It seems to be impossible to draw metaphysical conclusions about the world merely from our concepts or our language alone. After all, our concepts alone only concern how we aim to represent the world, not how the world in fact is. In this paper I argue that this is mistaken. We can sometimes draw substantial metaphysical conclusions simply from thinking about how we represent the world. But by themselves such conclusions can be flawed if the concepts from which they are (...) drawn are themselves flawed. I propose that we can overcome these limitations by focusing on a special class of concepts: inescapable concepts. Combining arguments about what the world is like from considerations about our concepts alone, together with an argument that the relevant concepts are inescapable, leads to a novel method for metaphysics, which is broadly neo-Kantian. (shrink)

This paper elaborates distinctions between a core and a periphery in the ontological and the conceptual domain associated with natural language. The ontological core-periphery distinction is essential for natural language ontology and is the basis for the central thesis of my 2013 book Abstract Objects and the Semantics of Natural Language, namely that natural language permits reference to abstract objects in its periphery, but not its core.

A realist view of numbers often rests on the following thesis: statements like ‘The number of moons of Jupiter is four’ are identity statements in which the copula is flanked by singular terms whose semantic function consists in referring to a number (henceforth: Identity). On the basis of Identity the realists argue that the assertive use of such statements commits us to numbers. Recently, some anti-realists have disputed this argument. According to them, Identity is false, and, thus, we may deny (...) that the relevant statements commit us to numbers. The present paper argues that the correct linguistic analysis of the relevant number statements supports the anti-realist view that Identity is false. However, as will further be shown, pace the anti-realist, this analysis does not establish that such statements do not commit us to numbers after all. (shrink)

In recent years philosophers have used results from cognitive science to formulate epistemologies of arithmetic :5–18, 2001). Such epistemologies have, however, been criticised, e.g. by Azzouni, for interpreting the capacities found by cognitive science in an overly numerical way. I offer an alternative framework for the way these psychological processes can be combined, forming the basis for an epistemology for arithmetic. The resulting framework avoids assigning numerical content to the Approximate Number System and Object Tracking System, two systems that have (...) so far been the basis of epistemologies of arithmetic informed by cognitive science. The resulting account is, however, only a framework for an epistemology: in the final part of the paper I argue that it is compatible with both platonist and nominalist views of numbers by fitting it into an epistemology for ante rem structuralism and one for fictionalism. Unsurprisingly, cognitive science does not settle the debate between these positions in the philosophy of mathematics, but I it can be used to refine existing epistemologies and restrict our focus to the capacities that cognitive science has found to underly our mathematical knowledge. (shrink)

This monograph examines truth in fiction by applying the techniques of a naturalized logic of human cognitive practices. The author structures his project around two focal questions. What would it take to write a book about truth in literary discourse with reasonable promise of getting it right? What would it take to write a book about truth in fiction as true to the facts of lived literary experience as objectivity allows? It is argued that the most semantically distinctive feature of (...) the sentences of fiction is that they are unambiguously true and false together. It is true that Sherlock Holmes lived at 221B Baker Street and also concurrently false that he did. A second distinctive feature of fiction is that the reader at large knows of this inconsistency and isn't in the least cognitively molested by it. Why, it is asked, would this be so? What would explain it? Two answers are developed. According to the no-contradiction thesis, the semantically tangled sentences of fiction are indeed logically inconsistent but not logically contradictory. According to the no-bother thesis, if the inconsistencies of fiction were contradictory, a properly contrived logic for the rational management of inconsistency would explain why readers at large are not thrown off cognitive stride by their embrace of those contradictions. As developed here, the account of fiction suggests the presence of an underlying three - or four-valued dialethic logic. The author shows this to be a mistaken impression. There are only two truth-values in his logic of fiction. The naturalized logic of Truth in Fiction jettisons some of the standard assumptions and analytical tools of contemporary philosophy, chiefly because the neurotypical linguistic and cognitive behaviour of humanity at large is at variance with them. Using the resources of a causal response epistemology in tandem with the naturalized logic, the theory produced here is data-driven, empirically sensitive, and open to a circumspect collaboration with the empirical sciences of language and cognition. (shrink)

