The Best Writing on Mathematics 2010

Mircea Pitici e William P. Thurston


Princeton University Press | 2011 | 434 páginas | pdf | 6,5 Mb

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This anthology brings together the year's finest writing on mathematics from around the world. Featuring promising new voices alongside some of the foremost names in mathematics, The Best Writing on Mathematics 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 readers will discover why Freeman Dyson thinks some mathematicians are birds while others are frogs; why Keith Devlin believes there's more to mathematics than proof; what Nick Paumgarten has to say about the timing patterns of New York City's traffic lights (and why jaywalking is the most mathematically efficient way to cross Sixty-sixth Street); what Samuel Arbesman can tell us about the epidemiology of the undead in zombie flicks; and much, much more.

In addition to presenting the year's most memorable writing on mathematics, this must-have anthology also includes a foreword by esteemed mathematician William Thurston and an informative introduction by Mircea Pitici. This book belongs on the shelf of anyone interested in where math has taken us--and where it's headed.

TABLE OF CONTENTS:
Foreword by William P. Thurston xiIntroduction by Mircea Pitici xv
Mathematics Alive

The Role of the Untrue in Mathematics by Chandler Davis 3

Desperately Seeking Mathematical Proof by Melvyn B. Nathanson 13

An Enduring Error by Branko Grunbaum 18

What Is Experimental Mathematics? By Keith Devlin 32

What Is Information-Based Complexity? By Henryk Woz´niakowski 37

What Is Financial Mathematics? By Tim Johnson 43

If Mathematics Is a Language, How Do You Swear in It? By David Wagner 47
Mathematicians and the Practice of Mathematics

Birds and Frogs by Freeman Dyson 57

Mathematics Is Not a Game But . . . by Robert Thomas 79

Massively Collaborative Mathematics by Timothy Gowers and Michael Nielsen 89

Bridging the Two Cultures: Paul Valery by Philip J. Davis 94

A Hidden Praise of Mathematics by Alicia Dickenstein 99
Mathematics and Its Applications

Mathematics and the Internet: A Source of Enormous Confusion and Great Potential by Walter Willinger, David L. Alderson, and John C. Doyle 109

The Higher Arithmetic: How to Count to a Zillion without Falling Off the End of the Number Line by Brian Hayes 134

Knowing When to Stop: How to Gamble If You Must--The Mathematics of Optimal Stopping by Theodore P. Hill 145

Homology: An Idea Whose Time Has Come by Barry A. Cipra 158
Mathematics Education

Adolescent Learning and Secondary Mathematics by Anne Watson 163

Accommodations of Learning Disabilities in Mathematics Courses by Kathleen Ambruso Acker, Mary W. Gray, and Behzad Jalali 175

Audience,Style and Criticism by David Pimm and Nathalie Sinclair 194

Aesthetics as a Liberating Force in Mathematics Education? By Nathalie Sinclair 206

Mathematics Textbooks and Their Potential Role in Supporting Misconceptions by Ann Kajander and Miroslav Lovric 236

Exploring Curvature with Paper Models by Howard T. Iseri 247

Intuitive vs Analytical Thinking: Four Perspectives by Uri Leron and Orit Hazzan 260
History and Philosophy of Mathematics

Why Did Lagrange "Prove" the Parallel Postulate? By Judith V. Grabiner 283

Kronecker's Algorithmic Mathematics by Harold M. Edwards 303

Indiscrete Variations on Gian-Carlo Rota's Themes by Carlo Cellucci 311

Circle Packing: A Personal Reminiscence by Philip L. Bowers 330Applying Inconsistent Mathematics by Mark Colyvan 346

Why Do We Believe Theorems? By Andrzej Pelc 358
Mathematics in the Media

Mathematicians Solve 45-Year-Old Kervaire Invariant Puzzle by Erica Klarreich 373

Darwin: The Reluctant Mathematician by Julie Rehmeyer 377

Loves Me, Loves Me Not (Do the Math) by Steven Strogatz 380

The Mysterious Equilibrium of Zombies and Other Things Mathematicians See at the Movies by Samuel Arbesman 383

