I discuss a puzzle that shows there is a need to develop a new metaphysical interpretation of mathematical theories, because all well-known interpretations conflict with important aspects of mathematical activities. The new interpretation, I argue, must authenticate the ontological commitments of mathematical theories without curtailing mathematicians' freedom and authority to creatively introduce mathematical ontology during mathematical problem-solving. Further, I argue that these two constraints are best met by a metaphysical interpretation of mathematics that takes mathematical entities to be constitutively constructed (...) by human activity in a manner similar to the constitutive construction of the US Supreme Court by certain legal and political activities. Finally, I outline some of the philosophical merits of metaphysical interpretations of mathematical theories of this type. (shrink)

Many contemporary philosophers rate error theories poorly. We identify the arguments these philosophers invoke, and expose their deficiencies. We thereby show that the prospects for error theory have been systematically underestimated. By undermining general arguments against all error theories, we leave it open whether any more particular arguments against particular error theories are more successful. The merits of error theories need to be settled on a case-by-case basis: there is no good general argument against error theories.

in Robin Le Poidevin (ed.) Being: Developments in Contemporary Metaphysics. Cambridge: Cambridge University Press. Peter van Inwagen claims that there are no tables or chairs. He also claims that sentences such as ‘There are chairs here’, which seem to imply their existence, are often true. This combination of views opens van Inwagen to a charge of self-contradiction. I explain the charge, and van Inwagen’s response to it, which involves the claim that sentences like ‘There are tables’ shift their truth-conditions between (...) different contexts of utterance. I present an alternative response which involves the negation of that claim, and argue that it is preferable to van Inwagen’s. (shrink)

Platonism in the philosophy of mathematics is the doctrine that there are mathematical objects such as numbers. John Burgess and Gideon Rosen have argued that that there is no good epistemological argument against platonism. They propose a dilemma, claiming that epistemological arguments against platonism either rely on a dubious epistemology, or resemble a dubious sceptical argument concerning perceptual knowledge. Against Burgess and Rosen, I show that an epistemological anti- platonist argument proposed by Hartry Field avoids both horns of their dilemma.

ABSTRACT According to edenic idealism, our ordinary object terms refer to items in the manifest world—the world of primitive objects and properties presented in experience. I motivate edenic idealism as a response to scenarios where it is difficult to match the objects in experience with corresponding items in the external world. I argue that edenic idealism has important semantic advantages over realism: it is the most intuitive view of what we are actually talking about when we use terms for objects.

This book is an exploration and defense of the coherence of classical theism’s doctrine of divine aseity in the face of the challenge posed by Platonism with respect to abstract objects. A synoptic work in analytic philosophy of religion, the book engages discussions in philosophy of mathematics, philosophy of language, metaphysics, and metaontology. It addresses absolute creationism, non-Platonic realism, fictionalism, neutralism, and alternative logics and semantics, among other topics. The book offers a helpful taxonomy of the wide range of options (...) available to the classical theist for dealing with the challenge of Platonism. It probes in detail the diverse views on the reality of abstract objects and their compatibility with classical theism. It contains a most thorough discussion, rooted in careful exegesis, of the biblical and patristic basis of the doctrine of divine aseity. Finally, it challenges the influential Quinean metaontological theses concerning the way in which we make ontological commitments. (shrink)

In this entry I will offer a survey of the contemporary debate on fic- tionalism, which is a distinctive anti-realist view about certain regions of discourse that are valued for their usefulness rather than their truth.

In the ‘ordinary business of life’, everyone makes claims about what there is. For instance, we say things like: ‘There are some beautiful chairs in my favourite furniture shop’. Within the context of philosophical debate, some philosophers also make claims about what there is. For instance, some ontologists claim that there are chairs; other ontologists claim that there are no chairs. What is the relation between ontologists’ philosophical claims about what there is and ordinary claims about what there is? According (...) to Cian Dorr, ontologists’ claims and denials of existence belong to ‘a sort of professional jargon’. Dorr claims, for example, that ‘There are prime numbers between 20 and 30’ can be used superficially or used fundamentally. Ordinary uses of are superficial: we use the sentence to assert a boring, well-known truth. But in the ontology room, is used fundamentally, to assert that numbers are ‘part of the ultimate furniture of reality’: and this is a substantial metaphysical doctrine, not a boring truth. In this paper, we will show that none of Dorr’s arguments for these claims succeeds. (shrink)

