Inexhaustibility

a non-exhaustive treatment

Author: Torkel Franzén

Publisher: A K Peters Ltd

ISBN: 9781568811741

Category: Mathematics

Page: 251

View: 6664

Gödel's Incompleteness Theorems are among the most significant results in the foundation of mathematics. These results have a positive consequence: any system of axioms for mathematics that we recognize as correct can be properly extended by adding as a new axiom a formal statement expressing that the original system is consistent. This suggests that our mathematical knowledge is inexhaustible, an essentially philosophical topic to which this book is devoted. Basic material in predicate logic, set theory and recursion theory is presented, leading to a proof of incompleteness theorems. The inexhaustibility of mathematical knowledge is treated based on the concept of transfinite progressions of theories as conceived by Turing and Feferman. All concepts and results necessary to understand the arguments are introduced as needed, making the presentation self-contained and thorough.
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Alan Turing: His Work and Impact

Author: S. Barry Cooper,J. van Leeuwen

Publisher: Elsevier

ISBN: 0123870127

Category: Mathematics

Page: 944

View: 1800

In this 2013 winner of the prestigious R.R. Hawkins Award from the Association of American Publishers, as well as the 2013 PROSE Awards for Mathematics and Best in Physical Sciences & Mathematics, also from the AAP, readers will find many of the most significant contributions from the four-volume set of the Collected Works of A. M. Turing. These contributions, together with commentaries from current experts in a wide spectrum of fields and backgrounds, provide insight on the significance and contemporary impact of Alan Turing's work. Offering a more modern perspective than anything currently available, Alan Turing: His Work and Impact gives wide coverage of the many ways in which Turing's scientific endeavors have impacted current research and understanding of the world. His pivotal writings on subjects including computing, artificial intelligence, cryptography, morphogenesis, and more display continued relevance and insight into today's scientific and technological landscape. This collection provides a great service to researchers, but is also an approachable entry point for readers with limited training in the science, but an urge to learn more about the details of Turing's work. 2013 winner of the prestigious R.R. Hawkins Award from the Association of American Publishers, as well as the 2013 PROSE Awards for Mathematics and Best in Physical Sciences & Mathematics, also from the AAP Named a 2013 Notable Computer Book in Computing Milieux by Computing Reviews Affordable, key collection of the most significant papers by A.M. Turing Commentary explaining the significance of each seminal paper by preeminent leaders in the field Additional resources available online
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Alan Turing's Systems of Logic

The Princeton Thesis

Author: Andrew W. Appel

Publisher: Princeton University Press

ISBN: 0691164738

Category: Computers

Page: 160

View: 4847

Between inventing the concept of a universal computer in 1936 and breaking the German Enigma code during World War II, Alan Turing (1912-1954), the British founder of computer science and artificial intelligence, came to Princeton University to study mathematical logic. Some of the greatest logicians in the world--including Alonzo Church, Kurt Gödel, John von Neumann, and Stephen Kleene--were at Princeton in the 1930s, and they were working on ideas that would lay the groundwork for what would become known as computer science. This book presents a facsimile of the original typescript of Turing's fascinating and influential 1938 Princeton PhD thesis, one of the key documents in the history of mathematics and computer science. The book also features essays by Andrew Appel and Solomon Feferman that explain the still-unfolding significance of the ideas Turing developed at Princeton. A work of philosophy as well as mathematics, Turing's thesis envisions a practical goal--a logical system to formalize mathematical proofs so they can be checked mechanically. If every step of a theorem could be verified mechanically, the burden on intuition would be limited to the axioms. Turing's point, as Appel writes, is that "mathematical reasoning can be done, and should be done, in mechanizable formal logic." Turing's vision of "constructive systems of logic for practical use" has become reality: in the twenty-first century, automated "formal methods" are now routine. Presented here in its original form, this fascinating thesis is one of the key documents in the history of mathematics and computer science.
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The History of Philosophical and Formal Logic

