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

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

The Unimaginable Mathematics of Borges' Library of Babel

William Goldbloom Bloch 

Oxford University Press | 2011 | 213 páginas | pdf | 1 Mb

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Written in the vein of Douglas R. Hofstadter's Pulitzer Prize-winning Gödel, Escher, Bach, this original and imaginative book sheds light on one of Borges' most complex, richly layered works. Bloch begins each chapter with a mathematical idea--combinatorics, topology, geometry, information theory--followed by examples and illustrations that put flesh on the theoretical bones. In this way, he provides many fascinating insights intoBorges' Library. He explains, for instance, a straightforward way to calculate how many books are in the Library--an easily notated but literally unimaginable number--and also shows that, if each book were the size of a grain of sand, the entire universe could only hold a fraction of the books in the Library. Indeed, if each book were the size of a proton, our universe would still not be big enough to hold anywhere near all the books.
Given Borges' well-known affection for mathematics, this exploration of the story through the eyes of a humanistic mathematician makes a unique and important contribution to the body of Borgesian criticism. Bloch not only illuminates one of the great short stories of modern literature but also exposes the reader--including those more inclined to the literary world--to many intriguing and entrancing mathematical ideas.
Written in the vein of Douglas R. Hofstadter's Pulitzer Prize-winning Gödel, Escher, Bach, this original and imaginative book sheds light on one of Borges' most complex, richly layered works. Bloch begins each chapter with a mathematical idea--combinatorics, topology, geometry, information theory--followed by examples and illustrations that put flesh on the theoretical bones. In this way, he provides many fascinating insights intoBorges' Library. He explains, for instance, a straightforward way to calculate how many books are in the Library--an easily notated but literally unimaginable number--and also shows that, if each book were the size of a grain of sand, the entire universe could only hold a fraction of the books in the Library. Indeed, if each book were the size of a proton, our universe would still not be big enough to hold anywhere near all the books.
Given Borges' well-known affection for mathematics, this exploration of the story through the eyes of a humanistic mathematician makes a unique and important contribution to the body of Borgesian criticism. Bloch not only illuminates one of the great short stories of modern literature but also exposes the reader--including those more inclined to the literary world--to many intriguing and entrancing mathematical ideas.
Given Borges' well-known affection for mathematics, this exploration of the story through the eyes of a humanistic mathematician makes a unique and important contribution to the body of Borgesian criticism. Bloch not only illuminates one of the great short stories of modern literature but also exposes the reader--including those more inclined to the literary world--to many intriguing and entrancing mathematical ideas.
"The Library of Babel" is arguably Jorge Luis Borges' best known story--memorialized along with Borges on an Argentine postage stamp. Now, inThe Unimaginable Mathematics of Borges' Library of Babel, William Goldbloom Bloch takes readers on a fascinating tour of the mathematical ideas hidden within one of the classic works of modern literature.

Contents
Acknowledgments vii
Preface xi
Introduction xvii
The Library of Babel 3
Chapter 1 Combinatorics: Contemplating Variations of the 23 Letters 11
Chapter 2 Information Theory: Cataloging the Collection 30
Chapter 3 Real Analysis: The Book of Sand 45
Chapter 4 Topology and Cosmology: The Universe (Which Others Call the Library) 57
Chapter 5 Geometry and Graph Theory: Ambiguity and Access 93
Chapter 6 More Combinatorics: Disorderings into Order 107
Chapter 7 A Homomorphism: Structure into Meaning 120
Chapter 8 Critical Points 126
Chapter 9 Openings 141
Appendix—Dissecting the 3-Sphere 148
Notations 157
Notes 159
Glossary 165
Annotated Suggested Readings 175
Bibliography 181
Index 187

The Best Writing on Mathematics 2013

 

