Results for 'Arithmetic Foundations'

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  1.  49
    The Foundations of Arithmetic.Gottlob Frege - 1953 - Evanston: Ill., Northwestern University Press.
    In arithmetic, if only because many of its methods and concepts originated in India, it has been the tradition to reason less strictly than in geometry, ...
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  2.  94
    The Foundations of Arithmetic: A Logico-Mathematical Enquiry Into the Concept of Number.Gottlob Frege - 1950 - 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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  3.  26
    A Note on Finiteness in the Predicative Foundations of Arithmetic.Fernando Ferreira - 1999 - Journal of Philosophical Logic 28 (2):165-174.
    Recently, Feferman and Hellman (and Aczel) showed how to establish the existence and categoricity of a natural number system by predicative means given the primitive notion of a finite set of individuals and given also a suitable pairing function operating on individuals. This short paper shows that this existence and categoricity result does not rely (even indirectly) on finite-set induction, thereby sustaining Feferman and Hellman's point in favor of the view that natural number induction can be derived from a very (...)
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  4. The Foundations of Arithmetic in Finite Bounded Zermelo Set Theory.Richard Pettigrew - 2010 - Cahiers du Centre de Logique 17:99-118.
    In this paper, I pursue such a logical foundation for arithmetic in a variant of Zermelo set theory that has axioms of subset separation only for quantifier-free formulae, and according to which all sets are Dedekind finite. In section 2, I describe this variant theory, which I call ZFin0. And in section 3, I sketch foundations for arithmetic in ZFin0 and prove that certain foundational propositions that are theorems of the standard Zermelian foundation for arithmetic are (...)
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  5. The Foundations of Arithmetic: A Logico-Mathematical Enquiry Into the Concept of Number.J. L. Austin (ed.) - 1950 - Northwestern University Press.
    _The Foundations of Arithmetic_ is undoubtedly the best introduction to Frege's thought; it is here that Frege expounds the central notions of his philosophy, subjecting the views of his predecessors and contemporaries to devastating analysis. The book represents the first philosophically sound discussion of the concept of number in Western civilization. It profoundly influenced developments in the philosophy of mathematics and in general ontology.
     
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  6. On the Foundations of Greek Arithmetic.Holger A. Leuz - 2009 - Logical Analysis and History of Philosophy 12:13-47.
    The aim of this essay is to develop a formal reconstruction of Greek arithmetic. The reconstruction is based on textual evidence which comes mainly from Euclid, but also from passages in the texts of Plato and Aristotle. Following Paul Pritchard’s investigation into the meaning of the Greek term arithmos, the reconstruction will be mereological rather than set-theoretical. It is shown that the reconstructed system gives rise to an arithmetic comparable in logical strength to Robinson arithmetic. Our reconstructed (...)
     
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  7.  32
    Challenges to Predicative Foundations of Arithmetic.Solomon Feferman - manuscript
    This is a sequel to our article “Predicative foundations of arithmetic” (1995), referred to in the following as [PFA]; here we review and clarify what was accomplished in [PFA], present some improvements and extensions, and respond to several challenges. The classic challenge to a program of the sort exemplified by [PFA] was issued by Charles Parsons in a 1983 paper, subsequently revised and expanded as Parsons (1992). Another critique is due to Daniel Isaacson (1987). Most recently, Alexander George (...)
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  8.  45
    A Variant to Hilbert's Theory of the Foundations of Arithmetic.G. Kreisel - 1953 - British Journal for the Philosophy of Science 4 (14):107-129.
    IN Hilbert's theory of the foundations of any given branch of mathematics the main problem is to establish the consistency (of a suitable formalisation) of this branch. Since the (intuitionist) criticisms of classical logic, which Hilbert's theory was intended to meet, never even alluded to inconsistencies (in classical arithmetic), and since the investigations of Hilbert's school have always established much more than mere consistency, it is natural to formulate another general problem in the foundations of mathematics: to (...)
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  9.  12
    Frege, Dedekind, and Peano on the Foundations of Arithmetic.J. P. Mayberry & D. A. Gillies - 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 (...)
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  10.  1
    Frege, Dedekind, and Peano on the Foundations of Arithmetic.Donald Gillies - 1982 - Wiley-Blackwell.
    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 (...)
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  11.  28
    Review of Gottlob Frege, Dale Jacquette (Tr.), The Foundations of Arithmetic[REVIEW]Michael Kremer - 2008 - Notre Dame Philosophical Reviews 2008 (1).