Strength in Numbers: On Mathematics and Musical Rhythm by Vijay Iyer 387

Math-hattan by Nick Paumgarten 391
Contributors 395

Acknowledgments 403

Credits 405

Sugestão de tibu

Letters to a Young Mathematician

Ian Stewart

Basic Books | 2006 | 224 páginas | rar - pdf | 469 kb

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Descrição: From the renowned mathematician and writer Ian Stewart, an insightful and lively exploration of why mathematics matters The first scientific entry in the acclaimed Art of Mentoring series from Basic Books, Letters to a Young Mathematician tells readers what Ian Stewart wishes he had known when he was a student and young faculty member. Subjects ranging from the philosophical to the practical--what mathematics is and why it's worth doing, the relationship between logic and proof, the role of beauty in mathematical thinking, the future of mathematics, how to deal with the peculiarities of the mathematical community, and many others--are dealt with in Stewart's much-admired style, which combines subtle, easygoing humor with a talent for cutting to the heart of the matter. In the tradition of G.H. Hardy's classic A Mathematician's Apology, this book is sure to be a perennial favorite with students at all levels, as well as with other readers who are curious about the frequently incomprehensible world of mathematics.

Contents
Preface ix
1 Why Do Math? 1
2 How I Almost Became a Lawyer 11
3 The Breadth of Mathematics 18
4 Hasn’t It All Been Done? 33
5 Surrounded by Math 45
6 How Mathematicians Think 53
7 How to Learn Math 62
8 Fear of Proofs 71
9 Can’t Computers Solve Everything? 82
10 Mathematical Storytelling 87
11 Going for the Jugular 95
12 Blockbusters 103
13 Impossible Problems 110
14 The Career Ladder 122
15 Pure or Applied? 131
16 Where Do You Get Those Crazy Ideas? 147
17 How to Teach Math 157
18 The Mathematical Community 168
19 Pigs and Pickup Trucks 178
20 Pleasures and Perils of Collaboration 188
21 Is God a Mathematician? 196

The Mathematical Experience, Study Edition

Philip J. Davis, Reuben Hersh

 Birkhäuser; 1st Reprint of the 1995 Edition ed. 2012. Updated with Epilogues by the Authors edition |  2011 | 527 páginas | rar - pdf | 14,1 Mb

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Winner of the 1983 National Book Award!

"…a perfectly marvelous book about the Queen of Sciences, from which one will get a real feeling for what mathematicians do and who they are. The exposition is clear and full of wit and humor..." - The New Yorker (1983 National Book Award edition)

Mathematics has been a human activity for thousands of years. Yet only a few people from the vast population of users are professional mathematicians, who create, teach, foster, and apply it in a variety of situations. The authors of this book believe that it should be possible for these professional mathematicians to explain to non-professionals what they do, what they say they are doing, and why the world should support them at it. They also believe that mathematics should be taught to non-mathematics majors in such a way as to instill an appreciation of the power and beauty of mathematics. Many people from around the world have told the authors that they have done precisely that with the first edition and they have encouraged publication of this revised edition complete with exercises for helping students to demonstrate their understanding. This edition of the book should find a new generation of general readers and students who would like to know what mathematics is all about. It will prove invaluable as a course text for a general mathematics appreciation course, one in which the student can combine an appreciation for the esthetics with some satisfying and revealing applications.

The text is ideal for 1) a GE course for Liberal Arts students 2) a Capstone course for perspective teachers 3) a writing course for mathematics teachers. A wealth of customizable online course materials for the book can be obtained from Elena Anne Marchisotto (elena.marchisotto@csun.edu) upon request.

