Phil. of maths

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!

Parsons’s Mathematical Thought: Secs 16, 17, Modalism

In Sec. 16, “Modalism”, Parsons considers the stratgegy of rescuing eliminativist structuralism from the vacuity problem by going modal.

To recap, we’re considering the schematic idea that an ordinary arithmetical statement is elliptical for something along the lines of

For any N, 0, S, if Ω(N, 0, S) then A(N, 0, S),

where Ω(N, 0, S) lays down the conditions for a set N (equipped with a distinguished element 0, and a mapping S: N -> N – {0}) to be “simply infinite”, and A(N, 0, S) is appropriately correlated with the ordinary statement. And the idea is, of course, that the quantifications here are restricted to kosher, unproblematic, collections of physical objects N, and mappings on them (such as arrays of space-time points, and a ‘go to the next point’ map): in this way, problematic purely abstract entities drop out of the picture. And the vacuity problem is: what if, as a matter of physical fact, ours is a finite, granular, world and there is no infinite physical collection and physically realized mapping for which the condition Ω is true? In that case, the quantified conditional holds vacuously, and all arithmetical statements come out as indiscriminately true.

Now, the obvious modal gambit is to respond: Ah, we should require the quantified conditional to hold necessarily. For, even if in this world there are no physically realized simply infinite systems, there could be other possible worlds (maybe where the physical laws are very different) which do realize simply infinite systems. Why not? So even if the plain unmodalized conditional is vacuously true for any A, the modalized version won’t be.

Note, though, that the natural way of construing the modality here is surely in that sense of necessity which is spelt out as truth in all possible worlds (or at last, truth in all worlds that are in some sense physical worlds lacking abstract denizens) — as some kind of metaphysical necessity, in other words. So I’m a bit stumped as to why Parsons says “I will assume that the modal operators are understood either in the sense of mathematical modality or of metaphysical modality.” I’m quite unclear what the notion of mathematical modality would do for us here.

Anyway, Parsons considers two objections to modal structuralism. The first, however, is not an objection to the modal version in particular: “It is falsifying the sense of discourse about natural numbers [to take] arithmetical statements to be really about every simply infinite system.” And surely a structuralist of any sort will think this a pretty feeble objection: even if the structuralist account seems prima facie to do some violence to our intuitive model of what we mean, its defender will just say that that shows we are gripped by a bad philosophical picture of what we are really up to in doing mathematics.

Parsons considers another objection, by pressing “whether the modalist’s apparatus really does offer an elimination of mathematical objects” in Sec. 17, “Difficulties of modalism”. But in fact the way he develops the point seems not to tell against a modal structuralist account of arithmetic or indeed applicable analysis (given we can construct applicable analysis in such weak systems of second-order arithmetic). Rather the worry seems to be whether there could be structures realized in some sort of alternative physical world which have anything like the richness to be a model of higher set theory. Well, maybe not (though I’m not sure how one goes about settling the issue!). But then why shouldn’t the structuralist just say, so much the worse for higher set theory — it’s just a fiction, or a jeux d’esprit?

Parsons’s Mathematical Thought: Sec. 15, Mathematical modality

Chapter 3 of Mathematical Thought and Its Objects is called “Modality and structuralism”. Before turning to discuss modal structuralism in Secs. 16 and 17, Parsons discusses what kind modality it might involve. Setting aside epistemic modalities as not to the present purpose, he considers (i) physical (or natural) necessity, (ii) metaphysical necessity (truth in all possible worlds), (iii) mathematical necessity, (iv) logical necessity (meant in a narrow sense that can be explicated model-theoretically).

Parsons argues that we don’t want to spell out a modal structuralism in terms of (i) natural modalities: “it demands too much to ask that the structures considered in mathematics be physically possible; indeed, in the case of higher set theory, there is every reason to believe that they are not physically possible.” I’ll buy that.

