Charming Proofs A Journey into Elegant Mathematics

Claudi Alsina e Roger B. Nelsen

Dolciani mathematical expositions, nº 42

The Mathematical Association of America | 2011 | 320 páginas | rar - pdf | 2,35 Mb

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Theorems and their proofs lie at the heart of mathematics. In speaking of the purely aesthetic qualities of theorems and proofs, G. H. Hardy wrote that in beautiful proofs 'there is a very high degree of unexpectedness, combined with inevitability and economy'. Charming Proofs presents a collection of remarkable proofs in elementary mathematics that are exceptionally elegant, full of ingenuity, and succinct. By means of a surprising argument or a powerful visual representation, the proofs in this collection will invite readers to enjoy the beauty of mathematics, and to develop the ability to create proofs themselves. The authors consider proofs from topics such as geometry, number theory, inequalities, plane tilings, origami and polyhedra. Secondary school and university teachers can use this book to introduce their students to mathematical elegance. More than 130 exercises for the reader (with solutions) are also included.

Contents
A garden of integers
Distinguished numbers
Points in the plane
The polygonal playground
A treasury of triangle theorems
The enchantment of the equilateral triangle
The quadrilaterals' corner
Squares everywhere
Curves ahead
Adventures in tiling and coloring
Geometry in three dimensions
Additional theorems, problems, and proofs.

Kurt Gödel and the Foundations of Mathematics: Horizons of Truth

Matthias Baaz, Christos H. Papadimitriou, Hilary W. Putnam e Dana S. Scott

Cambridge University Press | 2011 | páginas | pdf | 3,9 Mb

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This volume commemorates the life, work, and foundational views of Kurt Gödel (1906-1978), most famous for his hallmark works on the completeness of first-order logic, the incompleteness of number theory, and the consistency - with the other widely accepted axioms of set theory - of the axiom of choice and of the generalized continuum hypothesis. It explores current research, advances, and ideas for future directions not only in the foundations of mathematics and logic, but also in the fields of computer science, artificial intelligence, physics, cosmology, philosophy, theology, and the history of science. The discussion is supplemented by personal reflections from several scholars who knew Gödel personally, providing some interesting insights into his life. By putting his ideas and life's work into the context of current thinking and perceptions, this book will extend the impact of Gödel's fundamental work in mathematics, logic, philosophy, and other disciplines for future generations of researchers.

Contents

Contributors page xi

Foreword – Gaisi Takeuti xiii

Preface xv

Acknowledgments xvii

Short Biography of Kurt Gödel xix

I Historical Context: Gödel’s Contributions and Accomplishments Gödel’s Historical, Philosoph1ical, and Scientific Work

1 The Impact of Gödel’s Incompleteness Theorems on Mathematics 3

Angus Macintyre

2 Logical Hygiene, Foundations, and Abstractions: Diversity among Aspects and Options 27

Georg Kreisel

Gödel’s Legacy: A Historical Perspective

3 The Reception of Gödel’s 1931 Incompletability Theorems by Mathematicians, and Some Logicians, to the Early 1960s 57

Ivor Grattan-Guinness

4 “Dozent Gödel Will Not Lecture” 75

Karl Sigmund

5 Gödel’s Thesis: An Appreciation 95

Juliette Kennedy

6 Lieber Herr Bernays! Lieber Herr Gödel! Gödel on Finitism, Constructivity, and Hilbert’s Program 111

Solomon Feferman

7 Computation and Intractability: Echoes of Kurt Gödel37

Christos H. Papadimitriou

8 Fromthe Entscheidungsproblem to the Personal Computer – and Beyond 151

B. Jack Copeland

G¨odelian Cosmology

9 Gödel, Einstein, Mach, Gamow, and Lanczos: Gödel’s Remarkable Excursion into Cosmology 185

Wolfgang Rindler

10 Physical Unknowables 213

Karl Svozil

II A Wider Vision: The Interdisciplinary, Philosophical, and Theological Implications of Gödel’s Work

