Heck fleshes this out a bit, explaining that equinumerousity can be cashed out in terms of one-to-one correspondence. To say that there is a one-to-one correspondence between a set of Fs and a set of Gs is to say that there is a relation (R) between them such that two conditions hold:

“1. The relation is one-one, that is, no object bears R to more than one object, and no object is borne R by more than one object

2. Every F bears R to some G and every G is borne R by some F” (Heck, 64).

Equations (identities) are important for Frege because he wants to determine the concept of number by means of our definition of numerical identity.

The Julius Caesar problem shows a way in which Hume’s principle is inadequate to explain names of numbers. Hume’s principle “will not decide for us whether [Julius Caesar] is the same as the [number of Roman emperors]” (Qtd. in Heck, 65). We seem to intuitively understand that Caesar isn’t the type of thing that can be a number. Hume’s principle, however doesn’t rule this out. Therefore, it looks like there is something more to our concept of number than what Hume’s principle implies.

-- AnnieS - 26 Sep 2006