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Numbers

Edited by Rafal Urbaniak (University of Ghent, University of Gdansk)
Assistant editors: Sam Roberts, Pawel Pawlowski
About this topic
Summary Various theories concerned with numbers (arithmetic, real number theory, ...) are among the most often taught and applied mathematical theories. Accordingly, philosophers paid a significant amount of attention to considerations pertaining the status of such theories and the nature of numbers and number-theoretic discourse. Because of their relative simplicity, philosophical discussion surrounding such theories provide a neat proving ground for various wider philosophical accounts of mathematics, which makes this category fairly closely intertwined with other categories falling under Ontology of Mathematics.
Key works Frege 1980 is a seminal work on the philosophy of numbers (his approached has been further developed byWright 1983). A very good anthology of classic papers is Heijenoort 1967.
Introductions Potter 2000 is a nice book to start with. 
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  1. David Auerbach (1994). Saying It With Numerals. Notre Dame Journal of Formal Logic 35 (1):130-146.
    This article discusses the nature of numerals and the plausibility of their special semantic and epistemological status as proper names of numbers. Evidence is presented that minimizes the difference between numerals and other devices of direct reference. The availability of intensional contexts within formalised metamathematics is exploited to shed light on the relation between formal numerals and numerals.
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  2. Jeremy Avigad, Philosophy of Mathematics.
    The philosophy of mathematics plays an important role in analytic philosophy, both as a subject of inquiry in its own right, and as an important landmark in the broader philosophical landscape. Mathematical knowledge has long been regarded as a paradigm of human knowledge with truths that are both necessary and certain, so giving an account of mathematical knowledge is an important part of epistemology. Mathematical objects like numbers and sets are archetypical examples of abstracta, since we treat such objects in (...)
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  3. Jody Azzouni (2009). Empty de Re Attitudes About Numbers. Philosophia Mathematica 17 (2):163-188.
    I dub a certain central tradition in philosophy of language (and mind) the de re tradition. Compelling thought experiments show that in certain common cases the truth conditions for thoughts and public-language expressions categorically turn on external objects referred to, rather than on linguistic meanings and/or belief assumptions. However, de re phenomena in language and thought occur even when the objects in question don't exist. Call these empty de re phenomena. Empty de re thought with respect to numeration is explored (...)
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  4. Eric Temple Bell (1946). The Magic of Numbers. London, Mcgraw-Hill Book Company, Inc..
    It probes the work of Pythagoras, Galileo, Berkeley, Einstein, and others, exploring how "number magic" has influenced religion, philosophy, science, and mathematics.
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  5. Paul Benacerraf (1965). What Numbers Could Not Be. Philosophical Review 74 (1):47-73.
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  6. José A. Benardete (1991). Constructibility and Mathematical Existence. Review of Metaphysics 45 (1):114-115.
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  7. L. Berk (2013). Second-Order Arithmetic Sans Sets. Philosophia Mathematica 21 (3):339-350.
    This paper examines the ontological commitments of the second-order language of arithmetic and argues that they do not extend beyond the first-order language. Then, building on an argument by George Boolos, we develop a Tarski-style definition of a truth predicate for the second-order language of arithmetic that does not involve the assignment of sets to second-order variables but rather uses the same class of assignments standardly used in a definition for the first-order language.
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  8. Dougal Blyth (2000). Platonic Number in the Parmenides and Metaphysics XIII. International Journal of Philosophical Studies 8 (1):23 – 45.
    I argue here that a properly Platonic theory of the nature of number is still viable today. By properly Platonic, I mean one consistent with Plato's own theory, with appropriate extensions to take into account subsequent developments in mathematics. At Parmenides 143a-4a the existence of numbers is proven from our capacity to count, whereby I establish as Plato's the theory that numbers are originally ordinal, a sequence of forms differentiated by position. I defend and interpret Aristotle's report of a Platonic (...)
