Logic Matters

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.)

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.

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!