Arithmetic and Ontology. A Non-Realist Philosophy of Arithmetic

Charles Sayward, Philip Hugley

Editions Rodopi | 2006 |  393 páginas | DJVU | 1,21 Mb

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This volume documents a lively exchange between five philosophers of mathematics. It also introduces a new voice in one central debate in the philosophy of mathematics. Non-realism, i.e., the view supported by Hugly and Sayward in their monograph, is an original position distinct from the widely known realism and anti-realism. Non-realism is characterized by the rejection of a central assumption shared by many realists and anti-realists, i.e., the assumption that mathematical statements purport to refer to objects. The defense of their main argument for the thesis that arithmetic lacks ontology brings the authors to discuss also the controversial contrast between pure and empirical arithmetical discourse. Colin Cheyne, Sanford Shieh, and Jean Paul Van Bendegem, each coming from a different perspective, test the genuine originality of non-realism and raise objections to it. Novel interpretations of well-known arguments, e.g., the indispensability argument, and historical views, e.g. Frege, are interwoven with the development of the authors’ account. The discussion of the often neglected views of Wittgenstein and Prior provide an interesting and much needed contribution to the current debate in the philosophy of mathematics. Contents Acknowledgments Editor’s Introduction Philip HUGLY and Charles SAYWARD: Arithmetic and Ontology a Non-Realist Philosophy of Arithmetic Preface Analytical 

Table of Contents 
Chapter 1. Introduction Part One: Beginning with Frege 
Chapter 2. Notes to Grundlagen 
Chapter 3. Objectivism and Realism in Frege’s Philosophy of Arithmetic 
Part Two: Arithmetic and Non-Realism 
Chapter 4. The Peano Axioms 
Chapter 5. Existence, Number, and Realism 
Part Three: Necessity and Rules Chapter 6. Arithmetic and Necessity 
Chapter 7. Arithmetic and Rules 
Part Four: The Three Theses 
Chapter 8. Thesis One 
Chapter 9. Thesis Two 
Chapter 10. Thesis Three 
References 

The Search for Certainty: A Philosophical Account of Foundations of Mathematics

Marcus Giaquinto

Oxford University Press | 2002  | 330 páginas | PDF | 13 Mb

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"The Search for Certainty is a superb synoptic account of the intense and fruitful work that went into clarifying the foundations of mathematics...It fills the gap in the literature that Giaquinto reports to have noticed when a student of logic, and it does so in an excellent manner." -- The Review of Modern Logic
The nineteenth century saw a movement to make higher mathematics rigorous. This seemed to be on the brink of success when it was thrown into confusion by the discovery of the class paradoxes. That initiated a period of intense research into the foundations of mathematics, and with it the birth of mathematical logic and a new, sharper debate in the philosophy of mathematics.
The Search for Certainty examines this foundational endeavour from the discovery of the paradoxes to the present. Focusing on Russell's logicist programme and Hilbert's finitist programme, Giaquinto investigates how successful they were and how successful they could be. These questions are set in the context of a clear, non-technical exposition and assessment of the most important discoveries in mathematical logic, above all Godel's underivability theorems. More than six decades after those discoveries, Giaquinto asks what our present perspective should be on the question of certainty in mathematics. Taking recent developments into account, he gives reasons for a surprisingly positive response.

Livro do mesmo autor, disponível no blog

Visual Thinking in Mathematics (2007)

An Introduction to the Philosophy of Mathematics

Cambridge Introductions to Philosophy

Mark Colyvan

Cambridge University Press | 2012 | 200 páginas | PDF | 800 kb

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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.


