Numbers

Edited by Rafal Urbaniak (Uniwersytetu Gdanskiego, Uniwersytetu Gdanskiego)
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 1960 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 van Heijenoort 1967.
Introductions Potter 2000 is a nice book to start with. 
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421 found
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1 — 50 / 421
  1. The Cultural Phenomenology of Qualitative quantity - work in progress - Introduction autobiographical.Borislav Dimitrov - manuscript
    This study is about the Quality. Here I have dealt with the quality that differs significantly from the common understanding of quality /as determined quality/ that arise from the law of dialectics. This new quality is the quality of the quantity /quality of the quantitative changes/, noticed in philosophy by Plato as “quality of numbers”, and later developed by Hegel as “qualitative quantity. The difference between the known determined quality and qualitative quantity is evident in the exhibit form of these (...)
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  2. Phenomenological Objects & Meaning: A Fregean & Husserlian Discussion.Daniel Sierra - manuscript
    Gottlob Frege and Edmund Husserl are two seemingly different philosophers in their methodology. Both have significantly influenced Western philosophy in that their contributions established fields within philosophy that are of intensive study today. Still, their differences in methodology have, in certain instances, yielded similar or distinct results. Their results ranged from the distinction of sense and reference, objectivity, and the theory of mathematics: specifically, their definition of number. Frege and Husserl have such striking similarities in their theory of sense and (...)
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  3. From the History of the Concept of Number.Roman Murawski & Thomas Bedürftig - unknown - Poznan Studies in the Philosophy of the Sciences and the Humanities 98:95-122.
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  4. 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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  5. Weyl and Two Kinds of Potential Domains.Laura Crosilla & Øystein Linnebo - forthcoming - Noûs.
    According to Weyl, “‘inexhaustibility’ is essential to the infinite”. However, he distinguishes two kinds of inexhaustible, or merely potential, domains: those that are “extensionally determinate” and those that are not. This article clarifies Weyl's distinction and explains its enduring logical and philosophical significance. The distinction sheds lights on the contemporary debate about potentialism, which in turn affords a deeper understanding of Weyl.
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  6. Abstraction and Grounding.Louis deRosset & Øystein Linnebo - forthcoming - Philosophy and Phenomenological Research.
    The idea that some objects are metaphysically “cheap” has wide appeal. An influential version of the idea builds on abstractionist views in the philosophy of mathematics, on which numbers and other mathematical objects are abstracted from other phenomena. For example, Hume’s Principle states that two collections have the same number just in case they are equinumerous, in the sense that they can be correlated one-to-one: (HP) #xx=#yy iff xx≈yy. The principal aim of this article is to use the notion of (...)
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  7. Eric Snyder. Semantics and the Ontology of Number.Michael Glanzberg - forthcoming - Philosophia Mathematica.
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  8. On some properties of numbers.Jason Katzourakis - forthcoming - Eleutheria.
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  9. Numbers: No Dogs or Philosophers Allowed.Ken Knisely, Michael Moses, Ihran Izmirlih & Michael Stein - forthcoming - DVD.
    How is it that numbers can magically map the world around us? Are numbers really Real? Was Pythagoras completely crazy when he seemed to regard numbers as divine entities capable of revealing the truth about things around us? With Michael Moses, Ihran Izmirlih, and Michael Stein.
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  10. Hilbert mathematics versus (or rather “without”) Gödel mathematics: V. Ontomathematics!Vasil Penchev - forthcoming - Logic and Philosophy of Mathematics eJournal (Elsevier: SSRN).
    The paper is the final, fifth part of a series of studies introducing the new conceptions of “Hilbert mathematics” and “ontomathematics”. The specific subject of the present investigation is the proper philosophical sense of both, including philosophy of mathematics and philosophy of physics not less than the traditional “first philosophy” (as far as ontomathematics is a conservative generalization of ontology as well as of Heidegger’s “fundamental ontology” though in a sense) and history of philosophy (deepening Heidegger’s destruction of it from (...)
