Ontological realism and sentential form

The standard argument for the existence of distinctively mathematical objects like numbers has two main premises: (i) some mathematical claims are true, and (ii) the truth of those claims requires the existence of distinctively mathematical objects. Most nominalists deny (i). Those who deny (ii) typically reject Quine’s criterion of ontological commitment. I target a different assumption in a standard type of semantic argument for (ii). Benacerraf’s semantic argument, for example, relies on the claim that two sentences, one about numbers and the other about cities, have the same grammatical form. He makes this claim on the grounds that the two sentences are superficially similar. I argue that these grounds are not sufficient. Other sentences with the same superficial form appear to have different grammatical forms. I offer two plausible interpretations of Benacerraf’s number sentence that make use of plural quantification. These interpretations appear not to incur ontological commitments to distinctively mathematical objects, even assuming Quine’s criterion. Such interpretations open a new, plural strategy for the mathematical nominalist.