Contents
Acknowledgments page viii
1 Mathematics and its philosophy 1
1.1 Skipping through the big isms 2
1.2 Charting a course to contemporary topics 8
1.3 Planning for the trip 15
2 The limits of mathematics 21
2.1 The Löwenheim–Skolem Theorem 22
2.2 Gödel’s Incompleteness Theorems 27
2.3 Independent questions 30
3 Plato’s heaven 36
3.1 A menagerie of realisms 36
3.2 Indispensability arguments 41
3.3 Objections 46
4 Fiction, metaphor, and partial truths 55
4.1 Fictionalism 55
4.2 An easier route to nominalism? 62
4.3 Mathematics as metaphor 68
5 Mathematical explanation 75
5.1 Theories of explanation 76
5.2 Intra-mathematical explanation 77
5.3 Extra-mathematical explanation 90
6 The applicability of mathematics 98
6.1 The unreasonable effectiveness of mathematics 98
6.2 Towards a philosophy of applied mathematics 104
6.3 What’s maths got to do with it? 109
7 Who’s afraid of inconsistent mathematics? 118
7.1 Introducing inconsistency 118
7.2 Paraconsistent logic 123
7.3 Applying inconsistent mathematics 127
8 A rose by any other name 132
8.1 More than the language of science 133
8.2 Shakespeare’s mistake 140
8.3 Mathematical definitions 145
9 Epilogue: desert island theorems 151
9.1 Philosophers’ favourites 151
9.1.1 Tarski–Banach Theorem (1924) 152
9.1.2 Löwenheim–Skolem Theorem (1922) 152
9.1.3 Gödel’s Incompleteness Theorems (1931) 152
9.1.4 Cantor’s Theorem (1891) 153
9.1.5 Independence of continuum hypothesis (1963) 153
9.1.6 Four-Colour Theorem (1976) 153
9.1.7 Fermat’s Last Theorem (1995) 153
9.1.8 Bayes’s Theorem (1763) 155
9.1.9 Irrationality of √2 (c. 500 BCE) 156
9.1.10 Infinitude of the primes (c. 300 BCE) 157
9.2 The under-appreciated classics 157
9.2.1 Borsuk–Ulam Theorem (1933) 157
9.2.2 Riemann Rearrangement Theorem (1854) 158
9.2.3 Gauss’s Theorema Egregium (1828) 159
9.2.4 Residue Theorem (1831) 160
9.2.5 Poincaré conjecture (2002) 161
9.2.6 Prime Number Theorem (1849) 162
9.2.7 The Fundamental Theorems of Calculus (c. 1675) 163
9.2.8 Lindemann’s Theorem (1882) 164
9.2.9 Fundamental Theorem of Algebra (1816) 164
9.2.10 Fundamental Theorem of Arithmetic (c. 300 BCE) 165
9.3 Some famous open problems 167
9.3.1 Riemann hypothesis 167
9.3.2 The twin prime conjecture 167
9.3.3 Goldbach’s conjecture 168
9.3.4 Infinitude of the Mersenne primes 168
9.3.5 Is there an odd perfect number? 168
9.4 Some interesting numbers 169
Bibliography 173
Index 184

Outro livro do mesmo autor, disponível no blog:

- Colyvan, M. (2001). The Indispensability of Mathematics . Oxford University Press, New York.

Platonism, Naturalism, and Mathematical Knowledge

(Routledge Studies in the Philosophy of Science)

James Robert Brown

Routledge | 2011 | 194 páginas | rar - PDF | 1,2 Mb

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This study addresses a central theme in current philosophy: Platonism vs Naturalism and provides accounts of both approaches to mathematics, crucially discussing Quine, Maddy, Kitcher, Lakoff, Colyvan, and many others. Beginning with accounts of both approaches, Brown defends Platonism by arguing that only a Platonistic approach can account for concept acquisition in a number of special cases in the sciences. He also argues for a particular view of applied mathematics, a view that supports Platonism against Naturalist alternatives. Not only does this engaging book present the Platonist-Naturalist debate over mathematics in a comprehensive fashion, but it also sheds considerable light on non-mathematical aspects of a dispute that is central to contemporary philosophy.

Link para vídeo da conferência de James Brown:

Proofs and Pictures: The Role of Visualization in Mathematical and Scientific Reasoning

pirsa.org

Link para o livro:

Philosophy of Mathematics: An Introduction to a World of Proofs and Pictures

Realism in Mathematics

(Clarendon Paperbacks)

Penelope Maddy

Oxford University Press, USA  | 216 páginas | PDF | 9Mb

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Mathematicians tend to think of themselves as scientists investigating the features of real mathematical things, and the wildly successful application of mathematics in the physical sciences reinforces this picture of mathematics as an objective study. For philosophers, however, this realism about mathematics raises serious questions: What are mathematical things? Where are they? How do we know about them? Offering a scrupulously fair treatment of both mathematical and philosophical concerns, Penelope Maddy here delineates and defends a novel version of mathematical realism. She answers the traditional questions and poses a challenging new one, refocusing philosophical attention on the pressing foundational issues of contemporary mathematics.

Meaning in Mathematics

John Polkinghorne

Oxford University Press | 2011 | 192 páginas | PDF  |  850 kb 

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Is mathematics a highly sophisticated intellectual game in which the adepts display their skill by tackling invented problems, or are mathematicians engaged in acts of discovery as they explore an independent realm of mathematical reality? Why does this seemingly abstract discipline provide the key to unlocking the deep secrets of the physical universe? How one answers these questions will significantly influence metaphysical thinking about reality.