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  9. Andrew Boucher, The Existence of Numbers (Or: What is the Status of Arithmetic?) By V2.00 Created: 11 Oct 2001 Modified: 3 June 2002 Please Send Your Comments to Abo. [REVIEW]
    I begin with a personal confession. Philosophical discussions of existence have always bored me. When they occur, my eyes glaze over and my attention falters. Basically ontological questions often seem best decided by banging on the table--rocks exist, fairies do not. Argument can appear long-winded and miss the point. Sometimes a quick distinction resolves any apparent difficulty. Does a falling tree in an earless forest make noise, ie does the noise exist? Well, if noise means that an ear must be (...)
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  10. Otávio Bueno & Øystein Linnebo (eds.) (2009). New Waves in Philosophy of Mathematics. Palgrave Macmillan.
    Thirteen up-and-coming researchers in the philosophy of mathematics have been invited to write on what they take to be the right philosophical account of mathematics, examining along the way where they think the philosophy of mathematics is and ought to be going. A rich and diverse picture emerges. Some broader tendencies can nevertheless be detected: there is increasing attention to the practice, language and psychology of mathematics, a move to reassess the orthodoxy, as well as inspiration from philosophical logic.
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  11. Piotr Błaszczyk (2004). O przedmiocie matematycznym. Filozofia Nauki 2 (1):45-59.
    In this paper we show that the field of the real numbers is an intentional object in the sense specified by Roman Ingarden in his Das literarische Kunstwer and Der Streit um die Existenz der Welt. An ontological characteristics of a classic example of an intentional object, i.e. a literary character, is developed. There are three principal elements of such an object: the author, the text and the entity in which the literary character forms the content. In the case of (...)
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  12. Patrick Caldon & Aleksandar Ignjatović (2005). On Mathematical Instrumentalism. Journal of Symbolic Logic 70 (3):778 - 794.
    In this paper we devise some technical tools for dealing with problems connected with the philosophical view usually called mathematical instrumentalism. These tools are interesting in their own right, independently of their philosophical consequences. For example, we show that even though the fragment of Peano's Arithmetic known as IΣ₁ is a conservative extension of the equational theory of Primitive Recursive Arithmetic (PRA). IΣ₁ has a super-exponential speed-up over PRA. On the other hand, theories studied in the Program of Reverse Mathematics (...)
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  13. Gregory Chaitin (2011). How Real Are Real Numbers? Manuscrito 34 (1):115-141.
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  14. Colin Cheyne & Charles R. Pigden (1996). Pythagorean Powers or a Challenge to Platonism. Australasian Journal of Philosophy 74 (4):639 – 645.
    The Quine/Putnam indispensability argument is regarded by many as the chief argument for the existence of platonic objects. We argue that this argument cannot establish what its proponents intend. The form of our argument is simple. Suppose indispensability to science is the only good reason for believing in the existence of platonic objects. Either the dispensability of mathematical objects to science can be demonstrated and, hence, there is no good reason for believing in the existence of platonic objects, or their (...)
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  15. Justin Clarke-Doane (2008). Multiple Reductions Revisited. Philosophia Mathematica 16 (2):244-255.
    Paul Benacerraf's argument from multiple reductions consists of a general argument against realism about the natural numbers (the view that numbers are objects), and a limited argument against reductionism about them (the view that numbers are identical with prima facie distinct entities). There is a widely recognized and severe difficulty with the former argument, but no comparably recognized such difficulty with the latter. Even so, reductionism in mathematics continues to thrive. In this paper I develop a difficulty for Benacerraf's argument (...)
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  16. David Neil Corfield (2004). Mathematical Kinds, or Being Kind to Mathematics. Philosophica 74.
    In 1908, Henri Poincaré claimed that: ...the mathematical facts worthy of being studied are those which, by their analogy with other facts, are capable of leading us to the knowledge of a mathematical law, just as experimental facts lead us to the knowledge of a physical law. They are those which reveal to us unsuspected kinship between other facts, long known, but wrongly believed to be strangers to one another. Towards the end of the twentieth century, with many more mathematical (...)
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  17. Barbara F. Csima, Johanna N. Y. Franklin & Richard A. Shore (2013). Degrees of Categoricity and the Hyperarithmetic Hierarchy. Notre Dame Journal of Formal Logic 54 (2):215-231.