Mircea Pitici 

Princeton University Press | 2014 | 273 páginas | rar - pdf | 3,65 Mb

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This annual anthology brings together the year's finest mathematics writing from around the world. Featuring promising new voices alongside some of the foremost names in the field,The Best Writing on Mathematics 2013 makes available to a wide audience many articles not easily found anywhere else--and you don't need to be a mathematician to enjoy them. These writings offer surprising insights into the nature, meaning, and practice of mathematics today. They delve into the history, philosophy, teaching, and everyday occurrences of math, and take readers behind the scenes of today's hottest mathematical debates. Here Philip Davis offers a panoramic view of mathematics in contemporary society; Terence Tao discusses aspects of universal mathematical laws in complex systems; Ian Stewart explains how in mathematics everything arises out of nothing; Erin Maloney and Sian Beilock consider the mathematical anxiety experienced by many students and suggest effective remedies; Elie Ayache argues that exchange prices reached in open market transactions transcend the common notion of probability; and much, much more.

Contents
Foreword
Roger Penrose ix
Introduction
Mircea Pitici xv
The Prospects for Mathematics in a Multimedia Civilization
Philip J. Davis 1
Fearful Symmetry
Ian Stewart 23
E pluribus unum: From Complexity, Universality
Terence Tao 32
Degrees of Separation
Gregory Goth 47
Randomness
Charles Seife 52
Randomness in Music
Donald E. Knuth 56
Playing the Odds
Soren Johnson 62
Machines of the Infinite
John Pavlus 67
Bridges, String Art, and Bézier Curves
Renan Gross 77
Slicing a Cone for Art and Science
Daniel S. Silver 90
High Fashion Meets Higher Mathematics
Kelly Delp 109
The Jordan Curve Theorem Is Nontrivial
Fiona Ross and William T. Ross 120
Why Mathematics? What Mathematics?
Anna Sfard 130
Math Anxiety: Who Has It, Why It Develops, and How to Guard against It
Erin A. Maloney and Sian L. Beilock 143
How Old Are the Platonic Solids?
David R. Lloyd 149
Early Modern Mathematical Instruments
Jim Bennett 163
A Revolution in Mathematics? What Really Happened a Century Ago and Why It Matters Today
Frank Quinn 175
Errors of Probability in Historical Context
Prakash Gorroochurn 191
The End of Probability
Elie Ayache 213
An abc Proof Too Tough Even for Mathematicians
Kevin Hartnett 225
Contributors 231
Notable Texts 237
Acknowledgments 241
Credits 243

Outros livros da mesma coleção:

The best writing on mathematics 2011 por Mircea Pitici; Idioma: Inglês   Editora: Princeton : Princeton University Press, ©2012.

The best writing on mathematics 2010 por Mircea Pitici; Idioma: Inglês   Editora: Princeton : Princeton University Press, ©2011.

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

Philosophy of Mathematics in the Twentieth Century: Selected Essays

Charles Parsons

Harvard University Press | 2014 | 365 páginas | rar - pdf |950 kb

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In this illuminating collection, Charles Parsons surveys the contributions of philosophers and mathematicians who shaped the philosophy of mathematics over the course of the past century.
Parsons begins with a discussion of the Kantian legacy in the work of L. E. J. Brouwer, David Hilbert, and Paul Bernays, shedding light on how Bernays revised his philosophy after his collaboration with Hilbert. He considers Hermann Weyl's idea of a "vicious circle" in the foundations of mathematics, a radical claim that elicited many challenges. Turning to Kurt Gödel, whose incompleteness theorem transformed debate on the foundations of mathematics and brought mathematical logic to maturity, Parsons discusses his essay on Bertrand Russell's mathematical logic--Gödel's first mature philosophical statement and an avowal of his Platonistic view.
Philosophy of Mathematics in the Twentieth Century insightfully treats the contributions of figures the author knew personally: W. V. Quine, Hilary Putnam, Hao Wang, and William Tait. Quine's early work on ontology is explored, as is his nominalistic view of predication and his use of the genetic method of explanation in the late work The Roots of Reference. Parsons attempts to tease out Putnam's views on existence and ontology, especially in relation to logic and mathematics. Wang's contributions to subjects ranging from the concept of set, minds, and machines to the interpretation of Gödel are examined, as are Tait's axiomatic conception of mathematics, his minimalist realism, and his thoughts on historical figures