This book is intended to fill a gap between popular 'wonders of mathematics' books and the technical writings of the philosophers of mathematics. The chapters are written by some of the world's finest mathematicians, mathematical physicists and philosophers of mathematics, each giving their perspective on this fascinating debate. Every chapter is followed by a short response from another member of the author team, reinforcing the main theme and raising further questions.

Accessible to anyone interested in what mathematics really means, and useful for mathematicians and philosophers of science at all levels, Meaning in Mathematics offers deep new insights into a subject many people take for granted.

The Search for Certainty: On the Clash of Science and Philosophy of Probability

Krzysztof Burdzy

World Scientific Publishing Company | 2009 | 272 páginas | PDF | 1,9 Mb

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This volume represents a radical departure from the current philosophical duopoly in the area of foundations of probability, that is, the frequency and subjective theories. One of the main new ideas is a set of scientific laws of probability. The new laws are simple, intuitive and, last but not least, they agree well with the contents of current textbooks on probability. Another major new claim is that the "frequency statistics" has nothing in common with the "frequency philosophy of probability," contrary to popular belief. Similarly, contrary to the general perception, the "Bayesian statistics" shares nothing in common with the "subjective philosophy of probability."

The book is non-partisan on the scientific side -- it is supportive of both frequency statistics and Bayesian statistics. On the other hand, it contains well-documented and thoroughly-explained criticisms of the frequency and subjective philosophies of probability. Short reviews of other philosophical theories of probability and basic mathematical methods of probability and statistics are incorporated. The book includes substantial chapters on decision theory and teaching probability, and it is easily accessible to the general audience.

Contents:

• Main Philosophies of Probability
• The Science of Probability
• Decision Making
• The Frequency Philosophy of Probability
• Classical Statistics
• The Subjective Philosophy of Probability
• Bayesian Statistics
• Teaching Probability
• Abuse of Language
• What is Science?
• What is Philosophy?
• Concluding Remarks
• Mathematical Methods of Probability and Statistics
• Literature Review

The Mathematician's Brain. A Personal Tour Through the Essentials of Mathematics and Some of the Great Minds Behind Them

David Ruelle

Princeton University Press | 2007 | 172 páginas | PDF | 1,31 Mb

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The Mathematician's Brain poses a provocative question about the world's most brilliant yet eccentric mathematical minds: were they brilliant because of their eccentricities or in spite of them? In this thought-provoking and entertaining book, David Ruelle, the well-known mathematical physicist who helped create chaos theory, gives us a rare insider's account of the celebrated mathematicians he has known-their quirks, oddities, personal tragedies, bad behavior, descents into madness, tragic ends, and the sublime, inexpressible beauty of their most breathtaking mathematical discoveries.

Consider the case of British mathematician Alan Turing. Credited with cracking the German Enigma code during World War II and conceiving of the modern computer, he was convicted of "gross indecency" for a homosexual affair and died in 1954 after eating a cyanide-laced apple--his death was ruled a suicide, though rumors of assassination still linger. Ruelle holds nothing back in his revealing and deeply personal reflections on Turing and other fellow mathematicians, including Alexander Grothendieck, René Thom, Bernhard Riemann, and Felix Klein. But this book is more than a mathematical tell-all. Each chapter examines an important mathematical idea and the visionary minds behind it. Ruelle meaningfully explores the philosophical issues raised by each, offering insights into the truly unique and creative ways mathematicians think and showing how the mathematical setting is most favorable for asking philosophical questions about meaning, beauty, and the nature of reality.

The Mathematician's Brain takes you inside the world--and heads--of mathematicians. It's a journey you won't soon forget.

