 A number of nominalists have been persuaded by Benacerraf’s (1973) epistemological challenge about reference to abstract objects and concluded that sentences with terms making apparent reference to them—such as mathematical statements—are either false or lack a truth value. They argue that those sentences must be paraphrasable without vocabulary that commits one to any sort of abstract entity. These proposals sometimes suggest that statements about abstract objects are merely instrumental; they serve only to help us establish conclusions about concrete objects. Field’s fictionalism (1980, 1989) has been influential in this regard. Field reconstructed Newtonian physics using second-order logic and quantification over (concrete) regions of space-time. A completely different tactic for avoiding the commitment to abstract, mathematical objects is put forward in Putnam (1967) and Hellman (1989), who separately reconstructed various mathematical theories in second-order modal logic. On their view, abstract objects aren’t in the range of the existential quantifier at the actual world (hence, we can’t say that they exist), but they do occur in the range of the quantifier at other possible worlds, where the axioms of the mathematical theory in question are true.