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  1. added 2020-06-02
    Frege's Definition of Numbers.Edwin Martin - 1987 - Philosophical Papers 16 (1):59-73.
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  2. added 2020-06-02
    Frege's Definition of Number.Steven Wagner - 1983 - Notre Dame Journal of Formal Logic 24 (1):1-21.
    Frege believes (1) that his definition of number is (partly) arbitrary; (2) that it "makes" numbers of certain extensions; (3) that without such a definition we cannot even think or understand arithmetical propositions. this position is part of a view according to which mathematics in general involves the free construction of objects, their properties, and the very contents of mathematical propositions. frege tries to avoid excess subjectivism by the kantian device of treating alternative systems of arithmetic (e.g.) as different appearances (...)
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  3. added 2020-06-01
    Frege, Dedekind, and Peano on the Foundations of Arithmetic. [REVIEW]J. P. Mayberry - 1984 - Philosophical Quarterly 34 (136):424.
    First published in 1982, this reissue contains a critical exposition of the views of Frege, Dedekind and Peano on the foundations of arithmetic. The last quarter of the 19th century witnessed a remarkable growth of interest in the foundations of arithmetic. This work analyses both the reasons for this growth of interest within both mathematics and philosophy and the ways in which this study of the foundations of arithmetic led to new insights in philosophy and striking advances in logic. This (...)
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  4. added 2020-05-31
    Erkenntnistheorie der Zahldefinition Und Philosophische Grundlegung der Arithmetik Unter Bezugnahme Auf Einen Vergleich Von Gottlob Freges Logizismus Und Platonischer Philosophie (Syrian, Theon Von Smyrna U.A.).Markus Schmitz - 2001 - Journal for General Philosophy of Science / Zeitschrift für Allgemeine Wissenschaftstheorie 32 (2):271-305.
    The epistomology of the definition of number and the philosophical foundation of arithmetic based on a comparison between Gottlob Frege's logicism and Platonic philosophy (Syrianus, Theo Smyrnaeus, and others). The intention of this article is to provide arithmetic with a logically and methodologically valid definition of number for construing a consistent philosophical foundation of arithmetic. The – surely astonishing – main thesis is that instead of the modern and contemporary attempts, especially in Gottlob Frege's Foundations of Arithmetic, such a definition (...)
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  5. added 2020-05-31
    Frege's `Series of Natural Numbers'.T. J. Smiley - 1988 - Mind 97 (388):583-584.
  6. added 2020-05-30
    Frege: Evidence for Self-Evidence.Robin Jeshion - 2004 - Mind 113 (449):131-138.
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  7. added 2020-05-30
    Frege and the Rigorization of Analysis.William Demopoulos - 1994 - Journal of Philosophical Logic 23 (3):225 - 245.
    This paper has three goals: (i) to show that the foundational program begun in the Begriffsschroft, and carried forward in the Grundlagen, represented Frege's attempt to establish the autonomy of arithmetic from geometry and kinematics; the cogency and coherence of 'intuitive' reasoning were not in question. (ii) To place Frege's logicism in the context of the nineteenth century tradition in mathematical analysis, and, in particular, to show how the modern concept of a function made it possible for Frege to pursue (...)
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  8. added 2020-05-30
    Relativism and the Sociology of Mathematics: Remarks on Bloor, Flew, and Frege.Timm Triplett - 1986 - Inquiry: An Interdisciplinary Journal of Philosophy 29 (1-4):439-450.
    Antony Flew's ?A Strong Programme for the Sociology of Belief (Inquiry 25 {1982], 365?78) critically assesses the strong programme in the sociology of knowledge defended in David Bloor's Knowledge and Social Imagery. I argue that Flew's rejection of the epistemological relativism evident in Bloor's work begs the question against the relativist and ignores Bloor's focus on the social relativity of mathematical knowledge. Bloor attempts to establish such relativity via a sociological analysis of Frege's theory of number. But this analysis only (...)
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  9. added 2020-05-29
    Review of M. Dummett, Frege: Philosophy of Mathematics[REVIEW]Mary Tiles - 1993 - Philosophy 68 (265):405-411.
