I propose a deductive-nomological model for mathematical scientific explanation. In this regard, I modify Hempel’s deductive-nomological model and test it against some of the following recent paradigmatic examples of the mathematical explanation of empirical facts: the seven bridges of Königsberg, the North American synchronized cicadas, and Hénon-Heiles Hamiltonian systems. I argue that mathematical scientific explanations that invoke laws of nature are qualitative explanations, and ordinary scientific explanations that employ mathematics are quantitative explanations. I analyse the repercussions of this deductivenomological model (...) on causal explanations. (shrink)

In this paper I propose a position in the ontology of mathematics which is inspired mainly by a case study in the mathematical discipline if-theory. The main theses of this position are that mathematical objects are introduced by mathematicians and that after mathematical objects have been introduced, they exist as objectively accessible abstract objects.

This article takes as a starting point the current popular anti realist position, Fictionalism, with the intent to compare it with actual mathematical practice. Fictionalism claims that mathematical statements do purport to be about mathematical objects, and that mathematical statements are not true. Considering these claims in the light of mathematical practice leads to questions about how mathematical objects are handled, and how we prove that certain statements hold. Based on a case study on Riemann’s work on complex functions, I (...) propose that mathematicians deal with systems of representations and that truth—or what we can prove—depends on available representations in some context where the problem can be solved. (shrink)

The linguistic turn provided philosophers with a range of reasons for engaging in careful investigation into the nature and structure of language. However, the linguistic turn is dead. The arguments for it have been abandoned. This raises the question: why should philosophers take an interest in the minutiae of natural language semantics? I’ll argue that there isn’t much of a reason - philosophy of language has lost its way. Then I provide a suggestion for how it can find its way (...) again. (shrink)

What are the ontological commitments of a sentence? In this paper I offer an answer from the perspective of the truthmaker theorist that contrasts with the familiar Quinean criterion. I detail some of the benefits of thinking of things this way: they include making the composition debate tractable without appealing to a neo-Carnapian metaontology, making sense of neo-Fregeanism, and dispensing with some otherwise recalcitrant necessary connections.

I here defend a theory consisting of four claims about ‘property’ and properties, and argue that they form a coherent whole that can solve various serious problems. The claims are (1): ‘property’ is defined by the principles (PR): ‘F-ness/Being F/etc. is a property of x iff F’ and (PA): ‘F-ness/Being F/etc. is a property’; (2) the function of ‘property’ is to increase the expressive power of English, roughly by mimicking quantification into predicate position; (3) property talk should be understood at (...) face value: apparent commitments are real and our apparently literal use of ‘property’ is really literal; (4) there are no properties. In virtue of (1)–(2), this is a deflationist theory and in virtue of (3)–(4), it is an error theory. (1) is fleshed out as a claim about understanding conditions, and it is argued at length, and by going through a number of examples, that it satisfies a crucial constraint on meaning claims: all facts about ‘property’ can be explained, together with auxiliary facts, on its basis. Once claim (1) has been expanded upon, I argue that the combination of (1)–(3) provides the means for handling several problems: they help giving a happy-face solution to what I call the paradox of abstraction , they form part of a plausible account of the correctness of committive sentences, and, most importantly, they help respond to various indispensability arguments against nominalism. (shrink)

This article surveys recent literature by Parsons, McGee, Shapiro and others on the significance of categoricity arguments in the philosophy of mathematics. After discussing whether categoricity arguments are sufficient to secure reference to mathematical structures up to isomorphism, we assess what exactly is achieved by recent ‘internal’ renditions of the famous categoricity arguments for arithmetic and set theory.

Confirmational holism is central to a traditional formulation of the indispensability argument for mathematical realism (IA). I argue that recent strategies for defending scientific realism are incompatible with confirmational holism. Thus a traditional formulation of IA is incompatible with recent strategies for defending scientific realism. As a consequence a traditional formulation of IA will only have limited appeal.

