Abstract: One of the more distinctive features of neologicism of the variety advocated by Bob Hale and Crispin Wright is its invocation of Frege’s Constraint, or roughly the requirement that the empirical applications of a class of numbers be “built directly into” their formal characterization. It is thought that Frege’s Constraint adjudicates between different available characterizations of the natural numbers specifically in favor of Hume’s Principle. However, we show here first that the arguments offered by Hale and Wright only support a version of Frege’s Constraint according to which the relevant application of the naturals is transitive counting, or using numerals to answer ‘how many’-questions, and secondly that neither formal characterization of the naturals considered by these neologicists – viz. Hume’s Principle and the Dedekind-Peano postulates – actually satisfies this version of Frege’s Constraint.
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