Contents
Preface
Introduction
Part I: Some Mathematicians as Philosophers
1. The Kantian Legacy in Twentieth-Century Foundations of Mathematics
2. Realism and the Debate on Impredicativity, 1917–1944 
Postscript to Essay 2
3. Paul Bernays’ Later Philosophy of Mathematics
4. Kurt Gödel
5. Gödel’s “Russell’s Mathematical Logic” ~
Postscript to Essay 5
6. Quine and Gödel on Analyticity
Postscript to Essay 6
7. Platonism and Mathematical Intuition in Kurt Gödel’s Thought
Postscript to Essay 7
Part II: Contemporaries
8. Quine’s Nominalism
9. Genetic Explanation in The Roots of Reference
10. Hao Wang as Philosopher and Interpreter of Gödel
11. Putnam on Existence and Ontology
12. William Tait’s Philosophy of Mathematics
Bibliography
Copyright Acknowledgments
Index

A Mathematician's Apology

University of Alberta Mathematical Sciences Society | 2005 | 56 páginas | pdf | 174 kb

online: math.ualberta.ca

Cambridge University Press | 1967 | 80 páginas

online: archive.org

G. H. Hardy was one of this century's finest mathematical thinkers, renowned among his contemporaries as a 'real mathematician ... the purest of the pure'. He was also, as C. P. Snow recounts in his Foreword, 'unorthodox, eccentric, radical, ready to talk about anything'. This 'apology', written in 1940 as his mathematical powers were declining, offers a brilliant and engaging account of mathematics as very much more than a science; when it was first published, Graham Greene hailed it alongside Henry James's notebooks as 'the best account of what it was like to be a creative artist'. C. P. Snow's Foreword gives sympathetic and witty insights into Hardy's life, with its rich store of anecdotes concerning his collaboration with the brilliant Indian mathematician Ramanujan, his aphorisms and idiosyncrasies, and his passion for cricket. This is a unique account of the fascination of mathematics and of one of its most compelling exponents in modern times

Proof and Knowledge in Mathematics

Michael Detlefsen
Routledge | 1992 | 256 páginas | 

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These questions arise from any attempt to discover an epistemology for mathematics. This collection of essays considers various questions concerning the nature of justification in mathematics and possible sources of that justification. Among these are the question of whether mathematical justification is a priori or a posteriori in character, whether logical and mathematical differ, and if formalization plays a significant role in mathematical justification.

CONTENTS
1 PROOF AS A SOURCE OF TRUTH
Michael D.Resnik
2 REFLECTIONS ON THE CONCEPT OF A PRIORI TRUTH AND ITS CORRUPTION BY KANT
William W.Tait
3 LOGICISM
Steven J.Wagner
4 EMPIRICAL INQUIRY AND PROOF
Shelley Stillwell
5 ON THE CONCEPT OF PROOF IN ELEMENTARY GEOMETRY
Pirmin Stekeler-Weithofer
6 MATHEMATICAL RIGOR IN PHYSICS
Mark Steiner
7 FOUNDATIONALISM AND FOUNDATIONS OF MATHEMATICS
Stewart Shapiro
8 BROUWERIAN INTUITIONISM
Michael Detlefsen

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

Infinity and the Mind: The Science and Philosophy of the Infinite

 Rudy Rucker

Princeton University Press | 2005 | 368 páginas | rar - epub | 8,4 Mb

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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.