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  11. Jacob Bernoulli on Analysis, Synthesis, and the Law of Large Numbers.Edith Dudley Sylla - forthcoming - Boston Studies in the Philosophy of Science.
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  12. Numbers Game.Mitsuye Yamada - forthcoming - Feminist Studies.
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  13. Number Concepts: An Interdisciplinary Inquiry.Richard Samuels & Eric Snyder - 2024 - Cambridge University Press.
    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, (...)
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  14. Playing with infinity: turtles, patterns, and pictures.Hans Zantema - 2024 - Boca Raton: AK Peters/CRC Press.
    This is a book about infinity, specifically the infinity of numbers and sequences. Amazing properties arise, for instance, some kinds of infinity are argued to be greater than others. Along the way the author will demonstrate how infinity can be made to create beautiful 'art', guided by the development of underlying mathematics. This book will provide a fascinating read for anyone interested in number theory, infinity, math art, and/or generative art, and could be used a valuable supplement to any course (...)
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  15. Ordinals vs. Cardinals in ℕ and Beyond.Aviv Keren - 2023 - In Carl Posy & Yemima Ben-Menahem (eds.), Mathematical Knowledge, Objects and Applications: Essays in Memory of Mark Steiner. Springer. pp. 193-225.
    Ordinality and cardinality, in the finite domain, are ordinarily considered as mere aspects of the very same objects, the natural numbers. Yet Steiner (Mathematics – application and applicability. In: Shapiro S (ed) The Oxford handbook of philosophy of mathematics and logic. Oxford University Press, 2005) draws attention to the intricate interplay between them, which is made implicit by this conception of them. In this chapter, I present a fitting cognitive framework and use it to account for how this situation comes (...)
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  16. Wittgenstein, Russell, and Our Concept of the Natural Numbers.Saul A. Kripke - 2023 - In Carl Posy & Yemima Ben-Menahem (eds.), Mathematical Knowledge, Objects and Applications: Essays in Memory of Mark Steiner. Springer. pp. 137-155.
    Wittgenstein gave a clearly erroneous refutation of Russell’s logicist project. The errors were ably pointed out by Mark Steiner. Nevertheless, I was motivated by Wittgenstein and Steiner to consider various ideas about the natural numbers. I ask which notations for natural numbers are ‘buck-stoppers’. For us it is the decimal notation and the corresponding verbal system. Based on the idea that a proper notation should be ‘structurally revelatory’, I draw various conclusions about our own concept of the natural numbers.
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  17. On What Ground Do Thin Objects Exist? In Search of the Cognitive Foundation of Number Concepts.Markus Pantsar - 2023 - Theoria 89 (3):298-313.
    Linnebo in 2018 argues that abstract objects like numbers are “thin” because they are only required to be referents of singular terms in abstraction principles, such as Hume's principle. As the specification of existence claims made by analytic truths (the abstraction principles), their existence does not make any substantial demands of the world; however, as Linnebo notes, there is a potential counter-argument concerning infinite regress against introducing objects this way. Against this, he argues that vicious regress is avoided in the (...)
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  18. Logic, mathematics, physics: from a loose thread to the close link: Or what gravity is for both logic and mathematics rather than only for physics.Vasil Penchev - 2023 - Astrophysics, Cosmology and Gravitation Ejournal 2 (52):1-82.
    Gravitation is interpreted to be an “ontomathematical” force or interaction rather than an only physical one. That approach restores Newton’s original design of universal gravitation in the framework of “The Mathematical Principles of Natural Philosophy”, which allows for Einstein’s special and general relativity to be also reinterpreted ontomathematically. The entanglement theory of quantum gravitation is inherently involved also ontomathematically by virtue of the consideration of the qubit Hilbert space after entanglement as the Fourier counterpart of pseudo-Riemannian space. Gravitation can be (...)
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  19. The Fallacy called Language.Ilexa Yardley - 2023 - Medium.Com/the-Circular-Theory.
    Symbolic representation demonstrates, and proves, the conservation of a circle (is the basis for, and, thus, controls, language) (all disciplines).