From Aristotle to Tarski

Author: Alex Malpass,Marianna Antonutti Marfori

Publisher: Bloomsbury Publishing

ISBN: 1472505255

Category: Philosophy

Page: 256

View: 4623

The History of Philosophical and Formal Logic introduces ideas and thinkers central to the development of philosophical and formal logic. From its Aristotelian origins to the present-day arguments, logic is broken down into four main time periods: Antiquity and the Middle Ages (Aristotle and The Stoics) The early modern period (Bolzano, Boole) High modern period (Frege, Peano & Russell and Hilbert) Early 20th century (Godel and Tarski) Each new time frame begins with an introductory overview highlighting themes and points of importance. Chapters discuss the significance and reception of influential works and look at historical arguments in the context of contemporary debates. To support independent study, comprehensive lists of primary and secondary reading are included at the end of chapters, along with exercises and discussion questions. By clearly presenting and explaining the changes to logic across the history of philosophy, The History of Philosophical and Formal Logic constructs an easy-to-follow narrative. This is an ideal starting point for students looking to understand the historical development of logic.
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Logical Foundations of Mathematics and Computational Complexity

A Gentle Introduction

Author: Pavel Pudlák

Publisher: Springer Science & Business Media

ISBN: 3319001191

Category: Mathematics

Page: 695

View: 1042

The two main themes of this book, logic and complexity, are both essential for understanding the main problems about the foundations of mathematics. Logical Foundations of Mathematics and Computational Complexity covers a broad spectrum of results in logic and set theory that are relevant to the foundations, as well as the results in computational complexity and the interdisciplinary area of proof complexity. The author presents his ideas on how these areas are connected, what are the most fundamental problems and how they should be approached. In particular, he argues that complexity is as important for foundations as are the more traditional concepts of computability and provability. Emphasis is on explaining the essence of concepts and the ideas of proofs, rather than presenting precise formal statements and full proofs. Each section starts with concepts and results easily explained, and gradually proceeds to more difficult ones. The notes after each section present some formal definitions, theorems and proofs. Logical Foundations of Mathematics and Computational Complexity is aimed at graduate students of all fields of mathematics who are interested in logic, complexity and foundations. It will also be of interest for both physicists and philosophers who are curious to learn the basics of logic and complexity theory.
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Theorem Proving in Higher Order Logics

20th International Conference, TPHOLs 2007, Kaiserslautern, Germany, September 10-13, 2007, Proceedings

Author: Klaus Schneider,Jens Brandt

Publisher: Springer Science & Business Media

ISBN: 3540745904

Category: Computers

Page: 399

View: 7668

This book contains the refereed proceedings of the 20th International Conference on Theorem Proving in Higher Order Logics, TPHOLs 2007, held in Kaiserslautern, Germany, September 2007. Among the topics of this volume are formal semantics of specification, modeling, and programming languages, specification and verification of hardware and software, formalization of mathematical theories, advances in theorem prover technology, as well as industrial application of theorem provers.
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Model theory of stochastic processes

Author: Sergio Fajardo,H. Jerome Keisler

Publisher: A K Peters Ltd

ISBN: 9781568811673

Category: Mathematics

Page: 136

View: 879

This book presents new research in probability theory using ideas from mathematical logic. It is a general study of stochastic processes on adapted probability spaces, employing the concept of similarity of stochastic processes based on the notion of adapted distribution. The authors use ideas from model theory and methods from nonstandard analysis. The construction of spaces with certain richness properties, defined by insights from model theory, becomes easy using nonstandard methods, but remains difficult or impossible without them.
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Gödel's Theorem

An Incomplete Guide to Its Use and Abuse

Author: Torkel Franzén

Publisher: A K Peters/CRC Press

ISBN: 9781568812380

Category: Mathematics

Page: 182

View: 8516

"Among the many expositions of Gödel's incompleteness theorems written for non-specialists, this book stands apart. With exceptional clarity, Franzén gives careful, non-technical explanations both of what those theorems say and, more importantly, what they do not. No other book aims, as his does, to address in detail the misunderstandings and abuses of the incompleteness theorems that are so rife in popular discussions of their significance. As an antidote to the many spurious appeals to incompleteness in theological, anti-mechanist and post-modernist debates, it is a valuable addition to the literature." --- John W. Dawson, author of Logical Dilemmas: The Life and Work of Kurt Gödel
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Inexhaustibility

