2008

In sum, then, we might put things like this. Parsons has defended an ‘internalist’ argument — an argument from “within mathematics” — for the uniqueness of the numbers we are talking about in our arithmetic, whilst arguing against the need for (or perhaps indeed, the possibility of) an ‘externalist’ justification for our intuition of uniqueness.

Can we rest content with that? Some philosophers would say we can get more — and Parsons briefly discusses two, Hartry Field and Shaughan Lavine, though he gives fairly short shrift to both. Field has argued that we can appeal to a ‘cosmological hypothesis’ together with an assumption of the determinateness of our physical vocabulary to rule out non-standard models of our applicable arithmetic. Parsons reasonably enough worries: “If our powers of mathematical concept formation are not sufficient [to rule out nonstandard models], then why should our powers of physical concept formation do any better?” Lavine supposes that our arithmetic can be regimented as a “full schematic theory” which is in fact stronger than the sort of theory with open-ended induction that we’ve been considering, and for which a categoricity theorem can be proved. But Parsons finds some difficulty in locating a clear conception of exactly what counts as a full schematic theory — a difficulty on which, indeed, I’ve commented elsewhere on this blog.

In both cases, I think Parsons’s points are well taken: but his discussions of Field and Lavine are brief, and more probably needs to be said (though not here).

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 …

”There is a strongly held intuition that the natural numbers are a unique structure.” Parsons now begins to discuss whether this intuition — using ‘intuition’, of course, in the common-or-garden non-Kantian sense! — is warranted. He sets aside until the long Sec. 49 issues arising from arguments of Dummett’s: here he makes some initial points on the uniqueness question, arising from the consideration of nonstandard models of arithmetic.

It’s worth commenting first, however, on a certain ‘disconnect’ between the previous section and this one. For recall, Parsons has just been discussing how we might introduce a predicate ‘N‘ (‘… is a natural number’) governed by the rules (i) N0, and (ii) from Nx infer N(Sx), plus the extremal clause (iii) that nothing is a number that can’t be shown to be so by rules (i) and (ii). Together with the rules for the successor function, the extremal clause — interpreted as intended — ensures that the numbers will be unique up to isomorphism. Conversely, our naive intuition that the numbers form a unique structure is surely most naturally sustained by appeal to that very clause. The thought is that any structure for interpreting arithmetic as informally understood must take numbers to comprise a zero element, its successors (all different, by the successor rules), and nothing else. And of course the numbers in each structure will then have a natural isomorphism between them (which matches zeros with zeros, and n-th successors with n-th successors). So the obvious issue to take up at this point is: what does it take to grasp the intended content of the extremal clause? Prescinding from general worries about rule-following, is that any special problem about understanding that clause which might suggest that, after all, different arithmeticians who deploy that clause could still be talking of different, non-isomorphic, structures? However, obvious though these questions are given what has gone before, Parsons doesn’t raise them.

Given the ready availability of the informal argument just sketched, why should we doubt uniqueness? Ah, the skeptical response will go, regiment arithmetic however we like, there can still be rival interpretations (thanks to the Löwenheim/Skolem theorem). Even if we dress up the uniqueness argument — by putting our arithmetic into a set-theoretic setting and giving a formal treatment of the content of the extremal clause, and then running a full-dress version of the informal Dedekind categoricity theorem — that still can’t be used settle the uniqueness question. For the requisite background set theory itself, presented in the usual first-order way, can itself have nonstandard models: and we can construct cases where the unique-up-to-isomorphism structure formed by ‘the natural numbers’ inside such a nonstandard model won’t be isomorphic to the ‘real’ natural numbers. And going second-order doesn’t help either: we can still have non-isomorphic ”general models” of second-order theories, and the question still arises how we are to exclude {those}. In sum, the skeptical line runs, someone who starts off with worries about the uniqueness of the natural-number structure because of the possibilities of non-standard models of arithmetic, won’t be mollified by an argument that presupposes uniqueness elsewhere, e.g. in our background set theory.

Now, that skeptical line of thought will, of course, be met with equally familiar responses (familiar, that is, from discussions of the philosophical significance of the existence of nonstandard models as assured us by the Löwenheim/Skolem theorem). For example, it will be countered that things go wrong at the outset. We can’t keep squinting sideways at our own language — the language in which we do arithmetic, express extremal clauses, and do informal set theory — and then pretend that more and more of it might be open to different interpretations. At some point, as Wittgenstein insisted, there has to be understanding without further interpretation (and at that point, assuming we are still able to do informal arithmetical reasoning at all, we’ll be able to run the informal argument for the uniqueness of the numbers).