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)

Enhanced indispensability arguments claim that Scientific Realists are committed to the existence of mathematical entities due to their reliance on Inference to the best explanation. Our central question concerns this purported parity of reasoning: do people who defend the EIA make an appropriate use of the resources of Scientific Realism to achieve platonism? We argue that just because a variety of different inferential strategies can be employed by Scientific Realists does not mean that ontological conclusions concerning which things we should (...) be Scientific Realists about are arrived at by any inferential route which eschews causes, and nor is there any direct pressure for Scientific Realists to change their inferential methods. We suggest that in order to maintain inferential parity with Scientific Realism, proponents of EIA need to give details about how and in what way the presence of mathematical entities directly contribute to explanations. (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)

My first goal is to motivate a distinctively metaphysical approach to the problem of induction. I argue that there is a precise sense in which the only way that orthodox Humean and non-Humean views can justify induction is by appealing to extremely strong and unmotivated probabilistic biases. My second goal is to sketch what such a metaphysical approach could possibly look like. After sketching such an approach, I consider a toy case that illustrates the way in which such a metaphysics (...) can help us make progress on the problem of induction. (shrink)

I challenge a claim that seems to be made when nominalists offer reformulations of the content of mathematical beliefs, namely that these reformulations are accessible to everyone. By doing so, I argue that these theories cannot account for the mathematical knowledge that ordinary people have. In the first part of the paper I look at reformulations that employ the concept of proof, such as those of Mary Leng and Ottavio Bueno. I argue that ordinary people don’t have many beliefs about (...) proofs, and that they are not in a position to acquire knowledge about proofs autonomously. The second part of the paper is concerned with other reformulations of content, such as those of Hartry Field and Stephen Yablo. There too, the problem is that people are not able to acquire knowledge of the reformulated propositions autonomously. Ordinary people simply do not have beliefs with the kind of content that the nominalists need, for their theory to account for the mathematical knowledge of ordinary people. All in all then, the conclusion is that a large number of theories that suggest reformulations of mathematical content yield contents that are inaccessible for most people. Thus, these theories are limited, in that they cannot account for the mathematical knowledge of ordinary people. (shrink)

Most of our best scientific descriptions of the world employ rates of change of some continuous quantity with respect to some other continuous quantity. For instance, in classical physics we arrive at a particle’s velocity by taking the time-derivative of its position, and we arrive at a particle’s acceleration by taking the time-derivative of its velocity. Because rates of change are defined in terms of other continuous quantities, most think that facts about some rate of change obtain in virtue of (...) facts about those other continuous quantities. For example, on this view facts about a particle’s velocity at a time obtain in virtue of facts about how that particle’s position is changing at that time. In this paper we raise a puzzle for this orthodox reductionist account of rate of change quantities and evaluate some possible replies. We don’t decisively come down in favour of one reply over the others, though we say some things to support taking our puzzle to cast doubt on the standard view that spacetime is continuous. (shrink)

A first step is taken towards articulating a constructive empiricist philosophy of mathematics, thus extending van Fraassen's account to this domain. In order to do so, I adapt Field's nominalization program, making it compatible with an empiricist stance. Two changes are introduced: (a) Instead of taking conservativeness as the norm of mathematics, the empiricist countenances the weaker notion of quasi-truth (as formulated by da Costa and French), from which the formal properties of conservativeness are derived; (b) Instead of quantifying over (...) spacetime regions, the empiricist only admits quantification over occupied regions, since this is enough for his or her needs. (shrink)

A number of people have recently argued for a structural approach to accounting for the applications of mathematics. Such an approach has been called "the mapping account". According to this view, the applicability of mathematics is fully accounted for by appreciating the relevant structural similarities between the empirical system under study and the mathematics used in the investigation ofthat system. This account of applications requires the truth of applied mathematical assertions, but it does not require the existence of mathematical objects. (...) In this paper, we discuss the shortcomings of this account, and show how these shortcomings can be overcome by a broader view of the application of mathematics: the inferential conception. (shrink)

