Logic Matters

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.”

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.