    We study arithmetic and hyperarithmetic degrees of categoricity. We extend a result of E. Fokina, I. Kalimullin, and R. Miller to show that for every computable ordinal $\alpha$, $\mathbf{0}^{(\alpha)}$ is the degree of categoricity of some computable structure $\mathcal{A}$. We show additionally that for $\alpha$ a computable successor ordinal, every degree $2$-c.e. in and above $\mathbf{0}^{(\alpha)}$ is a degree of categoricity. We further prove that every degree of categoricity is hyperarithmetic and show that the index set of structures with degrees (...)
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  18. Boudewijn de Bruin (2008). Wittgenstein on Circularity in the Frege-Russell Definition of Cardinal Number. Philosophia Mathematica 16 (3):354-373.
    Several scholars have argued that Wittgenstein held the view that the notion of number is presupposed by the notion of one-one correlation, and that therefore Hume's principle is not a sound basis for a definition of number. I offer a new interpretation of the relevant fragments on philosophy of mathematics from Wittgenstein's Nachlass, showing that if different uses of ‘presupposition’ are understood in terms of de re and de dicto knowledge, Wittgenstein's argument against the Frege-Russell definition of number turns out (...)
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  19. Helen De Cruz & Pierre Pica (2008). Knowledge of Number and Knowledge of Language: Number as a Test Case for the Role of Language in Cognition. Philosophical Psychology 21 (4):437 – 441.
    The relationship between language and conceptual thought is an unresolved problem in both philosophy and psychology. It remains unclear whether linguistic structure plays a role in our cognitive processes. This special issue brings together cognitive scientists and philosophers to focus on the role of language in numerical cognition: because of their universality and variability across languages, number words can serve as a fruitful test case to investigate claims of linguistic relativism.
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  20. Stanislas Dehaene, Véronique Izard, Pierre Pica & Elizabeth Spelke (2009). Response to Comment on "Log or Linear? Distinct Intuitions on the Number Scale in Western and Amazonian Indigene Cultures&Quot;. Science 323 (5910):38.
    The performance of the Mundurucu on the number-space task may exemplify a general competence for drawing analogies between space and other linear dimensions, but Mundurucu participants spontaneously chose number when other dimensions were available. Response placement may not reflect the subjective scale for numbers, but Cantlon et al.'s proposal of a linear scale with scalar variability requires additional hypotheses that are problematic.
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  21. Katharina Felka (2014). Number Words and Reference to Numbers. 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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  22. Gottlob Frege (1980). The Foundations of Arithmetic: A Logico-Mathematical Enquiry Into the Concept of Number. Northwestern University Press.
    § i. After deserting for a time the old Euclidean standards of rigour, mathematics is now returning to them, and even making efforts to go beyond them. ...
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  23. Laurence Goldstein (2002). The Indefinability of “One”. Journal of Philosophical Logic 31 (1):29 - 42.
    Logicism is one of the great reductionist projects. Numbers and the relationships in which they stand may seem to possess suspect ontological credentials - to be entia non grata - and, further, to be beyond the reach of knowledge. In seeking to reduce mathematics to a small set of principles that form the logical basis of all reasoning, logicism holds out the prospect of ontological economy and epistemological security. This paper attempts to show that a fundamental logicist project, that of (...)
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  24. Jeremy Gwiazda, Infinite Numbers Are Large Finite Numbers.
    In this paper, I suggest that infinite numbers are large finite numbers, and that infinite numbers, properly understood, are 1) of the structure omega + (omega* + omega)Ө + omega*, and 2) the part is smaller than the whole. I present an explanation of these claims in terms of epistemic limitations. I then consider the importance, part of which is demonstrating the contradiction that lies at the heart of Cantorian set theory: the natural numbers are too large to be counted (...)
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  25. Edward C. Halper (2011). Klein on Aristotle on Number. New Yearbook for Phenomenology and Phenomenological Philosophy 11:271-281.
    Jacob Klein raises two important questions about Aristotle’s account of number: (1) How does the intellect come to grasp a sensible as an intelligible unit? (2) What makes a collection of these intelligible units into one number? His answer to both questions is “abstraction.” First, we abstract (or, better, disregard) a thing’s sensible characteristics to grasp it as a noetic unit. Second, after counting like things, we again disregard their other characteristics and grasp the group as a noetic entity composed (...)