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  20. Mathematical Internal Realism.Tim Button - 2022 - In Sanjit Chakraborty & James Ferguson Conant (eds.), Engaging Putnam. Berlin, Germany: De Gruyter. pp. 157-182.
    In “Models and Reality” (1980), Putnam sketched a version of his internal realism as it might arise in the philosophy of mathematics. Here, I will develop that sketch. By combining Putnam’s model-theoretic arguments with Dummett’s reflections on Gödelian incompleteness, we arrive at (what I call) the Skolem-Gödel Antinomy. In brief: our mathematical concepts are perfectly precise; however, these perfectly precise mathematical concepts are manifested and acquired via a formal theory, which is understood in terms of a computable system of proof, (...)
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  21. On Number-Set Identity: A Study.Sean C. Ebels-Duggan - 2022 - Philosophia Mathematica 30 (2):223-244.
    Benacerraf’s 1965 multiple-reductions argument depends on what I call ‘deferential logicism’: his necessary condition for number-set identity is most plausible against a background Quineanism that allows autonomy of the natural number concept. Steinhart’s ‘folkist’ sufficient condition on number-set identity, by contrast, puts that autonomy at the center — but fails for not taking the folk perspective seriously enough. Learning from both sides, we explore new conditions on number-set identity, elaborating a suggestion from Wright.
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  22. Fermat’s last theorem proved in Hilbert arithmetic. II. Its proof in Hilbert arithmetic by the Kochen-Specker theorem with or without induction.Vasil Penchev - 2022 - Logic and Philosophy of Mathematics eJournal (Elsevier: SSRN) 14 (10):1-52.
    The paper is a continuation of another paper published as Part I. Now, the case of “n=3” is inferred as a corollary from the Kochen and Specker theorem (1967): the eventual solutions of Fermat’s equation for “n=3” would correspond to an admissible disjunctive division of qubit into two absolutely independent parts therefore versus the contextuality of any qubit, implied by the Kochen – Specker theorem. Incommensurability (implied by the absence of hidden variables) is considered as dual to quantum contextuality. The (...)
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  23. 1221.Ilexa Yardley - 2022 - Medium.Com/the-Circular-Theory.
    There are only two numbers in Nature. 12 and 21. Where 12 and 21 are different, yet, very much, the same. Explaining identity, complementarity, everything in mathematics, everything in technology, everything in biology (therefore, everything in physics and philosophy) (ontology and epistemology) (all disciplines).
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  24. Preface to Forenames of God: Enumerations of Ernesto Laclau toward a Political Theology of Algorithms.Virgil W. Brower - 2021 - Internationales Jahrbuch Für Medienphilosophie 7 (1):243-251.
    Perhaps nowhere better than, "On the Names of God," can readers discern Laclau's appreciation of theology, specifically, negative theology, and the radical potencies of political theology. // It is Laclau's close attention to Eckhart and Dionysius in this essay that reveals a core theological strategy to be learned by populist reasons or social logics and applied in politics or democracies to come. // This mode of algorithmically informed negative political theology is not mathematically inert. It aspires to relate a fraction (...)
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  25. Objectivity in Mathematics, Without Mathematical Objects†.Markus Pantsar - 2021 - Philosophia Mathematica 29 (3):318-352.
    I identify two reasons for believing in the objectivity of mathematical knowledge: apparent objectivity and applications in science. Focusing on arithmetic, I analyze platonism and cognitive nativism in terms of explaining these two reasons. After establishing that both theories run into difficulties, I present an alternative epistemological account that combines the theoretical frameworks of enculturation and cumulative cultural evolution. I show that this account can explain why arithmetical knowledge appears to be objective and has scientific applications. Finally, I will argue (...)
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  26. Semantics and the Ontology of Number.Eric Snyder - 2021 - Cambridge University Press.
    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 (...)
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  27. The Billionaire.Ilexa Yardley - 2021 - Https://Medium.Com/the-Circular-Theory/.
