Computable Functions Logic and the Foundations of Math

Author: Richard L. Epstein,Walter Alexandr Carnielli

Publisher: Chapman and Hall/CRC

ISBN: 9780534103569

Category: Mathematics

Page: 297

View: 7154

An introduction to recursion theory and particularly to the theory of computing, including fourteen readings by Hilbert, Godel, Turing, Post, Church, and others along with a discussion of issues such as self-reference and infinite sets. Annotation copyrighted by Book News, Inc., Portland, OR
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Eine Einführung in Algorithmen, Sprachen und Kalküle unter besonderer Berücksichtigung ihrer Komplexität

Author: Egon Börger

Publisher: Springer-Verlag

ISBN: 3322877779

Category: Computers

Page: 470

View: 6157

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Author: B. Jack. Copeland

Publisher: Clarendon Press

ISBN: 9780191520280

Category: Science

Page: 622

View: 4373

Alan Turing, pioneer of computing and WWII codebreaker, is one of the most important and influential thinkers of the twentieth century. In this volume for the first time his key writings are made available to a broad, non-specialist readership. They make fascinating reading both in their own right and for their historic significance: contemporary computational theory, cognitive science, artificial intelligence, and artificial life all spring from this ground-breaking work, which is also rich in philosophical and logical insight. An introduction by leading Turing expert Jack Copeland provides the background and guides the reader through the selection. About Alan Turing Alan Turing FRS OBE, (1912-1954) studied mathematics at King's College, Cambridge. He was elected a Fellow of King's in March 1935, at the age of only 22. In the same year he invented the abstract computing machines - now known simply as Turing machines - on which all subsequent stored-program digital computers are modelled. During 1936-1938 Turing continued his studies, now at Princeton University. He completed a PhD in mathematical logic, analysing the notion of 'intuition' in mathematics and introducing the idea of oracular computation, now fundamental in mathematical recursion theory. An 'oracle' is an abstract device able to solve mathematical problems too difficult for the universal Turing machine. In the summer of 1938 Turing returned to his Fellowship at King's. When WWII started in 1939 he joined the wartime headquarters of the Government Code and Cypher School (GC&CS) at Bletchley Park, Buckinghamshire. Building on earlier work by Polish cryptanalysts, Turing contributed crucially to the design of electro-mechanical machines ('bombes') used to decipher Enigma, the code by means of which the German armed forces sought to protect their radio communications. Turing's work on the version of Enigma used by the German navy was vital to the battle for supremacy in the North Atlantic. He also contributed to the attack on the cyphers known as 'Fish'. Based on binary teleprinter code, Fish was used during the latter part of the war in preference to morse-based Enigma for the encryption of high-level signals, for example messages from Hitler and other members of the German High Command. It is estimated that the work of GC&CS shortened the war in Europe by at least two years. Turing received the Order of the British Empire for the part he played. In 1945, the war over, Turing was recruited to the National Physical Laboratory (NPL) in London, his brief to design and develop an electronic computer - a concrete form of the universal Turing machine. Turing's report setting out his design for the Automatic Computing Engine (ACE) was the first relatively complete specification of an electronic stored-program general-purpose digital computer. Delays beyond Turing's control resulted in NPL's losing the race to build the world's first working electronic stored-program digital computer - an honour that went to the Royal Society Computing Machine Laboratory at Manchester University, in June 1948. Discouraged by the delays at NPL, Turing took up the Deputy Directorship of the Royal Society Computing Machine Laboratory in that year. Turing was a founding father of modern cognitive science and a leading early exponent of the hypothesis that the human brain is in large part a digital computing machine, theorising that the cortex at birth is an 'unorganised machine' which through 'training' becomes organised 'into a universal machine or something like it'. He also pioneered Artificial Intelligence. Turing spent the rest of his short career at Manchester University, being appointed to a specially created Readership in the Theory of Computing in May 1953. He was elected a Fellow of the Royal Society of London in March 1951 (a high honour).
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Author: O. Bradley Bassler

Publisher: Springer

ISBN: 1137578890

Category: Philosophy

Page: 215

View: 2974

This book charts the shape of future philosophical investigation by posing the question: “What is the Matrix?” Guided by the example of the Matrix film trilogy, the author examines issues ranging from simulation, proof and action to value, culture and mythology, offering a progressively deeper diagnosis of modern philosophical conditions. In contrast to the contemporary focus upon cognitive science and a commitment to the distinction between appearance and reality, this book helps readers to explore the argument that such abstractions are inevitably displaced by a more concrete distinction between dreaming and waking, with the Matrix as the real and only world we inhabit. Researchers and scholars will find this work an engaging and enlightening examination of reality, via the medium of popular culture and film.
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Studies in Agent-Based Computational Modeling

