Philosophy of Mathematics

by Øystein Linnebo

This makes identities of the form “the number of Fs = the number of Gs”—or in symbols, “#xFx = #xGx”—especially important.7 Putting things together, the original problem of explaining how the numbers are “given to us” is thus transformed into the problem of explaining the meanings of identities of the form “#xFx = #xGx.” The meanings of other sentences involving number terms can be explained later. Frege proposes a brilliant analysis of the mentioned identities. He proposes that the meaning of “#xFx = #xGx” be “recarved” as the claim that the Fs and the Gs can be one-to-one correlated. What it means for the number of knives on a table to be identical with the number of forks, for example, is that each knife is correlated with a unique fork and vice versa. This analysis is attractive. It seems to avoid all use of the number terms whose meaning we are trying to explain. All that the analysis requires are the resources to express that the Fs and the Gs can be one-to-one correlated. And these resources are available in pure second-order logic, where a second-order quantifier can be used to state that there is a relation that one-to-one correlates the Fs and the Gs. Thus, Frege’s proposed analysis seems both noncircular and purely logical. The proposed analysis is encapsulated in the principle where ‘F ≈ G’ abbreviates the mentioned claim about one-to-one correlations, or, as we shall put it, that the Fs and the Gs are equinumerous. Because Frege claims to be taking a cue from ...