Frege on the Real Numbers

This paper is concerned with Gottlob Frege’s theory of the real numbers as sketched in the second volume of his masterpiece Grundgesetze der Arithmetik. It is perhaps unsurprising that Frege’s theory of the real numbers is intimately intertwined with and largely motivated by his metaphysics. The account raises interesting, and surprisingly underexplored, questions about Frege’s metaphysics: Can this metaphysics even accommodate mass quantities like water, gold, light intensity, or charge? Frege’s main complaint with his contemporaries Cantor and Dedekind is that their theories of the real numbers do not build the applicability of the real numbers directly into the construction. In taking Cantor and Dedekind’s Arithmetic theories to be insufficient, clearly Frege takes it to be a desideratum on a theory of the real numbers that their applicability be essential to their construction. We begin with a detailed review of Frege’s theory, one that mirrors Frege’s exposition in structure. This is followed by a critique, outlining Frege’s linguistic motivation for ontologically distinguishing the cardinal numbers from the real numbers. We briefly consider how Frege’s metaphysics might need to be developed, or amended, to accommodate some of the problems. Finally, we offer a detailed examination of Frege’s Application Constraint – that the reals ought to have their applicability built directly into their characterization. It bears on deeper questions concerning the relationship between sophisticated mathematical theories and their applications.