Science is often said to be committed to reals, because physics, for example, essentially makes use of sentences with real-quantifiers. But we have perfectly good countable, well-founded, constructive models of full second order arithmetic. So why can't physics, for example, simply explicitly embrace one of these as what they are working over and thereby radically simplify their alleged ontological commitments?
I think we should be worried about the suggestion that science commits us to arbitrary reals - or to the full structure of reals - because as I noted in Part 1, I think we have no coherent grasp of that full structure or of arbitrary complex reals. But if claims of science committed us to arbitrary reals, then we would have to understand the mathematical structures to understand the claims of the empirical science.
But as I noted, we have various minimal models - standard, well founded, constructive models of second order arithmentic - indeed of full ZFC. (See here for a super-quick intro to minimal models.)
So why can't the scientist simply explicitly work over one of these models. It would be a model and it thereby satisfies all the mathematical claims she could have any reason to make use of and would not involve at least some of the problematic commitments she is accused of having taken on.
Looking at the ordinal constructions of some such models might even be positively useful. (Yeah, I'm speculating wildly now.) While it seems completely bizarre to think that there are uncountable collections of physically significant elements, the idea that ordinal constructions of functional relations between physically significant elements might reveal something of physical significance doesn't seem bizarre at all - again, to this amateurish dabbler. (My colleague and collaborator Joe Mourad has applied some of our constructions of Sigma-1-2 reals to economics, claiming that the non-local impredicativity that pops up in such sentences can provide a model of the instability of elaborate commodity constructions. He also thinks there are applications to biological systems. I am not even an amateur wrt economics and biology. So you'll have to ask Joe about this.)
So why don't philosophers of science attend to the work stretching from Goedel and late - i.e. post-G incompleteness - Hilbert on to Schoenfield and beyond on constructive models of set theory and mathematics?
One answer to that question would be that the ontological parsimony goals are much stricter than avoiding commitment to uncountable totalities of reals. If one wants to defend fictionalism about math in general, this isn't helping. (fwiw, my own view is that given structuralism as an account of what it is to be committed to the existence of mathematical objects, there is zero hope for fictionalism. When one specifies how the fiction works - actually when one does anything, but that claim will be left for a possible third post in this series - one always commits oneself to a structure of ordinal complexity, that is to integers.)
But then I am inclined to think that this is just one of the contexts in which asking ontological questions about mathematical objects in general is missing something important. Perhaps there are important questions that pertain to all mathematical objects. But there are certainly also new problems introduced by non-constructive, impredicative mathematics. No one can deny that our grasp of the integers is importantly more epistemically secure than our grasp on the full intended model of the reals, for example. And the epistemological problems that arise at these levels of complexity have far more impact on actual mathematical practice than philosophical worries about, say, integers. (Harvey Friedman has noted that core mathematicians - those not working in foundations, but in things like number theory, analysis, geometry, etc. - do not take strictly Delta-1-3 claims seriously. He once went through some enormous stack of journals looking at proved claims and unsolved problems and showed that all are provably equivalent to simpler claims.)
But however that all shakes out, I want to make a minimal claim: the existence of standard constructive models shows that the use of real-quantifiers on its own can't imply commitment to uncountable structures.
Is that to disagree with Quine? He is often presented as saying that we can just read our commitments off the logical grammar of our claims. If we know which claims with quantifiers out front we are making, we know what we are committed to. But the existence of non-isomorphic models shows that to be wrong. Surely Quine knew that. That's the whole point of ontological relativity. But that discussion seems to suppose that there are models, and there is grammar governed by proof theory, and nothing else. (And since one model is as good as another, really all that matters is the proof theory - the rules for the use of the grammar.) But that's the part that I disagree with. There are also ordinal constructions of models. And these are more determinate than talk of arbitary models - because you can actually specify something determinate with such a construction - and yet go beyond more axiomatic proof theory, because you can determinately specify models of non-axiomatizable sets of sentences. (Of course maybe we now need a post on "determinate". Hmmm. That's the problem with blogging about substantive things. Blogs are short. Ideas are big.)