Math. Thought and Its Objects

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.

There’s now a version of my posts on the first five chapters of Parsons book: so the newly added pages are on Chapter 5 of his book, on “Intuition”. I found these sections unconvincing (when I didn’t find them baffling) — a reaction that seemed to be shared by other members of the reading group here which is working through the book. So again, all comments and suggestions will be very gratefully received!

Well, “blogging at a snail’s pace” is all well and good, but my posts about Parsons have recently ground to a complete halt. Sorry about that. Pressure of other things. But I’m back on the case, now with the pressure of a deadline, and so here is a significantly expanded/improved version of my posts on the first four chapters of his Mathematical Thought and Its Objects. I’ll post on the next three chapters over the coming week. And then comment on the last two chapters the following week.

All comments will be very gratefully received as I’m going to be mining these long ruminations for a critical notice of the book.

Suppose we accept that “it is not necessary to attribute to the agent perception or intuition of a set as a single object” in order to ground arithmetical beliefs. Still, we might wonder whether some such intuition of sets-as-objects might serve to “give an intuitive foundation to theories of finite sets“.

But Parsons finds problems with this suggestion too. One difficulty can be introduced like this. Suppose I perceive the following array:

$$$$$$

Then do I ‘intuit’ six dollar signs, a single set of six dollar signs, a set of three elements each a set of two signs, or even a set containing the empty set together with a set of six signs? Which way do I ‘bracket things up’?

$$$$$$
{$$$$$$}
{{$$}{$$}{$$}}
{{}{$$$$$$}}

The possibilities are many — indeed literally endless, if we are indeed allowed the empty set (and what is our intuition of that?). So it seems that the “intuition” here has to involve some representational ingredient to play the role of the brackets in the various possible bracketings. But then we are losing our grip on any putative analogy between intuition and perception (as Parsons puts it, “in a perceptual situation involving the application of certain concepts, we not expect that a linguistic of other embodiment of the concepts should be perceptually present in that very situation”).

Secondly, note that we can in fact give a theory of those “bracket terms” — putatively for hereditarily finite sets constructed from a given domain D of individuals — which uses a relative substitutional semantics. That is to say, we can start with a first-order language for which D is the domain, add terms for hereditarily finite sets of elements from D, and variables and quantifiers for them, which we then interpret substitutionally relative to D. Parsons spells this out in an Appendix, but the general idea will be familiar to readers of his old paper on ‘Sets and Classes’. And the upshot of this, Parsons says, “is that if we take the relative substitutional semantics as capturing a speaker’s understanding of the language of hereditarily finite sets … then we largely remove the motives for characterizing awareness of such sets as initution”. That’s a significant “if” of course: but we might indeed wonder why we should take elementary talk about finite sets (and sets of those, and so on) to be more committing than the substitutional interpretation allows.

Note that this isn’t to say that we have entirely eliminated a role for intuition. For on the relative substitutional interpretation we still need the idea of sequences of individuals from D. And we might suppose that that notion is grounded in intuition. But even if true, that still falls well short of the original thought that we could need intuitions of sets-as-objects to give a foundation to theories of finite sets.

So where have we got to in talking about Parsons’s book? Chapter 6, you’ll recall, is titled “Numbers as objects”. So our questions are: what are the natural numbers, how are they “given” to us, are they objects available to intuition in any good sense? I’ve already discussed Secs 31 and 32, the first two long sections of this chapter.

There then seems to be something of a grinding of the gears between those opening sections and the next one. As we saw, Sec. 32 outlines rather incompletely the (illuminating) project of describing a sequence of increasingly sophisticated but purely arithmetical language games, and considering just what we are committed to at each stage. But Sec. 33 turns to consider the theory of hereditarily finite sets, and considers how a theory of numbers could naturally be implemented as an adjunct to such a theory. I’m not sure just what the relation between these projects is (we get “another perspective on arithmetic”, but what exactly does that mean? — but, looking ahead, I think things will be brought together a bit more in Sec. 36).

Anyway, in Sec. 33 (and an Appendix to the Chapter) Parsons outlines a neat little theory of hereditary finite sets, taking a dyadic operation x + y (intuitively, x U {y}) as primitive alongside the membership relation. The theory proves the axioms of ZF without infinity and foundation. I won’t reproduce it here. In such a theory, we can define a relation x ~ y that holds between the finite sets x and y when they are equinumerous. We can also define a “successor” relation between sets along the following lines: Syx iff (Ez)(z is not in x and y ~ x + z).

Now, as it stands, S is not a functional relation. But we can conservatively add (finite) “cardinal numbers” to our theory by introducing a functor C, using an abstraction axiom Cx = Cy iff x ~ y — so here “numbers are types, where the tokens are sets and the relation ~ is that of being of the same type”. And then we can define a successor function on cardinals in terms of S in the obvious way (and go on to define addition and multiplication too).

So far so good. But quite how far does this take us? We’d expect the next step to be a discussion of just how much arithmetic can be constructed like this. For example, can we cheerfully quantify over these defined cardinals? We don’t get the answer here, however. Which is disappointing. Rather Parsons first considers a variant construction in which we start not with the hierarchical structure of hereditarily finite sets but with a “flatter” structure of finite sequences (I’m not too sure anything much is gained here). And then — in Sec. 34 — he turns to consider whether such a story about grounding an amount of arithmetic in the theory of finite sets/sequences might give us an account of an intuitive grounding for arithmetic, via a story about intuitions of sets.

Well, we can indeed wonder whether we “might reasonably speak of intuition of finite sets under somewhat restricted circumstances” (i.e. where we have the right kinds of objects, the objects are not too separated in space or time, etc.). And Penelope Maddy, for one, has at one stage argued that we can not only intuit but perceive some such sets — see e.g. the set of three eggs left in the box.

But Parsons resists at least Maddy’s one-time line, on familiar — and surely correct — kinds of grounds. For while it may be the case that we, so to speak, take in the eggs in the box as a threesome (as it might be) that fact in itself gives us no reason to suppose that this cognitive achievement involves “seeing” something other than the eggs (plural). As Parsons remarks, “it seems to me that the primary elements of a story [a rival to Maddy’s] would be the capacity to classify what one sees … and to recognize identities and differences” — capacities that could underpin an ability to judge small numerical quantifications at a glance, and “it is not necessary to attribute to attribute to the agent perception or intuition of a set as a single object”. I agree.

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.

Luca Incurvati has just pointed out to me that John Burgess has a review of Parsons forthcoming in Philosophia Mathematica, and an electronic pre-print is available here (if your library has a subscription). Burgess is very polite, but reading between the lines, maybe he had some of the problems I’m having. For example, “[Parsons’s] own version of structuralism is only rather sketchily indicated”, and Burgess is himself pretty sketchy about Parsons on intuition.

I hope to return to Parsons here tomorrow; but in fact the pressure is off for me. It turns out that Bob Hanna and Michael Potter here are going to be running a reading group on the book this coming term, so I’ve arranged for the delivery date of my critical notice to be delayed until the end of term, after I’ve had the benefit of hearing what others think about some of what I’m finding obscure.