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. (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)

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. (shrink)

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.

I argue for two major claims in this paper. First, I argue that the linguistic evidence best supports a certain form of contextualism about predicates of personal taste (PPTs) like ?fun? and ?tasty?. In particular, I argue that these adjectives are both individual-level predicates (ILPs) and anaphoric implicit argument taking predicates (IATPs). As ILPs, these naturally form generics. As anaphoric IATPs, PPTs show the same dependencies on context and distributional behavior as more familiar anaphoric IATPs, for example, ?local? and ?apply?. (...) Moreover, they are subject to important binding-related phenomena, for example, various kinds of anaphora, binding arguments, and strict-sloppy ambiguities. The latter are particularly problematic for relativism, contextualism's major competitor. I argue that relativism cannot account for these binding facts in a non-ad hoc manner. Second, I argue that disagreement is a predictable feature of dialogues involving PPTs in conjunction with the contextualist semantics defended. Appealing to ?off the shelf? analyses of genericity, I show how disagreement arises in just the cases we would expect, licensing just the kinds of responses we would expect. They also suggest promising explanations, unavailable to relativists, of various otherwise puzzling features of disagreements involving PPTs. (shrink)

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. (shrink)

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. (shrink)

There are multiple formal characterizations of the natural numbers available. Despite being inter-derivable, they plausibly codify different possible applications of the naturals – doing basic arithmetic, counting, and ordering – as well as different philosophical conceptions of those numbers: structuralist, cardinal, and ordinal. Nevertheless, some influential philosophers of mathematics have argued for a non-egalitarian attitude according to which one of those characterizations is more “legitmate” in virtue of being “more basic” or “more fundamental”. This paper addresses two related issues. First, (...) we review some of these non-egalitarian arguments, lay out a laundry list of different, legitimate, notions of relative priority, and suggest that these arguments plausibly employ different such notions. Secondly, we argue that given a metaphysical-cum-epistemological gloss suggested by Frege’s foundationalist epistomology, the ordinals are plausibly more basic than the cardinals. This is just one orientation to relative priority one could take, however. Ultimately, we subscribe to an egalitarian attitude towards these formal characterizations: they are, in some sense, equally “legitimate”. (shrink)

There are multiple formal characterizations of the natural numbers available. Despite being inter-derivable, they plausibly codify different possible applications of the naturals – doing basic arithmetic, counting, and ordering – as well as different philosophical conceptions of those numbers: structuralist, cardinal, and ordinal. Some influential philosophers of mathematics have argued for a non-egalitarian attitude according to which one of those characterizations is ‘more basic’ or ‘more fundamental’ than the others. This paper addresses two related issues. First, we review some of (...) these non-egalitarian arguments, lay out a laundry list of different, legitimate, notions of relative priority, and suggest that these arguments plausibly employ different such notions. Secondly, we argue that given a metaphysical-cum-epistemological gloss suggested by Frege's foundationalist epistemology, the ordinals are plausibly more basic than the cardinals. This is just one orientation to relative priority one could take, however. Ultimately, we subscribe to an egalitarian attitude towards these formal characterizations: they are, in some sense, equally ‘legitimate’. (shrink)

We address a puzzle about the meanings of fraction words due to Nathan Salmon. Counting 2 1/2 oranges seemingly requires enumerating a collection of objects with a non-whole cardinal number, which is incoherent by the lights of traditional analyses of cardinality expressions. Thus, some argue that counting 2 1/2 oranges really amounts to measuring quantities of orange-stuff . However, we claim that `2 1/2 oranges' is ambiguous between counting and measuring interpretations, and recognizing this affords a novel solution to Salmon's (...) puzzle which is consistent with traditional analyses. (shrink)

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. (shrink)

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. (shrink)

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. (shrink)

abstractWe address a puzzle about the meanings of fraction words, due to Nathan Salmon. Counting 212 oranges seemingly requires enumerating a collection of objects with a non-whole cardinal number,...

Thought: A Journal of Philosophy, EarlyView.

