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?

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.

Languages Don't Exist

Here's an argument that we ought not to postulate the existence of languages.  The argument isn't formally valid, but it could easily be made formally valid, and I think that it would then be sound:

1. What a philosophy of language needs to account for is communication.  Once we have explained communication there are no further facts about language that need to be explained.
2. In order to account for both communication failure (the many occasions in which we talk past each other) and special cases of communicative success (when the meaning of our utterance is more than, less than, or has no overlap with what we intend to convey—as in, e.g., certain cases of definite description use, and metaphor), we must reject decoding accounts of communication and adopt relevance accounts of communication.
3. Relevance accounts of communication explain communication in terms of ampliative inferences as to a speaker’s communicative intentions.
4. An account of ampliative inference as to a speaker’s communicative intentions need not involve languages, only particular facts about individual speakers.
5. We can explain all the facts about language-use without positing the existence of languages.
6. (In practicing philosophy of language ) we should only postulate those entities required to explain the facts about language.
7. Hence, we should not postulate the existence of languages.

Perhaps some doubt could be cast on 1, but I'm sceptical that any other facts a philosophy of language ought to account for would require postulating the existence of languages.