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.

Kit Fine; XIV*—Ontological Dependence, Proceedings of the Aristotelian Society, Volume 95, Issue 1, 1 June 1995, Pages 269–290, https://doi.org/10.1093/aristote.

Hartry Field has shown us a way to be nominalists: we must purge our scientific theories of quantification over abstracta and we must prove the appropriate conservativeness results. This is not a path for the faint hearted. Indeed, the substantial technical difficulties facing Field's project have led some to explore other, easier options. Recently, Jody Azzouni, Joseph Melia, and Stephen Yablo have argued that it is a mistake to read our ontological commitments simply from what the quantifiers of our best (...) scientific theories range over. In this paper, I argue that all three arguments fail and they fail for much the same reason; would-be nominalists are thus left facing Field's hard road. (shrink)

‘Grounding and the indispensability argument’ presents a number of ways in which nominalists can use the notion of grounding to rebut the indispensability argument for the existence of mathematical objects. I will begin by considering the strategy that puts grounding to the service of easy-road nominalists. I will give some support to this strategy by addressing a worry some may have about it. I will then consider a problem for the fast-lane strategy and a problem for easy-road nominalists willing to (...) accept Liggins’ grounding strategy. Both are related to the problem of formulating nominalistic explanations at the right level of generality. I will then consider a problem that Liggins only hints at. This problem has to do with mathematics’ function of providing the sort of covering generalizations we need in scientific explanations. (shrink)

Some philosophers have recently suggested that the reason mathematics is useful in science is that it expands our expressive capacities. Of these philosophers, only Stephen Yablo has put forward a detailed account of how mathematics brings this advantage. In this article, I set out Yablo’s view and argue that it is implausible. Then, I introduce a simpler account and show it is a serious rival to Yablo’s. 1 Introduction2 Yablo’s Expressionism3 Psychological Objections to Yablo’s Expressionism4 Introducing Belief Expressionism5 Objections and (...) Replies5.1 Yablo’s likely response5.2 Charity6 Conclusion. (shrink)

General discussion of grounding, including its formal features, relations to other notions, and applications.

There is currently an explosion of interest in grounding. In this article we provide an overview of the debate so far. We begin by introducing the concept of grounding, before discussing several kinds of scepticism about the topic. We then identify a range of central questions in the theory of grounding and discuss competing answers to them that have emerged in the debate. We close by raising some questions that have been relatively neglected but which warrant further attention.

This book offers a defence of mathematical fictionalism, according to which we have no reason to believe that there are any mathematical objects. Perhaps the most pressing challenge to mathematical fictionalism is the indispensability argument for the truth of our mathematical theories (and therefore for the existence of the mathematical objects posited by those theories). According to this argument, if we have reason to believe anything, we have reason to believe that the claims of our best empirical theories are (at (...) least approximately) true. But since claims whose truth would require the existence of mathematical objects are indispensable in formulating our best empirical theories, it follows that we have good reason to believe in the mathematical objects posited by those mathematical theories used in empirical science, and therefore to believe that the mathematical theories utilized in empirical science are true. Previous responses to the indispensability argument have focused on arguing that mathematical assumptions can be dispensed with in formulating our empirical theories. This book, by contrast, offers an account of the role of mathematics in empirical science according to which the successful use of mathematics in formulating our empirical theories need not rely on the truth of the mathematics utilized. (shrink)

A main thread of the debate over mathematical realism has come down to whether mathematics does explanatory work of its own in some of our best scientific explanations of empirical facts. Realists argue that it does; anti-realists argue that it doesn't. Part of this debate depends on how mathematics might be able to do explanatory work in an explanation. Everyone agrees that it's not enough that there merely be some mathematics in the explanation. Anti-realists claim there is nothing mathematics can (...) do to make an explanation mathematical ; realists think something can be done, but they are not clear about what that something is. I argue that many of the examples of mathematical explanations of empirical facts in the literature can be accounted for in terms of Jackson and Pettit's [1990] notion of program explanation, and that mathematical realists can use the notion of program explanation to support their realism. This is exactly what has happened in a recent thread of the debate over moral realism. I explain how the two debates are analogous and how moves that have been made in the moral realism debate can be made in the mathematical realism debate. However, I conclude that one can be a mathematical realist without having to be a moral realist. (shrink)

I defend Joseph Melia’s nominalist account of mathematics from an objection raised by Mark Colyvan.

