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BL
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| Record Number
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604000
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| Doc. No
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b433219
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| Main Entry
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Gauthier, Yvon
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| Title & Author
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Towards an arithmetical logic : : the arithmetical foundations of logic /\ by Yvon Gauthier
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| Publication Statement
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Cham :: Birkhäuser,, 2015
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| Series Statement
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Studies in Universal Logic
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| Page. NO
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1 online resource
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| ISBN
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331922087X
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: 9783319220871
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3319220861 (print)
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9783319220864 (print)
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| Contents
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Foreword; Contents; 1 Introduction: The Internal Logic of Arithmetic; 2 Arithmetization of Analysis and Algebra; 2.1 Cauchy and Weierstrass; 2.2 Dedekind and Cantor; 2.3 Frege; 2.4 Russell, Peano and Zermelo; 2.5 Kronecker and the Arithmetization of Algebra; 3 Arithmetization of Logic; 3.1 Hilbert after Kronecker; 3.2 Hilbert's Arithmetization of Logic and the Epsilon Calculus; 3.3 Herbrand's Theorem; 3.4 Tarski's Quantifier Elimination; 3.5 Gödel's Functional Interpretation; 3.6 Skolem and Brouwer; 3.7 Gödel and Turing; 3.8 Arithmetic; 3.9 Constructive Arithmetic and Analysis
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3.10 Complexity4 Kronecker's Foundational Programme in Contemporary Mathematics; 4.1 Introduction; 4.2 Grothendieck's Programme; 4.3 Descent; 4.4 Langlands' Programme; 4.5 Kronecker's and Hilbert's Programmes in Contemporary Mathematical Logic; 4.6 Conclusion: Finitism and Arithmetism; 5 Arithmetical Foundations for Physical Theories; 5.1 Introduction: The Notion of Analytical Apparatus; 5.2 Analytical and Empirical Apparatuses; 5.3 Models; 5.4 The Consistency of Physical Theories; 5.5 Quantum Mechanics; 5.5.1 Hilbert Space; 5.5.2 Probabilities; 5.5.3 Logics; 5.5.4 Local Complementation
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5.17.1 Principles for a Theory of Measurement in QM6 The Internal Logic of Constructive Mathematics; 6.1 Transcendental Versus Elementary: The Gel'fond-Schneider Theorem; 6.2 Transcendental Number Theory; 6.3 The Internal Logic; 6.4 Descent or Descending Induction; 6.5 Induction Principles; 6.6 Intuitionistic Logic and Transfinite Induction; 6.7 Transfinite Induction; 6.8 Conclusion: A Finitist Logic for Constructive Mathematics; 7 The Internal Consistency of Arithmetic with Infinite Descent: A Syntactical Proof; 7.1 Preamble; 7.2 Introduction; 7.3 Arithmetic; 7.4 Arithmetization of Syntax
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5.5.5 The Total Hilbert Space5.5.6 Finite Derivation of the Local Complement; 5.6 Riemannian Geometry; 5.7 Riemann's ̀̀Hypotheses''; 5.8 Physical Geometry; 5.9 Minkowski's Spacetime; 5.10 Geometry of Numbers; 5.11 Spacetime Diagrams; 5.12 Physical Axiomatics; 5.13 Hermann Weyl and the Free-Will Theorem; 5.14 The Conway-Kochen Free-Will Theorem; 5.15 A General No-Cloning Theorem in the Multiversal Cosmology; 5.15.1 The No-Cloning Theorem in QM; 5.15.2 A No-Cloning Theorem in the Multiverse Cosmology; 5.16 Conclusion; 5.17 Appendix to Chapter 5
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7.5 Reducibility and Divisibility7.6 Elimination of Logical Constants; 7.7 The Elimination of Implication; 7.8 The Elimination of the Effinite Quantifier Through Infinite Descent; 7.9 Conclusion: The Polynomial Extension from a Finitist Point of View; 8 Conclusion: Arithmetism Versus Logicism or Kronecker Contra Frege; 8.1 Introduction: Arithmetical Philosophy; 8.2 Kronecker Today; 8.3 Arithmetization of Geometry: From Algebraic Geometry to Arithmetic Geometry; 8.4 From Geometry of Numbers to Physical Geometry and Physics; 8.5 Arithmetization of Logic; References
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| Abstract
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This book offers an original contribution to the foundations of logic and mathematics, and focuses on the internal logic of mathematical theories, from arithmetic or number theory to algebraic geometry. Arithmetical logic is the term used to refer to the internal logic of classical arithmetic, here called Fermat-Kronecker arithmetic, and combines Fermat's method of infinite descent with Kronecker's general arithmetic of homogeneous polynomials. The book also includes a treatment of theories in physics and mathematical physics to underscore the role of arithmetic from a constructivist viewpoint. The scope of the work intertwines historical, mathematical, logical and philosophical dimensions in a unified critical perspective; as such, it will appeal to a broad readership from mathematicians to logicians, to philosophers interested in foundational questions. Researchers and graduate students in the fields of philosophy and mathematics will benefit from the author's critical approach to the foundations of logic and mathematics
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| Subject
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Logic, Symbolic and mathematical
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| Subject
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Mathematics-- Philosophy
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| Dewey Classification
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510
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| LC Classification
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QA1-939
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| Added Entry
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Ohio Library and Information Network
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