    Zero and one. Circumference and diameter. Intelligent anarchy. Explanation for abstraction (as a noun) (as a verb).
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  28. The Key to Complexity.Ilexa Yardley - 2021 - Https://Medium.Com/the-Circular-Theory/.
    Complexity is dependent on the circular-linear relationship between an individual and a group, meaning we cannot use 'observation' to tell us what we need to know (to explain complexity).
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  29. Projective Games on the Reals.Juan P. Aguilera & Sandra Müller - 2020 - Notre Dame Journal of Formal Logic 61 (4):573-589.
    Let Mn♯ denote the minimal active iterable extender model which has n Woodin cardinals and contains all reals, if it exists, in which case we denote by Mn the class-sized model obtained by iterating the topmost measure of Mn class-many times. We characterize the sets of reals which are Σ1-definable from R over Mn, under the assumption that projective games on reals are determined:1. for even n, Σ1Mn=⅁RΠn+11;2. for odd n, Σ1Mn=⅁RΣn+11.This generalizes a theorem of Martin and Steel for L, (...)
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  30. Invasive Weeds in Parmenides' Garden.Olga Ramirez Calle - 2020 - Croatian Journal of Philosophy 20 (60):391-412.
    The paper attempts to conciliate the important distinction between what-is, or exists, and what-is-not _thereby supporting Russell’s existential analysis_ with some Meinongian insights. For this purpose, it surveys the varied inhabitants of the realm of ‘non-being’ and tries to clarify their diverse statuses. The position that results makes it possible to rescue them back in surprising but non-threatening form, leaving our ontology safe from contradiction.
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  31. An unpublished theorem of Solovay on OD partitions of reals into two non-OD parts, revisited.Ali Enayat & Vladimir Kanovei - 2020 - Journal of Mathematical Logic 21 (3):2150014.
    A definable pair of disjoint non-OD sets of reals exists in the Sacks and ????0-large generic extensions of the constructible universe L. More specifically, if a∈2ω is eith...
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  32. The adverbial theory of numbers: some clarifications.Joongol Kim - 2020 - Synthese 197 (9):3981-4000.
    In a forthcoming paper in this journal, entitled “Bad company objection to Joongol Kim’s adverbial theory of numbers”, Namjoong Kim presents an ingenious Russell-style paradox based on an analogue of Kim’s definition of the number 1, and argues that Kim’s theory needs to provide a criterion of demarcation between acceptable and unacceptable definitions of adverbial entities. This paper addresses this ‘bad company’ objection and some other related issues concerning Kim’s adverbial theory by clarifying the purposes and uses of the formal (...)
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  33. Correction to: Bad company objection to Joongol Kim’s adverbial theory of numbers.Namjoong Kim - 2020 - Synthese 197 (3):1379-1379.
    Unfortunately, there is a typo in the author name. The correct spelling is Namjoong Kim. The author name was updated in the original publication.
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  34. Fermat’s Last Theorem Proved by Induction (and Accompanied by a Philosophical Comment).Vasil Penchev - 2020 - Metaphilosophy eJournal (Elsevier: SSRN) 12 (8):1-8.
    A proof of Fermat’s last theorem is demonstrated. It is very brief, simple, elementary, and absolutely arithmetical. The necessary premises for the proof are only: the three definitive properties of the relation of equality (identity, symmetry, and transitivity), modus tollens, axiom of induction, the proof of Fermat’s last theorem in the case of n = 3 as well as the premises necessary for the formulation of the theorem itself. It involves a modification of Fermat’s approach of infinite descent. The infinite (...)
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  35. Numbers, Empiricism and the A Priori.Olga Ramírez Calle - 2020 - Logos and Episteme 11 (2):149-177.
    The present paper deals with the ontological status of numbers and considers Frege ́s proposal in Grundlagen upon the background of the Post-Kantian semantic turn in analytical philosophy. Through a more systematic study of his philosophical premises, it comes to unearth a first level paradox that would unset earlier still than it was exposed by Russell. It then studies an alternative path, that departin1g from Frege’s initial premises, drives to a conception of numbers as synthetic a priori in a more (...)