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. (shrink)

A core commitment of Bob Hale and Crispin Wright’s neologicism 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. According to these neologicists, if legitimate, Frege’s Constraint adjudicates in favor of their preferred foundation—Hume’s Principle—and against alternatives, such as the Dedekind–Peano axioms. In this paper, we consider a recent argument for legitimating Frege’s Constraint due to Hale, according to which the primary empirical application of (...) the naturals is transitive counting, or answering ‘how many’-questions using numerals. We make two claims regarding Hale’s argument. First, it fails to legitimate Frege’s Constraint in virtue of resting on unsupported and highly contentious assumptions. Secondly, even if sound, Hale’s argument would vindicate a version of Frege’s Constraint which fails to adjudicate in favor of Hume’s Principle over alternative characterizations of the naturals. (shrink)

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. (shrink)

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. (shrink)

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. (shrink)

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. (shrink)

A number of recent accounts for vague terms postulate a kind of context-sensitivity, one that kicks in after the usual ‘external’ contextual factors like comparison class are established and held fixed. In a recent paper, ‘Vagueness without Context Change’: 275–92), Rosanna Keefe criticizes all such accounts. The arguments are variations on considerations that have been brought against context-sensitive accounts of knowledge, predicates of personal taste, epistemic modals, and the like. The issues are well known and there are variety of options (...) available in reply. More important, the arguments rely on an overly narrow conception of context-sensitivity, suggesting that one size fits all. If Keefe’s arguments were cogent, they would tell against the context-sensitivity of just about any expression, beyond the typical indexicals, including the variation of vague terms with comparison class. However, the criticisms raised by Keefe do highlight certain questions that must be answered by an advo... (shrink)

HorstenLeon* * _ The Metaphysics and Mathematics of Ordinary Objects. _Cambridge University Press, 2019. Pp. xviii + 231. ISBN: 978-1-107-03941-4 ; 978-1-10860177-1. doi: 10.1017/9781139600293.

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.

Call a term ‘pseudo-singular’ if it is syntactically singular but semantically plural. ‘The pair who wrote Principia’ is a good example, standing as it does for the two individuals, Whitehead and Russell. In this journal (2021), Eric Snyder and Stewart Shapiro launched an attack on the idea, calling it ‘linguistically and logically untenable.’ In this reply we rebut every one of their criticisms.

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.

Here I consider the two most venerated arguments about the existence of God: the Ontological Argument and the Argument from Evil. The Ontological Argument purports to show that God’s nature guarantees that God exists. The Argument from Evil purports to show that God’s nature, combined with some plausible facts about the way the world is, guarantees (or is very compelling grounds for thinking) that God does not exist. Obviously, both arguments cannot be sound. But I argue here that they are (...) both unsound for the very same reason. (shrink)

There are many alleged problems with trusting another person’s moral testimony, perhaps the most prominent of which is that it fails to deliver moral understanding. Without moral understanding, one cannot do the right thing for the right reason, and so acting on trusted moral testimony lacks moral worth. This chapter, however, argues that moral advice differs from moral testimony, differs from it in a way that enables a defender of moral advice to parry this worry about moral worth. The basic (...) idea is that an advisor and an advisee can together constitute a joint agent, and that this joint agent’s action can indeed have moral worth. So while the advisee himself might not do the right thing for the right reason (this because all alone he lacks the right reason), and while the advisor herself might not do the right thing for the right reason (this because all alone she does not do the right thing), they together do the right for the right reason. (shrink)

This paper argues that the classification of propositional attitudes into the de re, de dicto, and de se is incomplete. De se attitudes are widely agreed to be closely connected to de re attitudes. But there is a species of belief that is linked to agent-centered action in the way that de se beliefs are, but is also associated with entities, places, and especially times, under a description. These mark out a fourth kind. One way to think about what makes (...) them distinctive is that, despite being ‘essentially indexical’ they can be retained across different contexts of evaluation without having diagonal content. The paper also discusses the connection between such beliefs and the kinds of utterances (like ‘Don't put off for tomorrow what you can do today’) that are contested as examples of what David Kaplan called ‘monsters.’. (shrink)

No citizen should be denied the right to vote due solely to her age. We can see this by showing that all objections to it fail. It might be objected that it is not unjust to so deprive children because children as a group are unintelligent or irrational, have their interests already represented by the parents, or are justly deprived of many other rights, among other reasons. But all these objections fail because there is no evidence to support it, even (...) if true, this would not justify disenfranchisement, or, most prominently argued here, they deprive individuals of political rights based upon their membership in a group. The voting age should be abolished. (shrink)

