What I talk about when I talk about numbers

Here is a valid argument:

(1) The number of Front national MEPS is worrying. 

(2) The number of Front national MEPS is 24. 

(3) 24 is worrying.

At least it’s valid if you think, as almost all philosophers who think about mathematical language seem to, that (2) refers to a number.  Contrast (2) with

(2*) There are 24 Front national MEPS.

(2*) is a statement about Front national MEPS, but (2) and (2*) are treated as being equivalent; not in the sense that they have the same meaning (one refers only to a political party, the other refers to a number) but in the sense that given (2) we can always infer (2*) and given (2*) we can always infer (2).  We can do this because we accept the abstraction principle:

(*) There are n Fs if and only if the number of Fs is n.

I’m not sure what to make of this.  I used to think that claims like (2*) were true because they predicate a property of something real, whereas claims like (2) were literally false because they make reference to something that doesn’t really exist—a number.  Making inferences using (literally false) claims like (2) was, I thought, fine, because doing so wouldn’t lead us astray with respect to how things stood with what really existed.  Similarly we could accept (*), not as being literally true, but as being “nominalistically adequate”, i.e. unable to lead us astray with respect to how things stand with what really existed.  (Compare: we accept ‘There is a dent in the car’ not because dents really exist or because they are an extra bit of the furniture of reality over and above the car, but because saying this doesn’t lead us astray with respect to the topographical properties of the car.)  But here is a problem with this: (3) is absurd.  A convenient fiction that aids inference-making is one thing; an absurd convenient fiction that aids inference-making is something else.  This is disastrous for the platonist who thinks that numbers really exist.  For the platonist (3) is true.  But it’s also bad news for the fictionalist who accepts that (2) refers (or at least purports to refer) to a number, because although fictionalist take (3) to be false, they’re still left with a problem: (3) isn’t even nominalistically adequate.  (3) can be used to infer falsehoods about the concrete world:

(3) 24 is worrying. 

(4) The number of Tunnock’s Teacakes in a four-pack is 24. 

(5) The number of Tunnock’s Teacakes in a four-pack is worrying.

(5) is about the concrete world and is false.  Maybe the only option is to drop the claim that phrases of the form ‘The number of Fs is n refer to the number n.  In this case the ‘is’ can’t be the ‘is’ of identity; the phrase can’t mean ‘The number of Fs = n’.

Softening the Blow of Truth Fictionalism

In the last post I said that mathematical fictionalism has consequences that sound terrible, but really aren't worrisome at all, when you think about what they actually entail. Something similar could be said about alethic fictionalism, or fictionalism about truth. Consider the truth predicate.  It’s a widely held view that the purpose of having a truth predicate is to allow people to undertake commitments to certain claims without having to explicitly state those claims.  So, if a theory $\Gamma$ entails infinitely many claims that you want to endorse, you can say ‘Everything entailed by $\Gamma$ is true', rather than explicitly state every claim entailed by $\Gamma$, which would be impossible.  So the job of the truth predicate isn’t to ascribe a special property, TRUTH, to things, but to allow us to undertake commitments without having to articulate those commitments explicitly.

Nevertheless, you could consistently hold that, even though what explains why we have a truth predicate has nothing to do with ascribing the property TRUTH to things, the meaning of a predicate is always to ascribe a property to something.  And if, in addition, you held that there is no such property as TRUTH, then you would end up being a fictionalist about truth discourse.  Any claim of the form ' $\phi$ is true’ would be false, since nothing has the property of truth.  But this wouldn’t really matter, since the truth predicate would still allow us to do what it was designed to do.  (It would, however, sound like a really bad result.)

Softening the Blow of Mathematical Fictionalism

Mathematical fictionalists are representationalists about mathematical discourse.  Not (necessarily) in the sense that they think that the meaning of mathematical sentences is to be understood in terms of reference or truth-making relations—mathematical fictionalists might be, and often are, deflationists about truth and reference—but rather in the sense that they take mathematical discourse to describe mathematical objects.  If someone claims there are 88 narcissistic numbers in base ten then the content of this claim has to do with the way it stands with a domain of things.  Mathematical discourse is descriptive rather than, say, expressivist.  Mathematical fictionalists also think that mathematical objects don’t exist, and hence the claims that there are 88 narcissistic numbers in base ten, which mathematicians accept, are, strictly and literally speaking, false.

This is usually enough to put people off mathematical fictionalism; even if the arguments for the position seem well founded enough, the conclusion that mathematical claims are, strictly and literally speaking, false (or trivially true if, e.g., they make universal negative claims such as “there are no positive integers x, y and z such that x3 + y3 = z3”, or form the antecedent of a conditional claim) will be too much to bear.  That modus ponens from fictionalism to the falsehood of mathematical claims will always seem like a modus tollens against fictionalism.

