Philosophy of Mathematics

by Øystein Linnebo

One natural continuation would be to regard numbers as second-level concepts that ascribe cardinality properties to first-level concepts. For instance, “These are five trees” might be analyzed as “5x(TREE(x)),” much like “There is a tree” is analyzed as “∃x(TREE(x)).” Frege subjects this analysis to a barrage of objections. The most serious is that “[e]very individual number is an independent object” and thus not a second-level concept. As evidence, he points to the fact that we speak of “the number 1,” where the definite article serves to class it as an object. In arithmetic this self-subsistence comes out at every turn, as for example in the identity 1 + 1 = 2. (Ibid., §57) This argument has attracted much scholarly attention because of its unusual mix of linguistic and metaphysical considerations.5 The premises involve linguistic considerations about how we talk about numbers. But the desired conclusion is metaphysical, namely that numbers are objects, not second-level concepts. That is, the conclusion is an instance of what we earlier called “object realism.”