On the Unknowables

11 Gödel and Physics 255

John D. Barrow

12 Gödel, Thomas Aquinas, and the Unknowability of God 277

Denys A. Turner

Gödel and the Mathematics of Philosophy

13 Gödel ’s Mathematics of Philosophy 299

Piergiorgio Odifreddi

Gödel and Philosophical Theology

14 Gödel’s Ontological Proof and Its Variants 307

Petr H´ajek

Gödel and the Human Mind

15 The Gödel Theorem and Human Nature 325

Hilary W. Putnam

16 Gödel , the Mind, and the Laws of Physics 339

Roger Penrose

III New Frontiers: Beyond Gödel ’sWork in Mathematics and Symbolic Logic

Extending G¨odel’s Work

17 Gödel ’s Functional Interpretation and Its Use in Current Mathematics 361

Ulrich Kohlenbach

18 My Forty Years on His Shoulders 399

Harvey M. Friedman

The Realm of Set Theory

19 My Interaction with Kurt G¨odel: The Man and HisWork 435

Paul J. Cohen

Gödel  and the Higher Infinite

20 The Transfinite Universe 449

W. Hugh Woodin

Gödel  and Computer Science

21 The Gödel  Phenomenon in Mathematics: A Modern View 475

Avi Wigderson

Index 509

Goedel's Way: Exploits into an undecidable world

Gregory Chaitin, Francisco A Doria e Newton C.A. da Costa

CRC Press | 2011 | 162 páginas | pdf | 1,1 Mb

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Kurt Gödel (1906-1978) was an Austrian-American mathematician, who is best known for his incompleteness theorems. He was the greatest mathematical logician of the 20th century, with his contributions extending to Einstein’s general relativity, as he proved that Einstein’s theory admits time machines. 
The Gödel incompleteness theorem - one cannot prove nor disprove all true mathematical sentences in the usual formal mathematical systems- is frequently presented in textbooks as something that happens in the rarefied realm of mathematical logic, and that has nothing to do with the real world. Practice shows the contrary though; one can demonstrate the validity of the phenomenon in various areas, ranging from chaos theory and physics to economics and even ecology. In this lively treatise, based on Chaitin’s groundbreaking work and on the da Costa-Doria results in physics, ecology, economics and computer science, the authors show that the Gödel incompleteness phenomenon can directly bear on the practice of science and perhaps on our everyday life.
This accessible book gives a new, detailed and elementary explanation of the Gödel incompleteness theorems and presents the Chaitin results and their relation to the da Costa-Doria results, which are given in full, but with no technicalities. Besides theory, the historical report and personal stories about the main character and on this book’s writing process, make it appealing leisure reading for those interested in mathematics, logic, physics, philosophy and computer science.
See also: http://www.youtube.com/watch?v=REy9noY5Sg8

Contents
1. Gödel, Turing 
2. Complexity, randomness 
3. A list of problems 
4. The halting function and its avatars 
5. Entropy, P vs. NP
6. Forays into uncharted landscapes.

Mathematics of the Transcendental: Onto-logy and being-there

 

Alain Badiou, A.J. Bartlett e Alex Ling

Bloomsbury Academic | 2014 | 291 páginas | rar - pdf | 2,74 Mb

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In Mathematics of the Transcendental, Alain Badiou painstakingly works through the pertinent aspects of Category Theory, demonstrating their internal logic and veracity, their derivation and distinction from Set Theory, and the 'thinking of being'. In doing so he sets out the basic onto-logical requirements of his greater and transcendental logics as articulated in his magnum opus, Logics of Worlds. This important book combines both his elaboration of the disjunctive synthesis between ontology and onto-logy (the discourses of being as such and being-appearing) from the perspective of Category Theory and the categorial basis of his philosophical conception of 'being there'.
Hitherto unpublished in either French or English, Mathematics of the Transcendental provides Badiou's readers with a much-needed complete elaboration of his understanding and use of Category Theory. The book is an essential aid to understanding the mathematical and logical basis of his theory of appearing as elaborated in Logics of Worlds and other works and is essential reading for his many followers.

TABLE OF CONTENTS
Translators’ Introduction: The Categorial Imperative 1
PART ONE TOPOS, OR LOGICS OF ONTO-LOGY: AN INTRODUCTION FOR PHILOSOPHERS 11
1 General Aim 13
2 Preliminary Definitions 17
3 The Size of a Category 21
4 Limit and Universality 27
5 Some Fundamental Concepts 29
6 Duality 37
7 Isomorphism 41
8 Exponentiation 45
9 Universe, 1: Closed Cartesian Categories 51
10 Structures of Immanence, 1: Philosophical Considerations 55
11 Structures of Immanence, 2: Sub-Object 59
12 Structures of Immanence, 3: Elements of an Object 63
13 ‘Elementary’ Clarification of Exponentiation 67
14 Central Object (or Sub-Object Classifier) 71
15 The True, the False, Negation and More 77
16 The Central Object as Linguistic Power 85
17 Universe, 2: The Concept of Topos 89
18 Ontology of the Void and Difference 95
19 Mono., Epi., Equ., and Other Arrows 99
20 Topoi as Logical Places 113
21 Internal Algebra of 1 123
22 Ontology of the Void and Excluded Middle 141
23 A Minimal Classical Model 147
24 A Minimal Non-Classical Model 151
PART TWO BEING THERE: MATHEMATICS OF THE TRANSCENDENTAL 163
Introduction 165
A. Transcendental Structures 171
B. Transcendental Connections 183
B.1. Connections between the transcendental and set-theoretic ontology: Boolean algebras 183
B.2. Connections between the transcendental and logic in its ordinary sense (propositional logic and first order predicate logic) 195
B.3. Connection between the transcendental and the general theory of localizations: Topology 202
C. Theory of Appearing and Objectivity 217
D. Transcendental Projections: Theory of Localization 235
E. Theory of Relations: Situation as Universe 249
Appendix: On Three Different Concepts of Identity Between Two
Multiples or Two Beings 265
Translator’s Endnotes 269
Index 277