Number words seemingly function both as adjectives attributing cardinality properties to collections, as in Frege’s ‘Jupiter has four moons’, and as names referring to numbers, as in Frege’s ‘The number of Jupiter’s moons is four’. This leads to what Thomas Hofweber calls Frege’s Other Puzzle: How can number words function as modifiers and as singular terms if neither adjectives nor names can serve multiple semantic functions? Whereas most philosophers deny that one of these uses is genuine, we instead argue that (...) number words, like many related expressions, are polymorphic, having multiple uses whose meanings are systematically related via type shifting. (shrink)

This paper investigates a certain puzzling argument concerning number expressions and their meanings, the Easy Argument for Numbers. After finding faults with previous views, I offer a new take on what’s ultimately wrong with the Argument: it equivocates. I develop a semantics for number expressions which relates various of their uses, including those relevant to the Easy Argument, via type-shifting. By marrying Romero ’s :687–737, 2005) analysis of specificational clauses with Scontras ’ semantics for Degree Nouns, I show how to (...) extend Landman ’s Adjectival Theory to numerical specificational clauses. The resulting semantics can explain various contrasts observed by Moltmann, but only if Scontras’ contention that degrees and numbers are sortally distinct is correct. At the same time, the Easy Argument can establish its intended conclusion only if numbers and degrees are mistakenly assumed to be identical. (shrink)

Neo-Fregeans take their argument for arithmetical realism to depend on the availability of certain, so-called broadly syntactic tests for whether a given expression functions as a singular term. The broadly syntactic tests proposed in the neo-Fregean tradition are the so-called inferential test and the Aristotelian test. If these tests are to subserve the neo-Fregean argument, they must be at least adequate, in the sense of correctly classifying paradigm cases of singular terms and non-singular terms. In this paper, I pursue two (...) main goals. On the one hand, I show that the tests’ current state-of-the-art formulations are inadequate and, hence, cannot subserve the neo-Fregean argument. On the other hand, I propose revisions that are adequate and, hence, can subserve this argument. (shrink)

Frege proposed that sentences like ‘The number of planets is eight’ be analysed as identity statements in which the number words refer to numbers. Recently, Friederike Moltmann argued that, pace Frege, such sentences be analysed as so-called specificational sentences in which the number words have the same non-referring semantic function as the number word ‘eight’ in ‘There are eight planets’. The aim of this paper is two-fold. First, I argue that Moltmann fails to show that such sentences should be analysed (...) as specificational sentences. Second, I show that even if they are to be analysed in this way, Moltmann’s proposed specificational analysis is unsatisfactory. (shrink)

Recently, philosophers have tried to develop a version of truthmaker theory which ties the truthmaking relation closely to the notion of fundamentality. In fact, some of these truthmaker-fundamentalists, as I call them, assume that the notion of fundamentality is intelligible in part by citing, as central examples of fundamentals, truthmakers, which they understand necessarily as constituents of fundamental reality. The aim of this paper is first to bring some order and clarity to this discussion, sketching how far TF is compatible (...) with orthodox truthmaking, and then critically to evaluate the limits of TF. It will be argued that truthmaker theory cannot directly help with articulating the nature of fundamental reality and that T-REL does not necessarily relate truths with anything more fundamental, unless what is fundamental is what the truthbearers in question are about. I shall argue that TF faces a rather thorny dilemma and some general problems. I shall present two exhaustive types of fundamentalism on which a version of TF can be based: deflationary and inflationary. It will be argued that each version of TF runs into significant troubles accounting for all truth, specifically ordinary truths and metaphysical truths about the relations between ordinary facts and fundamental facts. I shall not attempt to solve these problems, but rather, at the end, diagnose the issues with TF as lying in the difficulties with reconciling the manifest image with the scientific and metaphysical images of reality. (shrink)