Second-order logic has a number of attractive features, in particular the strong expressive resources it offers, and the possibility of articulating categorical mathematical theories (such as arithmetic and analysis). But it also has its costs. Five major charges have been launched against second-order logic: (1) It is not axiomatizable; as opposed to first-order logic, it is inherently incomplete. (2) It also has several semantics, and there is no criterion to choose between them (Putnam, J Symbol Logic 45:464–482, 1980 ). Therefore, (...) it is not clear how this logic should be interpreted. (3) Second-order logic also has strong ontological commitments: (a) it is ontologically committed to classes (Resnik, J Phil 85:75–87, 1988 ), and (b) according to Quine (Philosophy of logic, Prentice-Hall: Englewood Cliffs, 1970 ), it is nothing more than “set theory in sheep’s clothing”. (4) It is also not better than its first-order counterpart, in the following sense: if first-order logic does not characterize adequately mathematical systems, given the existence of non - isomorphic first-order interpretations, second-order logic does not characterize them either, given the existence of different interpretations of second-order theories (Melia, Analysis 55:127–134, 1995 ). (5) Finally, as opposed to what is claimed by defenders of second-order logic [such as Shapiro (J Symbol Logic 50:714–742, 1985 )], this logic does not solve the problem of referential access to mathematical objects (Azzouni, Metaphysical myths, mathematical practice: the logic and epistemology of the exact sciences, Cambridge University Press, Cambridge, 1994 ). In this paper, I argue that the second-order theorist can solve each of these difficulties. As a result, second-order logic provides the benefits of a rich framework without the associated costs. (shrink)

Mathematical concepts play at least three roles in the application of mathematics: an inferential role, a representational role, and an expressive role. In this paper, I argue that, despite what has often been alleged, platonists do not fully accommodate these features of the application of mathematics. At best, platonism provides partial ways of handling the issues. I then sketch an alternative, anti-realist account of the application of mathematics, and argue that this account manages to accommodate these features of the application (...) process. In this way, a better account of mathematical applications is, in principle, available. (shrink)

A number of issues connected with the nature of applied mathematics are discussed. Among the claims are these: mathematics "hooks onto" the world by providing models or representations, not by describing the world; classic platonism is to be preferred to structuralism; and several issues in the philosophy of science are intimately connected to the nature of applied mathematics.

Though the social world is real and objective, the way that social facts arise out of other facts is in an important way shaped by human thought, talk and behaviour. Building on recent work in social ontology, I describe a mechanism whereby this distinctive malleability of social facts, combined with the possibility of basic human error, makes it possible for a consistent physical reality to ground an inconsistent social reality. I explore various ways of resisting the prima facie case for (...) social inconsistency. I conclude, however, that the prima facie case survives scrutiny, and draw out some of the ramifications. (shrink)

(1988). Platonic explanation: Or, what abstract entities can do for you. International Studies in the Philosophy of Science: Vol. 3, No. 1, pp. 51-67. doi: 10.1080/02698598808573324.

Quine, taking the molecular constitution of matter as a paradigmatic example, offers an account of the relation between theory confirmation and ontology. Elsewhere, he deploys a similar ontological methodology to argue for the existence of mathematical objects. Penelope Maddy considers the atomic/molecular theory in more historical detail. She argues that the actual ontological practices of science display a positivistic demand for “direct observation,” and that fulfillment of this demand allows us to distinguish molecules and other physical objects from mathematical abstracta. (...) However, the confirmation of the atomic/molecular theory and the development of scientists’ ontological attitudes towards atoms was more complicated and subtle than even Maddy supposes. The present paper argues that the history of the theory in fact supports neither Quine’s and Maddy’s accounts of scientific ontology. There was no general demand from scientists to “see” atoms before they were reckoned to be real; but neither did the indispensable appearance of atoms in the best theory of chemical combination suffice to convince scientists of their reality. (shrink)

Kant's argument from incongruent counterparts for substantival space is examined; it is concluded that the argument has no force against a relationist. The argument does suggest that a relationist cannot give an account of enantiomorphism, incongruent counterparts and orientability. The prospects for a relationist account of these notions are assessed, and it is found that they are good provided the relationist is some kind of modal relationist. An illustration and interpretation of these modal commitments is given.