Second, Parsons argues that logical possibility — in the sense explicated via the idea of there being a suitable model — reveals itself as itself a mathematical notion, given that models are (at least typically) mathematical entities. So(?), “It is very doubtful that a generous notion of logical possibility would be distinguishable in a principled way from … mathematical possibility.”

But there is surely something rather odd here. For the idea, to repeat, is that we explicate “it is logically possible that P” (in the generous sense of allowed-at-least-by-considerations-of-logical-form, that runs beyond metaphysical possibility) in terms of there being a mathematical model on which P can be interpreted as true. It seems we don’t have a modality in the explanans here. Indeed, Parsons himself remarks on the common view that a mathematical truth (falsehood) is necessarily true (false): and on that view the very idea of a kind of “mathematical possibility” distinguished from plain truth evaporates.

So I’m left puzzled when Parsons concludes that the two runners for the kind of modality that might be involved in a modal structuralism are metaphysical modality and mathematical modality: for I just don’t have a grip on the latter notion.

(Relatedly, Parsons reads Putnam as holding that “it is mathematically possible that there should be no sets of uncountable rank, although it is a theorem of ZF that there are such sets”. Again, I really just don’t know how to construe that “mathematically possible” if that is supposed to be neither epistemic nor equivalent to “true”.)

Parsons’s Mathematical Thought: Sec 14, Structuralism and application

We’re considering the schematic idea that an ordinary arithmetical statement is elliptical for something generalizing over structures, along the lines of

For any N, 0, S, if Ω(N, 0, S) then A(N, 0, S),

where Ω(N, 0, S) lays down the conditions for a set N (equipped with a distinguished element 0, and a mapping S: N -> N – {0}) to be “simply infinite”, and A(N, 0, S) is appropriately correlated with the ordinary statement. (Parsons, you’ll recall, associates such a view with Dedekind. That doesn’t seem historically correct. But let that pass.)

Does this “eliminative structuralist” view have a problem accounting for the application of numbers as cardinals? Recall Frege’s remark: “It is applicability alone that raises arithmetic from the rank of a game to that of a science. Applicability therefore belongs to it of necessity.” And Frege further takes it that an account of numbers should start from their use in counting (so a structuralist understanding that explains the nature of arithmetical truths prior to explaining their application is going wrong). But, Parsons argues, our structuralist in fact can resist that further thought.

I’m not sure I fully have the measure of Parsons thinking here. Part of the trouble is that he slips back into talking of numbers as objects (e.p. pp. 74–75), while I thought the attraction of the eliminative structuralism was to get rid of numbers as a special kind of object. But I take it the thought is something like this. Counting some objects involves putting them into one-one correspondence with an initial segment of some paradigm simply infinite system (of numerals, say). That involves setting up some external relations between some members of the relevant simply infinite system, over an above the internal relations which constitute their being a such a system. But now, via the Dedekind categoricity theorem, we see that these external relations will engender a one-one correspondence with an isomorphic initial segment of any simply infinite system. So, in counting, we automatically get an implicit generalization over simply infinite systems — which is what, according to the eliminative structuralist, talk of numbers amounts to. Hence, as Frege wanted, even on the structuralist view, we do after all have an essential connection between numbers and their application in counting.

That, I think, does deal with the supposed general problem. Now, Dummett has raised a more specific problem — roughly, defining a simply infinite system doesn’t tell us whether its initial element is to be treated as 0 or 1 (or indeed, I suppose, 42). But Parsons (rightly in my view) doesn’t find this worry a telling one for the structuralist. He can regard it as just a matter of pragmatic convention whether, in applications, we start counting at 0 or 1, depending on how much we care about having a number for empty collections.

One final comment on this section. Having quieted worries about the structuralist view, Parsons remarks that as well as the natural number 3, we have the integer 3, the rational 3, the real number 3 and the complex number 3 (not to mention more exotic constructions). And the structuralist can say that the use of “3” each time signifies not the same entity but the same structural role, a point congenial to his general account of the significance of number words. But, contra Parsons, I don’t see that the multiple use of “3” counts at all against the Fregean view that numbers are specific objects. The Fregean can just say that there are here a number of different terms, (“natural number 3”, “rational number 3”, etc.) with different objects as reference — with the common elements of the referring terms justified by the likeness of the role of the denoted objects in the respective families.

