The norms of mathematical discourse and inquiry

David Lewis once (influentially) commented that it would be ludicrous to expect mathematicians to change their ways on the basis of philosophical arguments that mathematical objects don’t exist. Why he thought mathematical practices would have to be emended in the light of ontological facts about the existence of mathematical objects, I’m not sure.

Here’s a thought experiment to make explicit your own implicit commitments about this. Imagine that, instead of a philosopher, an infallible oracle told the world that mathematical objects don’t exist. Would mathematics professors be obliged to hand in their resignations? Would their discipline have been exposed as a sham?

I think the answer to these questions is a, very obvious, “no”, and I suspect that almost everyone would agree. But notice what that means. If we don’t accept that mathematical practices ought to change in light of word from an infallible oracle that mathematical objects don’t exist, then we must also accept that the norms governing mathematical discourse are not representational, in the robust sense of that word as pertaining to mapping, tracking or picturing how things stand with a domain of mathematical objects. The standards of correctness and incorrectness in mathematics do not derive from mathematical objects, but from standards internal to the game (or perhaps “game”) of mathematics itself.

Call this view normative nominalism. But if one is committed to normative nominalism (as a “no” answer to the above questions would reveal), then what could possibly be the motivation for platonism?

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.