Noûs 54 (1):54-77 (
2018)
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Abstract
One of the more distinctive features of Bob Hale and Crispin Wright’s neologicism
about arithmetic is their invocation of Frege’s Constraint – roughly, the requirement
that the core empirical applications for a class of numbers be “built directly into”
their formal characterization. In particular, they maintain that, if adopted, Frege’s
Constraint adjudicates in favor of their preferred foundation – Hume’s Principle –
and against alternatives, such as the Dedekind-Peano axioms. In what follows
we establish two main claims. First, we show that, if sound, Hale and Wright’s
arguments for Frege’s Constraint at most establish a version on which the relevant
application of the naturals is transitive counting – roughly, the counting procedure
by which numerals are used to answer “how many”-questions. Second, we show that
this version of Frege’s Constraint fails to adjudicate in favor of Hume’s Principle.
If this is the version of Frege’s Constraint that a foundation for arithmetic must
respect, then Hume’s Principle no more – and no less – meets the requirement than
the Dedekind-Peano axioms do.