According to a familiar, simple argument, numbers exist because sentences like ‘Two is an even number’ are true. Whereas realists accept the argument as sound, anti-realists either reject that number words function referentially in such sentences (non-referentialism) or else that such sentences are true (fictionalism). We argue that this dialectic, though familiar, drastically underestimates the extent to which natural language supports realism. Indeed, if dominant accounts of number and measurement-related expressions within linguistic semantics are correct, then far more than just our overt talk about numbers as objects would not be true if numbers do not exist. The purpose of this paper is to sketch this strengthened argument for realism and to survey its consequences for extant forms of anti-realism.

Any comprehensive semantics for number words, such as ‘two’, must explain why they can take on such a wide range of related meanings. Extant analyses offer promising explanations of a wide range of these meanings. On the other hand, they face a number of substantial, previously unrecognized theoretical and empirical challenges: they threaten incoherence, they identify numbers with the wrong sorts of entities, and they seemingly cannot explain the full range of meanings number words can take on. I argue that recognizing numbers as kinds affords a new type of polymorphic analysis, one that not only avoids these challenges, but also improves on extant explanations of polysemy, by expanding their empirical coverage.

According to singularism, 'the students' refers to a single collective entity, e.g. a sum or set. In contrast, according to pluralism, 'the students' plurally refers to multiple students at once, through the primitive relation of plural reference. Although it was originally designed exclusively for plural nouns, this paper addresses whether plural reference can be extended so as to provide an empirically adequate semantics for mass nouns, such as 'the furniture', as well, as certain pluralists have maintained. Our contention is that extant pluralist analyses of mass nouns suffer from two major setbacks. First, while they may or may not explain the semantic commonalities between plural and mass nouns, they do not explain their semantic differences. Secondly, pluralism is inconsistent with the possibility that mass nouns are gunky, i.e. fail to have atomic parts. However, we provide three kinds of semantic arguments showing that the gunkiness of at least some mass nouns is not only possible, but plausible.

Hofweber (Ontology and the ambitions of metaphysics, Oxford University Press, 2016) argues for a thesis he calls “internalism” with respect to natural number discourse: no expressions purporting to refer to natural numbers in fact refer, and no apparent quantification over natural numbers actually involves quantification over natural numbers as objects. He argues that while internalism leaves open the question of whether other kinds of abstracta exist, it precludes the existence of natural numbers, thus establishing what he calls “restricted nominalism” about natural numbers. We argue that Hofweber’s internalism fails to establish restricted nominalism. Not only is his primary argument for restricted nominalism invalid, the analysis of quantification proposed threatens to collapse internalism into either a traditional form of error theory or realism.

This Element, written for researchers and students in philosophy and the behavioral sciences, reviews and critically assesses extant work on number concepts in developmental psychology and cognitive science. It has four main aims. First, it characterizes the core commitments of mainstream number cognition research, including the commitment to representationalism, the hypothesis that there exist certain number-specific cognitive systems, and the key milestones in the development of number cognition. Second, it provides a taxonomy of influential views within mainstream number cognition research, along with the central challenges these views face. Third, it identifies and critically assesses a series of core philosophical assumptions often adopted by number cognition researchers. Finally, the Element articulates and defends a novel version of pluralism about number concepts.

Perhaps the most pressing challenge for singularism-the predominant view that definite plurals like 'the students' singularly refer to a collective entity, such as a mereological sum or set-is that it threatens paradox. Indeed, this serves as a primary motivation for pluralism-the opposing view that definite plurals refer to multiple individuals simultaneously through the primitive relation of plural reference. Groups represent one domain in which this threat is immediate. After all, groups resemble sets in having a kind of membership-relation and iterating: we can have groups of groups, groups of groups of groups, etc. Yet there cannot be a group of all non-self-membered groups. In response, we develop a potentialist theory of groups according to which we always can, but do not have to, form a group from any sum. Modalizing group-formation makes it a species of potential, as opposed to actual or completed, infinity. This allows for a consistent, plausible, and empirically adequate treatment of natural language plurals, one which is motivated by the iterative nature of syntactic and semantic processes more generally .