Contents

Preface to the 2005 Edition ix

Preface to the Paperback Edition xvii

Preface xix

Chapter One: Infinity 1

A Short History of Infinity
Physical Infinities; 
Temporal Infinities; Spatial Infinities; Infinities in the Small; Conclusion

Infinities in the Mindscape 35

The Absolute Infinite 44

Connections 49

Puzzles and Paradoxes 51
Chapter Two: All the Numbers 53
From Pythagoreanism to Cantorism 53
Transfinite Numbers 64
From Omega to Epsilon-Zero; The Alefs
Infinitesimals and Surreal Numbers 78
Higher Physical Infinities 87
Puzzles and Paradoxes 91
Chapter Three: The Unnameable 93
The Berry Paradox 93
Naming Numbers; Understanding Names
Random Reals 107
Constructing Reals; The Library of Babel ; Richard’s Paradox; Coding the World 
What is Truth? 143
Conclusion 152
Puzzles and Paradoxes 155
Chapter Four: Robots and Souls 157
Gödel’s Incompleteness Theorem 157
Conversations with Gödel 164
Towards Robot Consciousness171
Formal Systems and Machines;  The Liar Paradox and the Non-Mechanizability of Mathematics; Artificial Intelligence via Evolutionary Processes; Robot Consciousness
Beyond Mechanism?185
Puzzles and Paradoxes187
Chapter Five: The One and the Many189
The Classical One/Many Problem189
What is a Set?191
The Universe of Set Theory196
Pure Sets and the Physical Universe; Proper Classes and Metaphysical Absolutes
Interface Enlightenment206
One/Many in Logic and Set Theory; Mysticism and Rationality; Satori
Puzzles and Paradoxes219
Excursion One: The Transfinite Cardinals 221
On and Alef-One 221
Cardinality 226
The Continuum 238
Large Cardinals 253
Excursion Two: Gödel’s Incompleteness Theorems 267
Formal Systems 267
Self-Reference 280
Gödel’s Proof 285
A Technical Note on Man-Machine Equivalence 292
Answers to the Puzzles and Paradoxes 295
Notes 307
Bibliography 329

Pluralism in Mathematics: A New Position in Philosophy of Mathematics

Michèle Friend

Springer | 2014 | 297 páginas | rar - pdf | 1,8 Mb

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This book is about philosophy, mathematics and logic, giving a philosophical account of Pluralism which is a family of positions in the philosophy of mathematics. There are four parts to this book, beginning with a look at motivations for Pluralism by way of Realism, Maddy’s Naturalism, Shapiro’s Structuralism and Formalism.

In the second part of this book the author covers: the philosophical presentation of Pluralism; using a formal theory of logic metaphorically; rigour and proof for the Pluralist; and mathematical fixtures. In the third part the author goes on to focus on the transcendental presentation of Pluralism, and in part four looks at applications of Pluralism, such as a Pluralist approach to proof in mathematics and how Pluralism works in regard to together-inconsistent philosophies of mathematics. The book finishes with suggestions for further Pluralist enquiry.

In this work the author takes a deeply radical approach in developing a new position that will either convert readers, or act as a strong warning to treat the word ‘pluralism’ with care.  

Contents
Introduction
Part I. Motivating the Pluralist Position from Familiar Positions
Chapter 1. Introduction. The Journey from Realism to Pluralism
Chapter 2. Motivating Pluralism. Starting from Maddy?s Naturalism
Chapter 3. From Structuralism to Pluralism
Chapter 4. Formalism and Pluralism Co-written with Andrea Pedeferri
Part II. Initial Presentation of Pluralism.-?Chapter 5. Philosophical Presentation of Pluralism
Chapter 6. Using a Formal Theory of Logic Metaphorically
Chapter 7. Rigour in Proof Co-written with Andrea Pedeferri
Chapter 8. Mathematical Fixtures
Part III. Transcendental Presentation of Pluralism
Chapter 9. The Paradoxes of Tolerance and the Transcendental Paradoxes
Chapter 10. Pluralism Towards Pluralism
Part IV. Putting Pluralism to Work. Applications
Chapter 11. A Pluralist Approach to Proof in Mathematics
Chapter 12. Pluralism and Together-Inconsistent Philosophies of Mathematics
Chapter 13. Suggestions for Further Pluralist Enquiry
Conclusion