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  26. Daesuk Han (2011). Wittgenstein and the Real Numbers. History and Philosophy of Logic 31 (3):219-245.
    When it comes to Wittgenstein's philosophy of mathematics, even sympathetic admirers are cowed into submission by the many criticisms of influential authors in that field. They say something to the effect that Wittgenstein does not know enough about or have enough respect for mathematics, to take him as a serious philosopher of mathematics. They claim to catch Wittgenstein pooh-poohing the modern set-theoretic extensional conception of a real number. This article, however, will show that Wittgenstein's criticism is well grounded. A real (...)
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  27. Mirja Hartimo (2006). Mathematical Roots of Phenomenology: Husserl and the Concept of Number. History and Philosophy of Logic 27 (4):319-337.
    The paper examines the roots of Husserlian phenomenology in Weierstrass's approach to analysis. After elaborating on Weierstrass's programme of arithmetization of analysis, the paper examines Husserl's Philosophy of Arithmetic as an attempt to provide foundations to analysis. The Philosophy of Arithmetic consists of two parts; the first discusses authentic arithmetic and the second symbolic arithmetic. Husserl's novelty is to use Brentanian descriptive analysis to clarify the fundamental concepts of arithmetic in the first part. In the second part, he founds the (...)
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  28. Clevis Headley (1997). Platonism and Metaphor in the Texts of Mathematics: GöDel and Frege on Mathematical Knowledge. [REVIEW] Man and World 30 (4):453-481.
    In this paper, I challenge those interpretations of Frege that reinforce the view that his talk of grasping thoughts about abstract objects is consistent with Russell's notion of acquaintance with universals and with Gödel's contention that we possess a faculty of mathematical perception capable of perceiving the objects of set theory. Here I argue the case that Frege is not an epistemological Platonist in the sense in which Gödel is one. The contention advanced is that Gödel bases his Platonism on (...)
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  29. Otto Hölder (2013). Review of Graßmann, Robert, Theory of Number or Arithmetic in Strict Scientific Presentation by Strict Use of Formulas (1891). [REVIEW] Philosophia Scientiæ 17 (17-1):57-70.
    The author of this book pursues the objective of treating the whole of pure mathematics [die ganze reine Mathematik] in four sections [Abtheilungen]. One half of the first of these sections is dedicated to arithmetic and is already available. The other half of the first section “A heuristic treatise on number [Zahlenlehre in freier Gedankenentwicklung]” which treats the same discipline is supposed to follow. The author may have opted for such an unusual separation [of the treatment of arithme..
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  30. Philip Hugly & Charles Sayward (2006). Arithmetic and Ontology: A Non-Realist Philosophy of Arithmetic. rodopi.
    In this book a non-realist philosophy of mathematics is presented. Two ideas are essential to its conception. These ideas are (i) that pure mathematics--taken in isolation from the use of mathematical signs in empirical judgement--is an activity for which a formalist account is roughly correct, and (ii) that mathematical signs nonetheless have a sense, but only in and through belonging to a system of signs with empirical application. This conception is argued by the two authors and is critically discussed by (...)
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  31. Philip Hugly & Charles Sayward (1999). Did the Greeks Discover the Irrationals? Philosophy 74 (2):169-176.
    A popular view is that the great discovery of Pythagoras was that there are irrational numbers, e.g., the positive square root of two. Against this it is argued that mathematics and geometry, together with their applications, do not show that there are irrational numbers or compel assent to that proposition.
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  32. Edmund Husserl (2005). Lecture on the Concept of Number (Ws 1889/90). New Yearbook for Phenomenology and Phenomenological Philosophy 5:279-309 recto.
    Among the various lecture courses that Edmund Husserl held during his time as a Privatdozent at the University of Halle (1887-1901), there was one on "Ausgewählte Fragen aus der Philosophie der Mathematik" (Selected Questions from the Philosophy of Mathematics), which he gave twice, once in the WS 1889/90 and again in WS 1890/91. As Husserl reports in his letter to Carl Stumpf of February 1890, he lectured mainly on “spatial-logical questions” and gave an extensive critique of the Riemann-Helmholtz theories. Indeed, (...)
