I critically examine some deflationary theories of properties, according to which properties are ‘shadows of predicates’ and quantification over them serves a mere quasi-logical function. I start by considering Hofweber’s internalist theory, and pose a problem for his account of inexpressible properties. I then introduce a theory of properties that closely resembles Horwich’s minimalist theory of truth. This theory overcomes the problem of inexpressible properties, but its formulation presupposes the existence of various kinds of abstract objects. I discuss some ways (...) to reduce these existence assumptions, but ultimately suggest that deflationists can hardly avoid quantification over abstract objects of one sort or another. I conclude that property deflationism is perhaps not as deflationary as some philosophers want it to be, but that it’s still apt to call the position ‘deflationary’. (shrink)

This paper outlines an account of numbers based on the numerical equivalence schema (NES), which consists of all sentences of the form ‘#x.Fx=n if and only if ∃nx Fx’, where # is the number-of operator and ∃n is defined in standard Russellian fashion. In the first part of the paper, I point out some analogies between the NES and the T-schema for truth. In light of these analogies, I formulate a minimalist account of numbers, based on the NES, which strongly (...) parallels the minimalist (deflationary) account of truth. One may be tempted to develop the minimalist account in a fictionalist direction, according to which arithmetic is useful but untrue, if taken at face value. In the second part, I argue that this suggestion is not as attractive as it may first appear. The NES suffers from a similar problem to the T-schema: it is deductively weak and does not enable the derivation of any non-trivial generalizations. In the third part of the paper, I explore some strategies to deal with the generalization problem, again drawing inspiration from the literature on truth. In closing this paper, I briefly compare the minimalist to some other accounts of numbers. (shrink)

Nominalists need some account of correctness for sentences committed to the existence of abstract objects. This paper proposes a new statement of such conditions specifically for properties. The account builds on an earlier proposal of mine, but avoids the counter-examples against the latter pointed out by Thomas Schindler, particularly, the sentence ‘There are inexpressible properties’. I argue that the new proposal is independently motivated and more faithful to the spirit of the kind of error-theoretic nominalism that the original proposal was (...) part of. Along the way, I also propose novel treatments of Arthur Pap’s and W.V.O. Quine’s notoriously hard-to-paraphrase sentences about abstracta. (shrink)

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)

This paper outlines an account of numbers based on the numerical equivalence schema, which consists of all sentences of the form ‘#x.Fx=n if and only if ∃nx Fx’, where # is the number-of operator and ∃n is defined in standard Russellian fashion. In the first part of the paper, I point out some analogies between the NES and the T-schema for truth. In light of these analogies, I formulate a minimalist account of numbers, based on the NES, which strongly parallels (...) the minimalist account of truth. One may be tempted to develop the minimalist account in a fictionalist direction, according to which arithmetic is useful but untrue, if taken at face value. In the second part, I argue that this suggestion is not as attractive as it may first appear. The NES suffers from a similar problem to the T-schema: it is deductively weak and does not enable the derivation of any non-trivial generalizations. In the third part of the paper, I explore some strategies to deal with the generalization problem, again drawing inspiration from the literature on truth. In closing this paper, I briefly compare the minimalist to some other accounts of numbers. (shrink)

We offer a novel picture of mathematical language from the perspective of speech act theory. There are distinct speech acts within mathematics, and, as we intend to show, distinct illocutionary force indicators as well. Even mathematics in its most formalized version cannot do without some such indicators. This goes against a certain orthodoxy both in contemporary philosophy of mathematics and in speech act theory. As we will comment, the recognition of distinct illocutionary acts within logic and mathematics and the incorporation (...) of illocutionary force indicators in the formal language for both goes back to Frege’s conception of these topics. We are, therefore, going back to a Fregean perspective. This paper is part of a larger project of applying contemporary speech act theory to the scientific language of mathematics in order to uncover the varieties and regular combinations of illocutionary acts present in it. For reasons of space, we here concentrate only on assertive and declarative acts within mathematics, leaving the investigation of other kinds of acts for a future occasion. (shrink)