Parsons’s Mathematical Thought: Sec 13, Nominalism and second-order logic

A general comment before proceeding. Parsons himself says that this book has been a very long time in the writing. And I suspect that what we are reading is in fact a multi-layered text with different passages added at different times, without the whole being finally reorganized and rewritten from beginning to end. This does make for a bumpy read, with the to-and-fro of argument not always ideally well signalled.

Anyway, Sec. 13 falls into two parts, both related to nominalist takes on second-order logic. First, Parsons offers some remarks on the Fieldian project of using mereology to do the work of second-order logic. The key thought is this. For mereology to do all the work Field wants, it needs an (impredicative) comprehension principle: “Given a predicate of individuals that is true of at least one individual, there is a sum of just the individuals of which the predicate is true, and moreover, the admissible predicates will be closed under quantification over all individuals, including those very sums.” (Cf. the principle “Cs” in Field’s “On Conservativeness and Incompleteness”.) But what entitles Field to such a strong comprehension principle? Well, Parsons notes that it’s not clear that Field can offer any direct a priori argument (but then, I wonder, would he want to?). The justification will be that “the comprehension principle is a hypothesis justified by its consequences in systematizing the geometrical basis of physics”. But then “Field’s view, on this reading, puts him in a position in which we have found other formulations of nominalism: making the justification of mathematics turn on some hypothesis about the physical world, which is more vulnerable to refutation than the mathematics.”

But how troubled will a Fieldian be by that complaint? Suppose we decide that our physical theory of the world doesn’t require such a strong comprehension principle (we can get away with recognizing a less wide-ranging plurality of regions). That’s not at all implausible, actually, given that (nearly) all the mathematics required for physics can be reconstructed in a weak second-order arithmetic like ACA_0 with only predicative comprehension. Then the Fieldian response will (surely?) be just to demote the full mathematical apparatus of the classical reals from its status in Science without Numbers as a supposedly justified tool for getting more nominalistically acceptable consequences out of our best physics. It is no longer so justified. In that sense, for the Fieldian, the “justification” of a bit of mathematics is wrapped up with our hypotheses about the physical world, and Parsons’s complaint will seem question-begging. [Or am I missing something here?]

The second part of Sec. 13 considers Boolos’s attempt to make second-order logic ontologically tame by giving a plural reading to the second-order quantifiers. The thought under scrutiny is that plural quantification is ontologically innocent because, in plurally quantifying over Fs, we are just committing ourselves to Fs (not to sets or to Fregean concepts). Parsons’s discussion [or again, am I missing something here?] initially advances familiar sorts of worries about this claim of innocence. But Parsons does make one point towards the end of the section that I find very congenial (i.e. I’ve argued similarly myself!).

Consider (say) the range of second-order arithmetics that Simpson discusses in SOSOA. As we advance through theories with stronger and stronger comprehension principles, then — on a standard platonist construal — we are countenancing more and more sets of numbers. If we reconstrue the second-order quantifiers plural-wise, then, as we go from theory to theory, we are countenancing more and more …. well, more what? It is tempting to say “pluralities”. And indeed it is convenient to give an informal gloss of the plural reading using talk of pluralities. But — if this isn’t to smuggle back reference to pluralities-as-single-entities, i.e. sets — this convenient way of talking needs to be eliminable (cf. Linnebo’s nice article on plural logic). So how do we eliminate it here? We might, I suppose, trade in talk of countenancing more and more pluralities for talk of allowing more and more different ways we can take numbers together: but this seems tantamount to re-instating Fregean concepts as the values of the second-order variables — which is fine by me, but then the supposed ontological gain of interpreting the second-order quantifiers via plurals is lost.