    Last spring, as I was beginning a graduate seminar on Frege, I received a complimentary copy of this new translation of his masterwork, The Foundations of Arithmetic . I had ordered Austin's famous translation, well-loved for the beauty of its English and the clarity with which it presents Frege's overall argument, but known to be less than literal, and to sometimes supplement translation with interpretation. I was intrigued by Dale Jacquette's promise "to combine literal accuracy and readability for (...)
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  12.  10
    A Philosophical Introduction to the Foundations of Elementary Arithmetic by V1.03 Last Updated: 1 Jan 2001 Created: 1 Sept 2000 Please Send Your Comments to Abo. [REVIEW]Andrew Boucher - manuscript
    As it is currently used, "foundations of arithmetic" can be a misleading expression. It is not always, as the name might indicate, being used as a plural term meaning X = {x : x is a foundation of arithmetic}. Instead it has come to stand for a philosophico-logical domain of knowledge, concerned with axiom systems, structures, and analyses of arithmetic concepts. It is a bit as if "rock" had come to mean "geology." The conflation of subject (...)
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  13.  2
    Frege, Dedekind, and Peano on the Foundations of Arithmetic.Donald Gillies - 2011 - Routledge.
    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 (...)
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  14.  38
    Cognitive Foundations of Arithmetic: Evolution and Ontogenisis.Susan Carey - 2001 - Mind and Language 16 (1):37–55.
    Dehaene articulates a naturalistic approach to the cognitive foundations of mathematics. Further, he argues that the ‘number line’ system of representation is the evolutionary and ontogenetic foundation of numerical concepts. Here I endorse Dehaene’s naturalistic stance and also his characterization of analog magnitude number representations. Although analog magnitude representations are part of the evolutionary foundations of numerical concepts, I argue that they are unlikely to be part of the ontogenetic foundations of the capacity to represent natural number. (...)
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  15.  9
    Poincaré on the Foundations of Arithmetic and Geometry. Part 1: Against “Dependence-Hierarchy” Interpretations.Katherine Dunlop - 2016 - Hopos: The Journal of the International Society for the History of Philosophy of Science 6 (2):274-308.
    The main goal of part 1 is to challenge the widely held view that Poincaré orders the sciences in a hierarchy of dependence, such that all others presuppose arithmetic. Commentators have suggested that the intuition that grounds the use of induction in arithmetic also underlies the conception of a continuum, that the consistency of geometrical axioms must be proved through arithmetical induction, and that arithmetical induction licenses the supposition that certain operations form a group. I criticize each of (...)
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  16. Cognitive Foundations of Arithmetic: Evolution and Ontogenisis.Carey Susan - 2001 - Mind and Language 16 (1):37-55.
    Dehaene articulates a naturalistic approach to the cognitive foundations of mathematics. Further, he argues that the ‘number line’ system of representation is the evolutionary and ontogenetic foundation of numerical concepts. Here I endorse Dehaene’s naturalistic stance and also his characterization of analog magnitude number representations. Although analog magnitude representations are part of the evolutionary foundations of numerical concepts, I argue that they are unlikely to be part of the ontogenetic foundations of the capacity to represent natural number. (...)
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  17.  29
    Set Theory, Arithmetic, and Foundations of Mathematics: Theorems, Philosophies.Juliette Kennedy & Roman Kossak (eds.) - 2012 - Cambridge University Press.
    Machine generated contents note: 1. Introduction Juliette Kennedy and Roman Kossak; 2. Historical remarks on Suslin's problem Akihiro Kanamori; 3. The continuum hypothesis, the generic-multiverse of sets, and the [OMEGA] conjecture W. Hugh Woodin; 4. [omega]-Models of finite set theory Ali Enayat, James H. Schmerl and Albert Visser; 5. Tennenbaum's theorem for models of arithmetic Richard Kaye; 6. Hierarchies of subsystems of weak arithmetic Shahram Mohsenipour; 7. Diophantine correct open induction Sidney Raffer; 8. Tennenbaum's theorem and recursive reducts (...)
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  18.  16
    The Foundations of Arithmetic?Thomas Donaldson - 2016 - Noûs.
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  19.  37
    The Foundations of Arithmetic?Thomas Donaldson - 2016 - Noûs 50 (4).
  20. Cantor on Frege's Foundations of Arithmetic : Cantor's 1885 Review of Frege's Die Grundlagen der Arithmetik.Marcus Rossberg & Philip A. Ebert - 2010 - History and Philosophy of Logic 30 (4):341-348.