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  33. Daniel C. Hyde & Elizabeth S. Spelke, All Numbers Are Not Equal: An Electrophysiological Investigation of Small and Large Number Representations.
    & Behavioral and brain imaging research indicates that human infants, humans adults, and many nonhuman animals represent large nonsymbolic numbers approximately, discriminating between sets with a ratio limit on accuracy. Some behavioral evidence, especially with human infants, suggests that these representations differ from representations of small numbers of objects. To investigate neural signatures of this distinction, event-related potentials were recorded as adult humans passively viewed the sequential presentation of dot arrays in an adaptation paradigm. In two studies, subjects viewed successive (...)
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  34. Carlo Ierna (2008). Concluding Remarks (Abschließende Stellungnahme / Zehnte Diskussionseinheit). Erwägen Wissen Ethik 19 (4):600-602.
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  35. Carlo Ierna (2008). Sigwart's Numbers in Context (Erweiterte Stellungnahme / Zehnte Diskussionseinheit). Erwägen Wissen Ethik 19 (4):585-587.
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  36. Carlo Ierna (2006). The Beginnings of Husserl's Philosophy. Part 2: Mathematical and Philosophical Background. New Yearbook for Phenomenology and Phenomenological Philosophy 6 (1):23-71.
    The article examines the development of Husserl’s early philosophy from his Habilitationsschrift (1887) to the Philosophie der Arithmetik (1891). -/- An attempt will be made at reconstructing the lost Habilitationsschrift (of which only the first chapter survives, which we know as Über den Begriff der Zahl). The examined sources show that the original version of the Habilitationsschrift was by far broader than the printed version, and included most topics of the PA. -/- The article contains an extensive and detailed comparison (...)
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  37. Carlo Ierna (2005). The Beginnings of Husserl's Philosophy. Part 1: From "Über den Begriff der Zahl" to "Philosophie der Arithmetik&Quot;. New Yearbook for Phenomenology and Phenomenological Philosophy 5:1-56.
    The article examines the development of Husserl’s early philosophy from his Habilitationsschrift (1887) to the Philosophie der Arithmetik (1891). -/- An attempt will be made at reconstructing the lost Habilitationsschrift (of which only the first chapter survives, which we know as Über den Begriff der Zahl). The examined sources show that the original version of the Habilitationsschrift was by far broader than the printed version, and included most topics of the PA. -/- The article contains an extensive and detailed comparison (...)
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  38. Véronique Izard, Stanislas Dehaene, Pierre Pica & Elizabeth Spelke (2008). Response to Nunez. Science 312 (5803):1310.
    We agree with Nuñez that the Mundurucu do not master the formal propreties of number lines and logarithms, but as the term "intuition" implies, they spontaneously experience a logarithmic mapping of number to space as natural and "feeling right.".
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  39. Véronique Izard, Pierre Pica, Elizabeth S. Spelke & Stanislas Dehaene (2008). Exact Equality and Successor Function: Two Key Concepts on the Path Towards Understanding Exact Numbers. Philosophical Psychology 21 (4):491 – 505.
    Humans possess two nonverbal systems capable of representing numbers, both limited in their representational power: the first one represents numbers in an approximate fashion, and the second one conveys information about small numbers only. Conception of exact large numbers has therefore been thought to arise from the manipulation of exact numerical symbols. Here, we focus on two fundamental properties of the exact numbers as prerequisites to the concept of EXACT NUMBERS : the fact that all numbers can be generated by (...)
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  40. Véronique Izard, Pierre Pica, Elizabeth Spelke & Stanislas Dehaene (2008). The Mapping of Numbers on Space : Evidence for a Logarithmic Intuition. Médecine/Science 24 (12):1014-1016.
  41. Stanisław Jaśkowski (1975). About Certain Groups of Classes of Sets and Their Application to the Definitions of Numbers. [REVIEW] Studia Logica 34 (2):133 - 144.