  10. added 2020-05-29
    Frege: The Last Logicist.Paul Benacerraf - 1981 - Midwest Studies in Philosophy 6 (1):17-36.
  11. added 2020-05-28
    Concepts and Counting.Ian Rumfitt - 2001 - Proceedings of the Aristotelian Society 102 (1):41-68.
    Frege's analysis of Zahlangaben is expounded and evaluated.
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  12. added 2020-05-28
    Critical Notice of Michael Dummett, Frege: Philosophy of Mathematics. [REVIEW]Richard Heck - 1993 - Philosophical Quarterly 43:223-33.
  13. added 2020-05-27
    Platonic and Fregean Numbers.N. White - 2012 - Philosophia Mathematica 20 (2):224-244.
    Rather than reading Plato's philosophy of arithmetic ‘charitably’, it is better to try to explain its failure to generate any fruitful ideas. Prominent in the explanation is Plato's focus on predicates assigning cardinalities and on ‘groups’ falling under them. This focus left Plato unable to envisage the possibility, emerging in Dedekind and Frege but which arithmetic in Plato's time would not easily have suggested, of regarding numbers as objects essentially ranged in the structure of a progression.
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  14. added 2020-05-27
    Liberté Et Vérité: Pensée Mathématique & Spéculation Philosophique.Imre Tóth - 2009 - Éclat.
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  15. added 2020-05-27
    Frege: Philosophy of Mathematics. [REVIEW]William Demopoulos - 1993 - Canadian Journal of Philosophy 23 (3):477-497.
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  16. added 2020-05-26
    Neologicism, Frege's Constraint, and the Frege‐Heck Condition.Eric Snyder, Richard Samuels & Stewart Shapiro - forthcoming - Noûs.
    One of the more distinctive features of Bob Hale and Crispin Wright’s neologicism about arithmetic is their invocation of Frege’s Constraint – roughly, the requirement that the core empirical applications for a class of numbers be “built directly into” their formal characterization. In particular, they maintain that, if adopted, Frege’s Constraint adjudicates in favor of their preferred foundation – Hume’s Principle – and against alternatives, such as the Dedekind-Peano axioms. In what follows we establish two main claims. First, we show (...)
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  17. added 2020-05-26
    Review of "Frege: Philosophy of Mathematics". [REVIEW]Marco Antonio Ruffino - forthcoming - Manuscrito.
    In this review I briefly explain the most important points of each chapter of Dummett's book, and critically discuss some of them. Special attention is given to the criticisms of Crispin Wright's interpretation of Frege's Platonism, and also to Dummett's interpretation of the role(s) of the context principle in Frege's thought.
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  18. added 2020-05-26
    Euclid Strikes Back at Frege.Joongol Kim - 2014 - Philosophical Quarterly 64 (254):20-38.
    Frege’s argument against the ancient Greek conception of numbers as 'multitudes of units’ has been hailed as one of the most successful in his "Grundlagen". The aim of this paper is to show that despite Frege’s best efforts, the Euclidean conception remains a viable alternative to the Fregean conception of numbers by arguing that neither a dilemma argument Frege brings against the Euclidean conception nor a possible argument against it based on the truth of what is known as "Hume’s Principle" (...)
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  19. added 2020-05-26
    Zamyšlení nad Fregovou definicí čísla.Marta Vlasáková - 2010 - Organon F: Medzinárodný Časopis Pre Analytickú Filozofiu 17 (3):339-353.
    In his treatise Die Grundlagen der Arithmetik, Gottlob Frege tries to find a definition of number. First he rejects the idea that number could be a property of external objects. Then he comes with a suggestion that a numerical statement expresses a property of a concept, namely it indicates how many objects fall under the concept. Subsequently Frege rejects, or at least essentially modifies, also this definition, because in his view that a number cannot be a property – it should (...)
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  20. added 2020-05-26
    A Reflection on Frege's Definition of the Number.Marta Vlasakova - 2010 - Organon F: Medzinárodný Časopis Pre Analytickú Filozofiu 17 (3):339-353.
