One of the debates in philosophy of math that I’m quite interested in is the question of whether mathematical objects actually exist or not. This debate seems to have been one of the most central ones in the field in the last several decades. However, mathematicians tend to dismiss this debate, though they do care about some others, about methods of proof, justification of axioms, the role of explanation, and the like. Many philosophers often feel the same way, both about this ontological debate, and about other debates in analytic metaphysics.
John MacFarlane has pressed me on several occasions with a worry about something like this, and a related point is discussed in the introduction to Hartry Field’s Reason, Mathematics, and Modality (I think). If someone asserts that a very skilled detective lived on Baker Street in London in the 19th century, but then says that this assertion was meant in a merely fictional way, then there is a clear change in my reaction to the assertion. Instead of verifying it by looking at real birth records and legal histories and the like, I investigate fictional works (in this case by Conan Doyle). Instead of basing historical arguments on the facts mentioned, I base arguments aobut the fictional world and the like. However, if a mathematician tells me that every continuous closed curve cuts a plane into two disjoint connected regions, and then a philosopher tells me that this assertion was made merely within the fiction of mathematics, it’s not clear what difference this could make in my relation to the assertion. In either case I would verify it by proofs using the axioms, and make the same physical and mathematical applications of the theorem. My acceptance or non-acceptance of this philosophical thesis will not be manifest in any of my actions (other than my bestowing or withholding the honorific “exists”, or “literally exists” or the like). So something about the debate seems potentially misguided.
Worries like this may be behind Jody Azzouni’s assertions in Deflating Existential Consequence that there is no rational argument in favor of any particular criterion for ontological commitment, though we as a community have chosen to adopt ontological independence as such a criterion. He is here primarily concerned with mathematical entities, but also theoretical entities of other kinds as well. These seem to be precisely the kinds of entities where the above worry is strongest. Once he has adopted this criterion, then claims of mathematical anti-realism can be manifested in the kind of free-wheeling postulation of entities that he claims is characteristic of mathematics (and other “ultrathin posits”, though I think there’s room to contest this claim about mathematics). This postulation is, I think, what he calls “ontological dependence”, and is characteristic of fictional entities and other paradigmatic examples of non-existent posits. If this is right, then realism about mathematics would be manifested by trying to establish what he calls either thick or thin epistemic access to mathematical entities.
This may not be the best way to cash out the distinction between a commitment to the existence and non-existence of entities in reality, but however this distinction is made, I think we can get implications for the fictionalist position. However, if a philosopher manifests her belief in the real existence or non-existence of objects in a certain way, then she should manifest her belief in the fictional existence or non-existence of objects in a similar way. For instance, on Azzouni’s criterion, a fictionalist about mathematical objects should look for fictional epistemic access of either a thick or thin nature. Since this doesn’t seem to be plausible, Azzouni would have to say that even in most reasonable fictions, mathematical objects don’t exist.
However, someone with a more Quinean criterion might be able to take a fictionalist position. If our existence criterion is “playing an explanatory role in our best theory of the world”, then realist truth about mathematics would make verification dependent on applicability in scientific theories. (We don’t obviously seem to do this, which is why Maddy and others reject a Quinean realism about mathematics, but I think that we may have done this for some small number of axioms, so that it’s not obvious whether or not we have manifested a commitment to realist truth in this way.) Fictionalist truth would be manifest in an attempt to show that mathematical entities fictionally explain our observations – and I think this is exactly Hartry Field’s project. This makes sense of the fact that Field seems to turn the Quinean arguments on their head to say that mathematical objects don’t exist actually, but merely fictionally.
On a more Hilbertian criterion, that every postulated set of axioms describes some objects, it seems that there could be no reasonably different fictional manifestation of acceptance of an existence claim. Thus, for someone like Ed Zalta, a worry like John MacFarlane’s would be relevant. But this seems to be ok for him, because he has described his view both as a sort of platonized naturalism (which I take to be realist), in “Naturalized Platonism vs. Platonized Naturalism”, with Bernard Linsky, and also as a sort of nominalism, in “A Nominalist’s Dilemma and its Solution”, with Otávio Bueno.
Thus, the burden is on any such philosopher to show that mathematicians do in fact manifest their acceptance of mathematical statements in the way that the philosopher says they should (whether realist, fictionalist, or other). The difference between Quine and Field is just such a debate, as is the disagreement between Azzouni and the platonist Zalta. However, these two debates are someone orthogonal to one another, as they take acceptance of a statement to be manifested in a different way, so their disagreements may be merely verbal, as someone like MacFarlane might worry. But at any rate, those involved in these debates do seem to be engaged in the project of showing that mathematicians behave the way their theories predict, so MacFarlane’s worry doesn’t seem to damage any of these projects directly.