    The aim of the paper is to give a new definition of real number. The logical type of any number defined is that of the function B = h(A) which assigns to a class of sets A a class of sets B. I give some conditions which the function h has to fulfill to be considered as number; an intuitive sense of the conditions is as follows: a function, which is number, assigns a class of sets of measure h·m to (...)
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  42. Mark Eli Kalderon (1996). What Numbers Could Be (and, Hence, Necessarily Are). Philosophia Mathematica 4 (3):238-255.
    This essay explores the commitments of modal structuralism. The precise nature of the modal-structuralist analysis obscures an unclarity of its import. As usually presented, modal structuralism is a form of anti-platonism. I defend an interpretation of modal structuralism that, far from being a form of anti-platonism, is itself a platonist analysis: The metaphysically significant distinction between (i) primitive modality and (ii) the natural numbers (objectually understood) is genuine, but the arithmetic facts just are facts about possible progressions. If correct, modal (...)
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  43. Christopher Kennedy & Jason Stanley (2009). On 'Average'. 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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  44. Joongol Kim (2014). Euclid Strikes Back at Frege. Philosophical Quarterly 64 (254):20-38.
  45. Joongol Kim (2013). What Are Numbers? Synthese 190 (6):1099-1112.
    This paper argues that (cardinal) numbers are originally given to us in the context ‘Fs exist n-wise’, and accordingly, numbers are certain manners or modes of existence, by addressing two objections both of which are due to Frege. First, the so-called Caesar objection will be answered by explaining exactly what kind of manner or mode numbers are. And then what we shall call the Functionality of Cardinality objection will be answered by establishing the fact that for any numbers m and (...)
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  46. Joongol Kim (2010). Yi on 2. Philosophia Mathematica 18 (3):329-336.
    Byeong-Uk Yi has argued that number words like ‘two’ primarily function as numerical predicates as in ‘Socrates and Hippias are two (in number)’, and other grammatical uses of number words can be paraphrased in terms of the predicative use. This paper critically examines Yi’s paraphrase scheme and also some other alternative schemes, and argues that the adjectival use of number words as in ‘The Scots and the Irish are two peoples’ cannot be paraphrased in terms of the predicative use.
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  47. M. Kondo (1952). Review: Zyoiti Suetuna, On Mathematical Existence. [REVIEW] Journal of Symbolic Logic 17 (1):63-63.
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  48. M. Krynicki & K. Zdanowski (2005). Theories of Arithmetics in Finite Models. Journal of Symbolic Logic 70 (1):1-28.
    We investigate theories of initial segments of the standard models for arithmetics. It is easy to see that if the ordering relation is definable in the standard model then the decidability results can be transferred from the infinite model into the finite models. On the contrary we show that the Σ₂—theory of multiplication is undecidable in finite models. We show that this result is optimal by proving that the Σ₁—theory of multiplication and order is decidable in finite models as well (...)
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  49. Wojciech Krysztofiak (2012). Indexed Natural Numbers in Mind: A Formal Model of the Basic Mature Number Competence. [REVIEW] Axiomathes 22 (4):433-456.
    The paper undertakes three interdisciplinary tasks. The first one consists in constructing a formal model of the basic arithmetic competence, that is, the competence sufficient for solving simple arithmetic story-tasks which do not require any mathematical mastery knowledge about laws, definitions and theorems. The second task is to present a generalized arithmetic theory, called the arithmetic of indexed numbers (INA). All models of the development of counting abilities presuppose the common assumption that our simple, folk arithmetic encoded linguistically in the (...)
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  50. Wojciech Krysztofiak (2012). Logiczna składnia liczebnika. Studium kognitywistyczne. Część I. Filozofia Nauki 1.
    In the paper there are presented main assumptions underlying the construction of theoretic models of mental processes of numeral reference in mathematical practice which comprises such abilities as counting, solving story-tasks, estimating cardinalities and comparing magnitudes. Numerals are understood as any expressions which enable mind to refer to numbers, cardinalities and magnitudes. The main research question formulated in the article sounds: What cognitive processes do there occur in the mind during execution of various numeral acts of reference?
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