Author: Joshua M. Epstein

Publisher: Princeton University Press

ISBN: 1400842875

Category: Business & Economics

Page: 384

View: 6873

Agent-based computational modeling is changing the face of social science. In Generative Social Science, Joshua Epstein argues that this powerful, novel technique permits the social sciences to meet a fundamentally new standard of explanation, in which one "grows" the phenomenon of interest in an artificial society of interacting agents: heterogeneous, boundedly rational actors, represented as mathematical or software objects. After elaborating this notion of generative explanation in a pair of overarching foundational chapters, Epstein illustrates it with examples chosen from such far-flung fields as archaeology, civil conflict, the evolution of norms, epidemiology, retirement economics, spatial games, and organizational adaptation. In elegant chapter preludes, he explains how these widely diverse modeling studies support his sweeping case for generative explanation. This book represents a powerful consolidation of Epstein's interdisciplinary research activities in the decade since the publication of his and Robert Axtell's landmark volume, Growing Artificial Societies. Beautifully illustrated, Generative Social Science includes a CD that contains animated movies of core model runs, and programs allowing users to easily change assumptions and explore models, making it an invaluable text for courses in modeling at all levels.
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Malvern After Ten Years

Author: Steven Pressman

Publisher: Routledge

ISBN: 1134784104

Category: Business & Economics

Page: 240

View: 2156

In recent years there has been a growing dissatisfaction with standard economic theorising which has fostered the development of alternative ways of understanding how economies actually work. Too often though these approaches have been developed in isolation, or even in opposition to each other. However, Interactions in Political Economy, demonstrates that the different heterodox approaches to economics have much to learn from each other. Economists working within different paradigms, including Post Keynesian, Marxism and Neo-Ricardian economics address a wide range of issues in methodology, the history of economics, theory and policy. The result is a wealth of insight into how economics ought to be done, how various theoretical approaches dovetail, and the efficiency of various approaches to economic theory. The volume reflects the diversity and quality of the annual Great Malvern Political Economy. Contributors include some of the leading names in heterodox economics John Cornwall, Paul Davidson, Kevin Hoover, Philip Mirowski and Ed Nell.
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Author: E.R. Griffor

Publisher: Elsevier

ISBN: 9780080533049

Category: Mathematics

Page: 724

View: 9442

The chapters of this volume all have their own level of presentation. The topics have been chosen based on the active research interest associated with them. Since the interest in some topics is older than that in others, some presentations contain fundamental definitions and basic results while others relate very little of the elementary theory behind them and aim directly toward an exposition of advanced results. Presentations of the latter sort are in some cases restricted to a short survey of recent results (due to the complexity of the methods and proofs themselves). Hence the variation in level of presentation from chapter to chapter only reflects the conceptual situation itself. One example of this is the collective efforts to develop an acceptable theory of computation on the real numbers. The last two decades has seen at least two new definitions of effective operations on the real numbers.
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A Guided Tour Through Alan Turing's Historic Paper on Computability and the Turing Machine

Author: Charles Petzold

Publisher: John Wiley & Sons

ISBN: 0470229055

Category: Computers

Page: 372

View: 2082

Provides an expansion of Turing's original paper, a brief look at his life, and information on the Turing machine and computability topics.
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Author: David Hilbert,Wilhelm Ackermann

Publisher: Springer-Verlag

ISBN: 3662119323

Category: Mathematics

Page: 190

View: 1034

Die theoretische Logik, auch mathematische oder symbolische Logik genannt, ist eine Ausdehnung der formalen Methode der Mathematik auf das Gebiet der Logik. Sie wendet für die Logik eine ähnliche Formel sprache an, wie sie zum Ausdruck mathematischer Beziehungen schon seit langem gebräuchlich ist. In der Mathematik würde es heute als eine Utopie gelten, wollte man beim Aufbau einer mathematischen Disziplin sich nur der gewöhnlichen Sprache bedienen. Die großen Fortschritte, die in der Mathematik seit der Antike gemacht worden sind, sind zum wesentlichen Teil mit dadurch bedingt, daß es gelang, einen brauchbaren und leistungsfähigen Formalismus zu finden. - Was durch die Formel sprache in der Mathematik erreicht wird, das soll auch in der theoretischen Logik durch diese erzielt werden, nämlich eine exakte, wissenschaftliche Behandlung ihres Gegenstandes. Die logischen Sachverhalte, die zwischen Urteilen, Begriffen usw. bestehen, finden ihre Darstellung durch Formeln, deren Interpretation frei ist von den Unklarheiten, die beim sprachlichen Ausdruck leicht auftreten können. Der Übergang zu logischen Folgerungen, wie er durch das Schließen geschieht, wird in seine letzten Elemente zerlegt und erscheint als formale Umgestaltung der Ausgangsformeln nach gewissen Regeln, die den Rechenregeln in der Algebra analog sind; das logische Denken findet sein Abbild in einem Logikkalkül. Dieser Kalkül macht die erfolgreiche Inangriffnahme von Problemen möglich, bei denen das rein inhaltliche Denken prinzipiell versagt. Zu diesen gehört z. B.
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Author: Donald A. R. George