This paper argues that under some conditions, when one person acts on the direction of another person, the two of them thereby act together, and that this explains why both the director and the directee can be responsible for what is done. In other words, a director and a directee can be a joint agent, one whose members are responsible for what they together do. This is most clearly so when the directive is a command. But it is also sometimes (...) so when the directive is a bit of advice. (shrink)

This Element introduces the philosophical literature on models, with an emphasis on normative considerations relevant to models for decision-making. Chapter 1 gives an overview of core questions in the philosophy of modeling. Chapter 2 examines the concept of model adequacy for purpose, using three examples of models from the atmospheric sciences to describe how this sort of adequacy is determined in practice. Chapter 3 explores the significance of using models that are not adequate for purpose, including the purpose of informing (...) public decisions. Chapter 4 provides a basic framework for values in modelling, using a case study to highlight the ethical challenges in building models for decision making. It concludes by establishing the need for strategies to manage value judgments in modelling, including the potential for public participation in the process. (shrink)

The concept of minimal risk has been used to regulate and limit participation by adolescents in clinical trials. It can be understood as setting an absolute standard of what risks are considered minimal or it can be interpreted as relative to the actual risks faced by members of the host community for the trial. While commentators have almost universally opposed a relative interpretation of the environmental risks faced by potential adolescent trial participants, we argue that the ethical concerns against the (...) relative standard may not be as convincing as these commentators believe. Our aim is to present the case for a relative standard of environmental risk in order to open a debate on this subject. We conclude by discussing how a relative standard of environmental risk could be defended in the specific case of an HIV vaccine trial among adolescents in South Africa. (shrink)

Autant d'études classiques qui accompagnent les grands ouvrages d'Eric Weil, Logique de la philosophie, Philosophie morale, Philosophie politique.

In a penetrating analysis of student unrest, the Dean for Institute Relations at M.I.T (and former chief of the psychiatric services both there and at Wellesley College) isolates a major cause of campus conflict: the overwhelming, nonproductive mass of unstated academic and social norms that divert the student from creative intellectual effort and from the attempt to define and reach his goals as independently as he is able.

We review and present a new meta‐analysis of research suggesting that ethicists in the United States appear to behave no morally better overall than do non‐ethicist professors. Measures include: returning library books, peer evaluation of overall moral behavior, voting participation, courteous and discourteous behavior at conferences, replying to student emails, paying conference registration fees and disciplinary society dues, staying in touch with one's mother, charitable giving, organ and blood donation, vegetarianism, and honesty in responding to survey questions. One multi‐measure study (...) found ethicists tending to embrace more stringent moral views, especially about meat eating and charitable donation. The same multi‐measure study found ethicists and other professors to show similarly small‐to‐medium correlations between their expressed normative attitudes and their self‐reported or directly measured behavior. (shrink)

According to the analysis of concessive conditionals suggested by Crupi and Iacona, a concessive conditional $$p{{\,\mathrm{\hookrightarrow }\,}}q$$ p ↪ q is adequately formalized as a conjunction of conditionals. This paper presents a sound and complete axiomatic system for concessive conditionals so understood. The soundness and completeness proofs that will be provided rely on a method that has been employed by Raidl, Iacona, and Crupi to prove the soundness and completeness of an analogous system for evidential conditionals.

In 1970 a coalition of student activists opposing the Vietnam War circulated documents revealing the involvement of several prominent social scientists in U.S. counterinsurgency activities in Thailand--activities that could cause harm to the people who were the subject of the scholars' research. The disclosure of these materials, which detailed meetings with the Agency for International Development and the Defense Department, prompted two members of the Ethics Committee of the American Anthropological Association to issue an unauthorized rebuke of the accused. Over (...) the next two years, the AAA agonized over the allegations and the appropriate response to them. Within an academic community already polarized by the war, political and professional acrimony reached unprecedented levels. Although the association ultimately passed a code of ethics, the key issues raised in the process were never fully resolved. Now back in print, Eric Wakin's Anthropology Goes to War is the first comprehensive study of what became known as the Thailand Controversy--and a timely reminder of a debate whose echoes may be heard in our own time. (shrink)