Naturalized metaphysics is based on the idea that philosophy should be guided by the sciences. The paradigmatic science that is relevant for metaphysics is physics because physics tells us what fundamental reality is ultimately like. There are other sciences, however, that de facto play a role in philosophical inquiries about what there is, one of them being the science of language, i.e. linguistics. In this paper I will be concerned with the question what role linguistics should and does play for (...) the metametaphysical question of how our views about fundamental reality can be reconciled with the everyday truisms about what there is. I will present several examples of two kinds of approaches to this question, linguistics-based accounts and purely philosophical accounts, and will discuss their respective methodological merits and shortcomings. In the end I will argue that even proponents of a purely philosophical answer to the metametaphysical question should take the results of linguistics seriously. (shrink)

As introduction to the special issue on the semantics of cardinals, we offer some background on the relevant literature, and an overview of the contributions to this volume. Most of these papers were presented in earlier form at an interdisciplinary workshop on the topic at The Ohio State University, and the contributions to this issue reflect that interdisciplinary character: the authors represent both fields in the title of this journal.

I propose a way of thinking aboout content, and a related way of thinking about ontological commitment. (This is part of a series of four closely related papers. The other three are ‘On Specifying Truth-Conditions’, ‘An Actualist’s Guide to Quantifying In’ and ‘An Account of Possibility’.).

This essay is a study of ontological commitment, focused on the special case of arithmetical discourse. It tries to get clear about what would be involved in a defense of the claim that arithmetical assertions are ontologically innocent and about why ontological innocence matters. The essay proceeds by questioning traditional assumptions about the connection between the objects that are used to specify the truth-conditions of a sentence, on the one hand, and the objects whose existence is required in order for (...) the truth-conditions thereby specified to be satisfied, on the other. This allows one to set forth an assignment of truth-conditions to arithmetical sentences whereby nothing is required of the world in order for the truth-conditions of a truth of pure arithmetic to be satisfied. The essay then argues that such an assignment can be used to account for the a priori knowability of certain arithmetical truths. (shrink)

Kit Fine (2007) outlines an account of semantic coordination, an account motivated by the role of semantic coordination in cognition. Actually, Fine outlines two accounts of semantic coordination, one in terms of co-reference and another in terms of synonymy. I argue, first, that Fine's two accounts are not equivalent, with one being logically stronger than the other, but second and more importantly, that neither account is correct. I outline an alternative account of semantic coordination – the epistemic conception of semantic (...) coordination – that links semantic and epistemic aspects of cognition more directly. The most surprising result of my alternative account of semantic coordination is that semantic coordination is a relation that can hold between elements of thought and language of different semantic types. I close the paper by briefly outlining some applications of the idea that semantic coordination is a relation that can hold across semantic types. (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)

Hartry Field’s epistemological challenge to the mathematical platonist is often cast as an improvement on Paul Benacerraf’s original epistemological challenge. I disagree. While Field’s challenge is more difficult for the platonist to address than Benacerraf’s, I argue that this is because Field’s version is a special case of what I call the ‘sociological challenge’. The sociological challenge applies equally to platonists and fictionalists, and addressing it requires a serious examination of mathematical practice. I argue that the non-sociological part of Field’s (...) challenge amounts to a minor reformulation of Benacerraf’s original challenge. So, I contend, Field’s challenge is not an improvement on Benacerraf’s. What is new to Field’s challenge is as much a problem for the fictionalist as it is for the platonist. (shrink)

