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?