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  21. added 2020-05-25
    Some Naturalistic Comments on Frege's Philosophy of Mathematics.Y. E. Feng - 2012 - Frontiers of Philosophy in China 7 (3):378-403.
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  22. added 2020-05-25
    Frege and the Philosophy of Mathematics. [REVIEW]J. E. Tiles - 1982 - Philosophical Books 23 (3):164-166.
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  23. added 2020-05-24
    Frege: Philosophy of Mathematics. [REVIEW]Crispin Wright - 1995 - Philosophical Books 36 (2):89-102.
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  24. added 2020-05-24
    Frege's Conception of Numbers as Objects. [REVIEW]J. E. Tiles - 1984 - Philosophical Books 25 (3):159-161.
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  25. added 2020-05-24
    Die Revisionbedürftigkeit der logischen Semantik Freges.Christian Thiel - 1983 - Anuario Filosófico 16 (1):293-302.
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  26. added 2020-05-23
    The Logicism of Frege, Dedekind, and Russell.William Demopoulos & Peter Clark - 2005 - In Stewart Shapiro (ed.), Oxford Handbook of Philosophy of Mathematics and Logic. Oxford University Press. pp. 129--165.
    The common thread running through the logicism of Frege, Dedekind, and Russell is their opposition to the Kantian thesis that our knowledge of arithmetic rests on spatio-temporal intuition. Our critical exposition of the view proceeds by tracing its answers to three fundamental questions: What is the basis for our knowledge of the infinity of the numbers? How is arithmetic applicable to reality? Why is reasoning by induction justified?
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  27. added 2020-05-23
    Frege's and Bolzano's Rationalist Conceptions of Arithmetic.Charles Chihara - 1999 - Revue d'Histoire des Sciences 52 (3):343-362.
    In this article, I compare Gottlob Frege's and Bernard Bolzano's rationalist conceptions of arithmetic. Each philosopher worked out a complicated system of propositions, all of which were set forth as true. The axioms, or basic truths, make up the foundations of the subject of arithmetic. Each member of the system which is not an axiom is related (objectively) to the axioms at the base. Even though this relation to the base may not yet be scientifically proven, the propositions of the (...)
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  28. added 2020-05-22
    Predicative Frege Arithmetic and ‘Everyday’ Mathematics.Richard Heck - 2014 - Philosophia Mathematica 22 (3):279-307.
    The primary purpose of this note is to demonstrate that predicative Frege arithmetic naturally interprets certain weak but non-trivial arithmetical theories. It will take almost as long to explain what this means and why it matters as it will to prove the results.
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  29. added 2020-05-22
    Numerical Abstraction Via the Frege Quantifier.G. Aldo Antonelli - 2010 - Notre Dame Journal of Formal Logic 51 (2):161-179.
    This paper presents a formalization of first-order arithmetic characterizing the natural numbers as abstracta of the equinumerosity relation. The formalization turns on the interaction of a nonstandard cardinality quantifier with an abstraction operator assigning objects to predicates. The project draws its philosophical motivation from a nonreductionist conception of logicism, a deflationary view of abstraction, and an approach to formal arithmetic that emphasizes the cardinal properties of the natural numbers over the structural ones.
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  30. added 2020-05-22
    Frege: Philosophy of Mathematics By Michael Dummett Duckworth 1991 Xiii + 331 Pp. [REVIEW]Mary Tiles - 1993 - Philosophy 68 (265):405-411.
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  31. added 2020-05-22
    Michael Resnik, Frege and the Philosophy of Mathematics Reviewed By. [REVIEW]Alan McMichael - 1982 - Philosophy in Review 2 (6):291-294.
  32. added 2020-05-21
    Frege on the Statement of Number.David Sullivan - 1990 - Philosophy and Phenomenological Research 50 (3):595-603.
  33. added 2020-05-20
    Frege on Numbers: Beyond the Platonist Picture.Erich H. Reck - 2005 - The Harvard Review of Philosophy 13 (2):25-40.