How hard is it to answer an ontological question? Ontological trivialism,, inspired by Carnap’s internal-external distinction among “questions of existence”, replies “very easy.” According to, almost every ontologically disputed entity trivially exists. has been defended by many, including Schiffer and Schaffer. In this paper, I will take issue with. After introducing the view in the context of Carnap-Quine dispute and presenting two arguments for it, I will discuss Hofweber’s argument against and explain why it fails. Next, I will introduce a (...) modified version of ontological trivialism, i.e. negative ontological trivialism,, defended by Hofweber, according to which some ontologically disputed entities, e.g. properties, trivially do not exist. I will show that fails too. Then I will outline a Meinongian answer to the original question, namely, ‘How hard is it to answer an ontological question?’ The Carnapian intuition of the triviality of internal questions can be saved by the Meinongian proposal that quantification and reference are not ontologically committing and the Quinean intuition of the legitimacy of interesting ontological questions can be respected by the Meinongian distinction between being and so-being. (shrink)

A common view is that natural language treats numbers as abstract objects, with expressions like the number of planets, eight, as well as the number eight acting as referential terms referring to numbers. In this paper I will argue that this view about reference to numbers in natural language is fundamentally mistaken. A more thorough look at natural language reveals a very different view of the ontological status of natural numbers. On this view, numbers are not primarily treated abstract objects, (...) but rather 'aspects' of pluralities of ordinary objects, namely number tropes, a view that in fact appears to have been the Aristotelian view of numbers. Natural language moreover provides support for another view of the ontological status of numbers, on which natural numbers do not act as entities, but rather have the status of plural properties, the meaning of numerals when acting like adjectives. This view matches contemporary approaches in the philosophy of mathematics of what Dummett called the Adjectival Strategy, the view on which number terms in arithmetical sentences are not terms referring to numbers, but rather make contributions to generalizations about ordinary (and possible) objects. It is only with complex expressions somewhat at the periphery of language such as the number eight that reference to pure numbers is permitted. (shrink)

In this paper we explore the interpretation of quantity expressions in Yudja, an indigenous language spoken in the Amazonian basin, showing that while the language allows reference to exact cardinalities, it does not generally allow reference to exact measure values. It does, however, allow non-exact comparison along continuous dimensions. We use this data to argue that the grammar of exact measurement is distinct from a grammar allowing the expression of exact cardinalities, and that the grammar of counting and the grammar (...) of measurement may use numerals with different, though related interpretations. As Yudja shows, the language of measurement is not automatically acquired along with the knowledge of exact numeral expressions. We show that the ‘gap’ between prelinguistic intuitions about quantity in terms of numerosity and counting, which is bridged by the learning of language expressing exact cardinality, is paralleled by a similar gap between prelinguistic intuitions about quantity on a continuous dimension and measuring: this gap too must be bridged by language which expresses exact measure values. Our results suggest that the enculturation process by which we develop skills to perform abstract operations in the domain of measurement is (1) language dependent and (2) distinct from the process by which we learn to perform abstract calculations in the cardinal domain. (shrink)

In §57 of the Foundations of Arithmetic, Frege famously turns to natural language to support his claim that numbers are ‘self-subsistent objects’:I have already drawn attention above to the fact th...

‘The following sentence is true only if numbers exist: The number of planets is eight. It is true; hence, numbers exist.’ So runs a familiar argument for realism about mathematical objects. But this argument relies on a controversial semantic thesis: that ‘The number of planets’ and ‘eight’ are singular terms standing for the number eight, and the copula expresses identity. This is the ‘Fregean analysis’.I show that the Fregean analysis is false by providing an analysis of sentences such as that (...) best explains the available linguistic data, and according to which no terms in purport to stand for numbers. (shrink)

In this paper I explore a position on which it is possible to eliminate the need for postulating abstract objects through abstraction principles by treating terms for abstracta as ‘incomplete symbols’, using Russell's no-classes theory as a template from which to generalize. I defend views of this stripe against objections, most notably Richard Heck's charge that syntactic forms of nominalism cannot correctly deal with non-first-orderizable quantifcation over apparent abstracta. I further discuss how number theory may be developed in a system (...) treating apparent terms for numbers using these definitions. (shrink)

