An Introduction to Gödel's Theorems

Author: Peter Smith

Publisher: Cambridge University Press

ISBN: 9781139465939

Category: Mathematics

Page: N.A

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In 1931, the young Kurt Gödel published his First Incompleteness Theorem, which tells us that, for any sufficiently rich theory of arithmetic, there are some arithmetical truths the theory cannot prove. This remarkable result is among the most intriguing (and most misunderstood) in logic. Gödel also outlined an equally significant Second Incompleteness Theorem. How are these Theorems established, and why do they matter? Peter Smith answers these questions by presenting an unusual variety of proofs for the First Theorem, showing how to prove the Second Theorem, and exploring a family of related results (including some not easily available elsewhere). The formal explanations are interwoven with discussions of the wider significance of the two Theorems. This book will be accessible to philosophy students with a limited formal background. It is equally suitable for mathematics students taking a first course in mathematical logic.
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Quantum Information and Consciousness

A Gentle Introduction

Author: Danko D. Georgiev

Publisher: CRC Press

ISBN: 1351401785

Category: Science

Page: 344

View: 3162

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"I loved the book! This book is not just interesting, it is exciting. I have probably read every significant book in the field, and this is the strongest and most convincing one yet. It is also one of the most comprehensive in its explanations. I shall most certainly recommend the book to colleagues." –Richard G. Petty, MD "a very good introduction to the basic theory of quantum systems.... Dr. Georgiev’s book aptly prepares the reader to confront whatever might be in store later." –from the Foreword by Prof. James F. Glazebrook, Eastern Illinois University This book addresses the fascinating cross-disciplinary field of quantum information theory applied to the study of brain function. It offers a self-study guide to probe the problems of consciousness, including a concise but rigorous introduction to classical and quantum information theory, theoretical neuroscience, and philosophy of the mind. It aims to address long-standing problems related to consciousness within the framework of modern theoretical physics in a comprehensible manner that elucidates the nature of the mind-body relationship. The reader also gains an overview of methods for constructing and testing quantum informational theories of consciousness.
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Philosophical and Mathematical Logic

Author: Harrie de Swart

Publisher: Springer

ISBN: 3030032558

Category: Philosophy

Page: 539

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This book was written to serve as an introduction to logic, with in each chapter – if applicable – special emphasis on the interplay between logic and philosophy, mathematics, language and (theoretical) computer science. The reader will not only be provided with an introduction to classical logic, but to philosophical (modal, epistemic, deontic, temporal) and intuitionistic logic as well. The first chapter is an easy to read non-technical Introduction to the topics in the book. The next chapters are consecutively about Propositional Logic, Sets (finite and infinite), Predicate Logic, Arithmetic and Gödel’s Incompleteness Theorems, Modal Logic, Philosophy of Language, Intuitionism and Intuitionistic Logic, Applications (Prolog; Relational Databases and SQL; Social Choice Theory, in particular Majority Judgment) and finally, Fallacies and Unfair Discussion Methods. Throughout the text, the author provides some impressions of the historical development of logic: Stoic and Aristotelian logic, logic in the Middle Ages and Frege's Begriffsschrift, together with the works of George Boole (1815-1864) and August De Morgan (1806-1871), the origin of modern logic. Since "if ..., then ..." can be considered to be the heart of logic, throughout this book much attention is paid to conditionals: material, strict and relevant implication, entailment, counterfactuals and conversational implicature are treated and many references for further reading are given. Each chapter is concluded with answers to the exercises.
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The Map and the Territory

Exploring the Foundations of Science, Thought and Reality

Author: Shyam Wuppuluri,Francisco Antonio Doria

Publisher: Springer

ISBN: 3319724789

Category: Science

Page: 641

View: 9541

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This volume presents essays by pioneering thinkers including Tyler Burge, Gregory Chaitin, Daniel Dennett, Barry Mazur, Nicholas Humphrey, John Searle and Ian Stewart. Together they illuminate the very foundations of science and the scientific enterprise. The authors examine, from various perspectives, the complex relations between reality per se and our human descriptions of the world, as deduced in the process of scientific discovery. Are we living in Plato’s Cave? If not, how far have we emerged from it? And what are the various layers of description we can apply to objects and phenomena? A simple apple, for example, can be analyzed from several viewpoints beginning with evolution and biology, all the way down its microscopic quantum mechanical components. Together, the cast of eminent scientists and philosophers shed light on how our various theories and constructs co-exist in a patchwork to produce a seemingly coherent reality. All aficionados of philosophy as well as budding researchers will find here much food for thought as well as new and stimulating perspectives.The book also includes a foreword by Sir Roger Penrose and an afterword by Dagfinn Follesdal
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The Handy Philosophy Answer Book

