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 by Walsh, it was noted that one traditional abstraction principle, namely Hume's Principle, had a certain relative categoricity property, which in this talk will be termed "natural relative categoricity". In this talk, it will be shown 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. Towards a better understanding of the precise demands of relative categoricity in the context of abstraction principles, these constraints are compared to and contrasted with (i) stability-like acceptability criteria on abstraction principles, (ii) the Tarski-Sher logicality requirements on abstraction principles studied by Antonelli and Fine, and (iii) supervaluational ideas coming out of Hodes' work. (This is joint work with Sean Ebels-Duggan.)  


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