Computability: Turing, Gödel, Church, and Beyond

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B. Jack Copeland, Carl J. Posy, Oron Shagrir
MIT Press, Jun 7, 2013 - Computers - 362 pages
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Computer scientists, mathematicians, and philosophers discuss the conceptual foundations of the notion of computability as well as recent theoretical developments. 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.ContributorsScott 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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Contents

1 Turing versus Gödel on Computability and the Mind
1
2 Computability and Arithmetic
35
3 About and around Computing over the Reals
55
4 The ChurchTuring Thesis as a Special Corollary of Gödels Completeness Theorem
77
5 Computability and Constructibility
105
6 After Gödel
141
7 The Open Texture of Computability
153
8 Gödels Philosophical Challenge to Turing
183
9 Interactive Computing and Relativized Computability
203
10 Why Philosophers Should Care about Computational Complexity
261
11 Is Quantum Mechanics Falsifiable? A Computational Perspective on the Foundations of Quantum Mechanics
329
About the Authors
351
Index
355
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About the author (2013)

B. Jack Copeland is Professor of Philosophy at the University of Canterbury, New Zealand, Director of the Turing Archive for the History of Computing, and Honorary Research Professor of Philosophy at the University of Queensland, Australia.

Carl J. Posy is Professor of Philosophy and Member of the Centers for the Study of Rationality and for Language, Logic, and Cognition at the Hebrew University of Jerusalem.

Oron Shagrir is Professor of Philosophy and Former Chair of the Cognitive Science Department at the Hebrew University of Jerusalem. He is currently the vice rector of the Hebrew University.

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