categoricity property, wh…Read more
  •  62
    The Nuisance Principle in Infinite Settings
    Thought: A Journal of Philosophy 4 (4): 263-268. 2015.
    Neo-Fregeans have been troubled by the Nuisance Principle, an abstraction principle that is consistent but not jointly satisfiable with the favored abstraction principle HP. We show that logically this situation persists if one looks at joint consistency rather than satisfiability: under a modest assumption about infinite concepts, NP is also inconsistent with HP
  •  25
    Identifying finite cardinal abstracts
    Philosophical Studies 178 (5): 1603-1630. 2021.
    Objects appear to fall into different sorts, each with their own criteria for identity. This raises the question of whether sorts overlap.ionists about numbers—those who think natural numbers are objects characterized by abstraction principles—face an acute version of this problem. Many abstraction principles appear to characterize the natural numbers. If each abstraction principle determines its own sort, then there is no single subject-matter of arithmetic—there are too many numbers. That is, …Read more
  •  16
    Abstraction Principles and the Classification of Second-Order Equivalence Relations
    Notre Dame Journal of Formal Logic 60 (1): 77-117. 2019.
    This article improves two existing theorems of interest to neologicist philosophers of mathematics. The first is a classification theorem due to Fine for equivalence relations between concepts definable in a well-behaved second-order logic. The improved theorem states that if an equivalence relation E is defined without nonlogical vocabulary, then the bicardinal slice of any equivalence class—those equinumerous elements of the equivalence class with equinumerous complements—can have one of only …Read more
  •  7
    Deductive cardinality results and nuisance-like principles
    Review of Symbolic Logic 1-33. forthcoming.
    The injective version of Cantor’s theorem appears in full second-order logic as the inconsistency of the abstraction principle, Frege’s Basic Law V (BLV), an inconsistency easily shown using Russell’s paradox. This incompatibility is akin to others—most notably that of a (Dedekind) infinite universe with the Nuisance Principle (NP) discussed by neo-Fregean philosophers of mathematics. This paper uses the Burali–Forti paradox to demonstrate this incompatibility, and another closely related, witho…Read more