Review of Symbolic Logic 8 (3):572-606 (2015)

Sean Ebels-Duggan
Northwestern University
Sean Walsh
University of California, Los Angeles
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, which here we term natural relative categoricity. In this paper, we show that most other abstraction principles are not naturally relatively categorical, so that there is in fact a large amount of incompatibility between these two recent trends in contemporary philosophy of mathematics. To better understand the precise demands of relative categoricity in the context of abstraction principles, we compare and contrast these constraints to stability-like acceptability criteria on abstraction principles, the Tarski-Sher logicality requirements on abstraction principles studied by Antonelli and Fine, and supervaluational ideas coming out of Hodes' work.
Keywords relative categoricity  internal categoricity  abstraction principles  logicism  determinacy of truth-value  stability criteria for abstraction principles  invariance criteria for logicality  Hodes and supervaluationalism  Frege  Dedekind's categoricity theorem
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DOI 10.1017/s1755020315000052
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References found in this work BETA

What Are Logical Notions?Alfred Tarski - 1986 - History and Philosophy of Logic 7 (2):143-154.
Logicism and the Ontological Commitments of Arithmetic.Harold T. Hodes - 1984 - Journal of Philosophy 81 (3):123-149.
Frege, Kant, and the Logic in Logicism.John MacFarlane - 2002 - Philosophical Review 111 (1):25-65.
How We Learn Mathematical Language.Vann McGee - 1997 - Philosophical Review 106 (1):35-68.
Philosophy of Mathematics.Stewart Shapiro - 2003 - In Peter Clark & Katherine Hawley (eds.), Philosophy of Science Today. Clarendon Press.

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Citations of this work BETA

Abstraction and Four Kinds of Invariance.Roy T. Cook - 2017 - Philosophia Mathematica 25 (1):3–25.
The Nuisance Principle in Infinite Settings.Sean C. Ebels-Duggan - 2015 - Thought: A Journal of Philosophy 4 (4):263-268.
Is Hume’s Principle Analytic?Eamon Darnell & Aaron Thomas-Bolduc - forthcoming - Synthese 198 (1):169-185.

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