    Gottlob Frege is often called a "platonist". In connection with his philosophy we can talk about platonism concerning three kinds of entities: numbers, or logical objects more generally; concepts, or functions more generally; thoughts, or senses more generally. I will only be concerned about the first of these three kinds here, in particular about the natural numbers. I will also focus mostly on Frege's corresponding remarks in The Foundations of Arithmetic (1884), supplemented by a few asides on Basic Laws of (...)
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  34. added 2020-05-20
    Frege's Reduction.Patricia A. Blanchette - 1994 - History and Philosophy of Logic 15 (1):85-103.
    This paper defends the view that Frege’s reduction of arithmetic to logic would, if successful, have shown that arithmetical knowledge is analytic in essentially Kant’s sense. It is argued, as against Paul Benacerraf, that Frege’s apparent acceptance of multiple reductions is compatible with this epistemological thesis. The importance of this defense is that (a) it clarifies the role of proof, definition, and analysis in Frege’s logicist works; and (b) it demonstrates that the Fregean style of reduction is a valuable tool (...)
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  35. added 2020-05-20
    Frege and Husserl on Number.Richard Tieszen - 1990 - Ratio 3 (2):150-164.
  36. added 2020-05-18
    Two Questions About the Revival of Frege's Programme.Jean-Jacques Szczeciniarz - 2004 - In Friedrich Stadler (ed.), Vienna Circle Institute Yearbook. Springer. pp. 195.
    The essential aim of Professor P. Clark’s paper is to show that good reasons exist for opening the way for a revival of Frege’s Programme. Frege’s Programme tend to show that numbers are given to us by explaining the meanings of identity statements in which numbers words occur.
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  37. added 2020-05-18
    Frege Versus Cantor and Dedekind: On the Concept of Number.W. W. Tait - 1996 - In Matthias Schirn (ed.), Frege: Importance and Legacy. Berlin: Walter de Gruyter. pp. 70-113.
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  38. added 2020-05-18
    Frege, Dedekind, and the Philosophy of Mathematics.Philip Kitcher - 1986 - In L. Haaparanta & J. Hintikka (eds.), Frege Synthesized. D. Reidel Publishing Co.. pp. 299--343.
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  39. added 2020-05-18
    Frege Critique de Kant.Jacques Bouveresse - 1979 - Revue Internationale de Philosophie 130 (130):739-60.
  40. added 2020-05-17
    Mill, Frege and the Unity of Mathematics.Madeline Muntersbjorn - 2008 - ProtoSociology 25:143-159.
    This essay discusses the unity of mathematics by comparing the philosophies of Mill and Frege. While Mill is remembered as a progressive social thinker, his contributions to the development of logic are less widely heralded. In contrast, Frege made important and lasting contributions to the development of logic while his social thought, what little is known of it, was very conservative. Two theses are presented in the paper. The first is that in order to pursue Mill’s progressive sociopolitical project, one (...)
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  41. added 2020-05-13
    Des Projets Fondationnels de Husserl Et de Frege a la Perspective de Wittgenstein.Mamadou Djibo - 1996 - Dissertation, University of Ottawa
    Cette these divisee en trois parties, comporte deux caracteristiques essentielles: ; Elle comporte une teneure mathematique: des progres accomplis entre 1870 et 1914 par les mathematiciens au sein des savoirs que sont l'analyse infinitesimale, l'algebre, la geometrie etc. ont conferre une place sui generis a l'algebre. Sa redefinition va integrer des objets quelconques et non plus seulement les simples techniques du calcul litteral. Aussi certains mathematiciens observerent un desinteret pour ces calculs au profit d'une reflexion theorique autour du mot d'ordre (...)
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  42. added 2020-05-12
    Logicism as Making Arithmetic Explicit.Vojtěch Kolman - 2015 - Erkenntnis 80 (3):487-503.
