Chapter 6 of Parsons’s book is titled ‘Numbers as objects’. So: what are the natural numbers, how are they “given” to us, are they objects available to intuition in the kinds of ways suggested in the previous chapter?
Sec. 31 tells us that a partial answer to its title question ‘What are the natural numbers?’ is that they are a progression (a Dedekind simply infinite system). But “might we distinguish one progression as being the natural numbers, or at least uncover constraints such that some progressions are eligible and others are not?”. The non-eliminative structuralism of Sec. 18 is Parsons’s preferred answer to that question, he tells us. Which would be fine except that I’m still not clear what that comes to — and since it is evidently important, I’ve backtracked and tried reading that section another time. Thus, Parsons earlier talks on p. 105 of “the conclusion that natural numbers are in the end roles rather than objects with a definite identity”, while on p. 107 he is “most concerned to reject the idea that we don’t have genuine reference to objects if the ‘objects’ are impoverished in the way in which elements of mathematical structures appear to be”. So the natural numbers are, in the space of three pages, things to which we can make genuine reference (hence are genuine objects, given that “speaking of objects just is using the linguistic devices of singular terms, predication, identity and quantification to make serious statements”), but also are only impoverished ‘objects’, and are roles. I’m puzzled. This does seem to be metaphysics done with too broad a brush.
Anyway, Parsons feels the pressure to say more: “our discussion of the natural numbers will be incomplete so long as we have not gone into the concepts of cardinal and ordinal”. So, cardinals first …
Sec. 32 ‘Cardinality and the genesis of numbers as objects’. This section outlines a project which is close to my heart — roughly, the project of describing a sequence of increasingly sophisticated arithmetical language games, and considering just what we are committed to at each stage. (As Parsons remarks, “The project of describing the genesis of discourse about numbers as a sequence of stages was quite foreign to [Frege]”, and, he might have added, oddly continues to remain foreign to many.)
We start, let’s suppose, with a grasp of counting and a handle on ‘there are n Fs’. And it would seem over-interpreting to suppose that, at the outset, grasp of the latter kind of proposition involves grasping the second-order thought ‘there is a 1-1 correspondence between the Fs and the numerals from 1 to n‘. Parsons — reasonably enough — takes ‘there are n Fs’ to carry no more ontological baggage than a first-order numerical quantification ‘∃nxFx‘ defined in the familiar way. Does that mean, though, that we are to suppose that counting-numerals enter discourse as indices to numerical quantifiers? Even if ontologically lightweight, that still seems conceptually too sophisticated a story. And in fact Parsons has a rather attractive little story that treats numerals as demonstratives (in counting the spoons, I point to them, saying ‘one’, ‘two’, ‘three’ and so on), and then takes the competent counter as implicitly grasping principles which imply that, if the demonstratives up to n are correctly applied to all the Fs in turn, then it will be true that ∃nxFx.
So far so good. But thus far, numerals refer (when they do refer, in a counting context) to the objects being counted, and then recur as indices to quantifiers. Neither use refers to numbers. So how do we advance to uses which are (at least prima facie) apt to be construed as so referring?
Well, here Parsons’s story gets far too sketchy for comfort. He talks first about “the introduction of variables and quantifiers ‘ranging over numbers'” — with the variables replacing quantifier indices — which we can initially construe substitutionally. But how are we to develop this idea? He mentions Dale Gottlieb’s book Ontological Economy, but also refers to the approach to substitutional quantification of Kripke’s well-known paper (and as far as I recall, those aren’t consistent with each other). And then there’s the key issue — as Parsons himself notes — of moving from a story where number-talk is construed substitutionally to a story where numbers appear as objects that themselves are available to be counted. So, as he asks, “in what would this further conceptual leap consist?”. A good question, but one that Parsons singularly fails to answer (see the middle para on p. 197).
At the end of the section, Parsons returns to the Fregean construal of ‘there are n Fs’ as saying that there is a one-one correlation between the Fs and the Gs (with ‘G‘ a canonical predicate such that there are n Gs). He wants the equivalence between the two kinds of claim to be a consequence of a good story about the numbers, rather than the fundamental explanation. I’m sympathetic to that: and if I recall, Neil Tennant has pushed the point.