Webster’s Timeline History, 1869 - 2004

Author: Icon Group International

Publisher: ICON Group International

ISBN: 9780546476439

Category:

Page: N.A

View: 2753

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Philosophical Problems in the Law

Author: David M. Adams

Publisher: Wadsworth Publishing Company

ISBN: 9780534584283

Category: Philosophy

Page: 626

View: 3171

PHILOSOPHICAL PROBLEMS IN THE LAW is the perfect introduction to the philosophy of law! This collection of articles and cases helps you consider philosophical problems associated with the law through examples, case studies, and decision scenarios. Case examples and recent decisions such as Pledge of allegiance case, Freedom of expression on the Internet, Supreme Court rulings on affirmative action, "Three-Strikes laws," and the death penalty help you see the real world relevance of what you are learning.
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Feminism and the Mastery of Nature

Author: Val Plumwood

Publisher: Routledge

ISBN: 1134916698

Category: Philosophy

Page: 250

View: 9075

Two of the most important political movements of the late twentieth century are those of environmentalism and feminism. In this book, Val Plumwood argues that feminist theory has an important opportunity to make a major contribution to the debates in political ecology and environmental philosophy. Feminism and the Mastery of Nature explains the relation between ecofeminism, or ecological feminism, and other feminist theories including radical green theories such as deep ecology. Val Plumwood provides a philosophically informed account of the relation of women and nature, and shows how relating male domination to the domination of nature is important and yet remains a dilemma for women.
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Elements of Differential Geometry

Author: Richard S. Millman,George D. Parker

Publisher: Prentice Hall

ISBN: N.A

Category: Mathematics

Page: 265

View: 5127

This text is intended for an advanced undergraduate (having taken linear algebra and multivariable calculus). It provides the necessary background for a more abstract course in differential geometry. The inclusion of diagrams is done without sacrificing the rigor of the material. For all readers interested in differential geometry.
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Social Paralysis and Social Change

British Working-Class Education in the Nineteenth Century

Author: Neil J. Smelser

Publisher: Univ of California Press

ISBN: 9780520911543

Category: History

Page: 540

View: 9205

Neil Smelser's Social Paralysis and Social Change is one of the most comprehensive histories of mass education ever written. It tells the story of how working-class education in nineteenth-century Britain—often paralyzed by class, religious, and economic conflict—struggled forward toward change. This book is ambitious in scope. It is both a detailed history of educational development and a theoretical study of social change, at once a case study of Britain and a comparative study of variations within Britain. Smelser simultaneously meets the scholarly standards of historians and critically addresses accepted theories of educational change—"progress," conflict, and functional theories. He also sheds new light on the process of secularization, the relations between industrialization and education, structural differentiation, and the role of the state in social change. This work marks a return for the author to the same historical arena—Victorian Britain—that inspired his classic work Social Change in the Industrial Revolution thirty-five years ago. Smelser's research has again been exhaustive. He has achieved a remarkable synthesis of the huge body of available materials, both primary and secondary. Smelser's latest book will be most controversial in its treatment of class as a primordial social grouping, beyond its economic significance. Indeed, his demonstration that class, ethnic, and religious groupings were decisive in determining the course of British working-class education has broad-ranging implications. These groupings remain at the heart of educational conflict, debate, and change in most societies—including our own—and prompt us to pose again and again the chronic question: who controls the educational terrain?
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The Joy of the Gospel

