Search results for 'Arithmetic Foundations' (try it on Scholar)

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  1. Gottlob Frege (1953/1968). The Foundations of Arithmetic. Evanston, Ill.,Northwestern University Press.score: 69.0
    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. Gottlob Frege (1980). The Foundations of Arithmetic: A Logico-Mathematical Enquiry Into the Concept of Number. Northwestern University Press.score: 66.0
    § 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. Fernando Ferreira (1999). A Note on Finiteness in the Predicative Foundations of Arithmetic. Journal of Philosophical Logic 28 (2):165-174.score: 60.0
    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. Richard Pettigrew (2010). The Foundations of Arithmetic in Finite Bounded Zermelo Set Theory. Cahiers du Centre de Logique 17:99-118.score: 54.0
    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. Michael Kremer (2008). Review of Gottlob Frege, Dale Jacquette (Tr.), The Foundations of Arithmetic. [REVIEW] Notre Dame Philosophical Reviews 2008 (1).score: 48.0
    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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  6. G. Kreisel (1953). A Variant to Hilbert's Theory of the Foundations of Arithmetic. British Journal for the Philosophy of Science 4 (14):107-129.score: 48.0
    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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  7. Solomon Feferman, Challenges to Predicative Foundations of Arithmetic.score: 48.0
    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. Andrew Boucher, 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]score: 48.0
    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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  9. J. P. Mayberry & D. A. Gillies (1984). Frege, Dedekind, and Peano on the Foundations of Arithmetic. Philosophical Quarterly 34 (136):424.score: 48.0
    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. Joongol Kim (forthcoming). A Logical Foundation of Arithmetic. Studia Logica:1-32.score: 45.0
    The aim of this paper is to shed new light on the logical roots of arithmetic by presenting a logical framework (ALA) that takes seriously ordinary locutions like ‘at least n Fs’, ‘n more Fs than Gs’ and ‘n times as many Fs as Gs’, instead of paraphrasing them away in terms of expressions of the form ‘the number of Fs’. It will be shown that the basic concepts of arithmetic can be intuitively defined in the language of (...)
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  11. Juliette Kennedy & Roman Kossak (eds.) (2012). Set Theory, Arithmetic, and Foundations of Mathematics: Theorems, Philosophies. Cambridge University Press.score: 39.0
    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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  12. B. R. Buckingham (1953). Elementary Arithmetic. Boston, Ginn.score: 39.0
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  13. William Demopoulos (2013). Generality and Objectivity in Frege's Foundations of Arithmetic. In Alex Miller (ed.), Logic, Language and Mathematics: Essays for Crispin Wright. Oxford University Press.score: 38.0
     
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  14. Edward N. Zalta, Frege's Logic, Theorem, and Foundations for Arithmetic. Stanford Encyclopedia of Philosophy.score: 36.0
    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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  15. Marcus Rossberg & Philip A. Ebert (2010). Cantor on Frege's Foundations of Arithmetic : Cantor's 1885 Review of Frege's Die Grundlagen der Arithmetik. History and Philosophy of Logic 30 (4):341-348.score: 36.0
    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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  16. Matthias Schirn (2003). Fregean Abstraction, Referential Indeterminacy and the Logical Foundations of Arithmetic. Erkenntnis 59 (2):203 - 232.score: 36.0
    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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  17. Solomon Feferman & Geoffrey Hellman (1995). Predicative Foundations of Arithmetic. Journal of Philosophical Logic 24 (1):1 - 17.score: 36.0
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  18. Peter Apostoli (2000). The Analytic Conception of Truth and the Foundations of Arithmetic. Journal of Symbolic Logic 65 (1):33-102.score: 36.0
  19. Susan Carey (2001). Cognitive Foundations of Arithmetic: Evolution and Ontogenisis. Mind and Language 16 (1):37–55.score: 36.0
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  20. W. H. Mccrea (1951). 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] Philosophy 26 (97):178-.score: 36.0
