Parsons’s Mathematical Thought: Sec. 49, Uniqueness and communication
Parsons now takes another pass at the question whether the natural numbers form a unique structure. And this time, he offers something like the broadly Wittgensteinian line which we mooted above as a riposte to skeptical worries — though I’m not sure that I have grasped all the twists and turns of Parsons’s intricate discussion.
We’ll start by following Parsons in considering the following scenario. Michael uses a first-order language for arithmetic with primitives 0, S, N, and Kurt uses a similar language with primitives 0′, S’, N’. Each accepts the basic Peano axioms, and each also stands ready to accept any instances of the first-order induction schema for predicates formulable in his respective language (or in an extension of that language which he can come to understand). And we now ask: how could Michael determine that his ‘numbers’ are isomorphic to Kurt’s?
We’ll assume that Michael is a charitable interpreter, and so he thinks that what Kurt says about his numbers is in fact true. And we can imagine that Michael recursively defines a function f from his numbers to Kurt’s in the obvious way, putting f(0) = 0′, and f(Sn) = S’f(n) (of course, to do this, Michael has to add Kurt’s vocabulary to his own, while shelving detailed questions of interpretation — but suppose that’s been done). Then trivially, each f(n) is an N’ by Kurt’s explicit principles which Michael is charitably adopting. And Michael can also show that f is one-one using his own induction principle.
In sum, then, Michael can show that f is an injection from the Ns into the N’s, whatever exactly the latter are. But, at least prescinding from the considerations in the previous section, that so far leaves it open whether — from Michael’s point of view — Kurt’s numbers are non-standard (i.e. it doesn’t settle for Michael whether there are also Kurt-numbers which aren’t f-images of Michael-numbers). How could Michael rule that out? Well, he could show that f is onto, and hence prove it a bijection, if he could borrow Kurt’s induction principle — which he is charitably assuming is sound in Kurt’s use — applied to the predicate ∃m(Nm & fm = ξ). But now, asks Parsons, what entitles Michael to suppose that that is indeed one of the predicates Kurt stands prepared to apply induction to? Why presume, for a start, that Kurt can get to understand Michael’s predicate N so as to bring it under the induction principle?
It would seem that, so long as Michael regards Kurt ‘from the outside’, trying to ‘radically interpret’ him as if an alien, then he has no obvious good reason to presume that. But on the other hand, that’s just not a natural way to regard a fellow human being. The natural presumption is that Kurt could learn to use N as Michael does, and so — since grasping meaning is grasping use — could come to understand that predicate, and likewise grasp Michael’s f, and hence come to understand the predicate ∃m(Nm & fm = ξ). Hence, taking for granted Kurt’s common humanity and his willingingness to extend the use of induction to new predicates, Michael can then complete the argument that his and Kurt’s numbers are isomorphic. Parsons puts it like this. If Michael just takes Kurt as a fellow speaker who can come to share a language, then
We now have a situation that was lacking when we viewed Michael’s understanding of Kurt as a case of radical interpretation; namely, he will take his own number predicate as a well-defined predicate according to Kurt, and so he will allow himself to use it in induction on Kurt’s numbers. That will enable him to complete the proof that his own numbers are isomorphic to Kurt’s.
And note, the availability of the proof here ”does not depend on any global agreement between them as to what counts as a well-defined predicate”, nor on Michael’s deploying a background set theory.
So far, then, so good. But how far does this take us? You might say: if Michael and Kurt in effect can come to belong to the same speech community, then indeed they might then reasonably take each other to be talking of the same numbers (up to isomorphism) — but that doesn’t settle whether what they share is a grasp of a standard model. But again, that is to look at them together ‘from the outside’, as aliens. If we converse with them as fellow humans, presume that they stand ready to use induction on our predicates which they can learn, then we can use the same argument as Michael to argue that they share our conception of the numbers. You might riposte that this still leaves it open whether we’ve all grasped a nonstandard model. But that is surely confused: as Dummett for one has stressed, in order to formulate the very idea of models of arithmetic — whether standard or nonstandard — we must already be making use of our notion of ‘natural number’ (or notions that swim in the same conceptual orbit like ‘finite’, or stronger notions like ‘set’). To cast put that notion into doubt is to saw off the branch we are sitting on in describing the models. Or as Parsons says, commenting on Dummett,
[I]n the end, we have to come down to mathematical language as used, and this cannot be made to depend on semantic reflection on that same language. We can see that two purported number sequences are isomorphic without strong set-theoretic premisses, but we cannot in the end get away from the fact that the result obtained is one ”within mathematics” (in Wittgenstein’s phrase). We can avoid the dogmatic view about the uniqueness of the natural numbers by showing that the principles of arithmetic lead to the Uniqueness Thesis …
So, there is indeed basic agreement here with the Wittgensteinian observation that in the end there has to be understanding without further interpretation. But Parsons continues,
… but this does not protect the language of arithmetic from an interpretation completely from the outside, that takes quantifiers over numbers as ranging over a non-standard model. One might imagine a God who constructs such an interpretation, and with whom dialogue is impossible, and with whom dialogue is impossible. But so far the interpretation is, in the Kantian phrase, ”nothing to us”. If we came to understand it (which would be an essential extension of our own linguistic resources) we would recognize it as unintended, as we would have formulated a predicate for which, on the interpretation, induction fails.
Well, yes and no. True, if we come to understand someone as interpreting us as thinking of the natural numbers as outstripping zero and its successors, then we would indeed recognize him as getting us wrong — for we could then formulate a predicate ‘is-zero-or-one-of-its-successors’ for which induction would have to fail (according to the interpretation), contrary to our open-ended commitment to induction. And further dialogue will reveal the mistake to the interpreter who gets us wrong. However, contra Parsons, we surely don’t have to pretend to be able to make any sense of the idea of a God who constructs such an interpretation and ‘with whom dialogue is impossible’: Davidson and Dummett, for example, would both surely reject that idea.
But where exactly does all this leave us on the uniqueness question? To be continued …