Attempts at solving what has been labeled as Eugene Wigner’s puzzle of applicability of mathematics are still far from arriving at an acceptable solution. The accounts developed to explain the “miracle” of applied mathematics vary in nature, foundation, and solution, from denying the existence of a genuine problem to designing structural theories based on mathematical formalism. Despite this variation, all investigations treated the problem in a unitary way with respect to the target, pointing to one or two ‘why’ or ‘how’ (...) questions to be answered. In this paper, I argue that two analyses, a semantic analysis ab initio and a metatheoretical analysis starting from the types of unreasonableness involved in this problem, will establish the interdisciplinary character of the problem and reveal many more targets, which may be addressed with different methodologies. In order to address objectively the philosophical problem of applicability of mathematics, a foundational revision of the problem is needed. (shrink)

Modal logic provides an elegant way to understand the notion of potential infinity. This raises the question of what the right modal logic is for reasoning about potential infinity. In this article I identify a choice point in determining the right modal logic: Can a potentially infinite collection ever be expanded in two mutually incompatible ways? If not, then the possible expansions are convergent; if so, then the possible expansions are branching. When possible expansions are convergent, the right modal logic (...) is S4.2, and a mirroring theorem due to Linnebo allows for a natural potentialist interpretation of mathematical discourse. When the possible expansions are branching, the right modal logic is S4. However, the usual box and diamond do not suffice to express everything the potentialist wants to express. I argue that the potentialist also needs an operator expressing that something will eventually happen in every possible expansion. I prove that the result of adding this operator to S4 makes the set of validities Pi-1-1 hard. This result makes it unlikely that there is any natural translation of ordinary mathematical discourse into the potentialist framework in the context of branching possibilities. (shrink)

Should philosophers prefer simpler theories? Huemer (Philos Q 59:216–236, 2009) argues that the reasons to prefer simpler theories in science do not apply in philosophy. I will argue that Huemer is mistaken—the arguments he marshals for preferring simpler theories in science can also be applied in philosophy. Like Huemer, I will focus on the philosophy of mind and the nominalism/Platonism debate. But I want to engage with the broader issue of whether simplicity is relevant to philosophy.

Many who take a dismissive attitude towards metaphysics trace their view back to Carnap’s ‘Empiricism, Semantics and Ontology’. But the reason Carnap takes a dismissive attitude to metaphysics is a matter of controversy. I will argue that no reason is given in ‘Empiricism, Semantics and Ontology’, and this is because his reason for rejecting metaphysical debates was given in ‘Pseudo-Problems in Philosophy’. The argument there assumes verificationism, but I will argue that his argument survives the rejection of verificationism. The root (...) of his argument is the claim that metaphysical statements cannot be justified; the point is epistemic, not semantic. I will argue that this remains a powerful challenge to metaphysics that has yet to be adequately answered. (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 maps the landscape for a range of views concerning the metaphysics of natural kinds. I consider a range of increasingly ontologically committed views concerning natural kinds and the possible arguments for them. I then ask how these relate to natural kind essentialism, arguing that essentialism requires commitment to kinds as entities. I conclude by examining the homeostatic property cluster view of kinds in the light of the general understanding of kinds developed.

In recent years, the so-calledindispensability argument has been given a lotof attention by philosophers of mathematics.This argument for the existence of mathematicalobjects makes use of the fact, neglected inclassical schools of philosophy of mathematics,that mathematics is part of our best scientifictheories, and therefore should receive similarsupport to these theories. However, thisobservation raises the question about the exactnature of the alleged connection betweenexperience and mathematics (for example: is itpossible to falsify empirically anymathematical theorems?). In my paper I wouldlike to address this (...) question by consideringthe explicit assumptions of different versionsof the indispensability argument. My primaryclaim is that there are at least three distinctversions of the indispensability argument (andit can be even suggested that a fourth,separate version should be formulated). I willmainly concentrate my discussion on thisvariant of the argument, which suggests thepossibility of empirical confirmation ofmathematical theories. A large portion of mypaper will focus on the recent discussion ofthis topic, starting from the paper by E.Sober, who in my opinion put reasonablerequirements on what is to be counted as anempirical confirmation of a mathematicaltheory. I will develop his model into threeseparate scenarios of possible empiricalconfirmation of mathematics. Using an exampleof Hilbert space in quantum mechanicaldescription I will show that the most promisingscenario of empirical verification ofmathematical theories has neverthelessuntenable consequences. It will be hypothesizedthat the source of this untenability lies in aspecific role which mathematical theories playin empirical science, and that what is subjectto empirical verification is not themathematics used, but the representabilityassumptions. Further I will undertake theproblem of how to reconcile the allegedempirical verification of mathematics withscientific practice. I will refer to thepolemics between P. Maddy and M. Resnik,pointing out certain ambiguities of theirarguments whose source is partly the failure todistinguish carefully between different sensesof the indispensability argument. For thatreason typical arguments used in the discussionare not decisive, yet if we take into accountsome metalogical properties of appliedmathematics, then the thesis that mathematicshas strong links with experience seems to behighly improbable. (shrink)