Deleuze and the History of Mathematics: In Defense of the 'New'

(Continuum Studies in Continental Philosophy)

 Simon Duffy 

Bloomsbury Academic | 2013 | 225 páginas | rar -pdf | 968 kb

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Gilles Deleuze's engagements with mathematics, replete in his work, rely upon the construction of alternative lineages in the history of mathematics, which challenge some of the self imposed limits that regulate the canonical concepts of the discipline. For Deleuze, these challenges are an opportunity to reconfigure particular philosophical problems - for example, the problem of individuation - and to develop new concepts in response to them. The highly original research presented in this book explores the mathematical construction of Deleuze's philosophy, as well as addressing the undervalued and often neglected question of the mathematical thinkers who influenced his work. 

In the wake of Alain Badiou's recent and seemingly devastating attack on the way the relation between mathematics and philosophy is configured in Deleuze's work, Simon Duffy offers a robust defence of the structure of Deleuze's philosophy and, in particular, the adequacy of the mathematical problems used in its construction. By reconciling Badiou and Deleuze's seeming incompatible engagements with mathematics, Duffy succeeds in presenting a solid foundation for Deleuze's philosophy, rebuffing the recent challenges against it.

Contents
Acknowledgments x
List of Abbreviations xii
Introduction 1
1 Leibniz and the Concept of the Infinitesimal 7
2 Maimon’s Critique of Kant’s Approach to Mathematics 47
3 Bergson and Riemann on Qualitative Multiplicity 89
4 Lautman’s Concept of the Mathematical Real 117
5 Badiou and Contemporary Mathematics 137
Conclusion 161
Notes 175
Bibliography 189
Index 201

The Emergence of Probability: A Philosophical Study of Early Ideas about Probability, Induction and Statistical Inference

(Cambridge Series on Statistical & Probabilistic Mathematics)

Ian Hacking | 2006 | 246 páginas | DjVu (3 mb)

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Historical records show that there was no real concept of probability in Europe before the mid-seventeenth century, although the use of dice and other randomizing objects was commonplace. Ian Hacking presents a philosophical critique of early ideas about probability, induction, and statistical inference and the growth of this new family of ideas in the fifteenth, sixteenth, and seventeenth centuries. Hacking invokes a wide intellectual framework involving the growth of science, economics, and the theology of the period. He argues that the transformations that made it possible for probability concepts to emerge have constrained all subsequent development of probability theory and determine the space within which philosophical debate on the subject is still conducted. First published in 1975, this edition includes an introduction that contextualizes his book in light of developing philosophical trends. Ian Hacking is the winner of the Holberg International Memorial Prize 2009.

