Is nominalism self-defeating?

Here's an objection to nominalism I've heard a few times.  Sometimes the 'access problem' to abstract objects is motivated by the idea that embodied creatures adapted to a particular environment, such as ourselves, need to interact with the world in order to gain knowledge of what it's like.  If we're to learn something about a given domain of objects, at some point we will require some kind of causal interaction with at least some members of that domain of objects.  Abstract objects, such as mathematical objects, are not like this: there is no method by which we could interact with anything in a domain of abstract objects at any time.  As a result, even if abstract objects exist, there is no means by which we could come to gain knowledge of which abstract objects exist or what properties they have.  This kind of minimal causal condition for knowledge is sometimes called the (or a) "eleatic principle".  But it's sometimes said that this eleatic principle is self-defeating.  Here's Sorin Bangu in his nice book The Applicability of Mathematics in Science (pp.18-9):

My naturalist’s reaction to the reformulated [eleatic principle] challenge is to point out that it is ultimately self-defeating.  That is, the naturalist notes that one cannot even formulate the challenge without actually making appeal to mathematics: one simply can’t grasp what the new naturalized [eleatic principle] actually says unless one understands the physical theories describing the abovementioned types of interactions.  But these theories are, of course, thoroughly mathematical!  So, anyone attempting to advance a challenge of the [eleatic principle] type in naturalistically acceptable terms finds herself engaged in the self-undermining enterprise of rejecting the very (mathematical) terms which allow the (acceptable naturalistic version of the) challenge to be meaningfully formulated in the first place.

This criticism seems wrong to me, on two counts.  Firstly, it ignores responses to the indispensability argument.  The nominalist will need some response to the indispensability argument.  If this response doesn't work then the nominalist is in trouble anyway.  If it does work—whether it involves doing without reference to or quantification over mathematical objects in scientific theories, like Field, Chihara etc., or offering some account of why it's acceptable for the nominalist to continue to refer to or quantify over mathematical objects in scientific theories, like Leng—then it will work here too: that we give mathematical models of how we (concrete) creatures interact with (concrete) parts of the world will pose no special problems.  Secondly, even in lieu of a response to the indispensability argument, the eleatic principle can be used to give a sort of reductio of mathematical platonism: (i) assume mathematical platonism is true, (ii) motivate the eleatic principle, (iii) our own mathematicized theories which describe how we interact with the world show that we cannot have knowledge of mathematical objects. So the assumption we began with is unknowable and rationally self-defeating.

Two Kinds of Indispensability Argument

Continuing on the theme of nominalism and pragmatism…

The Putnam of yore took it that mathematical objects exist and is credited along with Quine as being an early proponent of the indispensability argument.  There are though two very different kinds of indispensability argument that Putnam made.  The first runs like this:

[Q]uantification over mathematical entities is indispensable for science, both formal and physical; therefore we should accept such quantification; but this commits us to accepting the existence of the mathematical entities in question.  This type of argument stems, of course, from Quine, who has for years stressed both the indispensability of quantification over mathematical entities and the intellectual dishonesty of denying the existence of what one daily presupposes. [Philosophy of Logic: 347]

Why be a platonist?  Because, according to the argument, nominalism is inconsistent with physics.  One big problem with the argument is that there is plenty that gets quantified over in the sciences that we don’t take to exist; especially idealised versions of physical systems, the stock examples being frictionless surfaces, continuous fluids and the like.  So nominalism’s being “inconsistent with physics” in this sense isn’t a big deal, since the (clearly true) claims that fluids are not continuous, that there are no frictionless planes etc. are also “inconsistent with physics”.  (Penelope Maddy in Naturalism in Mathematics and Mary Leng in Mathematics and Reality both make this kind of point.)