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  36. Reflections on Frege’s Theory of Real Numbers†.Peter Roeper - 2020 - Philosophia Mathematica 28 (2):236-257.
    ABSTRACT Although Frege’s theory of real numbers in Grundgesetze der Arithmetik, Vol. II, is incomplete, it is possible to provide a logicist justification for the approach he is taking and to construct a plausible completion of his account by an extrapolation which parallels his theory of cardinal numbers.
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  37. Números naturales: distintas metodologías que convergen en el análisis de su naturaleza y de cómo los entendemos.Melisa Vivanco - 2020 - Critica 51 (153).
    José Ferreirós y Abel Lasalle Casanave, El árbol de los números: cognición, lógica y práctica matemática, Editorial Universidad de Sevilla, Sevilla, 2015, 256 pp.
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  38. Referential uses of arabic numerals.Melissa Vivanco - 2020 - Manuscrito 43 (4):142-164.
    Is the debate over the existence of numbers unsolvable? Mario Gómez-Torrente presents a novel proposal to unclog the old discussion between the realist and the anti-realist about numbers. In this paper, the strategy is outlined, highlighting its results and showing how they determine the desiderata for a satisfactory theory of the reference of Arabic numerals, which should lead to a satisfactory explanation about numbers. It is argued here that the theory almost achieves its goals, yet it does not capture the (...)
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  39. Abstract objects and semantics: An essay on prospects and problems with abstraction principles as a means of justifying reference to abstract objects.Gnatek Zuzanna - 2020 - Dissertation, Trinity College, Dublin
  40. On the Varieties of Abstract Objects.James E. Davies - 2019 - Australasian Journal of Philosophy 97 (4):809-823.
    I reconcile the spatiotemporal location of repeatable artworks and impure sets with the non-location of natural numbers despite all three being varieties of abstract objects. This is possible because, while the identity conditions for all three can be given by abstraction principles, in the former two cases spatiotemporal location is a congruence for the equivalence relation featuring in the relevant principle, whereas in the latter it is not. I then generalize this to other ‘physical’ properties like shape, mass, and causal (...)
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  41. In Defense of Benacerraf’s Multiple-Reductions Argument.Michele Ginammi - 2019 - Philosophia Mathematica 27 (2):276-288.
    I discuss Steinhart’s argument against Benacerraf’s famous multiple-reductions argument to the effect that numbers cannot be sets. Steinhart offers a mathematical argument according to which there is only one series of sets to which the natural numbers can be reduced, and thus attacks Benacerraf’s assumption that there are multiple reductions of numbers to sets. I will argue that Steinhart’s argument is problematic and should not be accepted.
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  42. Apophatic Finitism and Infinitism.Jan Heylen - 2019 - Logique Et Analyse 62 (247):319-337.
    This article is about the ontological dispute between finitists, who claim that only finitely many numbers exist, and infinitists, who claim that infinitely many numbers exist. Van Bendegem set out to solve the 'general problem' for finitism: how can one recast finite fragments of classical mathematics in finitist terms? To solve this problem Van Bendegem comes up with a new brand of finitism, namely so-called 'apophatic finitism'. In this article it will be argued that apophatic finitism is unable to represent (...)
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  43. Loose Talk, Scale Presuppositions and QUD.Daniel Hoek - 2019 - In Julian J. Schlöder, Dean McHugh & Floris Roelofsen (eds.), Proceedings of the 22nd Amsterdam Colloquium. pp. 171-180.
    I present a new pragmatic theory of loose talk, focussing on the loose use of numbers and measurement expressions. The account explains loose readings as arising from a pragmatic mechanism aimed at restoring relevance to the question under discussion (QUD), appealing to Krifka's notion of a measurement scale. The core motivating observation is that the loose reading of a claim need not be weaker than its literal content, as almost all pragmatic treatments of loose talk have assumed (e.g. Lasersohn). The (...)
