In a subsection entitled ‘Schemes are not reducible to quantification’, Lavine writes
Schematic letters and quantifiable variables have different inferential roles. If n is a schematic letter then one can infer S0 ≠ 0 from Sn ≠ 0, but that is not so if n is a quantifiable variable — in that case the inference is valid only if n did not occur free in any of the premisses of the argument.
But, in so far as that is true, how does it establish the non-reducibility claim?
Of course, one familiar way of using schemes is e.g. as in Sec. 8.1 of my Gödel book where I am describing a quantifier-free arithmetic I call Baby Arithmetic, and say “any sentence that you get from the scheme Sζ ≠ 0 by subsituting a standard numeral for the place-holder ‘ζ ‘ is an axiom”. And to be sure, the role of the metalinguistic scheme Sζ ≠ 0 is different from that of the object language Sx ≠ 0. Still, it would misleading to talk of inferring an instance like S0 ≠ 0 from the schema. And here the generality, signalled by ‘any’, can — at least pending further, independent, argument — be thought of as unproblematically quantificational (though not quantifying over numbers of course). So this sort of apparently anodyne use of numerical schemes doesn’t make Lavine’s point, unless he can offer some additional considerations. So what does he have in mind?
Lavine’s discussion is not wonderfully clear. But I think the important thought comes out here:
One who doubts that the natural numbers form an actually infinite class will not take the scheme φ(n) → φ(Sn) to have a well-circumscribed class of instances and hence will not be willing to infer φ(x) → φ(Sx) from it; for the latter formula involves a quantifiable variable with the actually infinite class of all numbers as its domain or the actually infinite class of all numerals included in its substitution class.
We seemingly get a related thought e.g. in Dummett’s paper ‘What is mathematics about?’, where he argues that understanding quantification over some class of abstract objects requires that we should ‘grasp’ the domain, that is, the totality of objects of that class — which seems to imply that if there is no totality to be grasped, then here there can be no universal quantification properly understood.
But do note two things about this. First, a generalization’s failing to have a well-circumscribed class of instances because we are talking in a rough and ready way and haven’t bothered to be precise because we don’t need to be, and its failing because we can’t circumscribe the class because there is no relevant completed infinity (e.g. because of considerations about indefinite extensibility), are surely quite different cases. Lavine’s moving from an initial example of the first kind when he talked about arm-waving generalizations we make in introductory logic lectures to his later consideration of cases of the second kind suggests an unwarranted slide. Second, I can see no reason at all to suppose that sophisticated schematic talk to avoid being committed to actual infinities is “more primitive” than quantificational generality. On the contrary.
Still, with those caveats, I guess I am sympathetic to Lavine’s core claim that there is room for issuing schematic generalizations which don’t commit us to a clear conception of a complete(able) domain. In fact, I’d better be sympathetic, because I actually use the same idea myself here (where I talk about ACA0‘s quantifications over subsets of numbers, and argue that the core motivation for ACA0 in fact only warrants a weaker schematic version of the theory). So, even though I don’t think he really makes the case in his Sect. 7, I’m going to grant that there is something in Lavine’s idea here, and move on next to consider what he does with idea in the rest of the paper.