The question then is this: if we accept the pluralist’s contention that we can treat second-order numerical quantifiers as ontologically committing just us to numbers, period, then how are we to think of the surely varying commitments we take on with varying strengths of comprehension principle. As Parsons puts it, “If there is no enlargement of ontological commitment [my emphasis] as one passes to less restricted versions of the comprehension schema, then perhaps that speaks against the importance of the notion.”

Parsons’s Mathematical Thought: Sec. 12, Nominalism

This is a short and rather insubstantial section, which I’m just taking separately to get out of the way, because the next section is weighty (and one of the longest in the book).

Parsons understands ‘nominalism’ Harvard-style — no surprise there, then! — to mean the rejection of abstract entities and the eschewing of (ineliminable) modality. What hope, then, for giving a response to the potential-vacuity problem for eliminative structuralism about arithmetic (say) which meets nominalist constraints? We can’t, by hypothesis, go modal: so what to do?

Well, as the physical world actually is (or so we might well now believe), there are in fact enough physical things — e.g. space time points — and suitable physical orderings on them to give us physically realized ‘simply infinite’ structures. But Parsons is unhappy with this way of meeting the vacuity worry, and for familiar reasons: “[S]hould it be taken as a presupposition of elementary mathematics that the real world instantiates a mathematical conception of the infinite? This would have the consequence that mathematics is hostage to the future possible development of physics.”

But (although I have no particular nominalist sympathies myself), I’m not sure how worried the nominalist eliminative structuralist should be about giving such hostages to fortune. As things are, given how we believe the world actually to be, he can reasonably continue to speak with the vulgar and treat arithmetical claims as true or false. Even if the worst happens, so we come to believe the world is ultimately grainy and finite in all respects, it’s not that ‘school-room’ arithmetic is going to get undermined. At most, it is the idealizing rounding out of school-room arithmetic which insists on an infinitude of numbers. And if it should emerge that the rounding out, construed the eliminative-structuralist way, collapses in vacuity — well, formal arithmetic can still be played as an intriguingly entertaining game. It’s just that then, after all, the nominalist eliminative structuralist who is relying on physical realizations for structures can no longer readily construe idealized arithmetic’s claims as true or false, and so the nominalist has to sound a bit more revisionary. But, he’ll say, so what? (Parsons says “a great deal of the historically given mathematics would have to be jettisoned in this case” — but that’s too quick. Talk of ‘jettisoning’ covers over a slide. For no longer thinking of arithmetic as construable as literally true by the eliminative structuralist manoeuvre is not the same as throwing arithmetic into the trash-can, as any fictionalist will insist.)

What about the other line that offered to the nominalist at the end of Sec. 11? — i.e. sidestep the vacuity problem by going modal in an anodyne way (“interpret the theories in an if-thenist way, but deal with the problem of possibility by appealing to consistency, nominalistically interpreted”). Well, again Parsons sees trouble, this time arising from the fact that there might be physical limitations in how big a proof-token could be, and so a theory could count as (nominalistically) consistent — because no proof of an inconsistency could be tokened — even if we can show that there is a process which, given world enough and time, would produce an inconsistency. But again, I’m not sure that the obstreperous nominalist couldn’t swallow that too.

At the end of this section, Parsons revisits the question of how to frame an eliminative structuralism for arithmetic. He looked at a move from a set-theoretic formulation to a more ‘logical’, second-order formulation. But could we go first-order, in a way more congenial no doubt to those of nominalist inclinations? The trouble is, of course, that we won’t get categoricity (whatever we build into the axioms), so the eliminative structuralist who goes first-order runs up against the intuition that the natural numbers have a unique structure. But how secure, in fact, is that intuition? Parsons raises that excellent question (too often passed over in silence), but only to shelve it until Ch. 8. So we’ll have to return to that later.

Parsons’s Mathematical Thought: Secs 8 – 11

Back, after rather a gap, to Charles Parsons’s book and on to the first half of his second chapter, “Structuralism and nominalism”.

