Logic and Analysis in Russell’s 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. This view is expressed by Russell himself (IMP, p. 12). The claim to independent discovery is probably true enough, but the claim that what was discovered was Frege’s definition might require some qualification. 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 correlated with value-ranges (classes), that is, are correlated with concepts whose extensions we apprehend as value-ranges,¹ Russell was concerned with the metaphysical status of abstracta resulting from definition by abstraction. James Levine writes:

Frege, 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 with regard to those fundamental questions, classes [value-ranges] are no different from other abstracta. (Levine, 2007, p. 71)