CONTENTS
Introduction 2006 page xi
1 An absent family of ideas 1
Although dicing is one of the oldest of human pastimes, there is no known
mathematics of randomness until the Renaissance. None of the explanations
of this fact is compelling.
2 Duality 11
Probability, as we now conceive it, came into being about 1660. It was
essentially dual, on the one hand having to do with degrees of belief, on the
other, with devices tending to produce stable long-run frequencies.
3 Opinion 18
In the Renaissance, what was then called 'probability' was an attribute of
opinion, in contrast to knowledge, which could only be obtained by
demonstration. A probable opinion was not one supported by evidence, but
one which was approved by some authority, or by the testimony of respected
judges.
4 Evidence 31
Until the end of the Renaissance, one of our concepts of evidence was lacking:
that by which one thing can indicate, contingently, the state of something else.
Demonstration, versimilitude and testimony were all familiar concepts, but
not this further idea of the inductive evidence of things,
5 Signs 39
Probability is a child of the low sciences, such as alchemy or medicine, which
had to deal in opinion, whereas the high sciences, such as astronomy or
mechanics, aimed at demonstrable knowledge. A chief concept of the low
sciences was, that of the sign, here described in some detail. Observation of
signs was conceived of as reading testimony. Signs were more or less reliable.
Thus on the one hand a sign made an opinion probable (in the old sense of
Chapter 3) because it was furnished by the best testimony of all. On the other
hand, signs could be assessed by the frequency with which they spoke truly. At
the end of the Renaissance, the sign was transformed into the concept of
evidence described in Chapter 4. This new kind of evidence conferred
Contents
probability on propositions, namely made them worthy of approval. But it did
so in virtue of the frequency with which it made correct predictions. This
transformation from sign into evidence is the key to the emergence of a
concept of probability that is dual in the sense of Chapter 2.
6 The first calculations 49
Some isolated calculations on chances, made before 1660, are briefly
described.
7 The Roannez circle (1654) 57
Some problems solved by Pascal set probability rolling. From here until
Chapter 17 Leibniz is used as a witness to the early days of probability theory.
8 The great decision (1658?) 63
'Pascal's wager' for acting as if one believed in God is the first well-understood
contribution to decision theory.
9 The art of thinking (1662) 73
Something actually called 'probability' is first measured in the Port Royal
Logic, which is also one of the first works to distinguish evidence, in the sense
of Chapter 4, from testimony. The new awareness of probability, evidence
and conventional (as opposed to natural) sign, is illustrated by work of
Wilkins, first in 1640, before the emergence of probability, and then in 1668,
after the emergence.
10 Probability and the law (1665) 85
While young and ignorant of the Paris developments Leibniz proposed to
measure degrees of proof and right in law on a scale between 0 and 1, subject
to a crude calculation of what he called 'probability'.
11 Expectation (1657) 92
Huygens wrote the first printed textbook of probability using expectation as
the central concept. His justification of this concept is still of interest.
12 Political arithmetic (1662) 102
Graunt drew the first detailed statistical inferences from the bills of mortality
for the city of London, and Petty urged the need for a central statistical office.
13 Annuities (1671) 111
Hudde and de Witt used Dutch annuity records to infer a mortality curve on
which to work out the fair price for an annuity.
14 Equipossibility (1678) 122
The definition of probability as a ratio among 'equally possible cases1
originates with Leibniz. The definition, unintelligible to us, was natural at the
time, for possibility was either de re (about things) or de dicto (about
propositions). Probability was likewise either about things, in the frequency
sense, or about propositions, in the epistemic sense. Thus the duality of
probability was preserved by the duality of possibility.
15 Inductive logic 134
Leibniz anticipated Carnap's notion of inductive logic. He could do so
because of the central place occupied by the concept of possibility in his
scheme of metaphysics. Within that scheme, a global system of inductive logic
makes more sense than Carnap's does in our modern metaphysics.
16 The art of conjecturing (1692[?] published 1713) 143
The emergence of probability is completed with Jacques Bernoulli's book,
which both undertakes a self-conscious analysis of the concept of probability,
and proves the first limit theorem.
17 The first limit theorem 154
The possible interpretations of Bernoulli's theorem are described.
18 Design 166
The English conception of probability in the early eighteenth century, guided
by the Newtonian philosophy espoused by members of the Royal Society,
interprets the stability of stochastic processes proven by the limit theorems as
evidence of divine design.
19 Induction (1737) 176
Hume's sceptical problem of induction could not have arisen much before
1660, for there was no concept of inductive evidence in terms of which to raise
it. Why did it have to wait until 1737? So long as it was still believed that
demonstrative knowledge was possible, a knowledge in which causes were
proved from first principles, then Hume's argument could always be stopped.
It was necessary that the distinction between opinion and knowledge should
become a matter of degree. That means that high and low science had to
collapse into one another. This had been an ongoing process throughout the
seventeenth century. It was formalized by Berkeley who said that all causes
were merely signs. Causes had been the prerogative of high science, and signs
the tool of the low. Berkeley identified them and Hume thereby became
possible.
Bibliography 186
Index 203