Three Views of Logic: Mathematics, Philosophy, and Computer Science

Donald W. Loveland, Richard E. Hodel e S. G. Sterrett

Princeton University Press | 2014 | 339 páginas | rar - pdf |1,85 Mb

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Demonstrating the different roles that logic plays in the disciplines of computer science, mathematics, and philosophy, this concise undergraduate textbook covers select topics from three different areas of logic: proof theory, computability theory, and nonclassical logic. The book balances accessibility, breadth, and rigor, and is designed so that its materials will fit into a single semester. Its distinctive presentation of traditional logic material will enhance readers' capabilities and mathematical maturity.

The proof theory portion presents classical propositional logic and first-order logic using a computer-oriented (resolution) formal system. Linear resolution and its connection to the programming language Prolog are also treated. The computability component offers a machine model and mathematical model for computation, proves the equivalence of the two approaches, and includes famous decision problems unsolvable by an algorithm. The section on nonclassical logic discusses the shortcomings of classical logic in its treatment of implication and an alternate approach that improves upon it: Anderson and Belnap's relevance logic. Applications are included in each section. The material on a four-valued semantics for relevance logic is presented in textbook form for the first time.

Aimed at upper-level undergraduates of moderate analytical background, Three Views of Logic will be useful in a variety of classroom settings.

• Gives an exceptionally broad view of logic
• Treats traditional logic in a modern format
• Presents relevance logic with applications
• Provides an ideal text for a variety of one-semester upper-level undergraduate courses