Mathematical realism asserts that mathematical objects exist in the abstract world, and that a mathematical sentence is true or false, depending on whether the abstract world is as the mathematical sentence says it is. I raise two objections against mathematical realism. First, the abstract world is queer in that it allows for contradictory states of affairs. Second, mathematical realism does not have a theoretical resource to explain why a sentence about a tricle is true or false. A tricle is an (...) object that changes its shape from a triangle to a circle, and then back to a triangle with every second. (shrink)

Callard (2007) argues that it is metaphysically possible that a mathematical object, although abstract, causally affects the brain. I raise the following objections. First, a successful defence of mathematical realism requires not merely the metaphysical possibility but rather the actuality that a mathematical object affects the brain. Second, mathematical realists need to confront a set of three pertinent issues: why a mathematical object does not affect other concrete objects and other mathematical objects, what counts as a mathematical object, and how (...) we can have knowledge about an unchanging object. (shrink)

The standard argument for the existence of distinctively mathematical objects like numbers has two main premises: some mathematical claims are true, and the truth of those claims requires the existence of distinctively mathematical objects. Most nominalists deny. Those who deny typically reject Quine’s criterion of ontological commitment. I target a different assumption in a standard type of semantic argument for. Benacerraf’s semantic argument, for example, relies on the claim that two sentences, one about numbers and the other about cities, have (...) the same grammatical form. He makes this claim on the grounds that the two sentences are superficially similar. I argue that these grounds are not sufficient. Other sentences with the same superficial form appear to have different grammatical forms. I offer two plausible interpretations of Benacerraf’s number sentence that make use of plural quantification. These interpretations appear not to incur ontological commitments to distinctively mathematical objects, even assuming Quine’s criterion. Such interpretations open a new, plural strategy for the mathematical nominalist. (shrink)

We observe that Putnam’s model-theoretic argument against determinacy of the concept of second-order quantification or that of the set is harmless to the nominalist. It serves as a good motivation for the nominalist philosophy of mathematics. But in the end it can lead to a serious challenge to the nominalist account of mathematical objectivity if some minimal assumptions about the relation between mathematical objectivity and logical objectivity are made. We consider three strategies the nominalist might take to meet this challenge, (...) and we argue that all these strategies are untenable. (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.

The debate concerning the ontological status of musical works is perhaps the most animated debate in contemporary analytic philosophy of music. In my view, progress requires a piecemeal approach. So in this article I hone in on one particular musical work concept – that of the classical Western art musical work; that is, the work concept that regulates classical art-musical practice. I defend a fictionalist analysis – a strategy recently suggested by Andrew Kania as potentially fruitful – and I develop (...) a version of such an analysis in line with a broad commitment to philosophical naturalism. (shrink)

Many have pointed out that the utility of mathematical objects is somewhat disconnected from their ontological status. For example, one might argue that arithmetic is useful whether or not numbers exist. We explore this phenomenon in the context of Divine Conceptualism, which claims that mathematical objects exist as thoughts in the divine mind. While not arguing against DC claims, we argue that DC claims can lead to epistemological uncertainty regarding the ontological status of mathematical objects. This weakens DC attempts to (...) explain the utility of mathematical objects on the basis of their existence. To address this weakness, we propose an appeal to Liggins’ theory of Belief Expressionism. Indeed, we point out that BE is amenable to the ontological claims of DC while also explaining the utility of mathematical objects apart from reliance upon their existence. We illustrate these themes via a case study of Peano Arithmetic. (shrink)

Questions regarding what exists are central to various forms of Buddhist philosophy, as they are to many traditions of philosophy. Interestingly, there is perhaps a clearer consensus in Buddhist thought regarding what does not exist than there may be regarding precisely what does exist, at least insofar as the doctrine of anātman (no self, absence of self) is taken to be a fundamental Buddhist doctrine. It may be noted that many forms of Mahāyāna Buddhist philosophy in particular are considered to (...) offer a quite austere ontology—a rather ‘empty’ account of what exists. Continuing in this vein of ontological austerity, here I will attempt to lay out a relatively novel approach to Buddhist ontology, viz. Buddhist fictionalism. (shrink)

Do numbers exist? Do properties? Do possible worlds? Do fictional characters? Many metaphysicians spend time and effort trying to answer these and other questions about the existence of various entities. These inquiries have recently encountered opposition: a group of philosophers, drawing inspiration from Aristotle, have argued that many or all of the existence questions debated by metaphysicians can be answered trivially, and so are not worth debating. Our task is to defend existence questions from the neo-Aristotelians' attacks.

