Computability:Turing, Gödel, Church, and Beyond

Cover Image Copyright Year: 2013
Author(s): B. Jack Copeland; Carl J. Posy; Oron Shagrir
Book Type: MIT Press
Content Type : Books
Topics: Computing & Processing ;  General Topics for Engineers
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Abstract

In the 1930s a series of seminal works published by Alan Turing, Kurt Gödel, Alonzo Church, and others established the theoretical basis for computability. This work, advancing precise characterizations of effective, algorithmic computability, was the culmination of intensive investigations into the foundations of mathematics. In the decades since, the theory of computability has moved to the center of discussions in philosophy, computer science, and cognitive science. In this volume, distinguished computer scientists, mathematicians, logicians, and philosophers consider the conceptual foundations of computability in light of our modern understanding. Some chapters focus on the pioneering work by Turing, Gödel, and Church, including the Church-Turing thesis and Gödel's response to Church's and Turing's proposals. Other chapters cover more recent technical developments, including computability over the reals, Gödel's influence on mathematical logic and on recursion theory and the impact of work by Turing and Emil Post on our theoretical understanding of online and interactive computing; and others relate computability and complexity to issues in the philosophy of mind, the philosophy of science, and the philosophy of mathematics. Contributors:Scott Aaronson, Dorit Aharonov, B. Jack Copeland, Martin Davis, Solomon Feferman, Saul Kripke, Carl J. Posy, Hilary Putnam, Oron Shagrir, Stewart Shapiro, Wilfried Sieg, Robert I. Soare, Umesh V. Vazirani

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      Frontmatter

      Copyright Year: 2013

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      This chapter contains sections titled: Half title, Title, Copyright, Contents, Introduction: The 1930s Revolution View full abstract»

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      Turing versus Gödel on Computability and the Mind

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      This chapter contains sections titled: 1.1 Gödel on Turing's “Philosophical Error”, 1.2 Two Approaches to the Analysis of Computability, 1.3 Gödel and Turing on the Mind, 1.4 Conclusion, Acknowledgments, Notes, References View full abstract»

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      Computability and Arithmetic

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      This chapter contains sections titled: 2.1 The Chinese Remainder Theorem, 2.2 Julia Robinson and Existential Definability, 2.3 My Own Early Work, 2.4 Enter Hilary Putnam, 2.5 Yuri Matiyasevich's Triumph, 2.6 Computable Sets in Formal Arithmetic, 2.7 Arithmetic in Q, 2.8 The Rosser—Smullyan Theorem, Notes View full abstract»

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      About and around Computing over the Reals

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      This chapter contains sections titled: 3.1 One Theory or Many?, 3.2 The BSS Model, 3.3 The Theory of Computability over ℝ by Effective Approximation, 3.4 The View from Generalized Recursion Theory; Two Theories of Computability on Arbitrary Structures, 3.5 The Higher Type Approach to Computation on Arbitrary Structures, 3.6 Explicit Mathematics and the Bishop Approach to Constructive Analysis, Notes, References View full abstract»

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      The Church-Turing “Thesis” as a Special Corollary of Gödel's Completeness Theorem

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      This chapter contains sections titled: 4.1 The Previously Received View and More Recent Challenges, 4.2 Computation as a Special Form of Mathematical Argument, 4.3 Von Neumann's Problem of Characterizing and Proving Unsolvability and Gödel's Theorem IX, 4.4 Some Clarificatory Remarks on the Present Characterization, 4.5 Conclusion, Notes, References View full abstract»

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      Computability and Constructibility

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      This chapter contains sections titled: 5.1 Constructive Mathematics, 5.2 Hilbert and Brouwer: Constructivism in the Foundations of Mathematics, 5.3 Kant, The Father of Modern Constructivism, 5.4 Closing the Loop, Notes, References View full abstract»

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      After Gödel

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      This chapter contains sections titled: 6.1 Diophantine Equations, 6.2 Model Theory, 6.3 Kripke's Proof, 6.4 Prime Numbers, 6.5 An Application of Gödel's Method: Can our “scientific competence” be simulated by a Turing Machine? And if yes, can we know that fact?, 6.6 An Anti-Chomskian Incompleteness Theorem: “COMPETENCE” can't be both true and justified, Notes, References, Acknowledgment View full abstract»

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      The Open Texture of Computability

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      This chapter contains sections titled: 7.1 Proving Things about Intuitive Notions, 7.2 Analyses of “Proof”?, 7.3 Theses Everywhere, 7.4 Open Texture, 7.5 Proofs and Refutations, 7.6 Turing's Theorem, Acknowledgments, Notes, References View full abstract»

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      Gödel's Philosophical Challenge (to Turing)

