Dummett's criticism of Dedekind is that the final product of abstraction, namely a structurally-specified progression isn't enough to indentify the natural numbers. The first member of such a progression, for example, could be taken to be any number (each of 0, 1, 2 have all been choosen to play this role). This means that there must be something more than a strutural-specified progression needed to adaquately identify a number. Dummett suggests that this is the fact that the number n applies to certain things when these things can be counted by a sequence ending in n (or n - 1 if the sequence begins with 0).

The reason why numbers had to be objects (for Frege) is that if the terms in Frege's formal system failed to refer, then there can be no guarantee that true premises will logically imply only true conclusions. This is because expressions which fail to refer lack truth value (for early Frege at least), so then a true proposition could imply something lacking truth value. Moreover, this truth valueless proposition could imply something false as the laws of logic don't preserve lack of truth value. This means that all the terms in (the axioms of) the formal theory must refer, thus the number terms/descriptions must refer to mathematical objects if logic is to guarantee the truth-preservingness of the formal system.

-- GlicK - 02 Oct 2006