Contents

Preface ix

Acknowledgments xiii

PART 1. Proof Theory 1

DONALD W. LOVELAND

1 Propositional Logic 3

1.1 Propositional Logic Semantics 5

1.2 Syntax: Deductive Logics 13

1.3 The Resolution Formal Logic 14

1.4 Handling Arbitrary PropositionalWffs 26

2 Predicate Logic 31

2.1 First-Order Semantics 32

2.2 Resolution for the Predicate Calculus 40

2.2.1 Substitution 41

2.2.2 The Formal System for Predicate Logic 45

2.2.3 Handling Arbitrary PredicateWffs 54

3 An Application: Linear Resolution and Prolog 61

3.1 OSL-Resolution 62

3.2 Horn Logic 69

3.3 Input Resolution and Prolog 77

Appendix A: The Induction Principle 81

Appendix B: First-Order Valuation 82

Appendix C: A Commentary on Prolog 84

References 91

PART 2. Computability Theory 93

RICHARD E. HODEL

4 Overview of Computability 95

4.1 Decision Problems and Algorithms 95

4.2 Three Informal Concepts 107

5 A Machine Model of Computability 123

5.1 RegisterMachines and RM-Computable Functions 123

5.2 Operations with RM-Computable Functions; Church-Turing Thesis; LRM-Computable Functions 136

5.3 RM-Decidable and RM-Semi-Decidable Relations; the Halting Problem 144

5.4 Unsolvability of Hilbert’s Decision Problem and Thue’sWord Problem 154

6 A Mathematical Model of Computability 165

6.1 Recursive Functions and the Church-Turing Thesis 165

6.2 Recursive Relations and RE Relations 175

6.3 Primitive Recursive Functions and Relations; Coding 187

6.4 Kleene Computation Relation Tn(e, a1, . . . , an, c) 197

6.5 Partial Recursive Functions; Enumeration Theorems 203

6.6 Computability and the Incompleteness Theorem 216

List of Symbols 219

References 220

PART 3. Philosophical Logic 221

S. G. STERRETT

7 Non-Classical Logics 223

7.1 Alternatives to Classical Logic vs. Extensions of

Classical Logic 223

7.2 From Classical Logic to Relevance Logic 228

7.2.1 The (So-Called) “Paradoxes of Implication” 228

7.2.2 Material Implication and Truth Functional Connectives 234

7.2.3 Implication and Relevance 238

7.2.4 Revisiting Classical Propositional Calculus: What to Save,What to Change, What to Add? 240

8 Natural Deduction: Classical and Non-Classical 243

8.1 Fitch’s Natural Deduction System for Classical

Propositional Logic 243

8.2 Revisiting Fitch’s Rules of Natural Deduction to Better Formalize the Notion of Entailment—Necessity 251

8.3 Revisiting Fitch’s Rules of Natural Deduction to Better Formalize the Notion of Entailment—Relevance 253

8.4 The Rules of System FE (Fitch-Style Formulation of the Logic of Entailment) 261

8.5 The Connective “Or,” Material Implication, and the Disjunctive Syllogism 281

9 Semantics for Relevance Logic: A Useful Four-Valued Logic 288

9.1 Interpretations, Valuations, and Many Valued Logics 288

9.2 Contexts in Which This Four-Valued Logic Is Useful 290

9.3 The Artificial Reasoner’s (Computer’s) “State of Knowledge” 291

9.4 Negation in This Four-Valued Logic 295

9.5 Lattices: A Brief Tutorial 297

9.6 Finite Approximation Lattices and Scott’s Thesis 302

9.7 Applying Scott’s Thesis to Negation, Conjunction, and Disjunction 304

9.8 The Logical Lattice L4 307

9.9 Intuitive Descriptions of the Four-Valued Logic Semantics 309

9.10 Inferences and Valid Entailments 312

10 Some Concluding Remarks on the Logic of

Entailment 315

References 316

Index 319

Infinity and Truth

Chitat Chong, Qi Feng, Theodore A Slaman e W Hugh Woodin 

World Scientific Publishing Company | 2014 | páginas | rar - pdf | 1,6 Mb

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This volume is based on the talks given at the Workshop on Infinity and Truth held at the Institute for Mathematical Sciences, National University of Singapore, from 25 to 29 July 2011. The chapters are by leading experts in mathematical and philosophical logic that examine various aspects of the foundations of mathematics. The theme of the volume focuses on two basic foundational questions: (i) What is the nature of mathematical truth and how does one resolve questions that are formally unsolvable within the Zermelo Fraenkel Set Theory with the Axiom of Choice, and (ii) Do the discoveries in mathematics provide evidence favoring one philosophical view over others? These issues are discussed from the vantage point of recent progresses in foundational studies.

The final chapter features questions proposed by the participants of the Workshop that will drive foundational research. The wide range of topics covered here will be of benefit to students, researchers and mathematicians interested in the foundations of mathematics.

Readership: Mathematicians, philosophers, scientists, graduate students, academic institutions, and research organizations interested in logic and the philosophy of mathematics.

CONTENTS

Foreword vii

Preface ix

Section I. Invited Lectures

Absoluteness, Truth, and Quotients 1

Ilijas Farah

A Multiverse Perspective on the Axiom of Constructibility 25

Joel David Hamkins

Hilbert, Bourbaki and the Scorning of Logic 47

A. R. D. Mathias

Toward Objectivity in Mathematics 157

Stephen G. Simpson

Sort Logic and Foundations of Mathematics 171

Jouko Vaananen

Reasoning about Constructive Concepts 187

Nik Weaver

Perfect Infinities and Finite Approximation 199

Boris Zilber

Section II. Special Session

An Objective Justification for Actual Infinity? 225

Stephen G. Simpson

Oracle Questions 229

Theodore Slaman and W. Hugh Woodin

To mock a mockingbird

Raymond M. Smullyan 

Knopf | 1985 | 257 páginas | rar - epub | 2,11 Mb

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In this entertaining and challenging new collection of logic puzzles, Raymond Smullyan—author of What Is the Name of This Book? And The Lady or the Tiger?—continues to delight and astonish us with his gift for making available, in the thoroughly pleasurable form of puzzles, some of the most important mathematical thinking of our time.
In the first part of the book, he transports us once again to that wonderful realm where knights, knaves, twin sisters, quadruplet brothers, gods, demons, and mortals either always tell the truth or always lie, and where truth-seekers are set a variety of fascinating problems. The section culminates in an enchanting and profound metapuzzle (a puzzle about a puzzle), in which Inspector Craig of Scotland Yard gets involved in a search of the Fountain of Youth on the Island of Knights and Knaves.
In the second and larger section, we accompany the Inspector on a summer-long adventure into the field of combinatory logic (a branch of logic that plays an important role in computer science and artificial intelligence). His adventure, which includes enchanted forests, talking birds, bird sociologists, and a classic quest, provides for us along the way the pleasure of solving puzzles of increasing complexity until we reach the Master Forest and—thanks to Gödel’s famous theorem—the final revelation.
To Mock a Mockingbird will delight all puzzle lovers—the curious neophytes as well as the serious students of logic, mathematics, or computer science.