The indispensability argument is a method for showing that abstract mathematical objects exist. Various versions of this argument have been proposed. Lately, commentators seem to have agreed that a holistic indispensability argument will not work, and that an explanatory indispensability argument is the best candidate. In this paper I argue that the dominant reasons for rejecting the holistic indispensability argument are mistaken. This is largely due to an overestimation of the consequences that follow from evidential holism. Nevertheless, the holistic indispensability (...) argument should be rejected, but for a different reason —in order that an indispensability argument relying on holism can work, it must invoke an unmotivated version of evidential holism. Such an argument will be unsound. Correcting the argument with a proper construal of evidential holism means that it can no longer deliver mathematical Platonism as a conclusion: such an argument for Platonism will be invalid. I then show how the reasons for rejecting the holistic indispensability argument importantly constrain what kind of account of explanation will be permissible in explanatory versions. (shrink)

Science Without Numbers caused a stir in 1980, with its bold nominalist approach to the philosophy of mathematics and science. It has been unavailable for twenty years and is now reissued in a revised edition with a substantial new preface presenting the author's current views and responses to the issues raised in subsequent debate.

Stephen Schiffer presents a groundbreaking account of meaning and belief, and shows how it can illuminate a range of crucial problems regarding language, mind, knowledge, and ontology. He introduces the new doctrine of 'pleonastic propositions' to explain what the things we mean and believe are. He discusses the relation between semantic and psychological facts, on the one hand, and physical facts, on the other; vagueness and indeterminacy; moral truth; conditionals; and the role of propositional content in information acquisition and explanation. (...) This radical new treatment of meaning will command the attention of everyone who works on fundamental questions about language, and will attract much interest from other areas of philosophy. (shrink)

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)

We defend Joseph Melia's thesis that the role of mathematics in scientific theory is to 'index' quantities, and that even if mathematics is indispensable to scientific explanations of concrete phenomena, it does not explain any of those phenomena. This thesis is defended against objections by Mark Colyvan and Alan Baker.

Mathematical fictionalism (or as I'll call it, fictionalism) is best thought of as a reaction to mathematical platonism. Platonism is the view that (a) there exist abstract mathematical objects (i.e., nonspatiotemporal mathematical objects), and (b) our mathematical sentences and theories provide true descriptions of such objects. So, for instance, on the platonist view, the sentence ‘3 is prime’ provides a straightforward description of a certain object—namely, the number 3—in much the same way that the sentence ‘Mars is red’ provides a (...) description of Mars. But whereas Mars is a physical object, the number 3 is (according to platonism) an abstract object. And abstract objects, platonists tell us, are wholly nonphysical, nonmental, nonspatial, nontemporal, and noncausal. Thus, on this view, the number 3 exists independently of us and our thinking, but it does not exist in space or time, it is not a physical or mental object, and it does not enter into causal relations with other objects. This view has been endorsed by Plato, Frege (1884, 1893-1903, 1919), Gödel (1964), and in some of their writings, Russell (1912) and Quine (1948, 1951), not to mention numerous more recent philosophers of mathematics, e.g., Putnam (1971), Parsons (1971), Steiner (1975), Resnik (1997), Shapiro (1997), Hale (1987), Wright (1983), Katz (1998), Zalta (1988), and Colyvan (2001). (shrink)

David Malament argued that Hartry Field's nominalisation program is unlikely to be able to deal with non-space-time theories such as phase-space theories. We give a specific example of such a phase-space theory and argue that this presentation of the theory delivers explanations that are not available in the classical presentation of the theory. This suggests that even if phase-space theories can be nominalised, the resulting theory will not have the explanatory power of the original. Phase-space theories thus raise problems for (...) nominalists that go beyond Malament's initial concerns. Thanks to Mark Steiner, Jens Christian Bjerring, Ben Fraser, John Mathewson, and two anonymous referees for helpful comments on an earlier draft of this paper. CiteULike Connotea Del.icio.us What's this? (shrink)

