Frege on the Real Numbers

Abstract: This paper is divided into two major sections. In the first, we give a detailed synopsis of Frege's incomplete theory of the real numbers as developed in the Grudgesetze. We overview his critique of his contemporaries' theories of the reals, the informal development of his own view, and the formal development of that view leading up to the end of Volume II. We critique Frege's theory in the second major section, focusing on two important issues. First, Frege makes the surprising claim that the cardinal numbers and real numbers are ontologically distinct, or form "completely separate domains". He suggests that linguistic evidence supports his view of the reals, namely as magnitude ratios. We give the kind of linguistic evidence (in English) which Frege might have supplied and then argue that this same evidence reveals a surprisingly neglected, and seemingly deep tension within Frege's metaphysics. It seems as though Frege's argument for the ontological distinction is inconsistent with his theory of cardinal numbers as developed in the Grundlagen. Secondly, we argue that if what has become known as "Frege's (Application) Constraint" is unjustified in the case of the real numbers, as certain Neo-Fregeans have argued, then it is also unjustified in the case of the cardinal numbers. 

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