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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 1950 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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353 found
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1 — 50 / 353
  1. added 2020-05-11
    Frege on Number Properties.Andrew D. Irvine - 2010 - Studia Logica 96 (2):239-260.
    In the Grundlagen , Frege offers eight main arguments, together with a series of more minor supporting arguments, against Mill’s view that numbers are “properties of external things”. This paper reviews all eight of these arguments, arguing that none are conclusive.
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  2. added 2020-04-16
    O conceito de número.Fernando Raul Neto & Bruno Bentzen - 2013 - Perspectiva Filosófica 2 (40):140-178.
    "The Concept of Number", by Ernst Cassirer, is the second chapter of his first systematic work, the "Substanzbegriff und Funktionsbegriff: Untersuchungen über die Grundfragen der Erkenntniskritik", originally published in German in 1910. The translation to English, in 1953, by Marie Collins Swabeyand William Curtis Swabey, under the title "Substance and Function and Einstein's Theory of Relativity", despite its importance for having widely disseminated the work, loses in its title the work's essence: the opposition between "concept-substance" and "concept-function", or rather, between (...)
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  3. added 2020-03-30
    How Can Mathematical Objects Be Real but Mind-Dependent?Hazhir Roshangar - manuscript
    Taking mathematics as a language based on empirical experience, I argue for an account of mathematics in which its objects are abstracta that describe and communicate the structure of reality based on some of our ancestral interactions with their environment. I argue that mathematics as a language is mostly invented, and it is mind-dependent in a specific sense. However, the bases of mathematics will characterize it as a real, non-fictional science of structures.
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  4. added 2020-03-07
    To Reduce Nothingness Into a Reference by Falsity.Hazhir Roshangar - manuscript
    Assuming the absolute nothingness as the most basic object of thought, I present a way to refer to this object, by reducing it onto a primitive object that supersedes and comes right after the absolute nothingness. The new primitive object that is constructed can be regarded as a formal system that can generate some infinite variety of symbols. [The PDF here is outdated, for a recent draft please contact me.].
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  5. added 2020-03-06
    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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  6. added 2020-01-28
    Number and Reality: Sources of Scientific Knowledge.Alex V. Halapsis - 2016 - ScienceRise 23 (6):59-64.
    Pythagoras’s number doctrine had a great effect on the development of science. Number – the key to the highest reality, and such approach allowed Pythagoras to transform mathematics from craft into science, which continues implementation of its project of “digitization of being”. Pythagoras's project underwent considerable transformation, but it only means that the plan in knowledge is often far from result.
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  7. added 2020-01-28
    Visa to Heaven: Orpheus, Pythagoras, and Immortality.Alex V. Halapsis - 2016 - ScienceRise 25 (8):60-65.
    The article deals with the doctrines of Orpheus and Pythagoras about the immortality of the soul in the context of the birth of philosophy in ancient Greece. Orpheus demonstrated the closeness of heavenly (divine) and earthly (human) worlds, and Pythagoras mathematically proved their fundamental identity. Greek philosophy was “an investment in the afterlife future”, being the product of the mystical (Orpheus) and rationalist (Pythagoras) theology.
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  8. added 2019-12-31
    Numbers and Manifolds.Peter Simons - 1982 - In Barry Smith (ed.), Parts and Moments. Studies in Logic and Formal Ontology. Munich: Philosophia. pp. 160-197.
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  9. added 2019-12-17
    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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  10. added 2019-11-16
    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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  11. added 2019-09-16
    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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  12. added 2019-09-15
    Introduction to Husserl’s Lecture On the Concept of Number.Carlo Ierna - 2005 - New Yearbook for Phenomenology and Phenomenological Philosophy 5:276-277.
    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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  13. added 2019-09-05
    Semantic Nominalism.Gabriel Uzquiano - 2005 - Dialectica 59 (2):265-282.
    The aim of the present paper is twofold. One task is to argue that our use of the numerical vocabulary in theory and applications determines the reference of the numerical terms more precisely than up to isomorphism. In particular our use of the numerical vocabulary in modal and counterfactual contexts of application excludes contingent existents as candidate referents for the numerical terms. The second task is to explore the impact of this conclusion on what I call semantic nominalism, which is (...)
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  14. added 2019-07-22
    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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  15. added 2019-06-06
    Beyond Pico Della Mirandola: John Dee’s ‘Formal Numbers’ and ‘Real Cabala’.Jean-Marc Mandosio - 2012 - Studies in History and Philosophy of Science Part A 43 (3):489-497.