    This paper aims to shed light on the broader significance of Frege’s logicism against the background of discussing and comparing Wittgenstein’s ‘showing/saying’-distinction with Brandom’s idiom of logic as the enterprise of making the implicit rules of our linguistic practices explicit. The main thesis of this paper is that the problem of Frege’s logicism lies deeper than in its inconsistency : it lies in the basic idea that in arithmetic one can, and should, express everything that is implicitly presupposed so that (...)
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  43. added 2020-05-12
    The Finite and the Infinite in Frege's Grundgesetze der Arithmetik.Richard Heck - 1998 - In Matthias Schirn (ed.), Philosophy of Mathematics Today. Oxford University Press.
    Discusses Frege's formal definitions and characterizations of infinite and finite sets. Speculates that Frege might have discovered the "oddity" in Dedekind's famous proof that all infinite sets are Dedekind infinite and, in doing so, stumbled across an axiom of countable choice.
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  44. added 2020-05-12
    The Natural Numbers From Frege to Hilbert.David Wells Bennett - 1961 - Dissertation, Columbia University
  45. added 2020-05-11
    Frege's Cardinals Do Not Always Obey Hume's Principle.Gregory Landini - 2017 - History and Philosophy of Logic 38 (2):127-153.
    Hume's Principle, dear to neo-Logicists, maintains that equinumerosity is both necessary and sufficient for sameness of cardinal number. All the same, Whitehead demonstrated in Principia Mathematica's logic of relations that Cantor's power-class theorem entails that Hume's Principle admits of exceptions. Of course, Hume's Principle concerns cardinals and in Principia's ‘no-classes’ theory cardinals are not objects in Frege's sense. But this paper shows that the result applies as well to the theory of cardinal numbers as objects set out in Frege's Grundgesetze. (...)
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  46. added 2020-05-11
    Fragments of Frege’s Grundgesetze and Gödel’s Constructible Universe.Sean Walsh - 2016 - Journal of Symbolic Logic 81 (2):605-628.
    Frege's Grundgesetze was one of the 19th century forerunners to contemporary set theory which was plagued by the Russell paradox. In recent years, it has been shown that subsystems of the Grundgesetze formed by restricting the comprehension schema are consistent. One aim of this paper is to ascertain how much set theory can be developed within these consistent fragments of the Grundgesetze, and our main theorem shows that there is a model of a fragment of the Grundgesetze which defines a (...)
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  47. added 2020-05-11
    Number Sentences and Specificational Sentences: Reply to Moltmann.Robert Schwartzkopff - 2016 - Philosophical Studies 173 (8):2173-2192.
    Frege proposed that sentences like ‘The number of planets is eight’ be analysed as identity statements in which the number words refer to numbers. Recently, Friederike Moltmann argued that, pace Frege, such sentences be analysed as so-called specificational sentences in which the number words have the same non-referring semantic function as the number word ‘eight’ in ‘There are eight planets’. The aim of this paper is two-fold. First, I argue that Moltmann fails to show that such sentences should be analysed (...)
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  48. added 2020-05-11
    Is Frege's Definition of the Ancestral Adequate.Richard Heck - 2016 - Philosophia Mathematica 24 (1):91-116.
    Why should one think Frege's definition of the ancestral correct? It can be proven to be extensionally correct, but the argument uses arithmetical induction, and that seems to undermine Frege's claim to have justified induction in purely logical terms. I discuss such circularity objections and then offer a new definition of the ancestral intended to be intensionally correct; its extensional correctness then follows without proof. This new definition can be proven equivalent to Frege's without any use of arithmetical induction. This (...)
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  49. 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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  50. added 2020-05-11
    Frege’s Cardinals as Concept-Correlates.Gregory Landini - 2006 - Erkenntnis 65 (2):207-243.
    In his "Grundgesetze", Frege hints that prior to his theory that cardinal numbers are objects he had an "almost completed" manuscript on cardinals. Taking this early theory to have been an account of cardinals as second-level functions, this paper works out the significance of the fact that Frege's cardinal numbers is a theory of concept-correlates. Frege held that, where n > 2, there is a one—one correlation between each n-level function and an n—1 level function, and a one—one correlation between (...)
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