    In 1885, Georg Cantor published his review of Gottlob Frege's Grundlagen der Arithmetik . In this essay, we provide its first English translation together with an introductory note. We also provide a translation of a note by Ernst Zermelo on Cantor's review, and a new translation of Frege's brief response to Cantor. In recent years, it has become philosophical folklore that Cantor's 1885 review of Frege's Grundlagen already contained a warning to Frege. This warning is said to concern the defectiveness (...)
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  21. Evolutionary and Ontogenetic Foundations of Arithmetic.Susan Carey - 2001 - Mind and Language 16 (1):37-55.
     
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  22. The Consistency of Frege's Foundations of Arithmetic.George Boolos - 1987 - In J. Thomson (ed.), On Being and Saying: Essays in Honor of Richard Cartwright. MIT Press. pp. 3--20.
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  23. Frege, Mill, and the Foundations of Arithmetic.Glenn Kessler - 1980 - Journal of Philosophy 77 (2):65-79.
  24.  4
    Frege's Theorem and Foundations for Arithmetic.Edward N. Zalta - Spring 2015 - In Stanford Encyclopedia of Philosophy.
    The principal goal of this entry is to present Frege's Theorem (i.e., the proof that the Dedekind-Peano axioms for number theory can be derived in second-order logic supplemented only by Hume's Principle) in the most logically perspicuous manner. We strive to present Frege's Theorem by representing the ideas and claims involved in the proof in clear and well-established modern logical notation. This prepares one to better prepared to understand Frege's own notation and derivations, and read Frege's original work (whether in (...)
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  25.  72
    Predicative Foundations of Arithmetic.Solomon Feferman & Geoffrey Hellman - 1995 - Journal of Philosophical Logic 24 (1):1 - 17.
  26.  43
    The Foundations of Arithmetic.Joan B. Quick - 1952 - Thought: A Journal of Philosophy 27 (2):303-304.
  27.  40
    Fregean Abstraction, Referential Indeterminacy and the Logical Foundations of Arithmetic.Matthias Schirn - 2003 - Erkenntnis 59 (2):203 - 232.
    In Die Grundlagen der Arithmetik, Frege attempted to introduce cardinalnumbers as logical objects by means of a second-order abstraction principlewhich is now widely known as ``Hume's Principle'' (HP): The number of Fsis identical with the number of Gs if and only if F and G are equinumerous.The attempt miscarried, because in its role as a contextual definition HP fails tofix uniquely the reference of the cardinality operator ``the number of Fs''. Thisproblem of referential indeterminacy is usually called ``the Julius Caesar (...)
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  28.  6
    On the Foundations of Geometry and Formal Theories of Arithmetic.Gottlob Frege & Eike-Henner W. Kluge - 1973 - Philosophical Review 82 (2):266-269.
  29.  2
    William Ewald and Wilfried Sieg , Michael Hallett , David Hilbert’s Lectures on the Foundations of Arithmetic and Logic 1917–1933. In Collaboration with Ulrich Majer and Dirk Schlimm , Springer-Verlag: Berlin and Heidelberg, 2013, Xxvi + 1062 Pp. ISBN 978-3-540-20578-4 $129.00, $86.37. [REVIEW]Volker Peckhaus - 2017 - Studia Logica 105 (3):669-671.
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  30.  78
    Frege's Logic, Theorem, and Foundations for Arithmetic.Edward N. Zalta - 2008 - Stanford Encyclopedia of Philosophy.
    In this entry, Frege's logic is introduced and described in some detail. It is shown how the Dedekind-Peano axioms for number theory can be derived from a consistent fragment of Frege's logic, with Hume's Principle replacing Basic Law V.
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  31.  27
    The Analytic Conception of Truth and the Foundations of Arithmetic.Peter Apostoli - 2000 - Journal of Symbolic Logic 65 (1):33-102.
  32.  35
    On the Foundations of Logic and Arithmetic.David Hilbert - 1905 - The Monist 15 (3):338-352.
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  33.  14
    The Foundations of Arithmetic.Brian Coffey - 1952 - Modern Schoolman 29 (2):157-157.
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  34. FREGE, G. "On the Foundations of Geometry and Formal Theories of Arithmetic". Translated and with an Introduction by E.-H. W. Kluge. [REVIEW]V. H. Dudman - 1974 - Mind 83:131.
     
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  35.  15
    William Ewald and Wilfried Sieg, Eds, David Hilbert's Lectures on the Foundations of Arithmetic and Logic, 1917–1933. Heidelberg: Springer, 2013. ISBN: 978-3-540-69444-1 ; 978-3-540-20578-4 . Pp. Xxv + 1062. [REVIEW]Paolo Mancosu - 2015 - Philosophia Mathematica 23 (1):126-135.
