Godel's Proof

Author: Ernest Nagel,James Roy Newman

Publisher: NYU Press

ISBN: 0814758371

Category: Mathematics

Page: 129

View: 2336

“Come to me, all you that labor and are heavy laden, and I will give you rest.”Mathew 11:28 (AKJV) In the early 1990s, a grassroots coalition of churches in Baltimore, Maryland helped launch what would become a national movement. Joining forces with labor and low-wage worker organizations, they passed the first municipal living wage ordinance. Since then, over 144 municipalities and counties as well as numerous universities and local businesses in the United States have enacted such ordinances. Although religious persons and organizations have been important both in the origins of the living wage movement and in its continuing success, they are often ignored or under analyzed. Drawing on participant observation in multiple cities, All You That Labor analyzes and evaluates the contributions of religious activists to the movement. The book explores the ways religious organizations do this work in concert with low-wage workers, the challenges religious activists face, and how people of faith might better nurture moral agency in relation to the political economy. Ultimately, C. Melissa Snarr provides clarity on how to continue to cultivate, renew, and expand religious resources dedicated to the moral agency of low-wage workers and their allies.

An Introduction to Gödel's Theorems

Author: Peter Smith

Publisher: Cambridge University Press

ISBN: 1107022843

Category: Mathematics

Page: 406

View: 2277

A clear and accessible treatment of Gödel's famous, intriguing, but much misunderstood incompleteness theorems, extensively revised in a second edition.

Incompleteness: The Proof and Paradox of Kurt Gödel (Great Discoveries)

Author: Rebecca Goldstein

Publisher: W. W. Norton & Company

ISBN: 0393327604

Category: Biography & Autobiography

Page: 296

View: 1643

A portrait of the eminent twentieth-century mathematician discusses his theorem of incompleteness, relationships with such contemporaries as Albert Einstein, and untimely death as a result of mental instability and self-starvation.

A World Without Time

The Forgotten Legacy of Godel and Einstein

Author: Palle Yourgrau

Publisher: Basic Books

ISBN: 078673700X

Category: Science

Page: 224

View: 5171

In 1942, the logician Kurt Godel and Albert Einstein became close friends; they walked to and from their offices every day, exchanging ideas about science, philosophy, politics, and the lost world of German science. By 1949, Godel had produced a remarkable proof: In any universe described by the Theory of Relativity, time cannot exist. Einstein endorsed this result reluctantly but he could find no way to refute it, since then, neither has anyone else. Yet cosmologists and philosophers alike have proceeded as if this discovery was never made. In A World Without Time, Palle Yourgrau sets out to restore Godel to his rightful place in history, telling the story of two magnificent minds put on the shelf by the scientific fashions of their day, and attempts to rescue the brilliant work they did together.

On Formally Undecidable Propositions of Principia Mathematica and Related Systems

Author: Kurt Gödel

Publisher: Courier Corporation

ISBN: 0486158403

Category: Mathematics

Page: 80

View: 7049

First English translation of revolutionary paper (1931) that established that even in elementary parts of arithmetic, there are propositions which cannot be proved or disproved within the system. Introduction by R. B. Braithwaite.

Routledge Encyclopedia of Philosophy: Genealogy to Iqbal

Author: Edward Craig

Publisher: Taylor & Francis

ISBN: 9780415187091

Category: Electronic books

Page: 33

View: 6569

Review: "Depth and breadth of coverage, clarity of presentation, impressive bibliographies, excellent use of cross references, and an extensive index combine to make this an impressive reference work. The contributors have addressed both current and past scholarship on world philosophy and religion and have produced a worthy successor to Macmillan's 1967 Encyclopedia of Philosophy. It will be read and understood by the educated public as well as scholars and will be a fine addition to academic and large public library reference collections."--"Outstanding Reference Sources : the 1999 Selection Sources Committee, RUSA, ALA

A First Course in Artificial Intelligence, 1e

Author: Deepak Khemani

Publisher: McGraw-Hill Education

ISBN: 9383286784


Page: N.A

View: 9540

As the title says, A First Course in Artificial Intelligence is aptly suited for both undergraduate and postgraduate courses involving Artificial Intelligence. The book explores from scratch the algorithms and representations needed for building intelligent systems. The textbook follows a bottom-up approach exploring the basic strategies - starting with search based methods & then moving on to knowledge based methods - needed for problem solving and focuses primarily on the intelligence part. The only background required to comprehend the book is familiarity with programming. Easy explanations for the concepts bolstered with illuminative illustrations make the book an essential reading for the budding artificial intelligence students, seasoned practitioners and researchers; all alike.