Invoking a form of quantifier variance promises to let us explain mathematicians’ freedom to introduce new kinds of mathematical objects in a way that avoids some problems for standard platonist and nominalist views. In this paper I’ll note that, despite traditional associations between quantifier variance and Carnapian rejection of metaphysics, Siderian realists about metaphysics can naturally be quantifier variantists. Unfortunately a variant on the Quinean indispensability argument concerning grounding seems to pose a problem for philosophers who accept this hybrid. However (...) I will charitably reconstruct this problem and then argue for optimism about solving it. (shrink)

In this paper, I will attempt to develop and defend a common form of intuitive resistance to the companions in guilt argument. I will argue that one can reasonably believe there are promising solutions to the access problem for mathematical realism that don’t translate to moral realism. In particular, I will suggest that the structuralist project of accounting for mathematical knowledge in terms of some form of logical knowledge offers significant hope of success while no analogous approach offers such hope (...) for moral realism. (shrink)

_ Source: _Page Count 12 In a previous article, I argued against the widespread reluctance of philosophers to treat skeptical challenges to our a priori knowledge of necessary truths with the same seriousness as skeptical challenges to our a posteriori knowledge of contingent truths. Hamid Vahid has recently offered several reasons for thinking the unequal treatment of these two kinds of skepticism is justified, one of which is a priori skepticism’s seeming dependence upon the widely scorned kk thesis. In the (...) present article, I defend a priori skepticism against Vahid’s criticisms. (shrink)

Hale proposes a neo-logicist definition of real numbers by abstraction as ratios defined on a complete ordered domain of quantities (magnitudes). I argue that Hale's definition faces insuperable epistemological and ontological difficulties. On the epistemological side, Hale is committed to an explanation of measurement applications of reals which conflicts with several theorems in measurement theory. On the ontological side, Hale commits himself to the necessary and a priori existence of at least one complete ordered domain of quantities, which is extremely (...) implausible because science treats the logical structure of quantities as subject to experimentally and theoretically motivated refinements and revisions. (shrink)

Putnam's ``model-theoretic'' argument against metaphysical realism presupposes that an ideal scientific theory is expressible in a first order language. The central aim of this paper is to show that Putnam's ``first orderization'' of science, although unchallenged by numerous critics, makes his argument unsound even for adequate theories, never mind an ideal one. To this end, I will argue that quantitative theories, which dominate the natural sciences, can be adequately interpreted and evaluated only with the help of so-called theories of measurement (...) whose epistemological and methodological purpose is to justify systematic assignments of quantitative values to objects in the world. And, in order to fulfill this purpose, theories of measurement must have an essentially higher order logical structure. As a result, Putnam's argument fails because much of science turns out to rest on essentially higher order theoretical assumptions about the world. (shrink)

Mathematics clearly plays an important role in scientific explanation. Debate continues, however, over the kind of role that mathematics plays. I argue that if pure mathematical explananda and physical explananda are unified under a common explanation within science, then we have good reason to believe that mathematics is explanatory in its own right. The argument motivates the search for a new kind of scientific case study, a case in which pure mathematical facts and physical facts are explanatorily unified. I argue (...) that it is possible for there to be such cases, and provide some toy examples to demonstrate this. I then identify a potential source of scientific case studies as a guide for future work. (shrink)

