MTO

Parsons’s Mathematical Thought: Sec. 47, Induction and the concept of natural number

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

Parsons’s Mathematical Thought: Sec. 35, Intuition of finite sets

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.

Parsons’s Mathematical Thought: Secs 33, 34, Finite sets and intuitions of them

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.

Parsons’s Mathematical Thought: Secs 31, 32, Numbers as objects

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.

Burgess reviews Parsons

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.

Parsons’s Mathematical Thought: A footnote on intuition

Qn: “You do seem increasingly out of sympathy with Parsons’s book. So why are you spending all this effort blogging about it?”

Ans: “Well, as I think I said at the outset, I have promised to write a review (indeed, a critical notice) of the book, so this is just my way of forcing myself to read the book pretty carefully. And I’m not so much unsympathetic as puzzled and disappointed: I’m finding the book a much harder read than I was expecting. The fault could well in large part be mine. However, I do think that the prose is too often obscure, and the organization of thoughts unclear, so a bit of impatience may by now be creeping in (and talking to one or two others, I don’t think my reaction to Parsons’s writing is in fact that unique). But these ideas are certainly worth wrestling with: so I’m battling on!”

One thing I didn’t comment on before was Parsons’s motivation for pushing the notion of intuition and intuitive knowledge. “Intuition that,” he says, “becomes a persuasive idea when one reflects on the obviousness of elementary truths of arithmetic. Two alternative views have had influential advocates in this century: conventionalism … and a form of empiricism according to which mathematics is continuous with science, and the axioms of mathematics have a status similar to high-level theoretical hypotheses.” Carnapian(?) conventionalism is, Parsons seems to think, a non-starter: and Quinean empiricism “seems subject to the objection that it leave unaccounted for precisely the obviousness of elementary mathematics.” An appeal to some kind of intuition offers the needed account.

But I’m not sure that the Quinean should be abashed by that quick jab. For the respect in which the axioms of mathematics are claimed to have a status similar to high-level theoretical hypotheses is in their remoteness from the observational periphery, in their central organizational roles in a regimentation of our web of belief by logical/confirmational connections. That kind of shared status is surely quite compatible with the second-nature “obviousness” that accrues to simple arithmetic — for some of us! — due to intense childhood drilling and daily use. Logical position in the web, a Quinean would surely say, and degree of entrenched obviousness something else.

Parsons’s Mathematical Thought: Secs 27-30, Intuition, continued

I’ve been trying to make good sense of the rest of Parsons’s chapter on intuition, and have to confess failure. We might reasonably have hoped that we’d get here a really clear definitive version of the position on intuition that he has been developing for the better part of 30 years; but I’m afraid not. Looking for some help, I’ve just been rereading James Page’s 1993 Mind discussion ‘Parsons on Mathematical Intuition’, which Parsons touches on, and David Galloway’s 1999 Philosophical and Phenomenological Research paper ‘Seeing Sequences’, which he doesn’t mention. Those papers show that it is possible to write crisply and clearly (though critically) about these matters: but Parsons doesn’t pull it off. Or at least, his chapter didn’t work for me. Although this is supposed to be a pivotal chapter of the book, I’m left rather bereft of useful things to say.

Sec. 27, ‘Toward a viable concept of intuition: perception and the abstract’ is intended to soften us up for the idea that we can have intuitions of abstracta (remember: intuitive knowledge that, whatever exactly that is, is supposed to be somehow founded in intuitions of, where these are somehow quasi-perceptual). There’s an initial, puzzling, and inconclusive discussion of supposed intuitions of colours qua abstract objects: but Parsons himself sets this case aside as raising too many complications, so I will too. Which leaves the supposed case of perceptions/intuitions of abstract types (letters, say): the claim is that “the talk of perception of types is something normal and everyday”. But even here I balk. True, we might well say that I see a particular squiggle as, for example, a Greek phi. We might equivalently say, in such a case, that I see the letter phi written there (but still meaning that we see something as an instance of the letter phi). But I just don’t find it at all normal or everyday to say that I see the letter phi (meaning the type itself). So I’m not softened up!

Sec. 28, ‘Hilbertian intuition’ rehashes Parsons’s familiar arguments about seeing strings of strokes. I won’t rehash the arguments of his critics. But I’m repeatedly puzzled. Take, just for one example, this claim:

What is distinctive of intuitions of types [now, types of stroke-strings] is that the perceptions and imaginings that found them play a paradigmatic role. It is through this that intuition of a type can give rise to propositional knowledge about the type, an instance of intuition that. I will in these cases use the term ‘intuitive knowledge’. A simple case is singular propositions about types, such as that ||| is the successor of ||. We see this to be true on the basis of a single intuition, but of course in its implications for tokens it is a general proposition.

A single intuition? Really? If I’m following at all, I’d have thought that we see that proposition to be true on the basis of an intuition of ||| and a separate intuition of || and something else, some kind of intuitive (??) recognition of the relation between them. What is the ‘single’ intuition here?

