This chapter examines the status of inference to the best explanation in naturalistic metaphysics. The methodology of inference to the best explanation in metaphysics is studied from the perspective of contemporary views on scientific explanation and explanatory inferences in the history and philosophy of science. This reveals serious shortcomings in prevalent attempts to vindicate metaphysical "explanationism" by reference to similarities between science and naturalistic metaphysics. This critique is brought out by considering a common gambit of methodological unity: (1) Both metaphysics (...) and science employ inference to the best explanation. (2) One has no reason to think that if explanationism is truth-conducive in science, it is not so in metaphysics. (3) One has a positive reason to think that if explanationism is truth-conducive in science, it is also so in metaphysics. (shrink)

This article elaborates the epistemic indispensability argument, which fully embraces the epistemic contribution of mathematics to science, but rejects the contention that such a contribution is a reason for granting reality to mathematicalia. Section 1 introduces the distinction between ontological and epistemic readings of the indispensability argument. Section 2 outlines some of the main flaws of the first premise of the ontological reading. Section 3 advances the epistemic indispensability argument in view of both applied and pure mathematics. And Sect. 4 (...) makes a case for the epistemic approach, which firstly calls into question the appeal to inference to the best explanation in the defense of the indispensability claim; secondly, distinguishes between mathematical and physical posits; and thirdly, argues that even though some may think that inference to the best explanation works in the postulation of physical posits, no similar considerations are available for postulating mathematicalia. (shrink)

According to the indispensability argument, scientific realists ought to believe in the existence of mathematical entities, due to their indispensable role in theorising. Arguably the crucial sense of indispensability can be understood in terms of the contribution that mathematics sometimes makes to the super-empirical virtues of a theory. Moreover, the way in which the scientific realist values such virtues, in general, and draws on explanatory virtues, in particular, ought to make the realist ontologically committed to abstracta. This paper shows that (...) this version of the indispensability argument glosses over crucial detail about how the scientific realist attempts to generate justificatory commitment to unobservables. The kind of role that the Platonist attributes to mathematics in scientific reasoning is compatible with nominalism, as far as scientific realist arguments are concerned. (shrink)

Indispensablists argue that when our belief system conflicts with our experiences, we can negate a mathematical belief but we do not because if we do, we would have to make an excessive revision of our belief system. Thus, we retain a mathematical belief not because we have good evidence for it but because it is convenient to do so. I call this view ‘ mathematical convenientism.’ I argue that mathematical convenientism commits the consequential fallacy and that it demolishes the Quine-Putnam (...) indispensability argument and Baker’s enhanced indispensability argument. (shrink)

In this paper I present an argument for belief in inconsistent objects. The argument relies on a particular, plausible version of scientific realism, and the fact that often our best scientific theories are inconsistent. It is not clear what to make of this argument. Is it a reductio of the version of scientific realism under consideration? If it is, what are the alternatives? Should we just accept the conclusion? I will argue (rather tentatively and suitably qualified) for a positive answer (...) to the last question: there are times when it is legitimate to believe in inconsistent objects. (shrink)

This paper defends the Quine-Putnam mathematical indispensability argument against two objections raised by Penelope Maddy. The objections concern scientific practices regarding the development of the atomic theory and the role of applied mathematics in the continuum and infinity. I present two alternative accounts by Stephen Brush and Alan Chalmers on the atomic theory. I argue that these two theories are consistent with Quine’s theory of scientific confirmation. I advance some novel versions of the indispensability argument. I argue that these new (...) versions accommodate Maddy’s history of the atomic theory. Counter-examples are provided regarding the role of the mathematical continuum and mathematical infinity in science. (shrink)

The traditional formulation of the indispensability argument for the existence of mathematical entities (IA) has been criticised due to its reliance on confirmational holism. Recently a formulation of IA that works without appeal to confirmational holism has been defended. This recent formulation is meant to be superior to the traditional formulation in virtue of it not being subject to the kind of criticism that pertains to confirmational holism. I shall argue that a proponent of the version of IA that works (...) without appeal to confirmational holism will struggle to answer a challenge readily answered by proponents of a version of IA that does appeal to confirmational holism. This challenge is to explain why mathematics applied in falsified scientific theories is not considered to be falsified along with the rest of the theory. In cases where mathematics seemingly ought to be falsified it is saved from falsification, by a so called 'Euclidean rescue'. I consider a range of possible answers to this challenge and conclude that each answer fails. (shrink)