Author: Naomi Zack

Publisher: Visible Ink Press

ISBN: 1578592771

Category: Philosophy

Page: 504

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Combining a basic history of philosophical thought with the often quirky personal stories of famous philosophers, this comprehensive introduction to the world of philosophy answers more than 1,000 questions, ranging from What was the Enlightenment? to Why did the Pythagorians avoid fava beans? Analyzing the collective effort of philosophers throughout history in the pursuit of truth and wisdom, the guide explores the tangible significance of philosophical thought to modern society and civilization as a whole. With a wide range of information suitable for various knowledge basesÑfrom junior high to junior collegeÑthis is an ideal resource for anyone looking to get a better grasp of the history of thought.
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An Introduction to Non-Classical Logic

From If to Is

Author: Graham Priest

Publisher: Cambridge University Press

ISBN: 9781139469678

Category: Science

Page: N.A

View: 7166

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This revised and considerably expanded 2nd edition brings together a wide range of topics, including modal, tense, conditional, intuitionist, many-valued, paraconsistent, relevant, and fuzzy logics. Part 1, on propositional logic, is the old Introduction, but contains much new material. Part 2 is entirely new, and covers quantification and identity for all the logics in Part 1. The material is unified by the underlying theme of world semantics. All of the topics are explained clearly using devices such as tableau proofs, and their relation to current philosophical issues and debates are discussed. Students with a basic understanding of classical logic will find this book an invaluable introduction to an area that has become of central importance in both logic and philosophy. It will also interest people working in mathematics and computer science who wish to know about the area.
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An Introduction to the Philosophy of Mathematics

Author: Mark Colyvan

Publisher: Cambridge University Press

ISBN: 0521826020

Category: Mathematics

Page: 188

View: 542

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This introduction to the philosophy of mathematics focuses on contemporary debates in an important and central area of philosophy. The reader is taken on a fascinating and entertaining journey through some intriguing mathematical and philosophical territory, including such topics as the realism/anti-realism debate in mathematics, mathematical explanation, the limits of mathematics, the significance of mathematical notation, inconsistent mathematics and the applications of mathematics. Each chapter has a number of discussion questions and recommended further reading from both the contemporary literature and older sources. Very little mathematical background is assumed and all of the mathematics encountered is clearly introduced and explained using a wide variety of examples. The book is suitable for an undergraduate course in philosophy of mathematics and, more widely, for anyone interested in philosophy and mathematics.
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An Introduction to Decision Theory

Author: Martin Peterson

Publisher: Cambridge University Press

ISBN: 1107151597

Category: Business & Economics

Page: 348

View: 8767

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Now revised and updated, this introduction to decision theory is both accessible and comprehensive, covering topics including decision making under ignorance and risk, the foundations of utility theory, the debate over subjective and objective probability, Bayesianism, causal decision theory, game theory, and social choice theory. No mathematical skills are assumed, with all concepts and results explained in non-technical and intuitive as well as more formal ways. There are now over 140 exercises with solutions, along with a glossary of key terms and concepts. This second edition includes a new chapter on risk aversion as well as updated discussions of numerous central ideas, including Newcomb's problem, prisoner's dilemmas, and Arrow's impossibility theorem. The book will appeal particularly to philosophy students but also to readers in a range of disciplines, from computer science and psychology to economics and political science.
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An Introduction to Probability and Inductive Logic

Author: Ian Hacking

Publisher: Cambridge University Press

ISBN: 1139643614

Category: Philosophy

Page: N.A

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This is an introductory 2001 textbook on probability and induction written by one of the world's foremost philosophers of science. The book has been designed to offer maximal accessibility to the widest range of students (not only those majoring in philosophy) and assumes no formal training in elementary symbolic logic. It offers a comprehensive course covering all basic definitions of induction and probability, and considers such topics as decision theory, Bayesianism, frequency ideas, and the philosophical problem of induction. The key features of this book are a lively and vigorous prose style; lucid and systematic organization and presentation of ideas; many practical applications; a rich supply of exercises drawing on examples from such fields as psychology, ecology, economics, bioethics, engineering, and political science; numerous brief historical accounts of how fundamental ideas of probability and induction developed; and a full bibliography of further reading.
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