Publisher: Wiley-Blackwell

ISBN: 9781405179614

Category: Business & Economics

Page: 292

View: 4542

Through contributions from leading authors, Issues in Heterodox Economics provides a critical analysis of the methodology of mainstream economics. Challenges economists to abandon sterile formalism and develop new intellectual rigors to contribute to pressing contemporary issues A series of cutting-edge articles provides a critical analysis of the dependence of mainstream economics on mathematical modelling and other methodologies Topics discussed include sustainable development, worker control of firms, evolutionary growth theory, and more Challenges economists to abandon sterile formalism and develop new intellectual rigors to contribute to pressing contemporary issues
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Part One of the Proceedings of the Fifth International Congress of Logic, Methodology and Philosophy of Science, London, Ontario, Canada-1975

Author: Robert E. Butts,Jaakko Hintikka

Publisher: Springer Science & Business Media

ISBN: 9401011389

Category: Science

Page: 416

View: 1837

The Fifth International Congress of Logic, Methodology and Philosophy of Science was held at the University of Western Ontario, London, Canada, 27 August to 2 September 1975. The Congress was held under the auspices of the International Union of History and Philosophy of Science, Division of Logic, Methodology and Philosophy of Science, and was sponsored by the National Research Council of Canada and the University of Western Ontario. As those associated closely with the work of the Division over the years know well, the work undertaken by its members varies greatly and spans a number of fields not always obviously related. In addition, the volume of work done by first rate scholars and scientists in the various fields of the Division has risen enormously. For these and related reasons it seemed to the editors chosen by the Divisional officers that the usual format of publishing the proceedings of the Congress be abandoned in favour of a somewhat more flexible, and hopefully acceptable, method of pre sentation. Accordingly, the work of the invited participants to the Congress has been divided into four volumes appearing in the University of Western Ontario Series in Philosophy of Science. The volumes are entitled, Logic, Foundations of Mathematics and Computability Theory, Foun dational Problems in the Special Sciences, Basic Problems in Methodol ogy and Linguistics, and Historical and Philosophical Dimensions of Logic, Methodology and Philosophy of Science.
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Author: Arnold Oberschelp

Publisher: N.A

ISBN: 9783411161713

Category: Recursion theory

Page: 339

View: 5489

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Turing, Gödel, Church, and Beyond

Author: B. Jack Copeland,Carl J. Posy,Oron Shagrir

Publisher: MIT Press

ISBN: 0262018993

Category: Computers

Page: 362

View: 3347

Computer scientists, mathematicians, and philosophers discuss the conceptual foundations of the notion of computability as well as recent theoretical developments.
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Author: George Tourlakis

Publisher: John Wiley & Sons

ISBN: 1118030699

Category: Mathematics

Page: 294

View: 2752

A comprehensive and user-friendly guide to the use of logic inmathematical reasoning Mathematical Logic presents a comprehensive introductionto formal methods of logic and their use as a reliable tool fordeductive reasoning. With its user-friendly approach, this booksuccessfully equips readers with the key concepts and methods forformulating valid mathematical arguments that can be used touncover truths across diverse areas of study such as mathematics,computer science, and philosophy. The book develops the logical tools for writing proofs byguiding readers through both the established "Hilbert" style ofproof writing, as well as the "equational" style that is emergingin computer science and engineering applications. Chapters havebeen organized into the two topical areas of Boolean logic andpredicate logic. Techniques situated outside formal logic areapplied to illustrate and demonstrate significant facts regardingthe power and limitations of logic, such as: Logic can certify truths and only truths. Logic can certify all absolute truths (completeness theorems ofPost and Gödel). Logic cannot certify all "conditional" truths, such as thosethat are specific to the Peano arithmetic. Therefore, logic hassome serious limitations, as shown through Gödel'sincompleteness theorem. Numerous examples and problem sets are provided throughout thetext, further facilitating readers' understanding of thecapabilities of logic to discover mathematical truths. In addition,an extensive appendix introduces Tarski semantics and proceeds withdetailed proofs of completeness and first incompleteness theorems,while also providing a self-contained introduction to the theory ofcomputability. With its thorough scope of coverage and accessible style,Mathematical Logic is an ideal book for courses inmathematics, computer science, and philosophy at theupper-undergraduate and graduate levels. It is also a valuablereference for researchers and practitioners who wish to learn howto use logic in their everyday work.
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