It would seem that some statements like ‘There are exactly four moons of Jupiter’ and ‘The number of moons of Jupiter is four’ have the same truth-conditions and yet differ in ontological commitment. One strategy to resolve this paradoxical phenomenon is to insist that the statements have not only the same truth-conditions but also the same ontological commitments; the other strategy is to reject the presumption that they have the same truth-conditions. This paper critically examines some popular versions of these (...) two strategies, and defends a new solution according to which the statements have the same ontological commitments and yet differ in truth-conditions. (shrink)

This article investigates the semantics of sentences that express numerical averages, focusing initially on cases such as 'The average American has 2.3 children'. Such sentences have been used both by linguists and philosophers to argue for a disjuncture between semantics and ontology. For example, Noam Chomsky and Norbert Hornstein have used them to provide evidence against the hypothesis that natural language semantics includes a reference relation holding between words and objects in the world, whereas metaphysicians such as Joseph Melia and (...) Stephen Yablo have used them to provide evidence that apparent singular reference need not be taken as ontologically committing. We develop a fully general and independently justified compositional semantics in which such constructions are assigned truth conditions that are not ontologically problematic, and show that our analysis is superior to all extant rivals. Our analysis provides evidence that a good semantics yields a sensible ontology. It also reveals that natural language contains genuine singular terms that refer to numbers. (shrink)

Pairs of sentences like the following pose a problem for ontology: (1) Jupiter has four moons. (2) The number of moons of Jupiter is four. (2) is intuitively a trivial paraphrase of (1). And yet while (1) seems ontologically innocent, (2) appears to imply the existence of numbers. Thomas Hofweber proposes that we can resolve the puzzle by recognizing that sentence (2) is syntactically derived from, and has the same meaning as, sentence (1). Despite appearances, the expressions ‘the number of (...) moons of Jupiter’ and ‘four’ do not function semantically as singular terms in (2). Hofweber’s primary evidence for this proposal concerns differences in the focus-related communicative functions of (1) and (2). In this paper I raise several serious problems for Hofweber’s proposal, and for his attempt to support it by appeal to focus-related phenomena. I conclude by offering independent evidence for an alternative, purely pragmatic resolution of the ontological puzzle. (shrink)

The problem of change plays a central role in the metaphysics of time and material objects, and whoever does best in solving this problem has a leg up when it comes to choosing a metaphysics of time and material objects. But whether this central role of the problem of change in metaphysics is legitimate is not at all clear. This is so in part since it is not clear what the problem of change is, and why it is a problem (...) in metaphysics. We will investigate what metaphysical problem the problem of change might be, and how it relates to various other problems related to change that are studied in the empirical sciences. The problem of change can thus be a case study of what makes a problem a metaphysical one and how metaphysics relates to other parts of inquiry. We conclude that the central role the problem of change is given in the contemporary metaphysics of time is not justified. (shrink)

I wonder which one in a series of characters Agustín Rayo really is, with an emphasis on objective correctness and semantics.

In their recent article ‘Resolving Frege’s other Puzzle’ Eric Snyder, Richard Samuels, and Stewart Shapiro defend a semantic type-shifting solution to Frege’s other Puzzle and criticize my own cognitive type-shifting solution. In this article I respond to their criticism and in turn point to several problems with their preferred solution. In particular, I argue that they conflate semantic function and semantic value, and that their proposal is neither based on general semantic type-shifting principles nor adequate to the data.