A formal theory of quantity T Q is presented which is realist, Platonist, and syntactically second-order (while logically elementary), in contrast with the existing formal theories of quantity developed within the theory of measurement, which are empiricist, nominalist, and syntactically first-order (while logically non-elementary). T Q is shown to be formally and empirically adequate as a theory of quantity, and is argued to be scientifically superior to the existing first-order theories of quantity in that it does not depend upon empirically (...) unsupported assumptions concerning existence of physical objects (e.g. that any two actual objects have an actual sum). The theory T Q supports and illustrates a form of naturalistic Platonism, for which claims concerning the existence and properties of universals form part of natural science, and the distinction between accidental generalizations and laws of nature has a basis in the second-order structure of the world. (shrink)

(1) The average Mum has 2.4 children. (2) The number of Argle’s fingers equals the number of Bargle’s toes. (3) There are two possible ways in which Joe could win this chess game. In the right contexts, and outside the philosophy room, all the above sentences may be completely uncontroversial. For instance, if we know that Joe could win either by exchanging queens and entering an endgame, or by initiating a kingside attack then, if ignorant of Quine’s work on ontology, (...) we would take (3) to be uncontroversially true. (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)

Conflicting accounts of the role of mathematics in our physical theories can be traced to two principles. Mathematics appears to be both (1) theoretically indispensable, as we have no acceptable non-mathematical versions of our theories, and (2) metaphysically dispensable, as mathematical entities, if they existed, would lack a relevant causal role in the physical world. I offer a new account of a role for mathematics in the physical sciences that emphasizes the epistemic benefits of having mathematics around when we do (...) science. This account successfully reconciles theoretical indispensability and metaphysical dispensability and has important consequences for both advocates and critics of indispensability arguments for platonism about mathematics. (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)

My aim is twofold: first, to root out the metaphysical assumptions that generate the problem of mental causation and to show that they preclude its solution; second, to dissolve the problem of mental causation by motivating rejection of one of the metaphysical assumptions that give rise to it. There are three features of this metaphysical background picture that are important for our purposes. The first concerns the nature of reality: all reality depends on physical reality, where physical reality consists of (...) a network of events.1 The second concerns the nature of causation, and the third concerns the conception of behavior. I try to vindicate a robust idea of mental causation. (shrink)

The paper is a critique of the widespread conception of logic as a neutral arbiter between metaphysical theories, one that makes no `substantive’ claims of its own (David Kaplan and John Etchemendy are two recent examples). A familiar observation is that virtually every putatively fundamental principle of logic has been challenged over the last century on broadly metaphysical grounds (however mistaken), with a consequent proliferation of alternative logics. However, this apparent contentiousness of logic is often treated as though it were (...) neutralized by the possibility of studying all these alternative logics within an agreed metalogical framework, typically that of first-order logic with set theory. In effect, metalogic is given the role of neutral arbiter. The paper will consider a variety of examples in which deep logical disputes re-emerge at the meta-level. One case is quantified modal logic, where some varieties of actualism require a modal meta-language (as opposed to the usual non-modal language of possible worlds model theory) in order not to make their denial of the Barcan formula self-defeating. Similarly, on some views the intended model theory for second-order logic can only be given in a second-order metalanguage—this may be needed to avoid versions of Russell’s paradox when the first-order quantifiers are read as absolutely unrestricted. It can be shown that the phenomenon of higher-order vagueness eventually forces fuzzy logical treatments of vagueness to use a fuzzy metalanguage, with consequent repercussions for what first-order principles are validated. The difficulty of proving the completeness of first-order intuitionistic logic on its intended interpretation by intuitionistically rather than just classically valid means is a more familiar example. These case studies will be discussed in some detail to reveal a variety of ways in which even metalogic is metaphysically contested, substantial and non-neutral. (shrink)