TABLE OF CONTENTS:
Preface vii
Chapter 1: Scientific Thinking 1
Chapter 2: What Is Mathematics? 5
Chapter 3: The Erlangen Program 11
Chapter 4: Mathematics and Ideologies 17
Chapter 5: The Unity of Mathematics 23
Chapter 6: A Glimpse into Algebraic Geometry and Arithmetic 29
Chapter 7: A Trip to Nancy with Alexander Grothendieck 34
Chapter 8: Structures 41
Chapter 9: The Computer and the Brain 46
Chapter 10: Mathematical Texts 52
Chapter 11: Honors 57
Chapter 12: Infinity: The Smoke Screen of the Gods 63
Chapter 13: Foundations 68
Chapter 14: Structures and Concept Creation 73
Chapter 15: Turing's Apple 78
Chapter 16: Mathematical Invention: Psychology and Aesthetics 85
Chapter 17: The Circle Theorem and an Infinite-Dimensional Labyrinth 91
Chapter 18: Mistake! 97
Chapter 19: The Smile of Mona Lisa 103
Chapter 20: Tinkering and the Construction of Mathematical Theories 108
Chapter 21: The Strategy of Mathematical Invention 113
Chapter 22: Mathematical Physics and Emergent Behavior 119
Chapter 23: The Beauty of Mathematics 127
Notes 131
Index 157

Revisão em: plus.maths.org

The Determinate World: Kant and Helmholtz on the Physical Meaning of Geometry

David Hyder

Walter de Gruyter | 2006 | 229 páginas | PDF | 1,48 Mb

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This book offers a new interpretation of Hermann von Helmholtz´s work on the epistemology of geometry. A detailed analysis of the philosophical arguments of Helmholtz´s Erhaltung der Kraft shows that he took physical theories to be constrained by a regulative ideal. They must render nature ""completely comprehensible"", which implies that all physical magnitudes must be relations among empirically given phenomena. This conviction eventually forced Helmholtz to explain how geometry itself could be so construed. Hyder shows how Helmholtz answered this question by drawing on the theory of magnitudes developed in his research on the colour-space. He argues against the dominant interpretation of Helmholtzs work by suggesting that for the latter, it is less the inductive character of geometry that makes it empirical, and rather the regulative requirement that the system of natural science be empirically closed.

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.

Philosophy of Mathematics: Structure and Ontology

Stewart Shapiro

Oxford University Press | 1997 | 296 páginas | PDF | 1 Mb

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Do numbers, sets, and so forth, exist? What do mathematical statements mean? Are they literally true or false, or do they lack truth values altogether? Addressing questions that have attracted lively debate in recent years, Stewart Shapiro contends that standard realist and antirealist accounts of mathematics are both problematic.

As Benacerraf first noted, we are confronted with the following powerful dilemma. The desired continuity between mathematical and, say, scientific language suggests realism, but realism in this context suggests seemingly intractable epistemic problems. As a way out of this dilemma, Shapiro articulates a structuralist approach. On this view, the subject matter of arithmetic, for example, is not a fixed domain of numbers independent of each other, but rather is the natural number structure, the pattern common to any system of objects that has an initial object and successor relation satisfying the induction principle. Using this framework, realism in mathematics can be preserved without troublesome epistemic consequences.

Shapiro concludes by showing how a structuralist approach can be applied to wider philosophical questions such as the nature of an "object" and the Quinean nature of ontological commitment. Clear, compelling, and tautly argued, Shapiro's work, noteworthy both in its attempt to develop a full-length structuralist approach to mathematics and to trace its emergence in the history of mathematics, will be of deep interest to both philosophers and mathematicians.

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. 

Mathematical Knowledge

Mary Leng, Alexander Paseau, Micheal Potter 

Oxford University Press | 2008 | 199 páginas | PDF | 789 kb

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What is the nature of mathematical knowledge? Is it anything like scientific knowledge or is it sui generis? How do we acquire it? Should we believe what mathematicians themselves tell us about it? Are mathematical concepts innate or acquired? Eight new essays offer answers to these and many other questions. Written by some of the world's leading philosophers of mathematics, psychologists, and mathematicians, Mathematical Knowledge gives a lively sense of the current state of debate in this fascinating field.

The Indispensability of Mathematics

Mark Colyvan

Oxford University Press | 2003 | 192 | PDF | 6,4 Mb

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The Quine-Putnam indispensability argument in the philosophy of mathematics urges us to place mathematical entities on the same ontological footing as other theoretical entities essential to our best scientific theories. Recently, the argument has come under serious scrutiny, with many influential philosophers unconvinced of its cogency. This book not only outlines the indispensability argument in considerable detail but also defends it against various challenges.