  21. Glenn Kessler (1980). Frege, Mill, and the Foundations of Arithmetic. Journal of Philosophy 77 (2):65-79.score: 36.0
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  22. David Hilbert (1905). On the Foundations of Logic and Arithmetic. The Monist 15 (3):338-352.score: 36.0
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  23. Michael J. Loux (1970). The Foundations of Arithmetic. New Scholasticism 44 (3):470-471.score: 36.0
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  24. Max Black (1951). Review: Gottlob Frege, J. L. Austin, The Foundations of Arithmetic. A Logico-Mathematical Enquiry Into the Concept of Number. [REVIEW] Journal of Symbolic Logic 16 (1):67-67.score: 36.0
  25. Die Grundlagen der Arithmetik & Grundgesetze der Arithmetik (forthcoming). Frege's Logic, Theorem, and Foundations for Arithmetic. Stanford Encyclopedia of Philosophy.score: 36.0
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  26. F. P. O'Gorman (1973). On the Foundations of Geometry and Formal Theories of Arithmetic. Philosophical Studies 22:270-272.score: 36.0
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  27. Toshiyasu Arai (2000). Buss Samuel R.. First-Order Proof Theory of Arithmetic. Handbook of Proof Theory, Edited by Buss Samuel R., Studies in Logic and the Foundations of Mathematics, Vol. 137, Elsevier, Amsterdam Etc. 1998, Pp. 79–147. [REVIEW] Bulletin of Symbolic Logic 6 (4):465-466.score: 36.0
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  28. Jean-Pierre Belna (2006). Objectivity and the Principle of Duality: Paragraph 26 of Frege's Foundations of Arithmetic. Revue d'Histoire des Sciences 59 (2):319.score: 36.0
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  29. George Boolos (1987). The Consistency of Frege's Foundations of Arithmetic. In J. Thomson (ed.), On Being and Saying: Essays in Honor of Richard Cartwright. Mit Press. 3--20.score: 36.0
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  30. K. Brad Wray (1995). Reinterpreting Section 56 of Frege's The Foundations of Arithmetic. Auslegung 20:76-82.score: 36.0
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  31. Susan Carey (2001). Evolutionary and Ontogenetic Foundations of Arithmetic. Mind and Language 16 (1):37-55.score: 36.0
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  32. Brian Coffey (1952). The Foundations of Arithmetic. The Modern Schoolman 29 (2):157-157.score: 36.0
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  33. G. J. Whitrow (1948). On the Foundations and Application of Finite Classical Arithmetic. Philosophy 23 (86):256 - 261.score: 36.0
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  34. David A. Hunter (1996). Definition in Frege's' Foundations of Arithmetic'. Pacific Philosophical Quarterly 77 (2):88-107.score: 36.0
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  35. Howard Jackson (1981). Review: Eike-Henner W. Kluge, Gottlob Frege, On the Foundations of Geometry and Formal Theories of Arithmetic. [REVIEW] Journal of Symbolic Logic 46 (1):175-179.score: 36.0
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  36. Edward A. Maziarz (1952). The Foundations of Arithmetic. New Scholasticism 26 (1):91-92.score: 36.0
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  37. Andrzej Mostowski (1957). Review: G. Kreisel, A Variant to Hilbert's Theory of the Foundations of Arithmetic. [REVIEW] Journal of Symbolic Logic 22 (3):304-306.score: 36.0
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  38. Joan B. Quick (1952). The Foundations of Arithmetic. Thought 27 (2):303-304.score: 36.0
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  39. Roman Murawski (1995). The Contribution of Zygmunt Ratajczyk to the Foundations of Arithmetic. Notre Dame Journal of Formal Logic 36 (4):502-504.score: 36.0
    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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  40. S. K. Thomason (1984). DA Gillies, Frege, Dedekind and Peano on the Foundations of Arithmetic Reviewed By. Philosophy in Review 4 (3):111-113.score: 36.0
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  41. Frode Bjørdal (2012). Librationist Closures of the Paradoxes. Logic and Logical Philosophy 21 (4):323-361.score: 31.0
    We present a semi-formal foundational theory of sorts, akin to sets, named librationism because of its way of dealing with paradoxes. Its semantics is related to Herzberger’s semi inductive approach, it is negation complete and free variables (noemata) name sorts. Librationism deals with paradoxes in a novel way related to paraconsistent dialetheic approaches, but we think of it as bialethic and parasistent. Classical logical theorems are retained, and none contradicted. Novel inferential principles make recourse to theoremhood and failure of theoremhood. (...)