Contents
Other Books by This Author
Title Page
Copyright
Dedication
Acknowledgments
Preface
PART I · LOGIC PUZZLES
1 The Prize—and Other Puzzles
2 The Absentminded Logician
3 The Barber of Seville
4 The Mystery of the Photograph
PART II · KNIGHTS, KNAVES, AND THE FOUNTAIN OF YOUTH
5 Some Unusual Knights and Knaves
6 Day-Knights and Night-Knights
7 Gods, Demons, and Mortals
8 In Search of the Fountain of Youth
PART III · TO MOCK A MOCKINGBIRD
9 To Mock a Mockingbird
10 Is There a Sage Bird?
11 Birds Galore
12 Mockingbirds, Warblers, and Starlings
13 A Gallery of Sage Birds
PART IV · SINGING BIRDS
14 Curry’s Lively Bird Forest
15 Russell’s Forest
16 The Forest Without a Name
17 Gödel’s Forest
PART V · THE MASTER FOREST
18 The Master Forest
19 Aristocratic Birds
20 Craig’s Discovery
PART VI · THE GRAND QUESTION!
21 The Fixed Point Principle
22 A Glimpse into Infinity
23 Logical Birds
24 Birds That Can Do Arithmetic
25 Is There an Ideal Bird?
Epilogue
Who’s Who Among the Birds

The Adventure of Reason: Interplay Between Philosophy of Mathematics and Mathematical Logic, 1900-1940

Paolo Mancosu

Oxford University Press | 2011 | 631 páginas | rar - pdf | 2,68 Mb

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Paolo Mancosu presents a series of innovative studies in the history and the philosophy of logic and mathematics in the first half of the twentieth century. The Adventure of Reason is divided into five main sections: history of logic (from Russell to Tarski); foundational issues (Hilbert's program, constructivity, Wittgenstein, Godel); mathematics and phenomenology (Weyl, Becker, Mahnke); nominalism (Quine, Tarski); semantics (Tarski, Carnap, Neurath). Mancosu exploits extensive untapped archival sources to make available a wealth of new material that deepens in significant ways our understanding of these fascinating areas of modern intellectual history. At the same time, the book is a contribution to recent philosophical debates, in particular on the prospects for a successful nominalist reconstruction of mathematics, the nature of finitist intuition, the viability of alternative definitions of logical consequence, and the extent to which phenomenology can hope to account for the exact sciences. 

CONTENTS

PART I. MATHEMATICAL LOGIC, 1900–1935

Introduction 2

1. The Development of Mathematical Logic from Russell to Tarski, 1900–1935 (with Richard Zach and Calixto Badesa) 5

PART II. FOUNDATIONS OF MATHEMATICS

Introduction 122

2. Hilbert and Bernays on Metamathematics 125

Addendum 155

3. Between Russell and Hilbert: Behmann on the Foundations of Mathematics 159

4. The Russellian Influence on Hilbert and His School 176

5. On the Constructivity of Proofs: A Debate among Behmann, Bernays, Godel, and Kaufmann 199

6. Wittgenstein’s Constructivization of Euler’s Proof of the Infinity of Primes (with Mathieu Marion) 217

7. Between Vienna and Berlin: The Immediate Reception of Godel’s Incompleteness Theorems 232

8. Review of Godel’s CollectedWorks, Vols. IV and V 240

PART III. PHENOMENOLOGY AND THE EXACT SCIENCES

Introduction 256

9. HermannWeyl: Predicativity and an Intuitionistic Excursion 259

10. Mathematics and Phenomenology: The Correspondence

between O. Becker and H.Weyl (with T. Ryckman) 277

11. Geometry, Physics, and Phenomenology: Four Letters of O. Becker to H.Weyl (with T. Ryckman) 308

12. “Das Abenteuer der Vernunft”: O. Becker and D. Mahnke on the Phenomenological Foundations of the Exact Sciences 346

