Solving the Caesar Problem Without Categorical Sortals

The neo-Fregean account of arithmetical knowledge is centered around the abstraction principle known as Hume’s Principle: for any concepts X and Y, the number of X’s is the same as the number of Y’s just in case there is a 1–1 correspondence between X and Y. The Caesar Problem, originally raised by Frege in §56 of Die Grundlagen der Arithmetik, emerges in the context of the neo-Fregean programme, because, though Hume’s Principle provides a criterion of identity for objects falling under the concept of Number–namely, 1–1 correspondence—the principle fails to deliver a criterion of application. That is, it fails to deliver a criterion that will tell us which objects fall under the concept Number, and so, leaves unanswered the question whether Caesar could be a number. Hale and Wright have recently offered a neo-Fregean solution to this problem. The solution appeals to the notion of a categorical sortal. This paper offers a reconstruction of their solution, which has the advantage over Hale and Wright’s original proposal of making clear what the structure of the background ontology is. In addition, it is shown that the Caesar Problem can be solved in a framework more minimal than that of Hale and Wright, viz. one that dispenses with categorical sortals. The paper ends by discussing an objection to the proposed neo-Fregean solutions, based on the idea that Leibniz’s Law gives a universal criterion of identity. This is an idea that Hale and Wright reject. However, it is shown that a solution very much in keeping with their own proposal is available, even if it is granted that Leibniz’s Law provides a universal criterion of identity