On the indispensability of theoretical terms and entities

Some realists claim that theoretical entities like numbers and electrons are indispensable for describing the empirical world. Motivated by the meta-ontology of Quine, I take this claim to imply that, for some first-order theory T and formula δ ( x ) such that T ⊢ ∃ x δ ∧ ∃ x ¬ δ , where δ ( x ) is intended to apply to all and only empirical entities, there is no first-order theory T ′ such that (a) T and T ′ describe the δ :s in the same way, (b) T ′ ⊢ ∀ x δ , and (c) T ′ is at least as attractive as T in terms of other theoretical virtues. In an attempt to refute the realist claim, I try to solve the general problem of nominalizing T (with respect to δ ), namely to find a theory T ′ satisfying conditions (a)–(c) under various precisifications thereof. In particular, I note that condition (a) can be understood either in terms of syntactic or semantic equivalence, where the latter is strictly stronger than the former. The results are somewhat mixed. On the positive side, even under the stronger precisification of (a), I establish that (1) if the vocabulary of T is finite, a nominalizing theory can always be found that is recursive if T is, and (2) if T postulates infinitely many δ :s, a nominalizing theory can always be found that is no more computationally complex than T . On the negative side, even under the weaker precisification of (a), I establish that (3) certain finite theories cannot be nominalized by a finite theory.