    It is well known that, in both the Monas hieroglyphica and the Mathematicall praeface, Dee drew a part of his inspiration from Pico della Mirandola’s works. However, the nature and extent of Dee’s borrowings has not yet been studied. In fact, the only work of Pico really read and used by Dee was the 900 conclusions, where he found the conception of ‘formal numbers’: that is, mystical numbers carrying magical and divinatory powers. This is very important, since Dee sees these (...)
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  16. added 2019-06-06
    Logicism and the Problem of Infinity: The Number of Numbers: Articles.Gregory Landini - 2011 - Philosophia Mathematica 19 (2):167-212.
    Simple-type theory is widely regarded as inadequate to capture the metaphysics of mathematics. The problem, however, is not that some kinds of structure cannot be studied within simple-type theory. Even structures that violate simple-types are isomorphic to structures that can be studied in simple-type theory. In disputes over the logicist foundations of mathematics, the central issue concerns the problem that simple-type theory fails to assure an infinity of natural numbers as objects. This paper argues that the problem of infinity is (...)
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  17. added 2019-06-06
    Gestures Expressing Numbers — or — Numbers Expressed by Gestures.Vilmos Voigt - 2010 - American Journal of Semiotics 26 (1/4):111-127.
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  18. added 2019-06-06
    The Completeness of the Real Line.Matthew E. Moore - 2007 - Critica 39 (117):61-86.
    It is widely taken for granted that physical lines are real lines, i.e., that the arithmetical structure of the real numbers uniquely matches the geometrical structure of lines in space; and that other number systems, like Robinson's hyperreals, accordingly fail to fit the structure of space. Intuitive justifications for the consensus view are considered and rejected. Insofar as it is justified at all, the conviction that physical lines are real lines is a scientific hypothesis which we may one day reject. (...)
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  19. added 2019-06-06
    Review of Real Numbers, Generalizations of the Reals, & Theories of Continua by Philip Ehrlich. [REVIEW]Colin McLarty - 1999 - Philosophy of Science 66 (3):500-501.
  20. added 2019-06-06
    Real Numbers, Generalizations of the Reals and Theories of Continua.Philip Ehrlich - 1996 - British Journal for the Philosophy of Science 47 (2):320-324.
  21. added 2019-06-06
    Jens Høyrup, In Measure, Number, and Weight: Studies in Mathematics and Culture. SUNY Series in Science, Technology and Society. Albany: State University of New York Press, 1994. Pp. Xviii + 430. ISBN 0-7914-1821-9. $16.95. [REVIEW]George Molland - 1996 - British Journal for the History of Science 29 (2):229-230.
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  22. added 2019-06-06
    On the Failure of Mathematics' Philosophy: Review of P. Maddy, Realism in Mathematics; and C. Chihara, Constructibility and Mathematical Existence.David Charles McCarty - 1993 - Synthese 96 (2):255-291.
  23. added 2019-06-06
    ¿Existen Numeros Fuera de la Matematica?Alfonso Avila del Palacio - 1993 - Theoria: Revista de Teoría, Historia y Fundamentos de la Ciencia 8 (1):89-112.
    Our aim in this paper is to propose an ontology for numbers that is compatible with an epistemology that does not invoke mysterious faculties. On the basis of my explanatory system, we find objects capable of being classified: horses, colors, etc. Once grouped, they can be reclassified in units, pairs, and so on. When they use expressions like “three horses”, in fact, I believe that what they mean is “a threesome of horses”. I call nonmathematical numbers those reclassification, and I (...)
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  24. added 2019-06-06
    Formalism, Hamilton and Complex Numbers.John O'Neill - 1986 - Studies in History and Philosophy of Science Part A 17 (3):351.
    The development and applicability of complex numbers is often cited in defence of the formalist philosophy of mathematics. This view is rejected through an examination of hamilton's development of the notion of complex numbers as ordered pairs of reals, And his later development of the quaternion theory, Which subsequently formed the basis of vector analysis. Formalism, By protecting informal assumptions from critical scrutiny, Constrained rather than encouraged the development of mathematics.
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  25. added 2019-06-06
    Science Without Numbers: A Defence of Nominalism. [REVIEW]Nino B. Cocchiarella - 1984 - International Studies in Philosophy 16 (1):93-95.
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  26. added 2019-06-06
    Bostock David. Logic and Arithmetic. Volume 1. Natural Numbers. The Clarendon Press, Oxford University Press, Oxford 1974, X + 219 Pp.Bostock David. Logic and Arithmetic. Volume 2. Rational and Irrational Numbers. The Clarendon Press, Oxford University Press, Oxford 1979, Ix + 307 Pp. [REVIEW]Michael D. Resnik - 1982 - Journal of Symbolic Logic 47 (3):708-713.
