Thursday, August 21, 2008

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.

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