PART IV. TARSKI AND QUINE ON NOMINALISM

Introduction 358

13. Harvard 1940–1941: Tarski, Carnap, and Quine on a Finitistic Language of Mathematics for Science 361

14. Quine and Tarski on Nominalism 387

PART V. TARSKI AND THE VIENNA CIRCLE ON TRUTH AND LOGICAL CONSEQUENCE

Introduction 412

15. Tarski, Neurath, and Kokoszy´nska on the Semantic Conception of Truth 415

16. Tarski on Models and Logical Consequence 440

Addendum 463

17. Tarski on Categoricity and Completeness: An Unpublished Lecture from 1940 469

18. Appendix: “On the Completeness and Categoricity of Deductive Systems” (1940) 485

Notes 493

Bibliography 571

Index 611

Outros livros do mesmo autor:

The philosophy of mathematical practice por Paolo Mancosu; Idioma: Inglês   Editora: Oxford : Oxford University Press, 2010.

Visualization, explanation and reasoning styles in mathematics por Paolo Mancosu; Klaus Frovin Jørgensen; Stig Andur Pedersen; Idioma: Inglês   Editora: Dordrecht ; Norwell, MA : Springer, ©2005.

Philosophy of mathematics and mathematical practice in the seventeenth century por Paolo Mancosu Idioma: Inglês   Editora: New York [u.a.] Oxford Univ. Press 1996

Rethinking Logic: Logic in Relation to Mathematics, Evolution, and Method

Carlo Cellucci

Springer | 2013 | 391 páginas | rar - pdf | 1,9 Mb

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This volume examines the limitations of mathematical logic and proposes a new approach to logic intended to overcome them. To this end, the book compares mathematical logic with earlier views of logic, both in the ancient and in the modern age, including those of Plato, Aristotle, Bacon, Descartes, Leibniz, and Kant. From the comparison it is apparent that a basic limitation of mathematical logic is that it narrows down the scope of logic confining it to the study of deduction, without providing tools for discovering anything new. As a result, mathematical logic has had little impact on scientific practice.

Therefore, this volume proposes a view of logic according to which logic is intended, first of all, to provide rules of discovery, that is, non-deductive rules for finding hypotheses to solve problems. This is essential if logic is to play any relevant role in mathematics, science and even philosophy. To comply with this view of logic, this volume formulates several rules of discovery, such as induction, analogy, generalization, specialization, metaphor, metonymy, definition, and diagrams. A logic based on such rules is basically a logic of discovery, and involves a new view of the relation of logic to evolution, language, reason, method and knowledge, particularly mathematical knowledge. It also involves a new view of the relation of philosophy to knowledge. This book puts forward such new views, trying to open again many doors that the founding fathers of mathematical logic had closed historically.

Contents

Preface.
Chapter 1. Introduction.
Part I. Ancient Perspectives.
Chapter 2. The Origin of Logic.
Chapter 3. Ancient Logic and Science
Chapter 4. The Analytic Method.
Chapter 5. The Analytic-Synthetic Method.
Chapter 6. Aristotle's Logic: The Deductivist View.
Chapter 7. Aristotle's Logic: The Heuristic View.
Part II. Modern Perspectives.
Chapter 8. The Method of Modern Science
Chapter 9. The Quest for a Logic of Discovery.
Chapter 10. Frege's Approach to Logic
Chapter 11. Gentzen's Approach to Logic.
Chapter 12. The Limitations of Mathematical Logic
Chapter 13. Logic, Method, and the Psychology of Discovery.
Part III: An Alternative Perspective.
Chapter 14. Reason and Knowledge.
Chapter 15. Reason, Knowledge and Emotion.
Chapter 16. Logic, Evolution, Language and Reason
Chapter 17. Logic, Method and Knowledge.
Chapter 18. Classifying and Justifying Inference Rules
Chapter 19. Philosophy and Knowledge.
Part IV: Rules of Discovery.
Chapter 20. Induction and Analogy.
hapter 21. Other Rules of Discovery.
Chapter 22. Conclusion.
References.
Name Index
Subject Index.

From Peirce to Skolem, Volume 4: A Neglected Chapter in the History of Logic

(Studies in the History and Philosophy of Mathematics)

Geraldine Brady 

North Holland | 2000 | 480 páginas | PDF | 20 Mb

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This book is an account of the important influence on the development of mathematical logic of Charles S. Peirce and his student O.H. Mitchell, through the work of Ernst Schrder, Leopold Lwenheim, and Thoralf Skolem. As far as we know, this book is the first work delineating this line of influence on modern mathematical logic.

