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  1. The Number of Moons Is Not a Number. Towards a Comprehensive Linguistic Approach to Frege's Commitment Puzzle.Borys Jastrzębski - 2016 - Filozofia Nauki 24 (2 (94)):31-49.
    Comprehensive Linguistic Approach to Frege's Commitment Puzzle There is a puzzle, noticed by Frege, about inferences from sentences like (F1) "Jupiter has four moons" to sentences like (F2) "The number of moons of Jupiter is four". They seem to be truth-conditionally equivalent but, apparently, they say something about completely different things. (F1) seems to be about moons, while (F2) about numbers. This phenomenon raises several puzzles about semantics, syntax, and is one of main tools of easy ontology. Recently, new linguistic (...)
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  • Mathematics as a Science of Non-abstract Reality: Aristotelian Realist Philosophies of Mathematics.James Franklin - 2022 - Foundations of Science 27 (2):327-344.
    There is a wide range of realist but non-Platonist philosophies of mathematics—naturalist or Aristotelian realisms. Held by Aristotle and Mill, they played little part in twentieth century philosophy of mathematics but have been revived recently. They assimilate mathematics to the rest of science. They hold that mathematics is the science of X, where X is some observable feature of the world. Choices for X include quantity, structure, pattern, complexity, relations. The article lays out and compares these options, including their accounts (...)
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  • Numbers and Cardinalities: What’s Really Wrong with the Easy Argument for Numbers?Eric Snyder - 2017 - Linguistics and Philosophy 40 (4):373-400.
    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 (...)
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  • Platonism in the Philosophy of Mathematics.Øystein Linnebo - forthcoming - Stanford Encyclopedia of Philosophy.
    Platonism about mathematics (or mathematical platonism) isthe metaphysical view that there are abstract mathematical objectswhose existence is independent of us and our language, thought, andpractices. Just as electrons and planets exist independently of us, sodo numbers and sets. And just as statements about electrons and planetsare made true or false by the objects with which they are concerned andthese objects' perfectly objective properties, so are statements aboutnumbers and sets. Mathematical truths are therefore discovered, notinvented., Existence. There are mathematical objects.
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  • Platonism in Metaphysics.Mark Balaguer - 2008 - Stanford Encyclopedia of Philosophy.
    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 (...)
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  • Platonism in Metaphysics.Markn D. Balaguer - 2016 - Stanford Encyclopedia of Philosophy 1 (1):1.
     
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  • Fictionalism in the Philosophy of Mathematics.Mark Balaguer - 2008 - Stanford Encyclopedia of Philosophy.
    Mathematical fictionalism (or as I'll call it, fictionalism) is best thought of as a reaction to mathematical platonism. Platonism is the view that (a) there exist abstract mathematical objects (i.e., nonspatiotemporal mathematical objects), and (b) our mathematical sentences and theories provide true descriptions of such objects. So, for instance, on the platonist view, the sentence ‘3 is prime’ provides a straightforward description of a certain object—namely, the number 3—in much the same way that the sentence ‘Mars is red’ provides a (...)
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  • Objectivity, Realism, and Proof. FilMat Studies in the Philosophy of Mathematics.Francesca Boccuni & Andrea Sereni (eds.) - 2016 - Cham, Switzerland: Springer International Publishing.
    This volume covers a wide range of topics in the most recent debates in the philosophy of mathematics, and is dedicated to how semantic, epistemological, ontological and logical issues interact in the attempt to give a satisfactory picture of mathematical knowledge. The essays collected here explore the semantic and epistemic problems raised by different kinds of mathematical objects, by their characterization in terms of axiomatic theories, and by the objectivity of both pure and applied mathematics. They investigate controversial aspects of (...)
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  • Building blocks for a cognitive science-led epistemology of arithmetic.Stefan Buijsman - 2021 - Philosophical Studies 179 (5):1-18.
    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 (...)
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  • How Do We Semantically Individuate Natural Numbers?†.Stefan Buijsman - forthcoming - Philosophia Mathematica.
    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 (...)
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  • Hale’s Argument From Transitive Counting.Eric Snyder, Richard Samuels & Stewart Shaprio - 2019 - Synthese 198 (3):1905-1933.
    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 (...)
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  • Mathematics as a Science of Non-Abstract Reality: Aristotelian Realist Philosophies of Mathematics.James Franklin - 2022 - Foundations of Science 27 (2):327-344.