In a lot of domains in metaphysics the tacit assumption has been that whichever metaphysical principles turn out to be true, these will be necessarily true. Let us call necessitarianism about some domain the thesis that the right metaphysics of that domain is necessary. Necessitarianism has flourished. In the philosophy of maths we find it held that if mathematical objects exist, then they do of necessity. Mathematical Platonists affirm the necessary existence of mathematical objects (see for instance Hale and Wright (...) 1992 and 1994; Wright 1983 and 1988; Schiffer 1996; Resnik 1997; Shapiro 1997 and Zalta 1988) while mathematical nominalists, usually in the form of fictionalists, hold that necessarily such objects fail to exist (see for instance Balaguer 1996 and 1998; Rosen 2001 and Yablo 2005). In metaphysics more generally, until recently it was more or less assumed that whatever the right account of composition—the account of under what conditions some xs compose a y—that account will be necessarily true (for a discussion of theories of composition see Simons 1987 and van Inwagen 1987 and 1990; the modal status of the composition relation is explicitly addressed in Schaffer 2007; Parsons 2006 and Cameron 2007). Similarly, it has generally been assumed that whatever the right account is of the nature of properties, whether they be universals, tropes, or whether nominalism is true, that account will be necessarily true (though see Rosen 2006 for a recent suggestion to the contrary). In considering theories of persistence it has been widely held that whether objects endure or perdure through time is a matter of necessity (Sider 2001; though see Lewis 1999 p227 who defends contingent perdurantism). And with respect to theories of time it is frequently held that whichever of the A- or B-theory is true is necessarily true. A-theorists often argue that there is time in a world only if the A-theory is true at that world (see for instance McTaggart 1903; Markosian 2004; Bigelow 1996; Craig 2001) while B-theorists often argue that the A-theory is internally inconsistent (Smart 1987; Mellor 1998; Savitt 2000 and Le Poidevin 1991). Once again, we find a few recent contingentist dissenters. Bourne (2006) suggests that it is a contingent feature of time that it is tensed, and thus that the A-theory is contingently true. Worlds in which there exist only B-theoretic properties are worlds with time, it is just that time in those worlds time is radically different to the way it is actually. Other defenders of the B-theory, though not expressly contingentists, do offer arguments against versions of the A-theory that try to show that such A-theories theories are inconsistent with the actual laws of nature (see for instance Saunders 2002 and Callender 2000); these arguments, at least, leave room for the possibility that the A-theories in question are contingently false (at least on the assumption that the laws of nature are themselves contingent, an assumption that not everyone accepts). Despite some notable exceptions, necessitarianism has flourished in many, if not most, domains in metaphysics. One such exception is Lewis’ famous defence of Humean supervenience as a contingent claim about our world. Lewis does not argue that necessarily, the supervenience base for all matters of fact in a world is nothing but a vast mosaic of local matters of particular fact. Rather, he thinks that we have reason to think that our world is one in which Humean supervenience holds (see Lewis 1986 p9-10 and 1994). Another exception to the necessitarian orthodoxy is to be found in the lively debate about the modal status of the laws of nature. Here, if anything, contingentism has been the dominating force, with it generally being held that there are possible worlds in which different laws of nature hold (this view is defended by, among others, Lewis 1986 and 2010; Schaffer 2005 and Sidelle 2002). Necessitarian dissenters hold that the laws of nature are necessary, frequently because they think it is necessary that fundamental properties have the causal or nomic profiles they do (see for instance Shoemaker 1980 and 1988; Swoyer 1982; Bird 1995; Ellis and Lierse 1994). Nevertheless, when it comes to thinking about the nature of the laws themselves, the necessitarian presumption is back on firm footing. Though there is disagreement about whether the laws are generalisations that feature in the most virtuous true axiomatisation of all the particular matters of fact (often known as the Humean view of laws and defended by Ramsey 1978; Lewis 1986 and Beebee 2000) or whether laws are relations of necessity that hold between universals (a view defended by Armstrong 1983; Dretske 1977; Tooley 1977 and Carroll 1990) no one has seriously suggested that it might be a contingent matter which of these is the right account of laws. The necessitarian orthodoxy is not surprising since metaphysics is largely an a priori process. While a priori reasoning may be used to determine whether a proposition is necessary or contingent, it is not well placed (in the absence of a posteriori evidence) to determine whether a contingent proposition is actually true or false. Since metaphysicians aim to tell us which principles are true in which worlds, on the face of it the discovery that metaphysical principles are contingent seems to make part of the task of metaphysics epistemically intractable. In what follows I consider two reasons one might end up embracing contingentism and whether this would lead one into epistemic difficulty. The following section considers a route to contingent metaphysical truths that proceeds via a combination of conceptual necessities and empirical discoveries. Section 3 considers whether there might be synthetic contingent metaphysical truths, and the final section raises the question of whether if there were such truths we would be well placed to come to know them. (shrink)