Hilbert’s program in the philosophy of mathematics comes in two parts. One part is a technical part. To carry out this part of the program one has to prove a certain technical result. The other part of the program is a philosophical part. It is concerned with philosophical questions that are the real aim of the program. To carry out this part one, basically, has to show why the technical part answers the philosophical questions one wanted to have answered. Hilbert (...) probably thought that he had completed the philosophical part of his program, maybe up to a few details. What was left to do was the technical part. To carry it out one, roughly, had to give a precise axiomatization of mathematics and show that it is consistent on purely finitistic grounds. This would come down to giving a relative consistency proof of mathematics in finitist mathematics, or to give a proof-theoretic reduction of mathematics on to finitist mathematics (we will look at these notions in more detail soon). It is widely believed that Gödel’s theorems showed that the technical part of Hilbert’s program could not be carried out. Gödel’s theorems show that the consistency of arithmetic can not even be proven in arithmetic, not to speak of by finitistic means alone. So, the technical part of Hilbert’s program is hopeless, and since Hilbert’s program essentially relied on both the technical and the philosophical part, Hilbert’s program as a whole is hopeless. Justified as this attitude is, it is a bit unfortunate. It is unfortunate because it takes away too much attention from the philosophical part of Hilbert’s program. And this is unfortunate for two reasons. (shrink)

Thomas Hofweber's well-known ontological project crucially involves inferring negative existential statements from statements of non-reference, i.e. statements that say that some term or terms do not refer. Here, after explaining the context of this move, I aim to show that it is fallacious, and that this vitiates Hofweber's ontological project.

Cognitive partitions are useful. The notion of numerical identity helps us induce them. Consider, for instance, the role of identity in representing an equivalence relation like taking the same train. This expressive function of identity has been largely overlooked. Other possible functions of the concept have been over-emphasized. It is not clear that we use identity to represent individual objects or quantify over collections of them. Understanding what the concept is good for looks especially urgent in light of the fact (...) that numerical identities themselves are tangibly trifling. (shrink)

ABSTRACT How do non-experts single out numbers for reference? Linnebo has argued that they do so using a criterion of identity based on the ordinal properties of numerals. Neo-logicists, on the other hand, claim that cardinal properties are the basis of individuation, when they invoke Hume’s Principle. I discuss empirical data from cognitive science and linguistics to answer how non-experts individuate numbers better in practice. I use those findings to develop an alternative account that mixes ordinal and cardinal properties to (...) provide a detailed answer to the question: how do we in fact semantically individuate numbers? (shrink)

Can concepts be acquired by testing hypotheses about these concepts? Fodor famously argued that this is not possible. Testing the correct hypothesis would require already possessing the concept. I argue that this does not generally hold for mathematical concepts. I discuss specific, empirically motivated, hypotheses for number concepts that can be tested without needing to possess the relevant number concepts. I also argue that one can test hypotheses about the identity conditions of other mathematical concepts, and then fix the application (...) conditions based on those hypotheses—under the assumption that the neo‐logicist view on abstraction principles is correct. (shrink)

In recent years philosophers have used results from cognitive science to formulate epistemologies of arithmetic :5–18, 2001). Such epistemologies have, however, been criticised, e.g. by Azzouni, for interpreting the capacities found by cognitive science in an overly numerical way. I offer an alternative framework for the way these psychological processes can be combined, forming the basis for an epistemology for arithmetic. The resulting framework avoids assigning numerical content to the Approximate Number System and Object Tracking System, two systems that have (...) so far been the basis of epistemologies of arithmetic informed by cognitive science. The resulting account is, however, only a framework for an epistemology: in the final part of the paper I argue that it is compatible with both platonist and nominalist views of numbers by fitting it into an epistemology for ante rem structuralism and one for fictionalism. Unsurprisingly, cognitive science does not settle the debate between these positions in the philosophy of mathematics, but I it can be used to refine existing epistemologies and restrict our focus to the capacities that cognitive science has found to underly our mathematical knowledge. (shrink)

This paper contributes to the debate over the so-called “easy argument for numbers”, an argument that uses evidence from natural language to support the metaphysically significant claim that numbers exist. It presents novel data showing that critical examples in the literature are ambiguous between two readings, contrary to previous assumptions. It then accounts for these data using independently motivated linguistic theory. The account developed rescues the easy argument from the primary challenges leveled against it in the literature and sets the (...) agenda for future work to determine whether or not the argument is valid. (shrink)