    There is a wide range of realist but non-Platonist philosophies of mathematics—naturalist or Aristotelian realisms. Held by Aristotle and Mill, they played little part in twentieth century philosophy of mathematics but have been revived recently. They assimilate mathematics to the rest of science. They hold that mathematics is the science of X, where X is some observable feature of the (physical or other non-abstract) world. Choices for X include quantity, structure, pattern, complexity, relations. The article lays out and compares these (...)
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  • Giving the Value of a Variable.Richard Lawrence - 2021 - Kriterion - Journal of Philosophy 35 (2):135-150.
    What does it mean to ‘give’ the value of a variable in an algebraic context, and how does giving the value of a variable differ from merely describing it? I argue that to answer this question, we need to examine the role that giving the value of a variable plays in problem-solving practice. I argue that four different features are required for a statement to count as giving the value of a variable in the context of solving an elementary algebra (...)
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  • Hofweber's Philosophy of Mathematics.AgustÍn Rayo - 2017 - Philosophy and Phenomenological Research 94 (2):474-480.
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  • Resolving Frege’s Other Puzzle.Eric Snyder, Richard Samuels & Stewart Shapiro - 2022 - Philosophica Mathematica 30 (1):59-87.
    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 (...)
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  • Number Words and Reference to Numbers.Katharina Felka - 2014 - Philosophical Studies 168 (1):261-282.
    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 (...)
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  • Hofweber’s Nominalist Naturalism.Eric Snyder, Richard Samuels & Stewart Shapiro - 2022 - In Gianluigi Oliveri, Claudio Ternullo & Stefano Boscolo (eds.), Objects, Structures, and Logics. Cham, Switzerland: pp. 31-62.
    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.
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  • Names, Light Nouns, and Countability.Friederike Moltmann - 2022 - Linguistic Inquiry.
    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 (...)
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  • Steps Towards a Minimalist Account of Numbers.Thomas Schindler - forthcoming - Mind.
    This paper outlines an account of numbers based on the numerical equivalence schema (NES), which consists of all sentences of the form ‘#x.Fx=n if and only if ∃nx Fx’, where # is the number-of operator and ∃n is defined in standard Russellian fashion. In the first part of the paper, I point out some analogies between the NES and the T-schema for truth. In light of these analogies, I formulate a minimalist account of numbers, based on the NES, which strongly (...)
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  • Focus on Numbers.Jefferson Barlew - 2017 - Linguistics and Philosophy 40 (4):401-426.
    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 (...)
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  • Ontology Via Semantics? Introduction to the Special Issue on the Semantics of Cardinals.Craige Roberts & Stewart Shapiro - 2017 - Linguistics and Philosophy 40 (4):321-329.
    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.
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  • Number Word Constructions, Degree Semantics and the Metaphysics of Degrees.Brendan Balcerak Jackson & Doris Penka - 2017 - Linguistics and Philosophy 40 (4):347-372.
    A central question for ontology is the question of whether numbers really exist. But it seems easy to answer this question in the affirmative. The truth of a sentence like ‘Seven students came to the party’ can be established simply by looking around at the party and counting students. A trivial paraphrase of is ‘The number of students who came to the party is seven’. But appears to entail the existence of a number, and so it seems that we must (...)
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  • Number Sentences and Specificational Sentences. Reply to Moltmann.Robert Schwartzkopff - 2015 - Philosophical Studies:1-20.
    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 (...)
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  • Defusing Easy Arguments for Numbers.Brendan Balcerak Jackson - 2013 - Linguistics and Philosophy 36 (6):447-461.
    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 (...)
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  • Counting, measuring, and the fractional cardinalities puzzle.Eric Snyder - 2021 - Linguistics and Philosophy 44 (3):513-550.
    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 (...)
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  • Why Care About What There Is?Daniel Z. Korman - forthcoming - In Javier Cumpa (ed.), The Question of Ontology: The Contemporary Debate. Oxford University Press.
    There’s the question of what there is, and then there’s the question of what ultimately exists. Many contend that, once we have this distinction clearly in mind, we can see that there is no sensible debate to be had about whether there are such things as properties or tables or numbers, and that the only ontological question worth debating is whether such things are ultimate (in one or another sense). I argue that this is a mistake. Taking debates about ordinary (...)
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  • On 'Average'.Christopher Kennedy & Jason Stanley - 2009 - Mind 118 (471):583 - 646.
    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 (...)
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  • Towards a Fictionalist Philosophy of Mathematics.Robert Knowles - unknown
    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 (...)