An emphasis on explanatory contribution is central to a recent formulation of the indispensability argument for mathematical realism. Because scientific realism is argued for by means of inference to the best explanation, it has been further argued that being a scientific realist entails a commitment to IA and thus to mathematical realism. It has, however, gone largely unnoticed that the way that IBE is argued to be truth conducive involves citing successful applications of IBE and tracing this success over time. (...) This in turn involves identifying those constituents of scientific theories that are responsible for their predictive success and showing that these constituents are retained across theory change in science. I argue that even if mathematics can be shown to feature in best explanations, the role of mathematics in scientific theories does not satisfy the condition that mathematics is always retained across theory change. According to a scientific realist, this condition needs to be met for making ontological claims on the basis of explanatory contribution. Thus scientific realists are not committed to mathematical realism on the basis of this recent formulation of IA. (shrink)

When the indispensability argument for mathematical entities (IA) is spelled out, it would appear confirmational holism is needed for the argument to work. It has been argued that confirmational holism is a dispensable premise in the argument if a construal of naturalism, according to which it is denied that we can take different epistemic attitudes towards different parts of our scientific theories, is adopted. I argue that the suggested variety of naturalism will only appeal to a limited number of philosophers. (...) I then suggest that if we allow for some degree of separation between different component parts of theories, IA can be formulated as an argument aimed at more than a limited number of philosophers, but in implementing this strategy the notion of indispensability needs spelling out. The best way of spelling out indispensability is in terms of theory contribution, but doing so requires adopting inference to the best explanation (IBE). IBE is however sufficient for establishing the conclusion that IA is supposed to establish. Thus, IA is a redundant argument. (shrink)

Indispensability arguments for mathematical realism are commonly traced back to Quine. We identify two different Quinean strands in the interpretation of IA, what we label the ‘logical point of view’ and the ‘theory-contribution’ point of view. Focusing on each of the latter, we offer two minimal versions of IA. These both dispense with a number of theoretical assumptions commonly thought to be relevant to IA. We then show that the attribution of both minimal arguments to Quine is controversial, and stress (...) the extent to which this is so in both cases, in order to attain a better appreciation of the Quinean heritage of IA. (shrink)

It is claimed that the indispensability argument for the existence of mathematical entities (IA) works in a way that allows a proponent of mathematical realism to remain agnostic with regard to how we establish that mathematical entities exist. This is supposed to be possible by virtue of the appeal to confirmational holism that enters into the formulation of IA. Holism about confirmation is supposed to be motivated in analogy with holism about falsification. I present an account of how holism about (...) falsification is supposed to be motivated. I argue that the argument for holism about falsification is in tension with how we think about confirmation and with two principles suggested by Quine for construing a plausible variety of holism. Finally, I show that one of Quine's principles does not allow a proponent of mathematical realism to remain agnostic with regard to how we establish that mathematical entities exist. (shrink)

Proponents of the explanatory indispensability argument for mathematical platonism maintain that claims about mathematical entities play an essential explanatory role in some of our best scientific explanations. They infer that the existence of mathematical entities is supported by way of inference to the best explanation from empirical phenomena and therefore that there are the same sort of empirical grounds for believing in mathematical entities as there are for believing in concrete unobservables such as quarks. I object that this inference depends (...) on a false view of how abductive considerations mediate the transfer of empirical support. More specifically, I argue that even if inference to the best explanation is cogent, and claims about mathematical entities play an essential explanatory role in some of our best scientific explanations, it doesn’t follow that the empirical phenomena that license those explanations also provide empirical support for the claim that mathematical entities exist. (shrink)

Easy-road mathematical fictionalists grant for the sake of argument that quantification over mathematical entities is indispensable to some of our best scientific theories and explanations. Even so they maintain we can accept those theories and explanations, without believing their mathematical components, provided we believe the concrete world is intrinsically as it needs to be for those components to be true. Those I refer to as “mathematical surrealists” by contrast appeal to facts about the intrinsic character of the concrete world, not (...) to explain why our best mathematically imbued scientific theories and explanations are acceptable in spite of having false components, but in order to replace those theories and explanations with parasitic, nominalistically acceptable alternatives. I argue that easy-road fictionalism is viable only if mathematical surrealism is and that the latter constitutes a superior nominalist strategy. Two advantages of mathematical surrealism are that it neither begs the question concerning the explanatory role of mathematics in science nor requires rejecting the cogency of inference to the best explanation. (shrink)

