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)

This paper is concerned with Noun Phrases (NPs, henceforth) occurring in two constructions: concealed question NPs and NP subjects of specificational sentences. The first type of NP is illustrated in (1). The underlined NPs in (1) have been called ‘concealed questions’ because sentences that embed them typically have the same truth-conditional meaning as the corresponding versions with a full-fledged embedded interrogative clause, as illustrated in (2) (Heim 1979).

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)

The complement of intensional transitive verbs, like any nonreferential complement, can be replaced by a ‘special quantifier’ or ‘special pronoun’ such as 'something', 'the same thing', or 'what'. In this paper, I will defend the ‘Nominalization Theory’ of special quantifiers against a range of apparent counterexamples involving intensional transitive verbs.

This is a brief response to Thomas Hofweber's "Extraction, Displacement and Focus: A Reply to Balcerak Jackson" (Linguistics and Philosophy 37.3 (2014)), which was a reply to my "Defusing Easy Arguments for Numbers" (Linguistics and Philosophy 36.6 (2013)).

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)

In his groundbreaking Grundlagen, Frege (1884) pointed out that number words like ‘four’ occur in ordinary language in two quite different ways and that this gives rise to a philosophical puzzle. On the one hand ‘four’ occurs as an adjective, which is to say that it occurs grammatically in sentences in a position that is commonly occupied by adjectives. Frege’s example was (1) Jupiter has four moons, where the occurrence of ‘four’ seems to be just like that of ‘green’ in (...) (2) Jupiter has green moons. On the other hand, ‘four’ occurs as a singular term, which is to say that it occurs in a position that is commonly occupied by paradigmatic cases of singular terms, like proper names: (3) The number of moons of Jupiter is four. Here ‘four’ seems to be just like ‘Wagner’ in (4) The composer of Tannhäuser is Wagner, and both of these statements seem to be identity statements, ones with which we claim that what two singular terms stand for is identical. But that number words can occur both as singular terms and as adjectives is puzzling. Usually adjectives cannot occur in a position occupied by a singular term, and the other way round, without resulting in ungrammaticality and nonsense. To give just one example, it would be ungrammatical to replace ‘four’ with ‘the number of moons of Jupiter’ in (1): (5) *Jupiter has the number of moons of Jupiter moons. This ungrammaticality results even though ‘four’ and ‘the number of moons of Jupiter’ are both apparently singular terms standing for the same object in (3). So, how can it be that number words can occur both as singular terms and as adjectives, while other adjectives cannot occur as singular terms, and other singular terms cannot occur as adjectives? (shrink)

Existence questions have been topics for heated debates in metaphysics, but this book argues that they can often be answered easily, by trivial inferences from uncontroversial premises. This 'easy' approach to ontology leads to realism about disputed entities, and to the view that metaphysical disputes about existence questions are misguided.

Die Grundlagen gehören zu den klassischen Texten der Sprachphilosophie, Logik und Mathematik. Frege stützt sein Programm einer Begründung von Arithmetik und Analysis auf reine Logik, indem er die natürlichen Zahlen als bestimmte Begriffsumfänge definiert. Die philosophische Fundierung des Fregeschen Ansatzes bilden erkenntnistheoretische und sprachphilosophische Analysen und Begriffserklärungen. Studienausgabe aufgrund der textkritisch herausgegebenen Jubiläumsausgabe (Centenarausgabe). Mit Einleitung, Anmerkungen, Literaturverzeichnis und Namenregister.

One puzzling feature of talk about properties, propositions and natural numbers is that statements that are explicitly about them can be introduced apparently without change of truth conditions from statements that don't mention them at all. Thus it seems that the existence of numbers, properties and propositions can be established`from nothing'. This metaphysical puzzle is tied to a series of syntactic and semantic puzzles about the relationship between ordinary, metaphysically innocent statements and their metaphysically loaded counterparts, statements that explicitly mention (...) numbers, properties and propositions, but nonetheless appear to be equivalent to the former. I argue that the standard solutions to the metaphysical puzzles make a mistaken assumption about the semantics of the loaded counterparts. Instead I propose a solution to the syntactic and semantic puzzles, and argue that this solution also gives us a new solution to the metaphysical puzzle. I argue that instead of containing more semantically singular terms that aim to refer to extra entities, the loaded counterparts are focus constructions. Their syntactic structure is in the service of presenting information with a focus, but not to refer to new entities. This will allow us to spell out Frege's metaphor of content carving. (shrink)