Hale's Argument from Transitive Counting

Abstract: We consider a recent argument from Hale (2016) for Frege's Constraint - roughly, the requirement that the primary empirical applications of a class of numbers such as the naturals ought to be "built in" to the formal characterization of those numbers. In effect, Hale argues that the primary empirical application of the naturals is what Benacerraf calls "transitive counting" - using numerals to answer 'how many'-questions - and that since Hume's Principle captures this application but the Dedekind-Peano Axioms do not, the neologicist characterization of the naturals has an advantage over the structuralist characterization. We dispute Hale's argument, however, suggesting that it fails to substantiate its intended conclusion, adding that if transitive counting is in fact the application relevatnt to satisfying Frege's Constraint, then neither Hume's Principle nor the Dedekind-Peano Axioms can satisfy it.

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