In ‘The varieties of indispensability arguments’ Marco Panza and Andrea Sereni argue that, for any clear notion of indispensability, either there is no conclusive argument for the thesis that mathematics is indispensable to science, or the notion of indispensability at hand does not support mathematical realism. In this paper, I shall not object to this main thesis directly. I shall instead try to assess in a naturalistic spirit a family of objections the authors make along the way to the use (...) of indispensability premises in indispensability arguments. (shrink)

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 ⊢ ∃xδ ∧ ∃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′ ⊢ ∀xδ, 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. (shrink)

In the discussion about Yablo’s Paradox, a debated topic is the status of infinitary proofs. It is usually considered that, although a realist could (with some effort) accept them, an anti-realist could not do it at all. In this paper I will argue that there are plausible reasons for an anti-realist to accept infinitary proofs and rules of inference. En la discusión sobre la Paradoja de Yablo, un tópico debatido es el estatus de las pruebas infinitarias. Se suele considerar que, (...) aunque un realista podría (con cierto esfuerzo) aceptarlas, un anti-realista no podría hacerlo en absoluto. En este artículo, argumento que hay razones plausibles para que un anti-realista acepte pruebas y reglas de inferencia infinitarias. (shrink)

In the philosophy of mathematics, indispensability arguments aim to show that we are justified in believing that abstract mathematical objects exist. I wish to defend a particular objection to such arguments that has become increasingly popular recently. It is called instrumental nominalism. I consider the recent versions of this view and conclude that it has yet to be given an adequate formulation. I provide such a formulation and show that it can be used to answer the indispensability arguments. -/- There (...) are two main indispensability arguments in the literature, though one has received nearly all of the attention. They correspond to two ways in which we use mathematics in science and in everyday life. We use mathematical language to help us describe non-mathematical reality; and we use mathematical reasoning to help us perform inferences concerning non-mathematical reality using only a feasible amount of cognitive power. The former use is the starting point of the Quine-Putnam indispensability argument ([Quine, 1980a], [Quine, 1980b], [Quine, 1981a], [Quine, 1981b], [Putnam, 1979a], [Putnam, 1979b]); the latter provides the basis for Ketland’s more recent argument ([Ketland, 2005]). I begin by considering the Quine-Putnam argument and introduce instrumental nominalism to defuse it. Then I show that Ketland’s argument can be defused in a similar way. (shrink)