Many idealists have thought that realism raises epistemological problems. The worry is that, if it is possible for truths about ordinary objects to outstrip our experiences in the ways that realists typically suppose, we could never be justified in our beliefs about objects. Few contemporary theorists find this argument convincing; philosophers have offered a variety of responses to defend the epistemology of our object judgments under the assumption of realism. But in this paper, I offer a new type of epistemic (...) argument against realism which is immune to the standard responses in the literature. In addition to raising a challenge for realism, the epistemology of our object judgments has implications for how the idealist should develop her own positive metaphysical view. So in the second half of this paper, I discuss how the idealist should understand the dependence between objects and our experiences if she is to secure epistemic advantages over the realist. (shrink)

Paraphrase is ubiquitous in philosophy, especially in discussions about ontological commitment. But should it be? Paraphrases are seldom accompanied by evidence that would convince, say, a linguist that the paraphrase and the paraphrased sentence have the same meaning. Indeed, from the perspective of linguistics, many paraphrases would seem to be nothing but bad jokes. For this reason, many philosophers have become deeply suspicious about paraphrase. I ague in this paper that this worry is misguided--that successful paraphrases do not need to (...) preserve semantic content, and that the way that paraphrase has been used in Quinean meta-ontology, and philosophical inquiry more generally, corresponds, at least roughly, to the way it should be used. (shrink)

Burgess and Rosen argue that Yablo’s figuralist account of mathematics fails because it says mathematical claims are really only metaphorical. They suggest Yablo’s view is implausible as an account of what mathematicians say and confused about literal language. I show their argument isn’t decisive, briefly exploring some questions in the philosophy of language it raises, and argue Yablo’s view may be amended to a kind of revolutionary fictionalism not refuted by Burgess and Rosen.

Philosophers often debate the existence of such things as numbers and propositions, and say that if these objects exist, they are abstract. But what does it mean to call something 'abstract'? And do we have good reason to believe in the existence of abstract objects? This Element addresses those questions, putting newcomers to these debates in a position to understand what they concern and what are the most influential considerations at work in this area of metaphysics. It also provides advice (...) on which lines of discussion promise to be the most fruitful. (shrink)

Analytic Philosophy, EarlyView. The method of paraphrasing away apparent ontological commitments is a familiar tool for trimming one's ontology. Even so, I argue that aiming to avoid commitment via paraphrase is unjustified. The reason is the standard motivations for paraphrase rest on implicit yet faulty principles regarding ontological categories and categorization- or so I argue. These results also provide indirect support for a permissivist approach to ontology.

The Doctrine of the Trinity says that there is one God, that there are three divine Persons, and that each divine Person is God. The Logical Problem of the Trinity is that these claims seem logically inconsistent. We argue that any coherent and orthodox solution to the Logical Problem must use the technique of paraphrase: a logically or metaphysically more perspicuous reformulation. If so, discussions of paraphrase deserve more prominence in the literature on the Doctrine of the Trinity. We also (...) show that such explicit discussion has important implications for theorizing about the Trinity. (shrink)

Hartry Field's formulation of an epistemological argument against platonism is only valid if knowledge is constrained by a causal clause. Contrary to recent claims (e.g. in Liggins (2006), Liggins (2010)), Field's argument therefore fails the very same criterion usually taken to discredit Benacerraf's earlier version.

Contemporary philosophy’s three main naturalisms are methodological, ontological and epistemological. Methodological naturalism states that the only authoritative standards are those of science. Ontological and epistemological naturalism respectively state that all entities and all valid methods of inquiry are in some sense natural. In philosophy of mathematics of the past few decades methodological naturalism has received the lion’s share of the attention, so we concentrate on this. Ontological and epistemological naturalism in the philosophy of mathematics are discussed more briefly in section (...) 6. (shrink)

This article attempts to motivate a new approach to anti-realism (or nominalism) in the philosophy of mathematics. I will explore the strongest challenges to anti-realism, based on sympathetic interpretations of our intuitions that appear to support realism. I will argue that the current anti-realistic philosophies have not yet met these challenges, and that is why they cannot convince realists. Then, I will introduce a research project for a new, truly naturalistic, and completely scientific approach to philosophy of mathematics. It belongs (...) to anti-realism, but can meet those challenges and can perhaps convince some realists, at least those who are also naturalists. (shrink)