There is a recent and growing trend in philosophy that involves deferring to the claims of certain disciplines outside of philosophy, such as mathematics, the natural sciences, and linguistics. According to this trend— deferentialism , as we will call it—certain disciplines outside of philosophy make claims that have a decisive bearing on philosophical disputes, where those claims are more epistemically justified than any philosophical considerations just because those claims are made by those disciplines. Deferentialists believe that certain longstanding philosophical problems (...) can be swiftly and decisively dispatched by appeal to disciplines other than philosophy. In this paper we will argue that such an attitude of uncritical deference to any non-philosophical discipline is badly misguided. With reference to the work of John Burgess and David Lewis, we consider deference to mathematics. We show that deference to mathematics is implausible and that main arguments for it fail. With reference to the work of Michael Blome-Tillmann, we consider deference to linguistics. We show that his arguments appealing to deference to linguistics are unsuccessful. We then show that naturalism does not entail deferentialism and that naturalistic considerations even motivate some anti-deferentialist views. Finally, we set out deferentialism’s failings and present our own anti-deferentialist approach to philosophical inquiry. (shrink)

Aim of the paper is to revise Boolos’ reinterpretation of second-order monadic logic in terms of plural quantification ([4], [5]) and expand it to full second order logic. Introducing the idealization of plural acts of choice, performed by a suitable team of agents, we will develop a notion of plural reference . Plural quantification will be then explained in terms of plural reference. As an application, we will sketch a structuralist reconstruction of second-order arithmetic based on the axiom of infinite (...) à la Dedekind, as the unique non-logical axiom. We will also sketch a virtual interpretation of the classical continuum involving no other infinite than a countable plurality of individuals. (shrink)

In To be is to be the object of a possible act of choice the authors defended Boolos’ thesis that plural quantification is part of logic. To this purpose, plural quantification was explained in terms of plural reference, and a semantics of plural acts of choice, performed by an ideal team of agents, was introduced. In this paper, following that approach, we develop a theory of concepts that—in a sense to be explained—can be labeled as a theory of logical concepts. (...) Within this theory, we propose a new logicist approach to natural numbers. Then, we compare our logicism with Frege’s traditional logicism. (shrink)

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)

Linsky and Zalta try to explain how we can refer to mathematical objects by saying that this happens through definite descriptions which may appeal to mathematical theories. I present two issues for their account. First, there is a problem of finding appropriate pre-conditions to reference, which are currently difficult to satisfy. Second, there is a problem of ensuring the stability of the resulting reference. Slight changes in the properties ascribed to a mathematical object can result in a shift of reference (...) and this leads to various problems, e.g., it makes inferring knowledge much harder than it is. (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)

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)

Critical notice of C. Pincock's Mathematics and Scientific Representation (2012).

Hermeneutic non-assertivism is a thesis that mathematical fictionalists might want to endorse in responding to a recent objection due to John Burgess. Brad Armour-Garb has argued that hermeneutic non-assertivism is false. A response is given here to Armour-Garb's argument.

This paper does two things. First, it argues for a metaphilosophical view of conceptual analysis questions; in particular, it argues that the facts that settle conceptual-analysis questions are facts about the linguistic intentions of ordinary folk. The second thing this paper does is argue that if this metaphilosophical view is correct, then experimental philosophy is a legitimate methodology to use in trying to answer conceptual-analysis questions.

In a recent paper in this journal, Mark Balaguer develops and defends a new version of mathematical fictionalism, what he calls ‘Hermeneutic non-assertivism’, and responds to some recent objections to mathematical fictionalism that were launched by John Burgess and others. In this paper I provide some fairly compelling reasons for rejecting Hermeneutic non-assertivism — ones that highlight an important feature of what understanding mathematics involves (or, as we shall see, does not involve).

In a recent paper, Mark Balaguer has responded to the argument that I launched against Hermeneutic Non-Assertivism, claiming that, as a matter of empirical fact, ‘when typical mathematicians utter mathematical sentences, they are doing something that differs from asserting in a pretty subtle way, so that the difference between [asserting] and this other kind of speech act is not obvious’. In this paper, I show the implausibility of this empirical hypothesis.

