Philosophy of Mathematics

by Øystein Linnebo

Frege’s fix was to provide an explicit definition of the numbers. To do so, he followed the lead of contemporary mathematicians such as Dedekind, who had discovered how to handle abstraction by equivalence classes rather than abstraction principles. Consider the case of directions. Instead of accepting directions as a distinct kind of object, we may identify the direction of a line l1 with an equivalence class, namely the class of all lines l2 that are parallel to l1. It is straightforward to show that the principle (Dir) can now be derived and is thus not needed as an axiom. Frege suggested we apply the same strategy to numbers. There is no need to regard Hume’s Principle as an axiom governing numbers understood as a distinct kind of object. It suffices to define the numbers as equivalence classes. We may for example define #xFx as the class of concepts that are equinumerous with F: Hume’s Principle can now be derived and need not figure as an axiom.