Nevertheless...

Nevertheless, even this sophisticated reading seems to conflict with a more conventionalist reading, and as Reck notes, there are certainly passages where Frege offers something very close to Carnap’s notion of explication (in lectures that Carnap actually attended).[^13] One way of approaching the issue is by comparing the Frege-Russell definition with alternative definitions such as those subsequently provided by John von Neumann and, more recently, by Crispin Wright and Bob Hale.

Taking these three cases, how do we decide whether to identify the natural numbers with the Frege-Russell numbers , the von Neumann numbers or the Wright-Hale numbers , as Reck calls them? Like the Frege-Russell numbers, the von Neumann numbers are classes (set-theoretic objects), which satisfy the Dedekind-Peano axioms, but they arguably do not do justice to the role of numbers in ‘bringing together’ equinumerous collections.

The Wright-Hale numbers, on the other hand, seem to do justice to the application of numbers, but do they really count as logical objects? Would Frege have been happy with Wright’s and Hale’s ‘neo-logicism’? Clearly, there are different constraints in different theoretical contexts, and the question of what the numbers ‘really’ are can only be answered in a particular conceptual framework.

As Reck suggests, this might help us in reconciling the Platonist and conventionalist strands in Frege’s thought, even if Frege himself may not have seen it in this way. Indeed, for any interpretation of Frege’s thought that might be offered, we might well be tempted to ask an analogous question. Does the interpretation offered count as an ‘analysis’ or an ‘explication’? Are there ‘facts of the matter’ as to what Frege really meant?

The question Reck addresses in his paper clearly has implications beyond the specific case of the natural numbers. Frege’s and Russell’s logicist definition of the natural numbers as equivalence classes of equinumerous classes is also the starting-point of James Levine’s paper, ‘Analysis and Abstraction Principles in Russell and Frege’.

Although they offered the same definition, however, Levine argues that they used that definition in quite different ways (providing a further illustration of the Carnapian message of Reck’s paper, we might add). For Frege, it played a role in his claim that numbers are ‘self-subsistent objects’, whereas for Russell, it was taken as showing that numbers can be dispensed with in giving an inventory of the world.