Contents
1 Mathematics and Its Applications 1
1.1 Realism and Anti-realism in Mathematics 2
1.2 Indispensability Arguments 6
1.3 The Road Ahead 17
2 The Quinean Backdrop 21
2.1 Introducing Naturalism 21
2.2 Quinean Naturalism 22
2.3 The Methodologies of Philosophy and Science 26
2.4 The Causal Version of Naturalism 32
2.5 Holism 33
2.6 The First Premise Revisited 37
3 The Eleatic Principle 39
3.1 The Inductive Argument 41
3.2 The Epistemic Argument 42
3.3 The Argument from Causal Explanation 45
3.4 Causal Relevance 51
3.5 Rejecting Inference to the Best Explanation 53
3.6 The Content of Scientific Theories 58
3.7 The Moral 62
3.8 Recapitulation 65
4 Field's Fictionalism 67
4.1 The Science without Numbers Project 68
4.2 What Is It to Be Indispensable? 76
4.3 The Role of Confirmation Theory 78
4.4 The Role of Mathematics in Physical Theories 81
4.5 Review of Field's Fictionalism 87
5 Maddy's Objections 91
5.1 The Objections 91
5.2 Maddy's Naturalism 96
5.3 Defending the Indispensability Argument 98
5.4 Review of Maddy's Objections 1ll
6 The Empirical Nature of Mathematical Knowledge 115
6.1 The Obviousness of Some Mathematical Truth 116
6.2 The Unfalsifiability of Mathematics 122
6.3 The Sober Objection 126
6.4 Is Mathematics Contingent? 134
7 Conclusion 141
7.1 What the Argument Doesn't Show 142
7.2 The Benacerraf Challenges 147
7.3 A Slippery Slope? 155
Bibliography 157
Index 169

Outro livro do mesmo autor, disponível no blog:

- Colyvan, M. (2012). An Introduction to the Philosophy of Mathematics. Cambridge University Press

The Conceptual Roots of Mathematics

J.R. Lucas

Routledge | 1999 | 492 páginas | PDF | 2 Mb

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The Conceptual Roots of Mathematics is a comprehensive study of the foundation of mathematics. J.R. Lucas, one of the most distinguished Oxford scholars, covers a vast amount of ground in the philosophy of mathematics, showing us that it is actually at the heart of the study of epistemology and metaphysics.

Routledge History of Philosophy, vol IX - Science, Logic and Mathematics

S. G. Shanker

Routledge | 1996 | 504 páginas | PDF | 4 Mb

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Volume 9 of the Routledge History of Philosophy surveys ten key topics in the philosophy of science, logic and mathematics in the twentieth century. Each of the essays is written by one of the world's leading experts in that field. Among the topics covered are the philosophy of logic, of mathematics and of Gottlob Frege; Ludwig Wittgenstein's Tractatus; a survey of logical positivism; the philosophy of physics and of science; probability theory, cybernetics and an essay on the mechanist/vitalist debates.
The volume also contains a helpful chronology to the major scientific and philosophical events in the twentieth century. It also provides an extensive glossary of technical terms in the notes on major figures in these fields.

A Structural Account of Mathematics

Charles S. Chihara

Oxford University Press | 2007 | 394 páginas | PDF | 4 Mb

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Charles Chihara's new book develops and defends a structural view of the nature of mathematics, and uses it to explain a number of striking features of mathematics that have puzzled philosophers for centuries. The view is used to show that, in order to understand how mathematical systems are applied in science and everyday life, it is not necessary to assume that its theorems either presuppose mathematical objects or are even true.

Chihara builds upon his previous work, in which he presented a new system of mathematics, the constructibility theory, which did not make reference to, or resuppose, mathematical objects. Now he develops the project further by analyzing mathematical systems currently used by scientists to show how such systems are compatible with this nominalistic outlook. He advances several new ways of undermining the heavily discussed indispensability argument for the existence of mathematical objects made famous by Willard Quine and Hilary Putnam. And Chihara presents a rationale for the nominalistic outlook that is quite different from those generally put forward, which he maintains have led to serious misunderstandings.