Reasoning about situations we take to be impossible is useful for a variety of theoretical purposes. Furthermore, using a device of impossible worlds when reasoning about the impossible is useful in the same sorts of ways that the device of possible worlds is useful when reasoning about the possible. This paper discusses some of the uses of impossible worlds and argues that commitment to them can and should be had without great metaphysical or logical cost. The paper then provides an (...) account of reasoning with impossible worlds, by treating such reasoning as reasoning employing counterpossible conditionals, and provides a semantics for the proposed treatment. (shrink)

Mathematics plays a central role in much of contemporary science, but philosophers have struggled to understand what this role is or how significant it might be for mathematics and science. In this book Christopher Pincock tackles this perennial question in a new way by asking how mathematics contributes to the success of our best scientific representations. In the first part of the book this question is posed and sharpened using a proposal for how we can determine the content of a (...) scientific representation. Several different sorts of contributions from mathematics are then articulated. Pincock argues that each contribution can be understood as broadly epistemic, so that what mathematics ultimately contributes to science is best connected with our scientific knowledge. In the second part of the book, Pincock critically evaluates alternative approaches to the role of mathematics in science. These include the potential benefits for scientific discovery and scientific explanation. A major focus of this part of the book is the indispensability argument for mathematical platonism. Using the results of part one, Pincock argues that this argument can at best support a weak form of realism about the truth-value of the statements of mathematics. The book concludes with a chapter on pure mathematics and the remaining options for making sense of its interpretation and epistemology. Thoroughly grounded in case studies drawn from scientific practice, this book aims to bring together current debates in both the philosophy of mathematics and the philosophy of science and to demonstrate the philosophical importance of applications of mathematics. (shrink)

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

This book expounds a system of ideas about the nature of mathematics which Michael Resnik has been elaborating for a number of years. In calling mathematics a science he implies that it has a factual subject-matter and that mathematical knowledge is on a par with other scientific knowledge; in calling it a science of patterns he expresses his commitment to a structuralist philosophy of mathematics. He links this to a defense of realism about the metaphysics of mathematics--the view that mathematics (...) is about things that really exist. (shrink)

First published in 1971, Professor Putnam's essay concerns itself with the ontological problem in the philosophy of logic and mathematics - that is, the issue of whether the abstract entities spoken of in logic and mathematics really exist. He also deals with the question of whether or not reference to these abstract entities is really indispensible in logic and whether it is necessary in physical science in general.

It is a popular view amongst some philosophers, most notably those with Quinean views about ontological commitment, that scientific theories are first-orderizable; that we can regiment all such theories in an extensional first-order language. I argue that this view is false, and that any acceptable account of science needs to take some modal notion as primitive.

A Commonplace of recent philosophy of mind is that intentional states are relations between thinkers and propositions. This thesis-call it the 'Relational Thesis'-does not depend on any specific theory of propositions. One can hold it whether one believes that propositions are Fregean Thoughts, ordered n-tuples of objects and properties or sets of possible worlds. An assumption that all these theories of propositions share is that propositions are abstract objects, without location in space or time...

The paper criticizes the truthmaker principle that every truth is made true by something. If we interpret ‘something’ as quantifying into sentence position, we can interpret the principle as a harmless logical truth, but that is not what advocates of the principle intend. They interpret ‘something’ as quantifying into name position, and the principle as requiring the existence of truthmaking individuals. The paper argues that we have no reason to believe the principle on this interpretation. Moreover, the converse Barcan formula (...) is inconsistent with the existence of truthmaking individuals for contingent truths. Considerations about our ability to count possible truthmaking individuals are used to argue that we should prefer the converse Barcan formula. (shrink)