  36.  10
    The Foundations of Arithmetic. A Logico-Mathematical Enquiry Into the Concept of Number. [REVIEW]N. E. - 1951 - Journal of Philosophy 48 (10):342-342.
  37.  3
    Wilfried Sieg.Hilbert's Programs and Beyond. Xii + 440 Pp., Illus., Bibl., Index. Oxford: Oxford University Press, 2013. $85 .William Ewald;, Wilfried Sieg ., Michael Hallett .David Hilbert's Lectures on the Foundations of Arithmetic and Logic, 1917–1933. Xxv + 1,062 Pp., Tables, Bibl., Indexes. Berlin: Springer, 2013. $139. [REVIEW]Tom Archibald - 2015 - Isis 106 (2):481-483.
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  38.  8
    Reviewed Work: David Hilbert's Lectures on the Foundations of Arithmetic and Logic 1917–1933 by William Ewald; Wilfried Sieg. [REVIEW]Review by: Jan von Plato - 2014 - Bulletin of Symbolic Logic 20 (3):363-365,.
  39.  14
    The Foundations of Arithmetic.Michael J. Loux - 1970 - New Scholasticism 44 (3):470-471.
  40.  7
    The Foundations of Arithmetic.Edward A. Maziarz - 1952 - New Scholasticism 26 (1):91-92.
  41.  7
    Frege's Logic, Theorem, and Foundations for Arithmetic.Die Grundlagen der Arithmetik & Grundgesetze der Arithmetik - forthcoming - Stanford Encyclopedia of Philosophy.
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  42.  1
    The Foundations of Arithmetic. A Logico-Mathematical Enquiry Into the Concept of Number.E. N., Gottlob Frege & J. L. Austin - 1951 - Journal of Philosophy 48 (10):342.
  43.  14
    Gottlob Frege: The Foundations of Arithmetic (Die Grundlagen der Arithmetik). Translation by J. L. Austin. (Oxford: Basil Blackwell. 1950. Pp. 132 (Xii + 119). Price 16s.). [REVIEW]W. H. Mccrea - 1951 - Philosophy 26 (97):178-.
  44.  5
    Reinterpreting § 56 of Frege's The Foundations of Arithmetic.K. Brad Wray - 1995 - Auslegung (2).
    I defend an alternative reading of §56 of Frege's Grundlagen, one that rescues Frege from Dummett's charge that this section is the weakest in the whole book. On my reading, Frege is not presenting arguments against the adjectival strategy. Rather, Frege presents the definitions in §55 in order to convince his reader that numbers must be objects. In §56 Frege suggests that these definitions contain two shortcomings that adequate definitions of numbers must overcome. And these short-comings, he argues, can only (...)
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  45.  6
    On the Foundations of Geometry and Formal Theories of Arithmetic.F. P. O'Gorman - 1973 - Philosophical Studies 22:270-272.
  46. Definition in Frege's' Foundations of Arithmetic'.David A. Hunter - 1996 - Pacific Philosophical Quarterly 77 (2):88-107.
     
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  47.  2
    Frege Gottlob, The Foundations of Arithmetic. A Logico-Mathematical Enquiry Into the Concept of Number. German with English Translation by Austin J. L.. Basil Blackwell, Oxford 1950; Philosophical Library, New York 1950; Pages I–Xii, I–XI, 1–119, and Parallel Pages Vie–Xiie, Ie–XIe, 1 E–119 E. [REVIEW]Max Black - 1951 - Journal of Symbolic Logic 16 (1):67.
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  48.  4
    The Contribution of Zygmunt Ratajczyk to the Foundations of Arithmetic.Roman Murawski - 1995 - Notre Dame Journal of Formal Logic 36 (4):502-504.
    Zygmunt Ratajczyk was a deep and subtle mathematician who, with mastery, used sophisticated and technically complex methods, in particular combinatorial and proof-theoretic ones. Walking always along his own paths and being immune from actual trends and fashions he hesitated to publish his results, looking endlessly for their improvement.
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  49.  5
    Review: Gottlob Frege, J. L. Austin, The Foundations of Arithmetic. A Logico-Mathematical Enquiry Into the Concept of Number. [REVIEW]Max Black - 1951 - Journal of Symbolic Logic 16 (1):67-67.
  50.  2
    D. A. Gillies, "Frege Dedekind, and Peano on the Foundations of Arithmetic". [REVIEW]J. P. Mayberry - 1984 - Philosophical Quarterly 34 (36):424.
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