But the function of a discourse—the reason that we go in for this kind of discourse in the first place—needn’t determine the meaning of that discourse.  So, a discourse can be representational, in the above sense, but it might exist in order to serve a purpose other than representing the way things are with the world.  There are many things we can do with language, other than picture the world as being a particular way.

The “blow” of fictionalism isn’t really a blow at all, so long as we think the following things:  Mathematical discourse is representational, but the point or purpose of engaging in mathematics is not to describe or picture how things stand with a realm of mathematical objects.  Moreover, mathematicians have objective standards of rightness and wrongness that determine which mathematical statements are correct or incorrect.  So although mathematical claims may be strictly false in the sense that they do not picture how things stand with a realm of abstract objects, this is wholly orthogonal to the goals of mathematics.  Here’s the litmus test: When you describe fictionalism replace every instance of “is true” with “correctly describes how things stand with a realm of abstract objects” and “is false” with “does not correctly describe how things stand with a realm of abstract objects”.  How bad does fictionalism sound now?

How to Eschew Metaphysics

Here’s how Blackburn describes pragmatism:

You will be a pragmatist about an area of discourse if you pose a Carnapian external question: how does it come about that we go in for this kind of discourse and thought?  What is the explanation of this bit of our language game?  And then you offer an account of what we are up to in going in for this discourse, and the account eschews any use of the referring expressions of the discourse; any appeal to anything that a Quinian would identify as the values of the bound variables if the discourse is regimented; or any semantic or ontological attempt to ‘interpret’ the discourse in a domain, to find referents for its terms, or truth-makers for its sentences … Instead the explanation proceeds by talking in different terms of what is done by so talking.  It offers a revelatory genealogy or anthropology or even a just-so story about how this mode of talking and thinking and practising might come about, given in terms of the functions it serves.  Notice that it does not offer a classical reduction, finding truth-makers in other terms.  It finds whatever plurality of functions it can lay its hands upon. [Simon Blackburn, Expressivism, Pragmatism and Representationalism: 75]

I'm interested in the claim often made by pragmatists, such as Simon Blackburn or Huw Price, that they are eschewing metaphysics, in contrast to platonists, fictionalists, error theorists and the like. Pragmatic accounts of a discourse provide a genealogy, or some consanguineous account, of why it is we go in for this way of talking and, as it may happen, this account may be metaphysically deflationary.  So it may be that the motivation for talking about, say, mathematical objects, does not involve representing how things stand with a domain of mathematical objects.  If there is some such story—if we can account for the uses of mathematical talk, without invoking mathematical objects—then we have an ontologically deflationary pragmatic account of mathematical discourse.

But so far, what’s been said about pragmatic accounts of mathematical discourse is open for the fictionalist to adopt.  The difference between the fictionalist (who is apparently engaged in metaphysics) and the pragmatist (who apparently eschews metaphysics) is that the fictionalist claims that mathematical talk is, strictly speaking, false, whereas the pragmatist does not.

Fictionalists and pragmatists then agree in methodology: provide an account of the usefulness of mathematical (or moral, or possible worlds) discourse that makes the existence of mathematical (or moral, or modal) objects orthogonal to the practice.  Their point of divergence is not methodological or ontological, but semantic: whether one opts for fictionalism or pragmatism depends on what one takes the meaning of existential quantification to be.  Here, the pragmatist reads the pragmatic purpose of quantification over mathematical objects back into the semantics of quantification over mathematical objects, and the fictionalist does not.  A truism: people can engage in ontological disputes.  There is something at stake between someone who claims that the Higgs boson exists and someone who claims that it does not, or between someone who claims that God exists and someone who claims that he does not.  The interlocutors in these debates are in disagreement over what the world is like.  So, sometimes at least, quantificational talk is used to express disagreements about what the world is like.  Ultimately then, the difference between the fictionalist and the pragmatist lies in what they take the meaning of existential quantification to be.  Fictionalists take existential quantification to be univocal: it always expresses claims about what the world is like.  Pragmatists (are committed to) taking existential quantification to be multivocal: within discourses whose purpose is to describe the world existential quantification expresses claims about what the world is like; within discourses whose purpose is not to describe the world, existential quantification does not express claims about what the world is like. (Note that the point of divergence is not, or need not be, over semantic minimalism. The person engaged in metaphysics need not couch what he is doing in terms of finding truth-makers or referents to be relata in substantive relations of truth or reference to given sentences; he can simply couch what she is doing in terms of whether such and such objects exist. Hartry Field is a case in point.)  

The take-away claim: whether one gets to eschew metaphysics depends on whether existential quantification is univocal or multivocal.

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.