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      This chapter contains sections titled: 8.1 Primitive and General Recursions, 8.2 Finite Machines and Computors, 8.3 Beyond Mechanisms and Discipline, 8.4 Finding Proofs (with Ingenuity), Acknowledgments, Notes, References View full abstract»

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      Interactive Computing and Relativized Computability

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      This chapter contains sections titled: 9.1 Introduction, 9.2 Computability and Incomputability, 9.3 Turing Breaks the Stalemate, 9.4 Turing's Oracle Machines, 9.5 Emil Post Introduces Relative Computability, 9.6 The Art of Classical Computability, 9.7 The Great Papers of Computability, 9.8 Online Computing, 9.9 Three Displacements in Computability Theory, 9.10 Displacement 1: “Recursive” = “Computable”, 9.11 Displacement 2: Church's vs. Turing's Thesis, 9.12 Displacement 3: Turing o-machines, 9.13 Evolution of Terminology, 9.14 Epilogue on Computability, 9.15 Conclusions, Acknowledgments, References View full abstract»

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      Why Philosophers Should Care about Computational Complexity

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      This chapter contains sections titled: 10.1 What This Essay Won't Cover, 10.2 Complexity 101, 10.3 The Relevance of Polynomial Time, 10.4 Computational Complexity and the Turing Test, 10.5 The Problem of Logical Omniscience, 10.6 Computationalism and Waterfalls, 10.7 PAC-Learning and the Problem of Induction, 10.8 Quantum Computing, 10.9 New Computational Notions of Proof, 10.10 Complexity, Space, and Time, 10.11 Economics, 10.12 Conclusions, Acknowledgments, Notes, References View full abstract»

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      Is Quantum Mechanics Falsifiable? A Computational Perspective on the Foundations of Quantum Mechanics

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      This chapter contains sections titled: 11.1 Polynomial Time and the Extended Church—Turing Thesis, 11.2 Interactive Proofs, 11.3 Interactive Proofs for Quantum Mechanics, 11.4 How Weak Verifiers Can Test Strong Machines: Proof Idea, 11.5 Further Complications: Can the Prover Apply the Gates?, 11.6 Summary, 11.7 Related Work, Acknowledgments, Notes, References View full abstract»

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      About the Authors

      Copyright Year: 2013

      MIT Press eBook Chapters

      In the 1930s a series of seminal works published by Alan Turing, Kurt Gödel, Alonzo Church, and others established the theoretical basis for computability. This work, advancing precise characterizations of effective, algorithmic computability, was the culmination of intensive investigations into the foundations of mathematics. In the decades since, the theory of computability has moved to the center of discussions in philosophy, computer science, and cognitive science. In this volume, distinguished computer scientists, mathematicians, logicians, and philosophers consider the conceptual foundations of computability in light of our modern understanding. Some chapters focus on the pioneering work by Turing, Gödel, and Church, including the Church-Turing thesis and Gödel's response to Church's and Turing's proposals. Other chapters cover more recent technical developments, including computability over the reals, Gödel's influence on mathematical logic and on recursion theory and the impact of work by Turing and Emil Post on our theoretical understanding of online and interactive computing; and others relate computability and complexity to issues in the philosophy of mind, the philosophy of science, and the philosophy of mathematics. Contributors:Scott Aaronson, Dorit Aharonov, B. Jack Copeland, Martin Davis, Solomon Feferman, Saul Kripke, Carl J. Posy, Hilary Putnam, Oron Shagrir, Stewart Shapiro, Wilfried Sieg, Robert I. Soare, Umesh V. Vazirani View full abstract»

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      Index

      Copyright Year: 2013

      MIT Press eBook Chapters

      In the 1930s a series of seminal works published by Alan Turing, Kurt Gödel, Alonzo Church, and others established the theoretical basis for computability. This work, advancing precise characterizations of effective, algorithmic computability, was the culmination of intensive investigations into the foundations of mathematics. In the decades since, the theory of computability has moved to the center of discussions in philosophy, computer science, and cognitive science. In this volume, distinguished computer scientists, mathematicians, logicians, and philosophers consider the conceptual foundations of computability in light of our modern understanding. Some chapters focus on the pioneering work by Turing, Gödel, and Church, including the Church-Turing thesis and Gödel's response to Church's and Turing's proposals. Other chapters cover more recent technical developments, including computability over the reals, Gödel's influence on mathematical logic and on recursion theory and the impact of work by Turing and Emil Post on our theoretical understanding of online and interactive computing; and others relate computability and complexity to issues in the philosophy of mind, the philosophy of science, and the philosophy of mathematics. Contributors:Scott Aaronson, Dorit Aharonov, B. Jack Copeland, Martin Davis, Solomon Feferman, Saul Kripke, Carl J. Posy, Hilary Putnam, Oron Shagrir, Stewart Shapiro, Wilfried Sieg, Robert I. Soare, Umesh V. Vazirani View full abstract»