The Existence of God

A Philosophical Introduction

Author: Yujin Nagasawa

Publisher: Routledge

ISBN: 1136737456

Category: Philosophy

Page: 208

View: 6450

Does God exist? What are the various arguments that seek to prove the existence of God? Can atheists refute these arguments? The Existence of God: A Philosophical Introduction assesses classical and contemporary arguments concerning the existence of God: the ontological argument, introducing the nature of existence, possible worlds, parody objections, and the evolutionary origin of the concept of God the cosmological argument, discussing metaphysical paradoxes of infinity, scientific models of the universe, and philosophers’ discussions about ultimate reality and the meaning of life the design argument, addressing Aquinas’s Fifth Way, Darwin’s theory of evolution, the concept of irreducible complexity, and the current controversy over intelligent design and school education. Bringing the subject fully up to date, Yujin Nagasawa explains these arguments in relation to recent research in cognitive science, the mathematics of infinity, big bang cosmology, and debates about ethics and morality in light of contemporary political and social events. The book also includes fascinating insights into the passions, beliefs and struggles of the philosophers and scientists who have tackled the challenge of proving the existence of God, including Thomas Aquinas, and Kurt Gödel - who at the end of his career as a famous mathematician worked on a secret project to prove the existence of God. The Existence of God: A Philosophical Introduction is an ideal gateway to the philosophy of religion and an excellent starting point for anyone interested in arguments about the existence of God.


Author: Cory Juhl,Eric Loomis

Publisher: Routledge

ISBN: 1135278407

Category: Philosophy

Page: 336

View: 1626

Analyticity, or the 'analytic/synthetic' distinction is one of the most important and controversial problems in contemporary philosophy. It is also essential to understanding many developments in logic, philosophy of language, epistemology and metaphysics. In this outstanding introduction to analyticity Cory Juhl and Eric Loomis cover the following key topics: The origins of analyticity in the philosophy of Hume and Kant Carnap's arguments concerning analyticity in the early twentieth century Quine's famous objections to analyticity in his classic 'Two Dogmas of Empiricism' essay The relationship between analyticity and central issues in metaphysics, such as ontology The relationship between analyticity and epistemology Analyticity in the context of the current debates in philosophy, including mathematics and ontology Throughout the book the authors show how many philosophical controversies hinge on the problem of analyticity. Additional features include chapter summaries, annotated further reading and a glossary of technical terms making the book ideal to those coming to the problem for the first time.

The New Wittgenstein

Author: Alice Crary,Rupert Read

Publisher: Routledge

ISBN: 1134689950

Category: Philosophy

Page: 416

View: 3655

This text offers major re-evaluation of Wittgenstein's thinking. It is a collection of essays that presents a significantly different portrait of Wittgenstein. The essays clarify Wittgenstein's modes of philosophical criticism and shed light on the relation between his thought and different philosophical traditions and areas of human concern. With essays by Stanley Cavell, James Conant, Cora Diamond, Peter Winch and Hilary Putnam, we see the emergence of a new way of understanding Wittgenstein's thought. This is a controversial collection, with essays by highly regarded Wittgenstein scholars that may change the way we look at Wittgenstein's body of work.

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: 3543

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.

Popular Lectures on Mathematical Logic

Author: Hao Wang

Publisher: Courier Corporation

ISBN: 0486171043

Category: Mathematics

Page: 292

View: 1952

Noted logician discusses both theoretical underpinnings and practical applications, exploring set theory, model theory, recursion theory and constructivism, proof theory, logic's relation to computer science, and other subjects. 1981 edition, reissued by Dover in 1993 with a new Postscript by the author.

Thinking about G”del and Turing

Essays on Complexity, 1970-2007

Author: Gregory J. Chaitin

Publisher: World Scientific

ISBN: 9812708952

Category: Mathematics

Page: 347

View: 5216

Dr Gregory Chaitin, one of the world's leading mathematicians, is best known for his discovery of the remarkable ê number, a concrete example of irreducible complexity in pure mathematics which shows that mathematics is infinitely complex. In this volume, Chaitin discusses the evolution of these ideas, tracing them back to Leibniz and Borel as well as G”del and Turing.This book contains 23 non-technical papers by Chaitin, his favorite tutorial and survey papers, including Chaitin's three Scientific American articles. These essays summarize a lifetime effort to use the notion of program-size complexity or algorithmic information content in order to shed further light on the fundamental work of G”del and Turing on the limits of mathematical methods, both in logic and in computation. Chaitin argues here that his information-theoretic approach to metamathematics suggests a quasi-empirical view of mathematics that emphasizes the similarities rather than the differences between mathematics and physics. He also develops his own brand of digital philosophy, which views the entire universe as a giant computation, and speculates that perhaps everything is discrete software, everything is 0's and 1's.Chaitin's fundamental mathematical work will be of interest to philosophers concerned with the limits of knowledge and to physicists interested in the nature of complexity.