I argue that it is metaphysically necessary that: (i) every fundamental entity is conscious, and (ii) every fundamental property is a phenomenal property.

Platonism is the view that there exist such things as abstract objects — where an abstract object is an object that does not exist in space or time and which is therefore entirely non-physical and nonmental. Platonism in this sense is a contemporary view. It is obviously related to the views of Plato in important ways, but it is not entirely clear that Plato endorsed this view, as it is defined here. In order to remain neutral on this question, the (...) term ‘platonism’ is spelled with a lower-case ‘p’. (See entry on Plato.) The most important figure in the development of modern platonism is Gottlob Frege (1884, 1892, 1893-1903, 1919). The view has also been endorsed by many others, including Kurt Gödel (1964), Bertrand Russell (1912), and W.V.O. Quine (1948, 1951). (shrink)

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)

This paper, written in Romanian, compares fictionalism, nominalism, and neo-Meinongianism as responses to the problem of objectivity in mathematics, and then motivates a fictionalist view of objectivity as invariance.

One of the important discussions in the philosophy of mathematics, is that centered on Benacerraf’s Dilemma. Benacerraf’s dilemma challenges theorists to provide an epistemology and semantics for mathematics, based on their favourite ontology. This challenge is the point on which all philosophies of mathematics are judged, and clarifying how we might acquire mathematical knowledge is one of the main occupations of philosophers of mathematics. In this thesis I argue that this discussion has overlooked an important part of mathematics, namely mathematics (...) as it is exercised by ordinary people. I do so by looking at the different theories that have been put forward in the recent debate, and showing for each of these that they are unable to account for the mathematical practices of ordinary people. In order to show that these practices do need to be accounted for, I also argue that ordinary people are doing mathematics, i.e. that they engage in properly mathematical practices. Because these practices are properly mathematical, they should be accounted for by any philosophy of mathematics. The conclusion of my thesis, then, is that current theories fail to do something that they should do, while remaining neutral on how well they perform when it comes to accounting for the practices of professional mathematicians. (shrink)

I defend a new position in philosophy of mathematics that I call mathematical inferentialism. It holds that a mathematical sentence can perform the function of facilitating deductive inferences from some concrete sentences to other concrete sentences, that a mathematical sentence is true if and only if all of its concrete consequences are true, that the abstract world does not exist, and that we acquire mathematical knowledge by confirming concrete sentences. Mathematical inferentialism has several advantages over mathematical realism and fictionalism.

This dissertation is centered around a set of apparently conflicting intuitions that we may have about mathematics. On the one hand, we are inclined to believe that the theorems of mathematics are true. Since many of these theorems are existence assertions, it seems that if we accept them as true, we also commit ourselves to the existence of mathematical objects. On the other hand, mathematical objects are usually thought of as abstract objects that are non-spatiotemporal and causally inert. This makes (...) it difficult to understand how we can have knowledge of them and how they can have any relevance for our mathematical theories. I begin by characterizing a realist position in the philosophy of mathematics and discussing two of the most influential arguments for that kind of view. Next, after highlighting some of the difficulties that realism faces, I look at a few alternative approaches that attempt to account for our mathematical practice without making the assumption that there exist abstract mathematical entities. More specifically, I examine the fictionalist views developed by Hartry Field, Mark Balaguer, and Stephen Yablo, respectively. A common feature of these views is that they accept that mathematics interpreted at face value is committed to the existence of abstract objects. In order to avoid this commitment, they claim that mathematics, when taken at face value, is false. I argue that the fictionalist idea of mathematics as consisting of falsehoods is counter-intuitive and that we should aim for an account that can accommodate both the intuition that mathematics is true and the intuition that the causal inertness of abstract mathematical objects makes them irrelevant to mathematical practice and mathematical knowledge. The solution that I propose is based on Rudolf Carnap's distinction between an internal and an external perspective on existence. I argue that the most reasonable interpretation of the notions of mathematical truth and existence is that they are internal to mathematics and, hence, that mathematical truth cannot be used to draw the conclusion that mathematical objects exist in an external/ontological sense. (shrink)