The Quine-Putnam Indispensability argument is the argument for treating mathematical entities on a par with other theoretical entities of our best scientific theories. This argument is usually taken to be an argument for mathematical realism. In this chapter I will argue that the proper way to understand this argument is as putting pressure on the viability of the marriage of scientific realism and mathematical nominalism. Although such a marriage is a popular option amongst philosophers of science and mathematics, in light (...) of the indispensability argument, the marriage is seen to be very unstable. Unless one is careful about how the Quine-Putnam argument is disarmed, one can be forced to either mathematical realism or, alternatively, scientific instrumentalism. I will explore the various options: (i) finding a way to reconcile the two partners in the marriage by disarming the indispensability argument (Jody Azzouni [2], Hartry Field [13, 14], Alan Musgrave [18, 19], David Papineau [21]); (ii) embracing mathematical realism (W.V.O. Quine [23], Michael Resnik [25], J.J.C. Smart [27]); and (iii) embracing some form of scientific instrumentalism (Ot´ avio Bueno [7, 8], Bas van Fraassen [30]). Elsewhere [11], I have argued for option (ii) and I won’t repeat those arguments here. Instead, I will consider the difficulties for each of the three options just mentioned, with special attention to option (i). In relation to the latter, I will discuss an argument due to Alan Musgrave [19] for why option (i) is a plausible and promising approach. From the discussion of Musgrave’s argument, it will emerge that the issue of holist versus separatist theories of confirmation plays a curious role in the realism–antirealism debate in the philosophy of mathematics. I will argue that if you take confirmation to be an holistic matter—it’s whole theories (or significant parts thereof) that are confirmed in any experiment—then there’s an inclination to opt for (ii) in order to resolve the marital tension outlined above.. (shrink)

This book not only outlines the indispensability argument in considerable detail but also defends it against various challenges.

In this book, Balaguer demonstrates that there are no good arguments for or against mathematical platonism. He does this by establishing that both platonism and anti-platonism are defensible views. Introducing a form of platonism ("full-blooded platonism") that solves all problems traditionally associated with the view, he proceeds to defend anti-platonism (in particular, mathematical fictionalism) against various attacks, most notably the Quine-Putnam indispensability attack. He concludes by arguing that it is not simply that we do not currently have any good argument (...) for or against platonism, but that we could never have such an argument and, indeed, that there is no fact of the matter as to whether platonism is correct. (shrink)

Many explanations in science make use of mathematics. But are there cases where the mathematical component of a scientific explanation is explanatory in its own right? This issue of mathematical explanations in science has been for the most part neglected. I argue that there are genuine mathematical explanations in science, and present in some detail an example of such an explanation, taken from evolutionary biology, involving periodical cicadas. I also indicate how the answer to my title question impacts on broader (...) issues in the philosophy of mathematics; in particular it may help platonists respond to a recent challenge by Joseph Melia concerning the force of the Indispensability Argument. (shrink)

According to the indispensability argument, the fact that we quantify over numbers, sets and functions in our best scientific theories gives us reason for believing that such objects exist. I examine a strategy to dispense with such quantification by simply replacing any given platonistic theory by the set of sentences in the nominalist vocabulary it logically entails. I argue that, as a strategy, this response fails: for there is no guarantee that the nominalist world that go beyond the set of (...) sentences in the nominalist language such theories entail. However, I argue that what such theories show is that mathematics can enable us to express possibilities about the concrete world that may not be expressible in nominalistically acceptable language. While I grant that this may make quantification over abstracta indispensable, I deny that such indispensability is a reason for accepting them into our ontology. I urge that the nominalist should be allowed to quantify over abstracta whilst denying their existence and I explain how this apparently contradictory practice (a practice I call 'weaseling') is in fact coherent, unproblematic and rational. Finally, I examine the view that platonistic theories are simpler or more attractive than their nominalistic reformulations, and thus that abstract ought to be accepted into our ontology for the same sorts of reasons as other theoretical objects. I argue that, at least in the case of numbers, functions and sets, such arguments misunderstand the kind of simplicity and attractiveness we seek. (shrink)

Mathematics has a great variety ofapplications in the physical sciences.This simple, undeniable fact, however,gives rise to an interestingphilosophical problem:why should physical scientistsfind that they are unable to evenstate their theories without theresources of abstract mathematicaltheories? Moreover, theformulation of physical theories inthe language of mathematicsoften leads to new physical predictionswhich were quite unexpected onpurely physical grounds. It is thought by somethat the puzzles the applications of mathematicspresent are artefacts of out-dated philosophical theories about thenature of mathematics. In this paper I argue (...) that this is not so.I outline two contemporary philosophical accounts of mathematics thatpay a great deal of attention to the applicability of mathematics and showthat even these leave a large part of the puzzles in question unexplained. (shrink)

This paper argues that it is scientific realists who should be most concerned about the issue of Platonism and anti-Platonism in mathematics. If one is merely interested in accounting for the practice of pure mathematics, it is unlikely that a story about the ontology of mathematical theories will be essential to such an account. The question of mathematical ontology comes to the fore, however, once one considers our scientific theories. Given that those theories include amongst their laws assertions that imply (...) the existence of mathematical objects, scientific realism, when construed as a claim about the truth or approximate truth of our scientific theories, implies mathematical Platonism. However, a standard argument for scientific realism, the 'no miracles' argument, falls short of establishing mathematical Platonism. As a result, this argument cannot establish scientific realism as it is usually defined, but only some weaker position. Scientific 'realists' should therefore either redefine their position as a claim about the existence of unobservable physical objects, or alternatively look for an argument for their position that does establish mathematical Platonism. (shrink)

This book does three main things. First, it defends mathematical platonism against the main objections to that view (most notably, the epistemological objection and the multiple-reductions objection). Second, it defends anti-platonism (in particular, fictionalism) against the main objections to that view (most notably, the Quine-Putnam indispensability objection and the objection from objectivity). Third, it argues that there is no fact of the matter whether abstract mathematical objects exist and, hence, no fact of the matter whether platonism or anti-platonism is true.

Achieving understanding of nature is one of the aims of science. In this paper we offer an analysis of the nature of scientific understanding that accords with actual scientific practice and accommodates the historical diversity of conceptions of understanding. Its core idea is a general criterion for the intelligibility of scientific theories that is essentially contextual: which theories conform to this criterion depends on contextual factors, and can change in the course of time. Our analysis provides a general account of (...) how understanding is provided by scientific explanations of diverse types. In this way, it reconciles conflicting views of explanatory understanding, such as the causal-mechanical and the unificationist conceptions. (shrink)

This essay aims to disentangle various types of anti-realism, and to disarm the considerations that are deployed to support them. I distinguish empiricist versions of anti-realism from constructivist versions, and, within each of these, semantic arguments from epistemological arguments. The centerpiece of my defense of a modest version of realism - real realism - is the thought that there are resources within our ordinary ways of talking about and knowing about everyday objects that enable us to extend our claims to (...) unobservable entities. This strategy, the Galilean strategy, is explained using the historical example of the telescope. (shrink)

There are almost as many versions of realism as there are antirealists, each ready to supply a preferred characterization before undertaking demolition. Even in the case of scientific realism, my topic here, I recognize two major antirealist themes.