Or for another example, consider Parsons’s wrestling with vagueness. You might initially have worried that intuitions which are “founded” in perceptions and imaginings will inherit the vagueness of those perceptions or imaginings (and how would that square with the claim that “mathematical intuition is of sharply delineated objects”?). But Parsons moves to block the worry, using the example of seeing letters again. The thought seems to be that we have some discrete conceptual pigeon-holes, and in seeing squiggles as a phi or a psi (say), we are pigeon-holing them. The fact that some squiggles might be borderline candidates for putting in this or that pigeon-hole doesn’t (so to speak) make the pigeon-holes less sharply delineated. Well, fair enough. I’m rather happy with a version of that sort of story. For I’m tempted by accounts of analog non-conceptual contents which are conceptually processed, “digitalizing” the information. But such accounts stress the differences between perceptions of squiggles and the conceptual apparatus which is brought to bear in coming to see the squiggles as e.g. instances of the letter phi. Certainly, on such a view, trying to understand our conceptual grip here in terms of a prior primitive notion of “perception of” the type phi is hopeless: but granted that, it is remains entirely unclear to me what a constructed notion of “perception of” types can do for us.

Sec. 29, ‘Intuitive knowledge: a step toward infinity’ Can we in any sense see or intuit that any stroke string can be extended? Parsons has discussed this before, and his discussions have been the subject of criticism. If anything — though I haven’t gone back to check my impression against a re-reading of his earlier papers — I think his claims may now be more cautious. Anyway, he now says (1) “If we imagine any [particular] string of strokes, it is immediately apparent that a new stroke can be added.” (2) “Although intuition yields one essential element of the idea that there are, at least potentially, infinitely many strings … more is involved in the idea, in particular that the operation of adding an additional stroke can be indefinitely iterated. The sense, if any, in which iteration tells us that is not obvious.” But (3) “Although it will follow from considerations advanced in Chapter 7 that it is intuitively known that every string can be extended by one of a different type, ideas connected with induction are needed to see it.” We could, I think, argue about (1). Also note the slide from “imagine” to “intuition” between (1) and (2): you might wonder about that too (Parsons is remarkably quiet about imagination). But obviously, the big issue is going to come later in trying to argue that ideas “connected with induction” can still be involved in what is “intuitively known”. We’ll see …

Finally, I took little away from Sec. 30, ‘The objections revisited’, so I won’t comment now.

Parsons’s Mathematical Thought: Secs 24-26, Intuition

Chapter 5 of Parsons’s book is called “Intuition”. And I guess I should declare an interest (or rather, lack of interest!) here. I’ve never really understood talk about intuition: and I’m certainly not helped when Parsons writes “I shall be concerned to develop a conception of mathematical intuition that is in a general way Kantian”, since Kant is pretty much a closed book to me. So perhaps I’m not the best reader for this chapter! But still, let’s proceed …

Sec. 24, “Intuition: Basic distinctions”. Parsons distinguishes supposed intuition of objects from intuition that such-and-such is the case. And he stresses that in his usage, intuition that isn’t factive. So is an intuition that such-and-such just a non-inferential belief? Well note, for example, that “knowledge without observation” of our own bodily movements is non-inferential, but is not normally counted as intuitive. So what differentiates intuition properly so-called? Parsons promises an answer by a “development of the concept … in the Kantian tradition”.

Sec. 25, “Intuition and perception”. Now, the headline suggestion here is that “It is hard to see what could make a cognitive relation to objects [intuition of] count as intuition if not some analogy with perception” (cf. e.g. Gödel). Further, intuition that is intimately connected with intuition of, rather as perception that is grounded in perception of. Well, fair enough: but that, of course, already does make claims about intuitions of mathematical objects very puzzling. Which leads to …

Sec. 26, “Objections to the very idea of mathematical intuition”. Start with the following point. Ordinary perception is (so to speak) evident to the subject — when I see an object, my computer screen say, “there is a phenomenological datum here”. But “it is hard to maintain that the case is the same for mathematical objects … [Are] there any experiences we can appeal to in the mathematical cases that are anywhere near as indisputed as my present experience of seeing the computer screen?” This seems to undermine any alleged analogy between “intuition of mathematical entities” and ordinary perception. So how are we to defend the analogy, given the different phenomenologies? Unfortunately, Parsons next remarks here are Kantian obscurities I can do nothing with. So I’m left stumped.

(Parsons also raises a question about the relation between structuralist thoughts and claims about intuition. The worry seems to be one about how a particular intuition can latch on to a particular object, if mathematical objects are indentified by their places in structures. The point, however, is rather rushed. But since I think Parsons is going to return to these matters, I won’t say more at the moment.)

Parsons’s Mathematical Thought: Secs 19-23, A problem about sets

These sections make up the short Chapter 4 of Parsons’s book (they are a slightly expanded version of a 1995 paper in a festschrifft for Ruth Barcan Marcus). The issue is whether there are special problems giving a broadly structuralist account of set theory. Since the last section of Chapter 3 left me puzzled about what, exactly, Parsons counted as a structuralist view, I’m not entirely sure I have the problem in sharp focus. But I’ll try to comment all the same.