Roads to Infinity: The Mathematics of Truth and Proof

John Stillwell

A K Peters | 2010 | 250 páginas | PDF | 1,3 Mb

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This popular account of set theory and mathematical logic introduces the reader to modern ideas about infinity and their implications for mathematics. It unifies ideas from set theory and mathematical logic and traces their effects on mainstream mathematical topics of today, such as number theory and combinatorics.The treatment is historical and partly informal, but with due attention to the subtleties of the subject. Ideas are shown to evolve from natural mathematical questions about the nature of infinity and the nature of proof, set against a background of broader questions and developments in mathematics. A particular aim of the book is to acknowledge some important but neglected figures in the history of infinity, such as Post and Gentzen, alongside the recognized giants Cantor and Goedel.

Logic: A Very Short Introduction

Graham Priest

Oxford University Press, USA | 2000 | 144 páginas | DJVU | 1,69 Mb

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Logic is often perceived as having little to do with the rest of philosophy, and even less to do with real life. In this lively and accessible introduction, Graham Priest shows how wrong this conception is. He explores the philosophical roots of the subject, explaining how modern formal logic deals with issues ranging from the existence of God and the reality of time to paradoxes of probability and decision theory. Along the way, the basics of formal logic are explained in simple, non-technical terms, showing that logic is a powerful and exciting part of modern philosophy.

Logic & mathematical paradoxes

Sindy Dunbar

White Word Publications | 2012 | 99 páginas | PDF | 629 kb

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Table of Contents

Chapter 1 - Accuracy Paradox & Apportionment Paradox

Chapter 2 - All Horses are the Same Color & Infinite Regress

Chapter 3 - Drinker Paradox & Lottery Paradox

Chapter 4 - Paradoxes of Material Implication

Chapter 5 - Raven Paradox

Chapter 6 - Unexpected Hanging Paradox

Chapter 7 - Banach–Tarski Paradox

Chapter 8 - Coastline Paradox & Paradoxical Set

Chapter 9 - Gabriel's Horn & Missing Square Puzzle

Chapter 10 - Smale's Paradox & Hausdorff Paradox

Chapter 11 - Borel–Kolmogorov Paradox & Berkson's Paradox

Chapter 12 - Boy or Girl Paradox & Burali-Forti Paradox

Chapter 13 - Elevator Paradox

Chapter 14 - Gödel's Incompleteness Theorems

Chapter 15 - Gambler's Fallacy

Sets, Functions, and Logic: An Introduction to Abstract Mathematics

Third Edition
(Chapman Hall/CRC Mathematics Series)

Keith Devlin

Chapman and Hall/CRC | 2003 | 160 páginas | PDF | 1,53 Mb

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Keith Devlin. You know him. You've read his columns in MAA Online, you've heard him on the radio, and you've seen his popular mathematics books. In between all those activities and his own research, he's been hard at work revising Sets, Functions and Logic, his standard-setting text that has smoothed the road to pure mathematics for legions of undergraduate students.

Now in its third edition, Devlin has fully reworked the book to reflect a new generation. The narrative is more lively and less textbook-like. Remarks and asides link the topics presented to the real world of students' experience. The chapter on complex numbers and the discussion of formal symbolic logic are gone in favor of more exercises, and a new introductory chapter on the nature of mathematics--one that motivates readers and sets the stage for the challenges that lie ahead. 

Students crossing the bridge from calculus to higher mathematics need and deserve all the help they can get. Sets, Functions, and Logic, Third Edition is an affordable little book that all of your transition-course students not only can afford, but will actually read…and enjoy…and learn from.

Mathematics and Logic

Mark Kac

Dover Publications | 1992 | 192 páginas

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Fascinating study considers the origins and nature of mathematics, its development and role in the history of scientific thinking, impact of high-speed computers, 20th-century changes in the foundations of mathematics and mathematical logic, mathematization of science and technology, much more. Compelling reading for anyone interested in the evolution of mathematical thought. Includes 34 illustrations. 1968 edition.

Intentional Mathematics

 (Studies in Logic and the Foundations of Mathematics)

Stewart Shapiro

Elsevier Science Ltd | 1985 | djvu | 1,38 Mb

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Platonism and intuitionism are rival philosophies of Mathematics, the former holding that the subject matter of mathematics consists of abstract objects whose existence is independent of the mathematician, the latter that the subject matter consists of mental construction...both views are implicitly opposed to materialistic accounts of mathematics which take the subject matter of mathematics to consist (in a direct way) of material objects...'' FROM THE INTRODUCTION Among the aims of this book are: - The discussion of some important philosophical issues using the precision of mathematics. - The development of formal systems that contain both classical and constructive components. This allows the study of constructivity in otherwise classical contexts and represents the formalization of important intensional aspects of mathematical practice. - The direct formalization of intensional concepts (such as computability) in a mixed constructive/classical context.