(Sec. 8) Parsons says that he himself thinks that “something close to the structuralist view is true”. But structuralist in what sense? It is often said, perhaps in a Bourbachiste spirit, that mathematics is the study of structures. But — as Parsons stresses — that leaves it wide open what picture we should adopt of the ontology of mathematical objects. He is more concerned with structuralism(s) with more ontological bite — something along the lines suggested by “the objects of mathematics are positions in structures, [and] have no identity or features outside of a structure” (to quote from Michael Resnik’s well-known 1981 Nous paper).

(Sec. 9) But what are structures? The usual modern mathematical story sees these as sets (or classes) with distinguished elements, equipped with relations and/or functions. So it looks as though an account of mathematical objects as positions in structures already presupposes familiar kinds of objects (sets, classes) to build structures out of, and explaining their nature in structuralist terms threatens circularity. But Parsons puts this worry on hold for the moment.

(Sec. 10) So go with the set-theoretic conception of structure, just pro tem, and consider as an exemplar Dedekind’s treatment of the natural numbers. Dedekind defines what it is for a set N, with distinguished element 0, and a mapping S: N -> N – {0} to be “simply infinite”. Abbreviate those (categorical) conditions Ω(N, 0, S). With some effort, an ordinary statement of arithmetic can be correlated with a version A(N, 0, S) whose primitives are again N, 0, S. And on one reading of Dedekind — the eliminative reading — the suggestion is that the ordinary statement can be treated as elliptical for

For any N, 0, S, if Ω(N, 0, S) then A(N, 0, S).

This is ‘eliminative’ in that a statement apparently about one kind of thing, numbers, is treated as in fact a disguised generalization about other kinds of things. The suggestion neatly sidesteps “multiple reduction” problems for more straightforward attemps to reduce arithmetic to set theory. But (on the face of it) it faces the worry that if there are no simply infinite systems then any ordinary arithmetical statement comes out as vacuously true and arithmetic is inconsistent. True, that first worry won’t be pressing if we already buy into a background universe with enough sets, but it will become more urgent when we try to repeat the trick and give an eliminative structuralist account of them. And there’s a related second worry. Ω(N, 0, S) will involve quantification over sets, as indeed will a typical A(N, 0, S) as we give explicit definitions of e.g. recursive arithmetical functions. Do we want really want a structuralist account of a particular familiar kind of mathematical object, numbers, to tells us that we’ve been generalizing about some other rather less familiar kind of object all along? (Parsons wonders: Maybe we need to generalize over structures to state structuralism as a general thesis: but does a structuralist account of a particular kind of object have to similarly generalize over structures?)

(Sec. 11) Well, we can sidestep the second of those worries, and the worries of Sec. 9, perhaps, by trading in an explicitly set-theoretic presentation of Dedekind’s eliminative structuralism for a version couched in second-order logical terms. We get a new second-order definition of being simply infinite, Ω'(N, 0, S), a new correlate of an ordinary arithmetical claim, A‘(N, 0, S), and correspondingly a new suggestion that the ordinary statement can be treated as elliptical for

For any N, 0, S, if Ω'(N, 0, S) then A’(N, 0, S).

where now ‘any N‘ and ‘any S‘ are treated as second-order. If we are relaxed enough about second-order quantification, we might find this easier to swallow that the previous version (though that’s quite a big “if”). However, this kind of ‘if-thenism’ is still threatened by the possibility of vacuity. What to do?

One option is to read the conditional as stronger-than-material, e.g. by discerning a governing modal operator. But that opens up another set of problems. What kind of modality is involved here? Can we e.g. give a modest possibility-as-consistency reading? Perhaps “we interpret the theories in an if-thenist way, but deal with the problem of possibility by appealing to consistency, nominalistically interpreted.” The suggestion is to be pursued critically in Sec. 12.

OK, so much by way of brisk summary of these sections (I didn’t find them entirely easy to follow, but I hope I’ve fairly represented the way the discussion develops). I don’t think I have much to add by way of commentary: in fact, the dialectic so far is a pretty familiar one.

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