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  44. The Problem of Fregean Equivalents.Joongol Kim - 2019 - Dialectica 73 (3):367-394.
    It would seem that some statements like ‘There are exactly four moons of Jupiter’ and ‘The number of moons of Jupiter is four’ have the same truth-conditions and yet differ in ontological commitment. One strategy to resolve this paradoxical phenomenon is to insist that the statements have not only the same truth-conditions but also the same ontological commitments; the other strategy is to reject the presumption that they have the same truth-conditions. This paper critically examines some popular versions of these (...)
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  45. Deflationary Nominalism and Puzzle Avoidance.David Mark Kovacs - 2019 - Philosophia Mathematica 27 (1):88-104.
    In a series of works, Jody Azzouni has defended deflationary nominalism, the view that certain sentences quantifying over mathematical objects are literally true, although such objects do not exist. One alleged attraction of this view is that it avoids various philosophical puzzles about mathematical objects. I argue that this thought is misguided. I first develop an ontologically neutral counterpart of Field’s reliability challenge and argue that deflationary nominalism offers no distinctive answer to it. I then show how this reasoning generalizes (...)
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  46. Hitting a Moving Target: Gödel, Carnap, and Mathematics as Logical Syntax.Gregory Lavers - 2019 - Philosophia Mathematica 27 (2):219-243.
    From 1953 to 1959 Gödel worked on a response to Carnap’s philosophy of mathematics. The drafts display Gödel’s familiarity with Carnap’s position from The Logical Syntax of Language, but they received a dismissive reaction on their eventual, posthumous, publication. Gödel’s two principal points, however, will here be defended. Gödel, though, had wished simply to append a few paragraphs to show that the same arguments apply to Carnap’s later views. Carnap’s position, however, had changed significantly in the intervening years, and to (...)
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  47. Antireductionism and Ordinals.Beau Madison Mount - 2019 - Philosophia Mathematica 27 (1):105-124.
    I develop a novel argument against the claim that ordinals are sets. In contrast to Benacerraf’s antireductionist argument, I make no use of covert epistemic assumptions. Instead, my argument uses considerations of ontological dependence. I draw on the datum that sets depend immediately and asymmetrically on their elements and argue that this datum is incompatible with reductionism, given plausible assumptions about the dependence profile of ordinals. In addition, I show that a structurally similar argument can be made against the claim (...)
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  48. Numbers, Their Occult Power and Mystic Virtue.William Wynn Westcott - 2019 - Independently Published.
    Numbers, Their Occult Power and Mystic Virtue, by English coroner, ceremonial magician, theosophist and Freemason, William Wynn Westcott, was originally published in 1911. The work surveys various numerical lore from the ages, and religions, and is a invaluable work for one studying numerical symbolism.
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  49. Maths, Logic and Language.Tetsuaki Iwamoto - 2018 - Geneva: Logic Forum.
    A work on the philosophy of mathematics (2017) -/- ‘Number’, such a simple idea, and yet it fascinated and absorbed the greatest proportion of human geniuses over centuries, not to mention the likes of Pythagoras, Euclid, Newton, Leibniz, Descartes and countless maths giants like Euler, Gauss and Hilbert, etc.. Einstein thought of pure maths as the poetry of logical ideas, the exactitude of which, although independent of experience, strangely seems to benefit the study of the objects of reality. And, interestingly (...)
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  50. Arbitrary reference, numbers, and propositions.Michele Palmira - 2018 - European Journal of Philosophy 26 (3):1069-1085.
    Reductionist realist accounts of certain entities, such as the natural numbers and propositions, have been taken to be fatally undermined by what we may call the problem of arbitrary identification. The problem is that there are multiple and equally adequate reductions of the natural numbers to sets (see Benacerraf, 1965), as well as of propositions to unstructured or structured entities (see, e.g., Bealer, 1998; King, Soames, & Speaks, 2014; Melia, 1992). This paper sets out to solve the problem by canvassing (...)
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1 — 50 / 421