In this paper, we outline and critically evaluate Thomas Hofweber's solution to a semantic puzzle he calls Frege's Other Puzzle. After sketching the Puzzle and two traditional responses to it—the Substantival Strategy and the Adjectival Strategy—we outline Hofweber's proposed version of Adjectivalism. We argue that two key components—the syntactic and semantic components—of Hofweber's analysis both suffer from serious empirical difficulties. Ultimately, this suggests that an altogether different solution to Frege's Other Puzzle is required.

Saul Kripke once noted that there is a tight connection between computation and de re knowledge of whatever the computation acts upon. For example, the Euclidean algorithm can produce knowledge of which number is the greatest common divisor of two numbers. Arguably, algorithms operate directly on syntactic items, such as strings, and on numbers and the like only via how the numbers are represented. So we broach matters of notation. The purpose of this article is to explore the relationship between the notations acceptable for computation, the usual idealizations involved in theories of computability, flowing from Alan Turing’s monumental work, and de re propositional attitudes toward numbers and other mathematical objects.

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.

What are the meanings of number expressions, and what can they tell us about questions of central importance to the philosophy of mathematics, specifically 'Do numbers exist?' This Element attempts to shed light on this question by outlining a recent debate between substantivalists and adjectivalists regarding the semantic function of number words in numerical statements. After highlighting their motivations and challenges, I develop a comprehensive polymorphic semantics for number expressions. I argue that accounting for the numerous meanings and how they are related leads to a strengthened argument for realism, one which renders familiar forms of nominalism highly implausible.

Definite group nouns, such as "the deck of cards," raise two important kinds of problems. Philosophically, they raise the ancient Problem of the Many: How can one deck be many cards? Linguistically, they threaten paradox: If such expressions singularly refer to groups as set-like entities, then analyses employing such entities threaten to be incoherent, due to Russell's paradox. On the other hand, no paradox is threatened if, per the suggestion of Alex Oliver and Timothy Smiley, "the deck of cards" is a pseudo-singular term, that is, a term which is syntactically singular but semantically plural, exploiting the primitive relation of plural reference. Against this, we argue that pseudo-singularity is linguistically and logically untenable. As such, it will not plausibly solve either kind of problem raised by definite group nouns.

Clarke and Beck import certain assumptions about the nature of numbers. Although these are widespread within research on number cognition, they are highly contentious among philosophers of mathematics. In this commentary, we isolate and critically evaluate one core assumption: the identity thesis.

The primary argument against mereological singularism-the view that definite plural noun phrases like 'the students' refer to "set-like entities"-is that it is ultimately incoherent. The most forceful form of this charge is due to Barry Schein, who argues that singularists must accept a certain comprehension principle which entails the existence of things having the contradictory property of being both atomic and non-atomic. The purpose of this paper is to defuse Schein's argument, by noting three necessary and independently motivated restrictions on the metalinguistic predicates 'atom' and 'non-atom': both are sort, property, and context-relative. With these restrictions in place, Schein's problematic assumption becomes evident: his presumed singularist analysis of 'non-atom' conflates the metalanguage with the meta-metalanguage, i.e. the language used to talk about the metalanguage.

Most philosophers are familiar with the metaphysical puzzle of the statue and the clay. A sculptor begins with some clay, eventually sculpting a statue from it. Are the clay and the statue one and the same thing? Apparently not, since they have different properties. For example, the clay could survive being squashed, but the statue could not. The statue is recently formed, though the clay is not, etc. Godehart Link 1983's highly influential analysis of the count/mass distinction recommends that English draws a distinction between uncountable "stuff" and countable "things". There are two mereological relations, related in specific ways. Our primary question here is whether an empirically adequate account of the mass/count distinction really does require distinguishing "things" from "stuff", and thus postulating two corresponding mereological relations, or if instead positing only one sort of entity and corresponding mereological relation is sufficient, as other semantic theories would have it. This question is meant to be one of what we call natural language mereology. We are asking about the mereological commitments of English, or perhaps competent speakers of English, and not about ultimate reality as such. There is no pretense that we will definitively solve the metaphysical puzzle of the statue and clay.