I argue that the most popular versions of naturalism imply nominalism in philosophy of mathematics. In particular, there is a conflict in Quine's philosophy between naturalism and realism in mathematics. The argument starts from a consequence of naturalism on the nature of human cognitive subjects, physicalism about cognitive subjects, and concludes that this implies a version of nominalism, which I will carefully characterize. The indispensability of classical mathematics for the sciences and semantic/confirmation holism does not affect the argument. The disquotational (...) theory of reference and truth is discussed but rejected. This argument differs from the Benacerrafian arguments against realism, because it does not rely on any specific assumption about the nature of knowledge or reference. It differs from the popular objections to the indispensability argument for realism as well, because it can admit both indispensability and holism. This argument motivates a new, radically naturalistic and nominalistic approach to philosophy of mathematics. (shrink)

The indispensability argument for abstract mathematical entities has been an important issue in the philosophy of mathematics. The argument relies on several assumptions. Some objections have been made against these assumptions, but there are several serious defects in these objections. Ameliorating these defects leads to a new anti-realistic philosophy of mathematics, mainly: first, in mathematical applications, what really exist and can be used as tools are not abstract mathematical entities, but our inner representations that we create in imagining abstract mathematical (...) entities; second, the thoughts that we create in imagining infinite mathematical entities are bounded by external conditions. (shrink)

Fictionalists claim that instead of believing certain controversial propositions they accept them nonseriously, as useful make-believe. In this way they present themselves as having an austere ontology despite the apparent ontological commitments of their discourse. Some philosophers object that this plays on a distinction without a difference: the fictionalist’s would-be nonserious acceptance is the most we can do for the relevant content acceptance-wise, hence such acceptance is no different from what we ordinarily call ‘belief’ and should be so called. They (...) conclude that it is subject to the norms applicable to paradigmatic empirical beliefs, and hence, pace fictionalists, ontological commitments must be taken seriously. I disentangle three strands in the objector’s thought: the ‘What more can you ask for?’ intuition, a linguistic/ conceptual claim, and a claim about norms. I argue that the former two are compatible with ontological deflationism, and therefore do not entail applicability of the norms. Nevertheless, if indeed there is no more robust acceptance with which to contrast the supposed nonserious acceptance, then the fictionalist’s claim to austere ontology must be abandoned. Is there a reason to suppose there is any merit to the distinction-without-a-difference charge? I argue that there is, clarify it, and defend against objections, focusing on Daly 2008. (shrink)

Yablo argued that some metaphors are representationally essential: they enable us to express contents that we would not be able to express without them. He defended a fictionalist view of mathematical language by making the case that it similarly serves as a representational aid. Against this, Colyvan argued that metaphorical/figurative language can never play an essential role in explanation and that mathematical language often does, hence concluding that Yablo’s fictionalism is untenable. I show that Colyvan’s thesis about explanation is highly (...) implausible in the absence of a challenge to Yablo’s position on representationally essential metaphors, which Colyvan does not attempt. I also briefly discuss other attempts to produce a simple knock-out argument against fictionalism and show them wanting. (shrink)

There is a puzzle about the use of paraphrase in philosophy, presented most famously in Alston's [1958] ‘Ontological Commitments’, but found throughout the literature. The puzzle arises from the fact that a symmetry required for a paraphrase to be successful seems to necessitate a symmetry sufficient for a paraphrase to fail, since any two expressions that stand in the means the same as relation must also stand in the has the same commitments as relation. I show that, while this problem (...) does undermine some conceptions of paraphrase, on a proper understanding of paraphrase the puzzle gets no purchase. Since paraphrase is an important component of Quinean approaches to meta-ontology, this paper constitutes a partial defence of Quinean meta-ontology. Since paraphrase is an important component of traditional methods of philosophical inquiry, this paper constitutes a partial defence of traditional modes of philosophizing as well. (shrink)

This is a survey of contemporary work on ‘fictionalism in metaphysics’, a term that is taken to signify both the place of fictionalism as a distinctive anti‐realist metaphysics in which usefulness rather than truth is the norm of acceptance, and the fact that philosophers have given fictionalist treatments of a range of specifically metaphysical notions.

In a 2005 paper, John Burgess and Gideon Rosen offer a new argument against nominalism in the philosophy of mathematics. The argument proceeds from the thesis that mathematics is part of science, and that core existence theorems in mathematics are both accepted by mathematicians and acceptable by mathematical standards. David Liggins (2007) criticizes the argument on the grounds that no adequate interpretation of “acceptable by mathematical standards” can be given which preserves the soundness of the overall argument. In this discussion (...) I offer a defense of the Burgess-Rosen argument against Liggins’s objection. I show how plausible versions of the argument can be constructed based on either of two interpretations of mathematical acceptability, and I locate the argument in the space of contemporary anti-nominalist views. (shrink)