What's in a Numeral? Frege's Answer

Mind 116 (463):677 - 716 (2007)
Abstract
Frege wanted to define the number 1 and the concept of number. What is required of a satisfactory definition? A truly arbitrary definition will not do: to stipulate that the number one is Julius Caesar is to change the subject. One might expect Frege to define the number 1 by giving a description that picks out the object that the numeral '1' already names; to define the concept of number by giving a description that picks out precisely those objects that are numbers. Yet Frege appears to do no such thing. Indeed, when he defends his definitions, he does not argue that they pick out objects that we have been talking about all along-the issue never comes up. The aim of this paper is to explain why. I argue that, on Frege's view, our numerals do not, antecedent to his work, name particular objects. This raises an obvious question: If (like 'Odysseus') the numerals do not name particular objects, how can Frege write (as he does) as if sentences in which numerals appear state truths? One central concern of this paper is exegetical-to answer these questions. But my aim is not solely exegetical. For these questions direct us to something that, I believe, creates only an apparent problem for Frege but an actual problem for many contemporary philosophers: the assumption that singular terms appearing in statements about the world must actually have referents. Another aim of this paper is to suggest that the problem-as well as a solution that can be found in Frege's writings-should be of import to contemporary philosophers
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