  27. added 2019-06-05
    Number Words as Number Names.Friederike Moltmann - 2017 - Linguistics and Philosophy 40 (4):331-345.
    This paper criticizes the view that number words in argument position retain the meaning they have on an adjectival or determiner use, as argued by Hofweber :179–225, 2005) and Moltmann :499–534, 2013a, 2013b). In particular the paper re-evaluates syntactic evidence from German given in Moltmann to that effect.
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  28. added 2019-06-05
    Negotiating Boundaries.Dennis T. Olson - 1997 - Interpretation: A Journal of Bible and Theology 51 (3):229-240.
    The story of Israel's apostasy in Numbers 25 marks a turning point in the wilderness narrative: a passing generation fails to find a faithful alternative to rigid obedience and rebellious resistance to authority. Yet a new generation of God's people emerges who work out a series of compromises between respect for old traditions and engagement with new realities. This new generation provides a model that negotiates between a hermeneutic of consent and a hermeneutic of suspicion.
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  29. added 2019-03-01
    On the Reality of the Continuum Discussion Note: A Reply to Ormell, ‘Russell's Moment of Candour’, Philosophy: Anne Newstead and James Franklin.Anne Newstead - 2008 - Philosophy 83 (1):117-127.
    In a recent article, Christopher Ormell argues against the traditional mathematical view that the real numbers form an uncountably infinite set. He rejects the conclusion of Cantor’s diagonal argument for the higher, non-denumerable infinity of the real numbers. He does so on the basis that the classical conception of a real number is mys- terious, ineffable, and epistemically suspect. Instead, he urges that mathematics should admit only ‘well-defined’ real numbers as proper objects of study. In practice, this means excluding as (...)
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  30. added 2019-02-05
    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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  31. added 2018-12-22
    Plato’s Philosophy of Cognition by Mathematical Modelling.Roman S. Kljujkov & Sergey F. Kljujkov - 2014 - Dialogue and Universalism 24 (3):110-115.
    By the end of his life Plato had rearranged the theory of ideas into his teaching about ideal numbers, but no written records have been left. The Ideal mathematics of Plato is present in all his dialogues. It can be clearly grasped in relation to the effective use of mathematical modelling. Many problems of mathematical modelling were laid in the foundation of the method by cutting the three-level idealism of Plato to the single-level “ideism” of Aristotle. For a long time, (...)
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  32. added 2018-12-08
    Approaching Infinity.Michael Huemer - 2016 - New York: Palgrave Macmillan.
  33. added 2018-12-08
    Aristotelian Finitism.Tamer Nawar - 2015 - Synthese 192 (8):2345-2360.
    It is widely known that Aristotle rules out the existence of actual infinities but allows for potential infinities. However, precisely why Aristotle should deny the existence of actual infinities remains somewhat obscure and has received relatively little attention in the secondary literature. In this paper I investigate the motivations of Aristotle’s finitism and offer a careful examination of some of the arguments considered by Aristotle both in favour of and against the existence of actual infinities. I argue that Aristotle has (...)
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  34. added 2018-11-23
    How Abstract Objects Strike Us.Michael Liston - 1994 - Dialectica 48 (1):3-27.
    SummaryBenacerraf challenges us to account for the reliability of our mathematical beliefs given that there appear to be no natural connections between mathematical believers and mathematical ontology. In this paper I try to do two things. I argue that the interactionist view underlying this challenge renders inexplicable not only the reliability of our mathematical beliefs, construed either platonistically or naturalistically , but also the reliability of most of our beliefs in physics. I attempt to counter Benacerraf's challenge by sketching an (...)
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  35. added 2018-11-22
    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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  36. added 2018-10-09
    Ontology and the Ambitions of Metaphysics.Thomas Hofweber - 2016 - Oxford University Press UK.
    Many significant problems in metaphysics are tied to ontological questions, but ontology and its relation to larger questions in metaphysics give rise to a series of puzzles that suggest that we don't fully understand what ontology is supposed to do, nor what ambitions metaphysics can have for finding out about what the world is like. Thomas Hofweber aims to solve these puzzles about ontology and consequently to make progress on four metaphysical debates tied to ontology: the philosophy of arithmetic, the (...)
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  37. added 2018-10-01
    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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  38. added 2018-09-21
    The Adverbial Theory of Numbers: Some Clarifications.Joongol Kim - forthcoming - Synthese:1-20.
