Abstract

In this paper, Bob Hale and Crispin Wright discuss the Caesar Problem, which raises the worry that abstraction principles––like the Direction Equivalence and Hume's Principle ––are subject to an indeterminacy of reference which leaves them incapable of excluding the possibility that Caesar is the Number 3. They review––and deem implausible––an attempt by Richard Heck to finesse the Caesar Problem by recasting Hume's Principle in a two‐sorted language. Drawing on considerations by Michael Dummett (_Frege: Philosophy of Mathematics_, 1991) and Gideon Rosen (”Refutation of Nominalism(?)”, 1993), they then address the question whether a coherent nominalist response to Fregean abstraction––which is supposed to involve a distinctively realist ontology––can be provided. A section is devoted to a discussion of two criticisms––launched by respectively Dummett and Potter and Sullivan––of Wright's solution to the Caesar Problem in _Frege's Conception of Numbers as Objects_ (1983), based on an inclusion principle for sortal concepts. While Hale and Wright disagree with Dummett, they find that a consideration close to the Potter‐Sullivan line of thought shows that Wright's earlier proposal stands in need of modification––and provide a modified solution centred around the notions of sortal concept, category, and criterion of identity.