Inexhaustibility

A Non-Exhaustive Treatment

Author: Torkel Franzén

Publisher: Cambridge University Press

ISBN: 1108641636

Category: Mathematics

Page: 251

View: 6706

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Since their inception, the Perspectives in Logic and Lecture Notes in Logic series have published seminal works by leading logicians. Many of the original books in the series have been unavailable for years, but they are now in print once again. This volume, the sixteenth publication in the Lecture Notes in Logic series, gives a sustained presentation of a particular view of the topic of Gödelian extensions of theories. It presents the basic material in predicate logic, set theory and recursion theory, leading to a proof of Gödel's incompleteness theorems. The inexhaustibility of mathematics is treated based on the concept of transfinite progressions of theories as conceived by Turing and Feferman. All concepts and results are introduced as needed, making the presentation self-contained and thorough. Philosophers, mathematicians and others will find the book helpful in acquiring a basic grasp of the philosophical and logical results and issues.

Alan Turing: His Work and Impact

Author: S. Barry Cooper,J. van Leeuwen

Publisher: Elsevier

ISBN: 0123870127

Category: Mathematics

Page: 944

View: 7108

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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

Model theory of stochastic processes

Author: Sergio Fajardo,H. Jerome Keisler

Publisher: A K Peters Ltd

ISBN: 9781568811673

Category: Mathematics

Page: 136

View: 6195

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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.

Alan Turing's Systems of Logic

The Princeton Thesis

Author: Andrew W. Appel

Publisher: Princeton University Press

ISBN: 0691164738

Category: Computers

Page: 160

View: 8620

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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.

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

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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.

Logical Foundations of Mathematics and Computational Complexity

A Gentle Introduction

Author: Pavel Pudlák

Publisher: Springer Science & Business Media

ISBN: 3319001191

Category: Mathematics

Page: 695

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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.

Forever Undecided

Author: Raymond M. Smullyan

Publisher: Knopf

ISBN: 0307962466

Category: Mathematics

Page: 257

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Forever Undecided is the most challenging yet of Raymond Smullyan’s puzzle collections. It is, at the same time, an introduction—ingenious, instructive, entertaining—to Gödel’s famous theorems. With all the wit and charm that have delighted readers of his previous books, Smullyan transports us once again to that magical island where knights always tell the truth and knaves always lie. Here we meet a new and amazing array of characters, visitors to the island, seeking to determine the natives’ identities. Among them: the census-taker McGregor; a philosophical-logician in search of his flighty bird-wife, Oona; and a regiment of Reasoners (timid ones, normal ones, conceited, modest, and peculiar ones) armed with the rules of propositional logic (if X is true, then so is Y). By following the Reasoners through brain-tingling exercises and adventures—including journeys into the “other possible worlds” of Kripke semantics—even the most illogical of us come to understand Gödel’s two great theorems on incompleteness and undecidability, some of their philosophical and mathematical implications, and why we, like Gödel himself, must remain Forever Undecided!

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: 4355

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"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

Inexhaustibility

Webster’s Timeline History, 1869 - 2004

Author: Icon Group International

Publisher: ICON Group International

ISBN: 9780546476439

Category:

Page: N.A

View: 6102

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The Condition of the Working-Class in England in 1844

Author: Frederick Engels

Publisher: BookRix GmbH & Company KG

ISBN: 3730964852

Category: History

Page: 466

View: 2474

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The Condition of the Working Class in England is one of the best-known works of Friedrich Engels. Originally written in German as Die Lage der arbeitenden Klasse in England, it is a study of the working class in Victorian England. It was also Engels' first book, written during his stay in Manchester from 1842 to 1844. Manchester was then at the very heart of the Industrial Revolution, and Engels compiled his study from his own observations and detailed contemporary reports. Engels argues that the Industrial Revolution made workers worse off. He shows, for example, that in large industrial cities mortality from disease, as well as death-rates for workers were higher than in the countryside. In cities like Manchester and Liverpool mortality from smallpox, measles, scarlet fever and whooping cough was four times as high as in the surrounding countryside, and mortality from convulsions was ten times as high as in the countryside. The overall death-rate in Manchester and Liverpool was significantly higher than the national average (one in 32.72 and one in 31.90 and even one in 29.90, compared with one in 45 or one in 46). An interesting example shows the increase in the overall death-rates in the industrial town of Carlisle where before the introduction of mills (1779-1787), 4,408 out of 10,000 children died before reaching the age of five, and after their introduction the figure rose to 4,738. Before the introduction of mills, 1,006 out of 10,000 adults died before reaching 39 years old, and after their introduction the death rate rose to 1,261 out of 10,000.

Elements of Differential Geometry

Author: Richard S. Millman,George D. Parker

Publisher: Prentice Hall

ISBN: N.A

Category: Mathematics

Page: 265

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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.

Teaching As a Subversive Activity

A No-Holds-Barred Assault on Outdated Teaching Methods-with Dramatic and Practical Proposals on How Education Can Be Made Relevant to Today's World

Author: Neil Postman

Publisher: Delta

ISBN: 0307491706

Category: Education

Page: 240

View: 8434

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A no-holds-barred assault on outdated teaching methods--with dramatic and practical proposals on how education can be made relevant to today's world. From the Trade Paperback edition.

Bitcoin and Cryptocurrency Technologies

A Comprehensive Introduction

Author: Arvind Narayanan,Joseph Bonneau,Edward Felten,Andrew Miller,Steven Goldfeder

Publisher: Princeton University Press

ISBN: 1400884152

Category: Computers

Page: 336

View: 6623

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Bitcoin and Cryptocurrency Technologies provides a comprehensive introduction to the revolutionary yet often misunderstood new technologies of digital currency. Whether you are a student, software developer, tech entrepreneur, or researcher in computer science, this authoritative and self-contained book tells you everything you need to know about the new global money for the Internet age. How do Bitcoin and its block chain actually work? How secure are your bitcoins? How anonymous are their users? Can cryptocurrencies be regulated? These are some of the many questions this book answers. It begins by tracing the history and development of Bitcoin and cryptocurrencies, and then gives the conceptual and practical foundations you need to engineer secure software that interacts with the Bitcoin network as well as to integrate ideas from Bitcoin into your own projects. Topics include decentralization, mining, the politics of Bitcoin, altcoins and the cryptocurrency ecosystem, the future of Bitcoin, and more. An essential introduction to the new technologies of digital currency Covers the history and mechanics of Bitcoin and the block chain, security, decentralization, anonymity, politics and regulation, altcoins, and much more Features an accompanying website that includes instructional videos for each chapter, homework problems, programming assignments, and lecture slides Also suitable for use with the authors' Coursera online course Electronic solutions manual (available only to professors)

Man and His Symbols

Author: C. G. Jung

Publisher: Dell

ISBN: 0307800555

Category: Social Science

Page: 320

View: 8216

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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

A Pluralistic Universe

Hibbert Lectures at Manchester College on the Present Situation in Philosophy

Author: William James

Publisher: The Floating Press

ISBN: 177556293X

Category: Philosophy

Page: 227

View: 5766

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Craving an intellectually stimulating read? Dive into A Pluralistic Universe by William James, an influential thinker and psychologist who also happened to be the brother of acclaimed novelist Henry James. This lucid, gripping account outlines some of James' critiques of standard methods of reasoning. It's definitely challenging, but much more appealing to a general audience than most philosophical tracts.