This paper investigates the nature of scientific realism. I begin by considering the anomalous fact that Bas van Fraassen’s account of scientific realism is strikingly similar to Arthur Fine’s account of scientific non-realism. To resolve this puzzle, I demonstrate how the two theorists understand the nature of truth and its connection to ontology, and how that informs their conception of the realism debate. I then argue that the debate is much better captured by the theory of truthmaking, and not by (...) any particular theory of truth. To be a scientific realist is to adopt a realism-relevant account of what makes true the scientific theories one accepts. The truthmaking approach restores realism’s metaphysical core—distancing itself from linguistic conceptions of the debate—and thereby offers a better characterization of what is at stake in the question of scientific realism. (shrink)

Proper names are generally taken to be count nouns. This paper argues that this is mistaken and that at least in some languages, for example German, names divide into mass and count. Making use of Kayne's (2005, 2010) theory of light nouns, this paper argues that light nouns are part of (simple) names and that a mass-count distinction among light nouns explains the behavior of certain types of names in German as mass rather than count. The paper elaborates the role (...) of light nouns with new generalizations regarding their linguistic behavior in quantificational and pronominal NPs, their selection of relative pronouns in German, and a general difference in the support of plural anaphora between English and German. (shrink)

In this thesis, I aim to motivate a particular philosophy of mathematics characterised by the following three claims. First, mathematical sentences are generally speaking false because mathematical objects do not exist. Second, people typically use mathematical sentences to communicate content that does not imply the existence of mathematical objects. Finally, in using mathematical language in this way, speakers are not doing anything out of the ordinary: they are performing straightforward assertions. In Part I, I argue that the role played by (...) mathematics in our scientific explanations is a purely expressive one, merely allowing us to say more about the physical world than we would otherwise be able to. Mathematical objects do not need to exist for mathematics to play this role. This proposal puts a normative constraint on our use of mathematical language: we ought to use mathematically presented theories to express belief only in the consequences they have for non-mathematical things. In Part II, I will argue that what the normative proposal recommends is in fact what people generally do in both pure and applied mathematical contexts. I motivate this claim by showing that it is predicted by our best general means of analysing natural language. I provide a semantic theory of applied arithmetical sentences that reveals they do not purport to refer to numbers, as well as a pragmatic theory for pure mathematical language use which reveals that pure mathematical utterances do not typically communicate content that implies the existence of mathematical objects. In conclusion, I show that the emerging hermeneutic fictionalist position is preferable to any alternative interpretation of mathematical discourse as aimed at describing a domain of independently existing abstract mathematical objects. (shrink)

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)

In many languages, the same particles build quantifier words and serve as connectives, additive and scalar particles, question markers, existential verbs, and so on. Do the roles of each particle form a natural class with a stable semantics? Are the particles aided by additional elements, overt or covert, in fulfilling their varied roles? I propose a unified analysis, according to which the particles impose partial ordering requirements (glb and lub) on the interpretations of their hosts and the immediate larger contexts, (...) but do not embody algebraic operations themselves. (shrink)

There is an emerging view according to which countability is not an integral part of the lexical meaning of singular count nouns, but is ‘added on’ or ‘made available’, whether syntactically, semantically or both. This view has been pursued by Borer and Rothstein among others in order to deal with classifier languages such as Chinese as well as challenges to standard views of the mass-count distinction such as object mass nouns such as furniture. I will discuss a range of data, (...) partly from German, that such a grammar-based view of countability receives support when applied to verbs with respect to the event argument position. Verbs themselves fail to specify events as countable in English and related languages; instead countability is made available only by the use of the event classifier time or else particular lexical items, such as frequency expressions, German beides ‘both’, or the nominalizing light noun -thing. The paper will not adopt or elaborate a particular version of the grammar-based view of countability, but rather critically discuss existing versions and present two semantic options of elaborating the view. (shrink)

