Underlying these two different philosophical approaches were...

Underlying these two different philosophical approaches were two different conceptions of analysis and propositional contents. Central to Russell’s philosophy from the time of his rejection of idealism, Levine argues, was the principle that every propositional content can be uniquely analyzed into ultimate simple constituents, a claim that Frege did not endorse.

This meant that, for Russell, every proposition had a privileged representation (even if no one had yet been able to give it), which mirrored its content at the ultimate level of analysis. If two sentences of different forms could be used to assert the same propositional content, therefore, then they could not both be privileged representations.

Frege, on the other hand, insisted throughout his life that one and the same content (‘thought’, in his later terminology) could be analyzed in indefinitely many ways, without assuming that there was some one way that was uniquely privileged. Consider, then, the case of the Cantor-Hume principle,[^14] asserting the equivalence between (Na) and (Nb):[^15] (Na) The concept F is equinumerous to (i.e., can be correlated one-one with) the concept G .

(Nb) The number of F s is equal to the number of G s. On Russell’s view, if (Na) and (Nb) have the same propositional content, then at most only one of them can offer a privileged representation of that content, since they are of different forms. So their equivalence suggests that talk of numbers can be ‘reduced’ to talk of the one-one correlation of concepts, so that we do need to suppose the existence of numbers in addition to that of concepts.

For Frege, on the other hand, the possibility of contextually defining numbers in this way does not imply that numbers are not objects. On the contrary, the fact that number statements can be true and that constituent number terms such as ‘the number of F s’ are proper names is enough to show that numbers are objects. The issue is how we can apprehend such objects, given (as Frege himself stressed) that they are not actual objects, i.e., spatio-temporal objects that have causal effects.

It was here that he appealed to the equivalence between (Na) and (Nb).