One of the more distinctive features of Bob Hale and Crispin Wright's neologicism about arithmetic is their invocation of Frege's Constraint - roughly, the requirement that the core empirical applications for a class of numbers be "built directly into" their formal characterization. In particular, they maintain that, if adopted, Frege's Constraint adjudicates in favor of their preferred foundation - Hume's Principle - and against alternatives, such as the Dedekind-Peano axioms. In what follows we establish two main claims. First, we show that, if sound, Hale and Wright's arguments for Frege's Constraint at most establish a version on which the relevant application of the naturals is transitive counting - roughly, the counting procedure by which numerals are used to answer "how many"-questions. Second, we show that this version of Frege's Constraint fails to adjudicate in favor of Hume's Principle. If this is the version of Frege's Constraint that a foundation for arithmetic must respect, then Hume's Principle no more - and no less - meets the requirement than the Dedekind-Peano axioms do.

According to what I call the Traditional View, there is a fundamental semantic distinction between counting and measuring, which is reflected in two fundamentally different sorts of scales: discrete cardinality scales and dense measurement scales. Opposed to the Traditional View is a thesis known as the Universal Density of Measurement: there is no fundamental semantic distinction between counting and measuring, and all natural language scales are dense. This paper considers a new argument for the latter, based on a puzzle I call the Fractional Cardinalities Puzzle: if answers to ‘how many’-questions always designate cardinalities, and if cardinalities are necessarily discrete, then how can e.g. ‘2.38’ be a correct answer to the question ‘How many ounces of water are in the beaker?’? If cardinality scales are dense, then the answer is obvious: ‘2.38’ designates a fractional cardinality, contra the Traditional View. However, I provide novel evidence showing that ‘many’ is not uniformly associated with the dimension of cardinality across contexts, and so ‘how many’-questions can ask about other kinds of measures, including e.g. volume. By combining independently motivated analyses of cardinal adjectives, measure phrases, complex fractions, and degrees, I develop a semantics intended to defend the Traditional View against purported counterexamples like this and others which have received a fair amount of recent philosophical attention.

Leon Horsten’s The Metaphysics and Mathematics of Arbitrary Objects is an important contribution to a relatively neglected area of analytic metaphysics: arbitrary objects. Building on Fine’s seminal work [1985], Horsten develops a new theory of arbitrary objects which not only aims to improve upon Fine’s original theory in various ways, but also to shed new light on disparate philosophical issues. These include the coherence of the nineteenth-century conception of variable mathematical quantities, realist/anti-realist debates about structure within the philosophy of mathematics, and the conception of random variables within statistics. Indeed, Horsten shows how his conception of arbitrary objects fruitfully extends to the natural numbers, mathematical structures more generally, and, at least to a certain extent, random variables. The book is dense but well written and relatively self-contained. I should mention, however, that certain technical parts of the book presuppose prior familiarity with basic concepts in proof theory, model theory, quantified modal logic, set theory, number theory, and probability theory. This material may be daunting for readers without these backgrounds. Nevertheless, the book is strongly recommended for philosophers with serious interests in any of the following topics:

What is the relation between some things and the set of these things? Mathematical practice does not provide a univocal answer. On the one hand, it relies on ordinary plural talk, which is implicitly committed to a traditional form of plural logic. On the other hand, mathematical practice favors a liberal view of definitions which entails that traditional plural logic must be restricted. We explore this predicament and develop a “critical” alternative to traditional plural logic.

This paper is concerned with Gottlob Frege’s theory of the real numbers as sketched in the second volume of his masterpiece Grundgesetze der Arithmetik. It is perhaps unsurprising that Frege’s theory of the real numbers is intimately intertwined with and largely motivated by his metaphysics. The account raises interesting, and surprisingly underexplored, questions about Frege’s metaphysics: Can this metaphysics even accommodate mass quantities like water, gold, light intensity, or charge? Frege’s main complaint with his contemporaries Cantor and Dedekind is that their theories of the real numbers do not build the applicability of the real numbers directly into the construction. In taking Cantor and Dedekind’s Arithmetic theories to be insufficient, clearly Frege takes it to be a desideratum on a theory of the real numbers that their applicability be essential to their construction. We begin with a detailed review of Frege’s theory, one that mirrors Frege’s exposition in structure. This is followed by a critique, outlining Frege’s linguistic motivation for ontologically distinguishing the cardinal numbers from the real numbers. We briefly consider how Frege’s metaphysics might need to be developed, or amended, to accommodate some of the problems. Finally, we offer a detailed examination of Frege’s Application Constraint – that the reals ought to have their applicability built directly into their characterization. It bears on deeper questions concerning the relationship between sophisticated mathematical theories and their applications.