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  • Nominalization, Specification, and Investigation.Richard Lawrence - 2017 - Dissertation, University of California, Berkeley
    Frege famously held that numbers play the role of objects in our language and thought, and that this role is on display when we use sentences like "The number of Jupiter's moons is four". I argue that this role is an example of a general pattern that also encompasses persons, times, locations, reasons, causes, and ways of appearing or acting. These things are 'objects' simply in the sense that they are answers to questions: they are the sort of thing we (...)
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  • Should Metaphysics Care About Linguistics?Tobias Rosefeldt - 2018 - Journal for General Philosophy of Science / Zeitschrift für Allgemeine Wissenschaftstheorie 49 (2):161-178.
    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 (...)
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  • Ontological Trivialism?Seyed N. Mousavian - 2017 - Grazer Philosophische Studien 94 (1-2):38-68.
    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 (...)
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  • Intensional Relative Clauses and the Semantics of Variable Objects.Friederike Moltmann - 2018 - In Manfred Krifka & Schenner Mathias (eds.), Reconstruction Effects in Relative Clauses. Berlin: De Gruyter. pp. 427-453..
    NPs with intensional relative clauses such as 'the book John needs to write' pose a significant challenge for semantic theory. Such NPs act like referential terms, yet they do not stand for a particular actual object. This paper will develop a semantic analysis of such NPs on the basis of the notion of a variable object. The analysis avoids a range of difficulties that a more standard analysis based on the notion of an individual concept would face. Most importantly, unlike (...)
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  • Unwarranted Philosophical Assumptions in Research on ANS.John Opfer, Richard Samuels, Stewart Shapiro & Eric Snyder - 2021 - Behavioral and Brain Sciences 44.
    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.
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  • Beta-Conversion and the Being Constraint.Agustín Rayo - 2021 - Aristotelian Society Supplementary Volume 95 (1):253-286.
    Modal contingentists face a dilemma: there are two attractive principles of which they can only accept one. In this paper I show that the most natural way of resolving the dilemma leads to expressive limitations. I then develop an alternative resolution. In addition to overcoming the expressive limitations, the alternative picture allows for an attractive account of arithmetic and for a style of semantic theorizing that can be helpful to contingentists.
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  • Events and Countability.Friederike Moltmann - manuscript
    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, (...)
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  • Extraction, Displacement, and Focus: A Reply to Balcerak Jackson.Thomas Hofweber - 2014 - Linguistics and Philosophy 37 (3):263-267.
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  • Number Sentences and Specificational Sentences: Reply to Moltmann.Robert Schwartzkopff - 2016 - Philosophical Studies 173 (8):2173-2192.
    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 (...)
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  • Talking About Numbers: Easy Arguments for Mathematical Realism. [REVIEW]Richard Lawrence - 2017 - History and Philosophy of Logic 38 (4):390-394.
  • Reply to Critics.Agustín Rayo - 2014 - Inquiry: An Interdisciplinary Journal of Philosophy 57 (4):498-534.
    Cameron, Eklund, Hofweber, Linnebo, Russell and Sider have written critical essays on my book, The Construction of Logical Space (Oxford: Oxford University Press, 2013). Here I offer some replies.
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  • Extraction, Displacement, and Focus: A Reply to Balcerak Jackson.Thomas Hofweber - 2014 - Linguistics and Philosophy 37 (3):263-267.
    On the one hand they seem to be quite obviously truth conditionally equivalent, but on the other hand they seem to be about different things. Whereas (1) is about Jupiter and its moons, (2) is about numbers. In particular, the word ‘four’ appears in (1) in the position of an adjective or determiner, whereas it seems to be a name for a number in (2). Furthermore, (2) appears to be an identity statement claiming that what two number terms stand for (...)
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  • A Numeral Oddity.Luca Gasparri - 2019 - Journal of Semantics 36 (3):563-571.
    Natural language appears to allow the ascription of properties of numeral symbols to the denotation of number referring phrases. The paper describes the phenomenon and presents two alternative explanations for why it obtains. One combining an intuitive semantics for number referring phrases and a predicate-shifting mechanism, the other assigning number referring phrases a structured denotation consisting of two parts: a mathematical object (the number) and a contextually determined numeral symbol. Some preliminary observations in favor of the second analysis are offered.
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  • Cardinals, Ordinals, and the Prospects for a Fregean Foundation.Eric Snyder, Stewart Shapiro & Richard Samuels - 2018 - Royal Institute of Philosophy Supplement 82:77-107.
    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 (...)
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