9781402033346

This book contains groundbreaking contributions to the philosophical analysis of mathematical practice. Several philosophers of mathematics have recently called for an approach to philosophy of mathematics that pays more attention to mathematical practice. Questions concerning concept-formation, understanding, heuristics, changes in style of reasoning, the role of analogies and diagrams, etc. have become the subject of intense interest. The historians and philosophers in this book agree that there is more to understanding mathematics than a study of its logical structure. How are mathematical objects and concepts generated? How does the process tie up with justification? What role do visual images and diagrams play in mathematical activity? What are the different epistemic virtues (explanatoriness, understanding, visualizability, etc.) which are pursued and cherished by mathematicians in their work? The reader will find here systematic philosophical analyses as well as a wealth of philosophically informed case studies ranging from Babylonian, Greek, and Chinese mathematics to nineteenth century real and complex analysis.

Contributing Authors

ix

Introduction

1

(12)

P. Mancosu

K.F. Jørgensen

S.A. Pedersen

PART I. MATHEMATICAL REASONING AND VISUALIZATION

Visualization in Logic and Mathematics

13

(18)

P. Mancosu

Diagrams and Images in the Late Nineteenth Century

13

(4)

The Return of the Visual as a Change in Mathematical Style

17

(4)

New Directions of Research and Foundations of Mathematics

21

(10)

Acknowledgements

26

(1)

Notes

27

(1)

References

28

(3)

From Symmetry Perception to Basic Geometry

31

(26)

M. Giaquinto

Introduction

31

(1)

Perceiving a Figure as a Square

31

(8)

A Geometrical Concept for Squares

39

(5)

Getting the Belief

44

(2)

Is It Knowledge?

46

(4)

Summary

50

(7)

Notes

51

(2)

References

53

(4)

Naturalism, Pictures, and Platonic Intuitions

57

(18)

J.R. Brown

Naturalism

57

(2)

Platonism

59

(1)

Godel's Platonism

60

(2)

The Concept of Observable

62

(2)

Proofs and Intuitions

64

(2)

Maddy's Naturalism

66

(1)

Refuting the Continuum Hypothesis

67

(8)

Acknowledgements

70

(1)

Appendix: Freiling's ``Philosophical'' Refutation of CH

71

(1)

References

72

(3)

Mathematical Activity

75

(16)

M. Giaquinto

Discovery

76

(1)

Explanation

77

(4)

Justification

81

(3)

Refining and Extending the List of Activities

84

(2)

Concluding Remarks

86

(5)

Notes

86

(1)

References

87

(4)

PART II. MATHEMATICAL EXPLANATION AND PROOF STYLES

Tertium Non Datur: On Reasoning Styles in Early Mathematics

91

(32)

J. Høyrup

Two Convenient Scapegoats

91

(1)

Old Babylonian Geometric Proto-algebra

92

(11)

Euclidean Geometry

103

(2)

Stations on the Road

105

(2)

Other Greeks

107

(2)

Proportionality -- Reasoning and its Elimination

109

(14)

Notes

113

(5)

References

118

(5)

The Interplay Between Proof and Algorithm in 3rd Century China: The Operation as Prescription of Computation and the Operation as Argument

123

(24)

K. Chemla

Elements of Context

125

(1)

Sketch of the Proof

126

(5)

First Remarks on the Proof

131

(1)

The Operation as Relation of Transformation

132

(2)

The Essential Link Between Proof and Algorithm

134

(5)

Conclusion

139

(8)

Appendix

141

(1)

Notes

141

(3)

References

144

(3)

Proof Style and Understanding in Mathematics I: Visualization, Unification and Axiom Choice

147

(68)

J. Tappenden

Introduction -- a ``New Riddle'' of Deduction

147

(2)

Understanding and Explanation in Mathematical Methodology: The Target

149

(9)

Understanding, Unification and Explanation -- Friedman

158

(10)

Kitcher: Patterns of Argument

168

(12)

Artin and Axiom Choice: ``Visual Reasoning'' Without Vision

180

(7)

Summary -- the ``new Riddle of Deduction''

187

(28)

Notes

188

(18)

References

206

(9)

The Varieties of Mathematical Explanations

215

(36)

J. Hafner

P. Mancosu

Back to the Facts Themselves

215

(1)

Mathematical Explanation or Explanation in Mathematics?

216

(2)

The Search for Explanation within Mathematics

218

(3)

Some Methodological Comments on the General Project

221

(1)

Mark Steiner on Mathematical Explanation

222

(2)

Kummer's Convergence Test

224

(6)

A Test Case for Steiner's Theory

230

(21)

Appendix

241

(5)

Notes

246

(3)

References

249

(2)

The Aesthetics of Mathematics: A Study

251

(44)

R. Netz

The Problem Motivated

251

(3)

Sources of Beauty in Mathematics

254

(27)

Conclusion

281

(14)

Notes

289

(3)

References

292

(3)

Index

295

What is included with this book?

The New copy of this book will include any supplemental materials advertised. Please check the title of the book to determine if it should include any access cards, study guides, lab manuals, CDs, etc.

The Used, Rental and eBook copies of this book are not guaranteed to include any supplemental materials. Typically, only the book itself is included. This is true even if the title states it includes any access cards, study guides, lab manuals, CDs, etc.

Amazon no longer offers textbook rentals. We do!

Amazon no longer offers textbook rentals. We do!

We're the #1 textbook rental company. Let us show you why.

Visualization, Explanation And Reasoning Styles in Mathematics

1402033346

Supplemental Materials

Summary

Table of Contents

Supplemental Materials

Rewards Program