Some proponents of the indispensability argument for mathematical realism maintain that the empirical evidence that confirms our best scientific theories and explanations also confirms their pure mathematical components. I show that the falsity of this view follows from three highly plausible theses, two of which concern the nature of mathematical application and the other the nature of empirical confirmation. The first is that the background mathematical theories suitable for use in science are conservative in the sense outlined by Hartry Field. (...) The second is that the empirical relevance of mathematical statements suitable for use in science is mediated by their non-mathematical consequences. The third is that statements receive additional empirical confirmation only by way of generating additional empirical expectations. Since each of these is a thesis we have good reason to endorse, my argument poses a challenge to anyone who argues that science affords empirical grounds for mathematical realism. (shrink)

This paper argues that scientific realism commits us to a metaphysical determination relation between the mathematical entities that are indispensible to scientific explanation and the modal structure of the empirical phenomena those entities explain. The argument presupposes that scientific realism commits us to the indispensability argument. The viewpresented here is that the indispensability of mathematics commits us not only to the existence of mathematical structures and entities but to a metaphysical determination relation between those entities and the modal structure of (...) the physical world. The no-miracles argument is the primary motivation for scientific realism. It is a presupposition of this argument that unobservable entities are explanatory only when they determine the empirical phenomena they explain. I argue that mathematical entities should also be seen as explanatory only when they determine the empirical facts they explain, namely, the modal structure of the physical world. Thus, scientific realism commits us to a metaphysical determination relation between mathematics and physical modality that has not been previously recognized. The requirement to account for the metaphysical dependence of modal physical structure on mathematics limits the class of acceptable solutions to the applicability problem that are available to the scientific realist. (shrink)

The indispensability argument seeks to establish the existence of mathematical objects. The success of the indispensability argument turns on finding cases of genuine extra- mathematical explanation. In this paper, I identify a new case of extra- mathematical explanation, involving the search patterns of fully-aquatic marine predators. I go on to use this case to predict the prevalence of extra- mathematical explanation in science.

When considering mathematical realism, some scientific realists reject it, and express sympathy for the opposite view, mathematical nominalism; moreover, many justify this option by invoking the causal inertness of mathematical objects. The main aim of this note is to show that the scientific realists’ endorsement of this causal mathematical nominalism is in tension with another position some of them also accept, the doctrine of methodological naturalism. By highlighting this conflict, I intend to tip the balance in favor of a rival (...) of mathematical nominalism, the mathematical realist position supported by the ‘Indispensability Argument’ – but I do this indirectly, by showing that the road toward it is not blocked by considerations from causation. (shrink)

The main purpose of this paper is to refute the metaphysicians ‘methodological continuation’ argument supporting epistemic realism in metaphysics. This argument aims to show that scientific realists have to accept that metaphysics is as rationally justified as science given that they both employ inference to the best explanation, i.e. that metaphysics and science are methodologically continuous. I argue that the reasons given by scientific realists as to why inference to the best explanation is reliable in science do not constitute a (...) reason to believe that it is reliable in metaphysics. The justification of IBE in science and the justification of IBE in metaphysics are two distinct issues with only superficial similarities, and one cannot rely on one for the other. This becomes especially clear when one analyses the debate about the legitimacy of IBE that has taken place between realists and empiricists. The metaphysician seeking to piggyback on the realist defense of IBE in science by the methodological continuation argument presupposes that the defense is straightforwardly applicable to metaphysics. I will argue that it is, in fact, not. The favoured defenses of IBE in scientific realism make extensive use of empirical considerations, predictive power and inductive evidence, all of which are paradigmatically absent in the metaphysical context. Furthermore, I argue that the metaphysician, even if the realist would concede to the methodological continuation argument, fails to offer any agreed upon conclusions resulting from its application in metaphysics. As a result, the scientific realist is not committed to believing that there is metaphysical knowledge. (shrink)

Saatsi’s minimal realism holds that science makes theoretical progress. It is designed to get around the pessimistic induction, to fall between scientific realism and instrumentalism, and to explain the success of scientific theories. I raise the following two objections to it. First, it is not clear whether minimal realism lies between realism and instrumentalism, given that minimal realism does not entail instrumentalism. Second, it is not clear whether minimal realism can explain the success of scientific theories, given that it is (...) doubtful that theoretical progress makes success likely. In addition to raising these two objections, I develop and criticize a new position that truly falls between realism and instrumentalism. (shrink)