It’s perhaps clear enough what the problem is for the eliminative structuralist (whether or not he modalizes). His idea is that an ordinary mathematical claim A is to be read as disguising a quantified claim of the form for all …., if Ω(…) then A*(…), where Ω is an appropriate set of axioms for the relevant mathematical domain, A* is a suitable formal rendering of A, and where the quantification is over kosher non-mathematical whatnots, and perhaps possible world indices too. This account escapes making A vacuously true only if Ω is satisfied somewhere (at some index). Now if Ω is suitably modest — axioms for arithmetic say — we might conceive of it being satisfied by some physical realization at this (or at least, at some not-too-remote) world. I’m not sure this is right because of issues about theories Ω with full second-order quantification (which Parsons himself touches on); but let that pass. For certainly, if Ω is a rich set theory, then it cetainly doesn’t seem so plausible to say that the relevant structure is realized somewhere. Unless, that is, we allow into our possible worlds abstracta to do the job — in which case the point of the eliminative structuralist manoeuvre is undermined. (The structuralist could just bite the bullet of course, as I remarked before, and say so much the worse for set theory. After all, what’s so great about something like ZFC? — we certainly don’t need it anything as exotic to construct applicable mathematics.)

But suppose we do want to endorse ZFC, and remain broadly structuralist. Even if we eschew eliminativist ambitions, presumably the idea will be at least that there isn’t a given unique universe of determinately identified objects, the sets, which set theory aims to describe. And on the face of it, this runs against the motivating stories told at the beginning of typical set theory texts, which do (it seems) purport to describe a unique universe of sets. For example, in the case of pure set theory without urelemente, take the empty set (isn’t that determinately unique?); now form its singleton; now form the sets whose members are what we have already; now do that again at the next level; keep on going … Thus iterative story is a familiar one, and seems (or so the authors of many texts apparently suppose) to fix a unique universe.

The main burden of Parsons’s discussion is to argue that familiar story isn’t in as good order as we might like to think. For a start, the metaphors of “forming” and “levels” don’t bear the weight that is put on them: “when we come to [a set] of sufficiently high rank, it is difficult to take seriously the idea that all the intermediate sets that arise in the construction of this set … can be formed by us”. And then there are problems wrapped up in the temporal metaphor of “keeping on going”, when the relevant ordinal structure we are supposed to grasp is much richer than that of time. Further, it is aguable that additional thoughts, over and above the basic conception of an iterative hierarchy, are needed to underpin all the axioms of ZFC — that’s arguably the case for replacement, and possibly even for the full powerset axiom.

I’m not going to try to assess Parsons’s arguments here. The idea that the iterative story is problematic and doesn’t get us everything we want is by now a familiar one; there are interesting and important discussions by George Boolos, Alex Paseau, Michael Potter and others, and I don’t have anything to add. But let’s suppose he is right. What then? Parsons writes that his “discussion of the arguments that are actually in the literature should make plausible that there is not a set of persuasive, direct and “intuitive” considerations in favour of the axioms of ZF that are incompatible with a structuralist conception of what talk of sets is.” But that seems too sanguine. For it isn’t that there are multiple lines of thought in the literature which, each taken separately, give us a conception of some structure that satisfies the ZF axioms (first or second order), indicating — perhaps — the kind of multiple realizability that is grist to the structuralist argument. No, the worry is that no familiar line of thought (e.g. the iterative conception, the idea of “limitation of size”, not to mention the ideas shaping NF) warrants all the axioms. So it isn’t, after all, clear we have an intuitive grasp of any structure that satisfies the axioms. Hence, the worry continues, for all we know maybe there is no structure that satisfies them. Which seems to take us back to vacuity worries for structuralism.

Parsons’s Mathematical Thought: Sec. 18, A noneliminative structuralism

The previous two sections critically discussed a modal version of eliminative structuralism (though to my mind, the objections raised weren’t particularly telling). Parsons now moves on characterize his own preferred “noneliminative structuralism”, and responds to some potential obections.

I wish I could give a sharp characterization of the position Parsons wants to occupy here in the longest section of his book. But I do have to confess bafflement. “We have emphasized the point going back to Bernays that reference to mathematical objects is relative to a background structure.” Further, structures aren’t themselves objects, and “[Parsons’s] structuralist account of a particular kind of mathematical object does not view statements about that kind of object as about structures at all”. But surely there’s thus far nothing that e.g. the Fregean need dissent from. The Fregean can agree that numbers, for example, don’t come (so to speak) independently, but come all together forming an intrinsically order structured: and in identifying the number 42 as such, we necessarily give its position in relation to other numbers. So what more is Parsons saying about (say) numbers that distinguishes his position? Well, I’ve read the section three times and I’m still rather lost, and won’t ramble here. If any other reader of the book can offer some crisp clarifying comments, I for one would be very grateful!

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