Putnam also made a very different kind of indispensability argument, often conflated with the first, that goes like this:

I believe that the positive argument for realism has an analogue in the case of mathematical realism.  Here too, I believe, realism is the only philosophy that doesn’t make a success of science a miracle.  [Philosophy of Logic: 73]

There is an important shift from looking flatly to what entities are quantified over in our best scientific theories to looking at what quantification over these entities can be used to achieve.  This pragmatic spin is in fact necessary because quantification over mathematical objects is not indispensable simpliciter (if such a notion even makes sense), but indispensable for certain ends.  We could do without quantification over mathematical objects; we might just also have to do without iPhones, air travel and so on, if we did.  As Sellars famously said (in ‘A Semantical Solution to the Mind-Body Problem’) ‘[c]learly human beings could dispense with all discourse, though only at the expense of having nothing to say’.  So, if the indispensability of quantification over mathematical objects is supposed to be a problem for nominalism, it must be because talk of mathematical objects must be made use of to achieve certain ends; in which case what is at issue are mathematical practices.  The best explanation for the success of mathematical practices must involve the existence of mathematical objects, or so the thought goes.  But here’s the kicker: mathematical objects, because they are acausal, changeless and not subject to any events, cannot be invoked to explain any practices.  So the best explanation of the success of science needn’t invoke mathematical objects.

Fixing Possible Worlds Fictionalism: The Incompleteness Worry

There is a problem in the standard formulation of possible worlds fictionalism, which goes as follows:

(*) Possibly p iff, according to the possible worlds fiction [p is true at some possible world]

Call the possible worlds fiction $\Theta$, then we can say that possibly p iff $\Theta \vDash$ `p is true at some possible world'.  What is $\Theta$?  $\Theta$ is a set of sentences closed under deduction, but what sentences are to be included in $\Theta$?  The problem for the possible worlds fictionalist, in brief, is this: unless she can specify a possible worlds fiction that for every proposition p, $\Theta \vDash$ p or $\neg$p, then bivalence will fail.  If we endorse the biconditional (*), this bivalence will bleed over into claims regarding what is possible.

Contrast this with Lewis extreme modal realism, according to which:

(**) Possibly p iff p is true at some possible world.

One can endorse (**) without having a complete theory of which possible worlds exist, because (**) leaves it up to Nature (as it were) to decide what is possible and what is not.  So that’s the incompleteness worry in brief: without a complete, Final Theory of what is possible and what is not, $\Theta$ will not settle every fact regarding what is true at some possible world, and, via (*) will entail that bivalence fails regarding modal propositions.

I think that there is a simple-minded fix for the possible worlds fictionalist.  The problem can be circumvented if the fictionalist holds instead:

(***) according to the possible worlds fiction [possibly p iff p is true at some possible world].

Here, $\Theta$ simply is this: Possibly p iff p is true at some possible world.  Stating the fiction thus doesn’t require that we know all the modal facts in advance.  The facts about what is possible are what they are (Nature decides), and, according to the fiction, to each of these modal facts corresponds a possible world.  Another, (I think) more perspicuous, way to put this, is that we should treat the claim

(**) Possibly p iff p is true at some possible world.

as being modally adequate; i.e. that (**) gets things right with respect to the modal facts (“modal facts” are not here thought of as facts about possible worlds, but facts about what is possible, necessary, impossible, and so on).  Intuitively, a claim is modally adequate when the modal facts are the way they would have to be for the claim to be true.  (The analogy here is with the empirical adequacy of Bas van Fraassen in The Scientific Image, or the nominalistic adequacy of Mary Leng in Mathematics and Reality.  There are various ways of fleshing out nominalistic adequacy, and modal adequacy, but I won’t do so here.)

Problem: if the fiction operator is placed at the beginning of the sentence, if we treat (**) as merely modally adequate, how can we ever make use of possible world fictions to come to know modal truths?  Won’t it be the case that we will only be able to make claims about what is possible, impossible or necessary according to the possible worlds fiction?  To see that this is not the case, recall that a claim is modally adequate just in case it gets things right regarding the modal facts.  As such, claims that pertain only to the modal facts which are modally adequate are, by definition, true simpliciter.  Derivations can be carried out which quantify over possible worlds.   From modally adequate premises regarding possible worlds, modally adequate conclusions regarding possible worlds can be obtained.  We can then make use of the biconditional (**) to “translate” these into modally adequate modal claims.  But, since modally adequate modal claims are true simpliciter, we can legitimately infer the truth (simpliciter) of modal claims by making use of merely modally adequate claims quantifying over possible worlds.  The trick is to treat (**) as merely modally adequate rather than true simpliciter.  Both fictionalism regarding possible worlds, and bivalence are retained.