This thesis articulates the resonances between J. M. Coetzee's lifelong engagement with mathematics and his practice as a novelist, critic, and poet. Though the critical discourse surrounding Coetzee's literary work continues to flourish, and though the basic details of his background in mathematics are now widely acknowledged, his inheritance from that background has not yet been the subject of a comprehensive and mathematically- literate account. In providing such an account, I propose that these two strands of his intellectual trajectory not (...) only developed in parallel, but together engendered several of the characteristic qualities of his finest work. The structure of the thesis is essentially thematic, but is also broadly chronological. Chapter 1 focuses on Coetzee's poetry, charting the increasing involvement of mathematical concepts and methods in his practice and poetics between 1958 and 1979. Chapter 2 situates his master's thesis alongside archival materials from the early stages of his academic career, and thus traces the development of his philosophical interest in the migration of quantificatory metaphors into other conceptual domains. Concentrating on his doctoral thesis and a series of contemporaneous reviews, essays, and lecture notes, Chapter 3 details the calculated ambivalence with which he therein articulates, adopts, and challenges various statistical methods designed to disclose objective truth. Chapter 4 explores the thematisation of several mathematical concepts in Dusklands and In the Heart of the Country. Chapter Five considers Waiting for the Barbarians and Foe in the context provided by Coetzee's interest in the attempts of Isaac Newton to bridge the gap between natural language and the supposedly transparent language of mathematics. Finally, Chapter 6 locates in Elizabeth Costello and Diary of a Bad Year a cognitive approach to the use of mathematical concepts in ethics, politics, and aesthetics, and, by analogy, a central aspect of the challenge Coetzee's late fiction poses to the contemporary literary landscape. (shrink)

The subject of this paper is the epistemology of identity: a general theory of knowledge, evidence and justification for the claim that one thing is identical to another. Although identity figures significantly in our epistemic lives, this is a topic that, to the best of my knowledge, has gone entirely unexplored. Initial attempts to integrate such an epistemology into existing theories of evidence---many of which are tailor-made for contingent propositions---are confounded by the necessity of identity. I defend a restricted form (...) of skepticism according to which the only knowable identity claims are trivial. (shrink)

I develop a technique for specifying truth-conditions.

Words have meaning. Sentences also have meaning, but their meaning is different in kind from any collection of the meanings of the words they contain. I discuss two puzzles related to this difference. The first is how the meanings of the parts of a sentence combine to give rise to a unified sentential meaning, as opposed to a mere collection of disparate meanings (UP1). The second is why the formal ontology of linguistic meaning changes when grammatical structure is built up (...) (UP2). For example, the meaning of a sentence is a proposition evaluable for truth and falsity. In contrast, a collection of the meanings of its parts does not constitute a proposition and is not evaluable for truth. These two puzzles are closely related, since change in formal ontology is the clearest sign of the unity of meaning. The most popular strategy for answering them is taking the meanings of the parts as abstractions from primitive sentence meanings. However, I argue that, given plausible psychological constraints, sentence meanings cannot be taken as explanatory primitives. Drawing on recent work in Generative Grammar and its philosophy, I suggest that the key to both unity questions is to distinguish strictly between lexical and grammatical meaning. The latter is irreducible and determines how lexical content is used in referential acts. I argue that these referential properties determine a formal ontology, which explains why and how formal ontology changes when grammatical structure is built up (UP2). As for UP1, I suggest that, strictly speaking, lexical meanings never combine. Instead, whenever grammar specifies a formal ontology for the lexical meanings entering a grammatical derivation, further lexical (or phrasal) meanings can only specify aspects of this recursive grammatical process. In this way, contemporary grammatical theory can be used to address old philosophical problems. (shrink)