The Best Wrinting on Mathematics 2011

Mircea Pitici e Freeman Dyson

Princeton University Press | 2011 | 412 páginas | djvu | 9,7 Mb

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This anthology brings together the year's finest mathematics writing from around the world. Featuring promising new voices alongside some of the foremost names in the field, The Best Writing on Mathematics 2011 makes available to a wide audience many articles not easily found anywhere else--and you don't need to be a mathematician to enjoy them. These writings offer surprising insights into the nature, meaning, and practice of mathematics today. They delve into the history, philosophy, teaching, and everyday occurrences of math, and take readers behind the scenes of today's hottest mathematical debates. Here Ian Hacking discusses the salient features that distinguish mathematics from other disciplines of the mind; Doris Schattschneider identifies some of the mathematical inspirations of M. C. Escher's art; Jordan Ellenberg describes compressed sensing, a mathematical field that is reshaping the way people use large sets of data; Erica Klarreich reports on the use of algorithms in the job market for doctors; and much, much more.

In addition to presenting the year's most memorable writings on mathematics, this must-have anthology includes a foreword by esteemed physicist and mathematician Freeman Dyson. This book belongs on the shelf of anyone interested in where math has taken us--and where it is headed.

Contents

Foreword: Recreational Mathematics

Freeman Dyson xi

Introduction

Mircea Pitici xvii

What Is Mathematics For?

Underwood Dudley 1

A Tisket, a Tasket, an Apollonian Gasket

Dana Mackenzie 13

The Quest for God’s Number

Rik van Grol 27

Meta-morphism: From Graduate Student to Networked Mathematician

Andrew Schultz 35

One, Two, Many: Individuality and Collectivity in Mathematics

Melvyn B. Nathanson 43

Reflections on the Decline of Mathematical Tables

Martin Campbell-Kelly 51

Under-Represented Then Over-Represented: A Memoir of Jews in American Mathematics

Reuben Hersh 55

Did Over-Reliance on Mathematical Models for Risk Assessment  Create the Financial Crisis?

David J. Hand 67

Fill in the Blanks: Using Math to Turn Lo-Res Datasets into Hi-Res Samples

Jordan Ellenberg 75

The Great Principles of Computing

Peter J. Denning 82

Computer Generation of Ribbed Sculptures

James Hamlin and Carlo H. Séquin 93

Lorenz System Offers Manifold Possibilities for Art

Barry A. Cipra 115

The Mathematical Side of M. C. Escher

Doris Schattschneider 121

Celebrating Mathematics in Stone and Bronze

Helaman Ferguson and Claire Ferguson 150

Mathematics Education: Theory, Practice, and Memories over 50 Years

John Mason 169

Thinking and Comprehending in the Mathematics Classroom

Douglas Fisher, Nancy Frey, and 

Heather Anderson 188

Teaching Research: Encouraging Discoveries

Francis Edward Su 203

Reflections of an Accidental Theorist

Alan H. Schoenfeld 219

The Conjoint Origin of Proof and Theoretical Physics

Hans Niels Jahnke 236

What Makes Mathematics Mathematics?

Ian Hacking 257

What Anti-realism in Philosophy of Mathematics Must Offer

Feng Ye 286

Seeing Numbers

Ivan M. Havel 312

Autism and Mathematical Talent

Ioan James 330

How Much Math is Too Much Math?

Chris J. Budd and Rob Eastaway 336

Hidden Dimensions

Marianne Freiberger 347

Playing with Matches

Erica Klarreich 356

Notable Texts 367

Contributors 371

Acknowledgments 379

Credits 381

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