The Oxford Handbook of Philosophy of Mathematics and Logic

Stewart Shapiro

O U P | 2005 | 856 Páginas | PDF | 3 MB
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Mathematics and logic have been central topics of concern since the dawn of philosophy. Since logic is the study of correct reasoning, it is a fundamental branch of epistemology and a priority in any philosophical system. Philosophers have focused on mathematics as a case study for general philosophical issues and for its role in overall knowledge- gathering. Today, philosophy of mathematics and logic remain central disciplines in contemporary philosophy, as evidenced by the regular appearance of articles on these topics in the best mainstream philosophical journals; in fact, the last decade has seen an explosion of scholarly work in these areas.

This volume covers these disciplines in a comprehensive and accessible manner, giving the reader an overview of the major problems, positions, and battle lines. The 26 contributed chapters are by established experts in the field, and their articles contain both exposition and criticism as well as substantial development of their own positions. The essays, which are substantially self-contained, serve both to introduce the reader to the subject and to engage in it at its frontiers. Certain major positions are represented by two chapters--one supportive and one critical.

The Oxford Handbook of Philosophy of Math and Logic is a ground-breaking reference like no other in its field. It is a central resource to those wishing to learn about the philosophy of mathematics and the philosophy of logic, or some aspect thereof, and to those who actively engage in the discipline, from advanced undergraduates to professional philosophers, mathematicians, and historians. 

Logical Labyrinths

Raymond Smullyan

AK Peters | 2009 |  275 páginas | PDF | 3,5 Mb

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This book features a unique approach to the teaching of mathematical logic by putting it in the context of the puzzles and paradoxes of common language and rational thought. It serves as a bridge from the author's puzzle books to his technical writing in the fascinating field of mathematical logic. Using the logic of lying and truth-telling, the author introduces the readers to informal reasoning preparing them for the formal study of symbolic logic, from propositional logic to first-order logic, a subject that has many important applications in philosophy, mathematics, and computer science. The book includes a journey through the amazing labyrinths of infinity, which have stirred the imagination of mankind as much, if not more, than any other subject. As much as a textbook for undergraduate courses in logic, in particular to a liberal- arts audience, this book will succeed as a trade book for anyone who has an interest in a more rigorous understanding of rational thought.

Logic For Dummies

Mark Zegarelli

For Dummies | 2006 | 388 páginas | PDF

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Logic concepts are more mainstream than you may realize. There's logic every place you look and in almost everything you do, from deciding which shirt to buy to asking your boss for a raise, and even to watching television, where themes of such shows as CSI and Numbers incorporate a variety of logistical studies. Logic For Dummies explains a vast array of logical concepts and processes in easy-to-understand language that make everything clear to you, whether you're a college student of a student of life. You'll find out about:

* Formal Logic
* Syllogisms
* Constructing proofs and refutations
* Propositional and predicate logic
* Modal and fuzzy logic
* Symbolic logic
* Deductive and inductive reasoning

Logic For Dummies tracks an introductory logic course at the college level. Concrete, real-world examples help you understand each concept you encounter, while fully worked out proofs and fun logic problems encourage you students to apply what you've learned.

Essays in the Philosophy and History of Logic and Mathematics

(Poznan Studies in the Philosophy of the Sciences & the Humanities)

Roman Murawski

Rodopi | 2010 | 344 páginas | PDF | 1,1 Mb

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The book is a collection of the author's selected works in the philosophy and history of logic and mathematics. Papers in Part I include both general surveys of contemporary philosophy of mathematics as well as studies devoted to specialized topics, like Cantor's philosophy of set theory, the Church thesis and its epistemological status, the history of the philosophical background of the concept of number, the structuralist epistemology of mathematics and the phenomenological philosophy of mathematics. Part II contains essays in the history of logic and mathematics. They address such issues as the philosophical background of the development of symbolism in mathematical logic, Giuseppe Peano and his role in the creation of contemporary logical symbolism, Emil L. Post's works in mathematical logic and recursion theory, the formalist school in the foundations of mathematics and the algebra of logic in England in the 19th century. The history of mathematics and logic in Poland is also considered. This volume is of interest to historians and philosophers of science and mathematics as well as to logicians and mathematicians interested in the philosophy and history of their fields.