    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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  39. added 2018-09-20
    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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  40. added 2018-09-06
    Frege Numbers and the Relativity Argument.Christopher Menzel - 1988 - Canadian Journal of Philosophy 18 (1):87-98.
    Textual and historical subtleties aside, let's call the idea that numbers are properties of equinumerous sets ‘the Fregean thesis.’ In a recent paper, Palle Yourgrau claims to have found a decisive refutation of this thesis. More surprising still, he claims in addition that the essence of this refutation is found in the Grundlagen itself – the very masterpiece in which Frege first proffered his thesis. My intention in this note is to evaluate these claims, and along the way to shed (...)
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  41. added 2018-08-17
    Numbers and Propositions: Reply to Melia.Tim Crane - 1992 - Analysis 52 (4):253-256.
    Is the way we use propositions to individuate beliefs and other intentional states analogous to the way we use numbers to measure weights and other physical magnitudes? In an earlier paper [2], I argued that there is an important disanalogy. One and the same weight can be 'related to' different numbers under different units of measurement. Moreover, the choice of a unit of measurement is arbitrary,in the sense that which way we choose doesn't affect the weight attributed to the object. (...)
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  42. added 2018-07-16
    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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  43. added 2018-04-02
    Beyond Witches, Angels and Unicorns. The Possibility of Expanding Russell´s Existential Analysis.Olga Ramirez - 2018 - E-Logos Electronic Journal for Philosophy 25 (1):4-15.
    This paper attempts to be a contribution to the epistemological project of explaining complex conceptual structures departing from more basic ones. The central thesis of the paper is that there are what I call “functionally structured concepts”, these are non-harmonic concepts in Dummett’s sense that might be legitimized if there is a function that justifies the tie between the inferential connection the concept allows us to trace. Proving this requires enhancing the russellian existential analysis of definite descriptions to apply to (...)
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  44. added 2018-03-31
    Our Incorrigible Ontological Relations and Categories of Being.Julian M. Galvez Bunge (ed.) - 2017 - USA: Amazon.
    The purpose of this book is to address the controversial issues of whether we have a fixed set of ontological categories and if they have some epistemic value at all. Which are our ontological categories? What determines them? Do they play a role in cognition? If so, which? What do they force to presuppose regarding our world-view? If they constitute a limit to possible knowledge, up to what point is science possible? Does their study make of philosophy a science? Departing (...)
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  45. added 2018-02-19
    Time and the Russell Definition of Number.Charles Byron Cross - 1979 - Southwestern Journal of Philosophy 10 (2):177-180.
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  46. added 2018-02-17
    Where Do the Natural Numbers Come From?Harold T. Hodes - 1990 - Synthese 84 (3):347-407.
  47. added 2018-02-16
    The Indefinability of €œOne”.Laurence Goldstein - 2002 - 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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  48. added 2018-02-16
    Why Numbers Are Sets.Eric Steinhart - 2002 - Synthese 133 (3):343-361.
    I follow standard mathematical practice and theory to argue that the natural numbers are the finite von Neumann ordinals. I present the reasons standardly given for identifying the natural numbers with the finite von Neumann's. I give a detailed mathematical demonstration that 0 is {} and for every natural number n, n is the set of all natural numbers less than n. Natural numbers are sets. They are the finite von Neumann ordinals.
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  49. added 2017-11-28
    What We Talk About When We Talk About Numbers.Richard Pettigrew - 2018 - Annals of Pure and Applied Logic 169 (12):1437-1456.
    In this paper, I describe and motivate a new species of mathematical structuralism, which I call Instrumental Nominalism about Set-Theoretic Structuralism. As the name suggests, this approach takes standard Set-Theoretic Structuralism of the sort championed by Bourbaki and removes its ontological commitments by taking an instrumental nominalist approach to that ontology of the sort described by Joseph Melia and Gideon Rosen. I argue that this avoids all of the problems that plague other versions of structuralism.
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  50. added 2017-11-09
    Fourteen Arguments in Favour of a Formalist Philosophy of Real Mathematics.Karlis Podnieks - 2015 - Baltic Journal of Modern Computing 3 (1):1-15.
    The formalist philosophy of mathematics (in its purest, most extreme version) is widely regarded as a “discredited position”. This pure and extreme version of formalism is called by some authors “game formalism”, because it is alleged to represent mathematics as a meaningless game with strings of symbols. Nevertheless, I would like to draw attention to some arguments in favour of game formalism as an appropriate philosophy of real mathematics. For the most part, these arguments have not yet been used or (...)
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1 — 50 / 353