Enhanced indispensability arguments seek to establish realism about mathematics based on the explanatory role that mathematics plays in science. Idealizations pose a problem for such arguments. Idealizations, in a similar way to mathematics, boost the explanatory credentials of our best scientific theories. And yet, idealizations are not the sorts of things that are supposed to attract a realist attitude. I argue that the explanatory symmetry between idealizations and mathematics can potentially be broken as follows: although idealizations contribute to the explanatory (...) power of our best theories, they do not carry the explanatory load. It is at least open however that mathematics is load-carrying. To give this idea substance, I offer an analysis of what it is to carry the explanatory load in terms of difference-making and counterfactuals. (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.

It has seemed, to many, that there is an important connection between the ways in which some theoretical posits explain our observations, and our reasons for being ontologically committed to those posits. One way to spell out this connection is in terms of what has become known as the explanatory criterion of ontological commitment. This is, roughly, the view that we ought to posit only those entities that are indispensable to our best explanations. Our primary aim is to argue that (...) the moral nonnaturalist places herself in an invidious position if she simply accepts that the nonnatural moral facts that she posits are not explanatory. Instead, we offer the nonnaturalist an alternative strategy for dealing with moral explanations. The strategy is to retain the explanatory criterion of ontological commitment and maintain that moral properties are, in fact, explanatory. The explanations they provide are not causal explanations; instead, moral properties make a non-causal difference to the physical facts. (shrink)

In this article I consider what it would take to combine a certain kind of mathematical Platonism with serious presentism. I argue that a Platonist moved to accept the existence of mathematical objects on the basis of an indispensability argument faces a significant challenge if she wishes to accept presentism. This is because, on the one hand, the indispensability argument can be reformulated as a new argument for the existence of past entities and, on the other hand, if one accepts (...) the indispensability argument for mathematical objects then it is hard to resist the analogous argument for the existence of the past. (shrink)

In this article I consider what it would take to combine a certain kind of mathematical Platonism with serious presentism. I argue that a Platonist moved to accept the existence of mathematical objects on the basis of an indispensability argument faces a significant challenge if she wishes to accept presentism. This is because, on the one hand, the indispensability argument can be reformulated as a new argument for the existence of past entities and, on the other hand, if one accepts (...) the indispensability argument for mathematical objects then it is hard to resist the analogous argument for the existence of the past. (shrink)

Recently, nominalists have made a case against the Quine–Putnam indispensability argument for mathematical Platonism by taking issue with Quine’s criterion of ontological commitment. In this paper I propose and defend an indispensability argument founded on an alternative criterion of ontological commitment: that advocated by David Armstrong. By defending such an argument I place the burden back onto the nominalist to defend her favourite criterion of ontological commitment and, furthermore, show that criterion cannot be used to formulate a plausible form of (...) the indispensability argument. (shrink)

There is an old meta-philosophical worry: very roughly, metaphysical theories have no observational consequences and so the study of metaphysics has no value. The worry has been around in some form since the rise of logical positivism in the early twentieth century but has seen a bit of a renaissance recently. In this paper, I provide an apology for metaphysics in the face of this kind of concern. The core of the argument is this: pure mathematics detaches from science in (...) much the same manner as metaphysics and yet it is valuable nonetheless. The source of value enjoyed by pure mathematics extends to metaphysics as well. Accordingly, if one denies that metaphysics has value, then one is forced to deny that pure mathematics has value. The argument places an added burden on the sceptic of metaphysics. If one truly believes that metaphysics is worthless (as some philosophers do), then one must give up on pure mathematics as well. (shrink)

Arguing for mathematical realism on the basis of Field’s explanationist version of the Quine–Putnam Indispensability argument, Alan Baker has recently claimed to have found an instance of a genuine mathematical explanation of a physical phenomenon. While I agree that Baker presents a very interesting example in which mathematics plays an essential explanatory role, I show that this example, and the argument built upon it, begs the question against the mathematical nominalist.