Introduction to Logic

Author: Harry J Gensler

Publisher: Taylor & Francis

ISBN: 1317436113

Category: Philosophy

Page: 418

View: 364

Introduction to Logic is clear and concise, uses interesting examples (many philosophical in nature), and has easy-to-use proof methods. Its key features, retained in this Third Edition, include: simpler ways to test arguments, including an innovative proof method and the star test for syllogisms; a wide scope of materials, suiting it for introductory or intermediate courses; engaging examples, from philosophy and everyday life; useful for self-study and preparation for standardized tests, like the LSAT; a reasonable price (a third the cost of some competitors); and exercises that correspond to the free LogiCola instructional program. This Third Edition: improves explanations, especially on areas that students find difficult; has a fuller explanation of traditional Copi proofs and of truth trees; and updates the companion LogiCola software, which now is touch friendly (for use on Windows tablets and touch monitors), installs more easily on Windows and Macintosh, and adds exercises on Copi proofs and on truth trees. You can still install LogiCola for free (from http://www.harryhiker.com/lc or http://www.routledge.com/cw/gensler).

LPL Software Manual

Author: Gerald Allwein

Publisher: N.A

ISBN: 9781575863740


Page: 56

View: 8030

Proofs and Computations

Author: Helmut Schwichtenberg,Stanley S. Wainer

Publisher: Cambridge University Press

ISBN: 1139504169

Category: Mathematics

Page: N.A

View: 4182

Driven by the question, 'What is the computational content of a (formal) proof?', this book studies fundamental interactions between proof theory and computability. It provides a unique self-contained text for advanced students and researchers in mathematical logic and computer science. Part I covers basic proof theory, computability and Gödel's theorems. Part II studies and classifies provable recursion in classical systems, from fragments of Peano arithmetic up to Π11–CA0. Ordinal analysis and the (Schwichtenberg–Wainer) subrecursive hierarchies play a central role and are used in proving the 'modified finite Ramsey' and 'extended Kruskal' independence results for PA and Π11–CA0. Part III develops the theoretical underpinnings of the first author's proof assistant MINLOG. Three chapters cover higher-type computability via information systems, a constructive theory TCF of computable functionals, realizability, Dialectica interpretation, computationally significant quantifiers and connectives and polytime complexity in a two-sorted, higher-type arithmetic with linear logic.

Popper Selections

Author: Karl R. Popper,David W. Miller,Sir Karl Raimund Popper

Publisher: N.A

ISBN: 9780691020310

Category: Philosophy

Page: 479

View: 8515

These excerpts from the writings of Sir Karl Popper are an outstanding introduction to one of the most controversial of living philosophers, known especially for his devastating criticisms of Plato and Marx and for his uncompromising rejection of inductive reasoning. David Miller, a leading expositor and critic of Popper's work, has chosen thirty selections that illustrate the profundity and originality of his ideas and their applicability to current intellectual and social problems. Miller's introduction demonstrates the remarkable unity of Popper's thought and briefly describes his philosophy of critical rationalism, a philosophy that is distinctive in its emphasis on the way in which we learn through the making and correcting of mistakes. Popper has relentlessly challenged both the authority and the appeal to authority of the most fashionable philosophies of our time. This book of selections from his nontechnical writings on the theory of knowledge, the philosophy of science, metaphysics, and social philosophy is imbued with his emphasis on the role and by reason in exposing and eliminating the errors among them.

Godel's Theorem in Focus

Author: S.G. Shanker

Publisher: Routledge

ISBN: 1134947976

Category: Philosophy

Page: 272

View: 6437

A layman's guide to the mechanics of Gödel's proof together with a lucid discussion of the issues which it raises. Includes an essay discussing the significance of Gödel's work in the light of Wittgenstein's criticisms.

The Proof is in the Pudding

The Changing Nature of Mathematical Proof

Author: Steven G. Krantz

Publisher: Springer Science & Business Media

ISBN: 9780387487441

Category: Mathematics

Page: 264

View: 9489

This text explores the many transformations that the mathematical proof has undergone from its inception to its versatile, present-day use, considering the advent of high-speed computing machines. Though there are many truths to be discovered in this book, by the end it is clear that there is no formalized approach or standard method of discovery to date. Most of the proofs are discussed in detail with figures and equations accompanying them, allowing both the professional mathematician and those less familiar with mathematics to derive the same joy from reading this book.