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  42. Alfred Tarski (1946/1995). Introduction to Logic and to the Methodology of Deductive Sciences. Dover Publications.score: 30.0
    This classic undergraduate treatment examines the deductive method in its first part and explores applications of logic and methodology in constructing mathematical theories in its second part. Exercises appear throughout.
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  43. Alfred Tarski (1994). Introduction to Logic and to the Methodology of the Deductive Sciences. Oxford University Press.score: 30.0
    Now in its fourth edition, this classic work clearly and concisely introduces the subject of logic and its applications. The first part of the book explains the basic concepts and principles which make up the elements of logic. The author demonstrates that these ideas are found in all branches of mathematics, and that logical laws are constantly applied in mathematical reasoning. The second part of the book shows the applications of logic in mathematical theory building with concrete examples that draw (...)
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  44. Crispin Wright (1983). Frege's Conception of Numbers as Objects. Aberdeen University Press.score: 30.0
  45. Dennis Sentilles (1975). A Bridge to Advanced Mathematics. Baltimore,Williams & Wilkins.score: 30.0
     
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  46. Stewart Shapiro (1991). Foundations Without Foundationalism: A Case for Second-Order Logic. Oxford University Press.score: 21.0
    The central contention of this book is that second-order logic has a central role to play in laying the foundations of mathematics. In order to develop the argument fully, the author presents a detailed description of higher-order logic, including a comprehensive discussion of its semantics. He goes on to demonstrate the prevalence of second-order concepts in mathematics and the extent to which mathematical ideas can be formulated in higher-order logic. He also shows how first-order languages are often insufficient to (...)
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  47. Bob Hale (2005). Real Numbers and Set Theory – Extending the Neo-Fregean Programme Beyond Arithmetic. Synthese 147 (1):21 - 41.score: 21.0
    It is known that Hume’s Principle, adjoined to a suitable formulation of second-order logic, gives a theory which is almost certainly consistent4 and suffices for arithmetic in the sense that it yields the Dedekind-Peano axioms as theorems. While Hume’s Principle cannot be taken as a definition in any strict sense requiring that it provide for the eliminative paraphrase of its definiendum in every admissible type of occurrence, we hold that it can be viewed as an implicit definition of a (...)
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  48. Andrew Boucher, Systems for a Foundation of Arithmetic.score: 21.0
    A new second-order axiomatization of arithmetic, with Frege's definition of successor replaced, is presented and compared to other systems in the field of Frege Arithmetic. The key in proving the Peano Axioms turns out to be a proposition about infinity, which a reduced subset of the axioms proves.
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  49. Mojżesz Presburger & Dale Jabcquette (1991). On the Completeness of a Certain System of Arithmetic of Whole Numbers in Which Addition Occurs as the Only Operation. History and Philosophy of Logic 12 (2):225-233.score: 21.0
    Presburger's essay on the completeness and decidability of arithmetic with integer addition but without multiplication is a milestone in the history of mathematical logic and formal metatheory. The proof is constructive, using Tarski-style quantifier elimination and a four-part recursive comprehension principle for axiomatic consequence characterization. Presburger's proof for the completeness of first order arithmetic with identity and addition but without multiplication, in light of the restrictive formal metatheorems of Gödel, Church, and Rosser, takes the foundations of (...) in mathematical logic to the limits of completeness and decidability. (shrink)
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  50. Ernst von Glasersfeld (2006). A Constructivist Approach to Experiential Foundations of Mathematical Concepts Revisited. Constructivist Foundations 1 (2):61-72.score: 21.0
    Purpose: The paper contributes to the naturalization of epistemology. It suggests tentative itineraries for the progression from elementary experiential situations to the abstraction of the concepts of unit, plurality, number, point, line, and plane. It also provides a discussion of the question of certainty in logical deduction and arithmetic. Approach: Whitehead’s description of three processes involved in criticizing mathematical thinking (1956) is used to show discrepancies between a traditional epistemological stance and the constructivist approach to knowing and communication. Practical (...)
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