How does Parsons stand with respect to this sort of dialectic? He outlines the skeptical take on the Dedekind argument at some length, explaining how to parlay a certain kind of nonstandard model of set theory into a nonstandard model of arithmetic. And his response isn’t the very general one just mooted but rather he claims that the way the construction works ”witnesses the fact the model is nonstandard” — and he means, in effect, that our grasp of the constructed model which provides a deviant interpretation of arithmetic piggy-backs on a prior grasp of the standard interpretation — so the idea that we might have deviantly cottoned on to the nonstandard model from the outset is undermined. Yet a bit later he says he is not going to attempt to directly answer skeptical arguments based on the L-S theorem. And he finishes the section by saying the theorem ”seems still to cast doubt on whether we have really ‘captured’ the ‘standard’ model of arithmetic”. So I’m left puzzled.

Parsons does, however, touch on one interesting general point along the way, noting the difference between those cases where we get deviant interpretations that we can understand but which piggy-back on a prior understanding of the theory in question, and those cases where we know there are alternative models because of the countable elementary submodel version of the L-S theorem. Since the existence of such submodels is given to us by the axiom of choice, these resulting interpretations are, in a sense, unsurveyable by us, so — for a different reason — are also not available as alternative interpretations we might have cottoned on to from the outset. The point is worth further exploration which it doesn’t receive here.

Why does the principle of mathematical induction hold for the natural numbers? Well, arguably, “induction falls out of an explanation of the meaning of the term ‘natural number’”.

How so? Well, the thought can of course be developed along Frege’s lines, by simply defining the natural numbers to be those objects which have all the properties of zero which are hereditary with respect to the successor function. But it seems that we don’t need to appeal to impredicative second-order reasoning in this way. Instead, and more simply, we can develop the idea as follows.

Put ‘N’ for ‘. . . is a natural number’. Then we have the obvious ‘introduction’ rules, (i) N0, and (ii) from Nx infer N(Sx), together with the extremal clause (iii) that nothing is a number that can’t be shown to be so by rules (i) and (ii).

Now suppose that for some predicate φ we are given both φ(0) and φ(x) → φ(Sx). Then plainly, by repeated instances of modus ponens, φ is true of 0, S0, SS0, SSS0, . . .. Hence, by the extremal clause (iii), φ is true of all the natural numbers. So it is immediate that the induction principle holds for φ – e.g. in the form of this elimination rule for N:
Thus far, then, Parsons.

So: two initial issues about this, one of which Parsons himself touches on, the other of which he seems to ignore.

First, as an argument warranting induction doesn’t this go round in a circle? For doesn’t the observation that each and every instance φ(SS . . . S0) is derivable given φ(0) and φ(x) → φ(Sx) itself depend on an induction? Parsons says that, yes, “As a proof of induction, this is circular. . . . Nonetheless, . . . it is no worse than arguments for the validity of elementary logical rules.” This of course doesn’t count against the claim that “induction falls out of an explanation of the meaning of the term ‘natural number’” – it is just that the “falling out” is so immediate that we can’t count as fully grasping the idea of a natural number while not finding inductive arguments primitively compelling (in something like Peacocke’s sense). I’m minded to agree with Parsons here.

But, second, some will complain that Parsons’s preferred way of seeing induction as given to us in the very notion of ‘natural number’ is actually not significantly different from Frege’s way, because the extremal clause (iii) is essentially second order. It will be said: the idea in (iii) is that something is a natural number if belongs to all sets which contain 0 and are closed under applications of the successor function – which is just Frege’s second-order definition put in set terms. Now, Parsons doesn’t address this familiar line of thought. However, I in fact agree with his implicit assumption that his preferred line of thought does not presuppose second-order ideas. In headline terms, just because the notion of transitive closure can be defined defined in second-order terms, that doesn’t make it a second-order notion (compare: we can define identity in second-order terms, but that surely doesn’t make identity a second-order notion!). And it is arguable that the child who picks up the notion of an ancestor doesn’t thereby exhibit a grasp of second-order quantification. But more really needs to be said about this (for a little more, see my Introduction to Gödel’s Theorems, §23.5).

To be continued

Here, as promised, are some comments on Chapter 7 of Parsons’s book. They are quite lengthy, and since in writing them I found myself going back to revise/improve some of my discussions of earlier sections, I’m just posting a single composite version of all my comments on the first seven chapters. I’m afraid that is already over 20K words and 36 single-spaced pages (start at p.31 for the substantially new stuff). So I am sounding off at some length: but it seems to me that the topics tackled in Mathematical Thought are so very central as to be well worth extended discussion.

I’ve still two more chapters to go: next up is a fifty page chapter on induction, which I think can be discussed fairly independently from what’s gone before. So I’ll revert to section-by-section blogging here.