Relative Categoricity and Abstraction Principles
Abstract
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 (Parsons1990a, Parsons2008 49, McGee1997aa, Lavine1999aa, Vaananen2014aa). Another great enterprise in contemporary philosophy of mathematics has been Wright's and Hale's project of founding mathematics on abstraction principles (Hale2001, Cook2007aa). In Walsh2012aa, 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 (i) stability-like acceptability criteria on abstraction principles (cf. Cook2012aa), (ii) the Tarski-Sher logicality requirements on abstraction principles studied by Antonelli Antonelli2010aa and Fine~Fine2002, and (iii) supervaluational ideas coming out of Hodes' work Hodes1984, Hodes1990aa, Hodes1991.
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