Evangelii Gaudium

Author: Pope Francis

Publisher: Image

ISBN: 0553419544

Category: Religion

Page: 224

View: 3736

The perfect gift! A specially priced, beautifully designed hardcover edition of The Joy of the Gospel with a foreword by Robert Barron and an afterword by James Martin, SJ. “The joy of the gospel fills the hearts and lives of all who encounter Jesus… In this Exhortation I wish to encourage the Christian faithful to embark upon a new chapter of evangelization marked by this joy, while pointing out new paths for the Church’s journey in years to come.” – Pope Francis This special edition of Pope Francis's popular message of hope explores themes that are important for believers in the 21st century. Examining the many obstacles to faith and what can be done to overcome those hurdles, he emphasizes the importance of service to God and all his creation. Advocating for “the homeless, the addicted, refugees, indigenous peoples, the elderly who are increasingly isolated and abandoned,” the Holy Father shows us how to respond to poverty and current economic challenges that affect us locally and globally. Ultimately, Pope Francis demonstrates how to develop a more personal relationship with Jesus Christ, “to recognize the traces of God’s Spirit in events great and small.” Profound in its insight, yet warm and accessible in its tone, The Joy of the Gospel is a call to action to live a life motivated by divine love and, in turn, to experience heaven on earth. Includes a foreword by Robert Barron, author of Catholicism: A Journey to the Heart of the Faith and James Martin, SJ, author of Jesus: A Pilgrimage From the Hardcover edition.
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Man and His Symbols

Author: C. G. Jung

Publisher: Dell

ISBN: 0307800555

Category: Social Science

Page: 320

View: 8076

Man and His Symbols owes its existence to one of Jung's own dreams. The great psychologist dreamed that his work was understood by a wide public, rather than just by psychiatrists, and therefore he agreed to write and edit this fascinating book. Here, Jung examines the full world of the unconscious, whose language he believed to be the symbols constantly revealed in dreams. Convinced that dreams offer practical advice, sent from the unconscious to the conscious self, Jung felt that self-understanding would lead to a full and productive life. Thus, the reader will gain new insights into himself from this thoughtful volume, which also illustrates symbols throughout history. Completed just before his death by Jung and his associates, it is clearly addressed to the general reader. Praise for Man and His Symbols “This book, which was the last piece of work undertaken by Jung before his death in 1961, provides a unique opportunity to assess his contribution to the life and thought of our time, for it was also his firsat attempt to present his life-work in psychology to a non-technical public. . . . What emerges with great clarity from the book is that Jung has done immense service both to psychology as a science and to our general understanding of man in society, by insisting that imaginative life must be taken seriously in its own right, as the most distinctive characteristic of human beings.”—Guardian “Straighforward to read and rich in suggestion.”—John Barkham, Saturday Review Syndicate “This book will be a resounding success for those who read it.”—Galveston News-Tribune “A magnificent achievement.”—Main Currents “Factual and revealing.”—Atlanta Times
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The Logica Yearbook

Author: N.A

Publisher: N.A

ISBN: N.A

Category: Logic, Symbolic and mathematical

Page: N.A

View: 6698

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Infinity and the Mind

The Science and Philosophy of the Infinite

Author: Rudy Rucker

Publisher: Princeton University Press

ISBN: 1400849047

Category: Mathematics

Page: 368

View: 7046

In Infinity and the Mind, Rudy Rucker leads an excursion to that stretch of the universe he calls the "Mindscape," where he explores infinity in all its forms: potential and actual, mathematical and physical, theological and mundane. Rucker acquaints us with Gödel's rotating universe, in which it is theoretically possible to travel into the past, and explains an interpretation of quantum mechanics in which billions of parallel worlds are produced every microsecond. It is in the realm of infinity, he maintains, that mathematics, science, and logic merge with the fantastic. By closely examining the paradoxes that arise from this merging, we can learn a great deal about the human mind, its powers, and its limitations. Using cartoons, puzzles, and quotations to enliven his text, Rucker guides us through such topics as the paradoxes of set theory, the possibilities of physical infinities, and the results of Gödel's incompleteness theorems. His personal encounters with Gödel the mathematician and philosopher provide a rare glimpse at genius and reveal what very few mathematicians have dared to admit: the transcendent implications of Platonic realism.
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