Why There Is No Frege-Russell Definition of Number

Abstract

It is generally agreed that in defining the cardinals as classes of equinumerous classes in 1901, Russell had independently discovered Frege’s definition of the cardinals (IMP: 12). The extent to which Russell’s conception of the cardinals should be viewed as akin to Frege’s is a matter of historical importance, insofar as points of divergence between Frege’s and Russell’s definitions of the cardinals illuminate more fundamental differences in their logicist projects on the very point on which they are supposed to agree, namely, the logicization of arithmetic. It has been argued that while Frege simply accepted that numbers as logical objects are apprehended as the value-ranges (classes) correlated with concepts whose extensions we apprehend, Russell was concerned with the metaphysical status of abstracta resulting from abstraction principles. James Levine writes:

Frege, unlike Russell, does not introduce such definitions in order to address fundamental questions regarding the metaphysical status of abstracta or our knowledge of them, [hence] Frege, unlike Russell (in POM), is in a position to hold that ... classes are no different from other abstracta.¹