Mathematics As a Science of Patterns

This book expounds a system of ideas about the nature of mathematics which Michael Resnik has been elaborating for a number of years. In calling mathematics a science he implies that it has a factual subject-matter and that mathematical knowledge is on a par with other scientific knowledge; in calling it a science of patterns he expresses his commitment to a structuralist philosophy of mathematics. He links this to a defense of realism about the metaphysics of mathematics--the view that mathematics is about things that really exist.

PART ONE: PROBLEMS AND POSITIONS

3

(96)

1. Introduction

3

(7)

2. What is Mathematical Realism?

10

(31)

1. To Characterize Realism

10

(4)

2. Immanent Truth

14

(16)

3. Realism and Immanent Truth

30

(9)

4. Some Concluding Remarks

39

(2)

3. The Case for Mathematical Realism

41

(11)

1. The Prima Facie Case for Realism

41

(2)

2. The Quine-Putnam View of Applied Mathematics

43

(1)

3. Indispensability Arguments for Mathematical Realism

44

(5)

4. Indispensability and Fictionalism about Science

49

(1)

5. Conclusion

50

(2)

4. Recent Attempts at Blunting the Indispensability Thesis

52

(30)

1. Synthetic Science: Field

53

(6)

2. Saving the Mathematical Formalism while Changing its Interpretation: Chihara and Kitcher

59

(8)

3. An Intermediate Approach: Hellman's Modal-Structuralism

67

(8)

4. What Has Introducing Modalities Gained?

75

(6)

5. Conclusion

81

(1)

5. Doubts about Realism

82

(17)

1. How Can We Know Mathematical Objects?

82

(5)

2. How Can We Refer to Mathematical Objects?

87

(2)

3. The Incompleteness of Mathematical Objects

89

(3)

4. Some Morals for Realists

92

(1)

5. An Aside: Penelope Maddy's Perceivable Sets

93

(6)

PART TWO: NEUTRAL EPISTEMOLOGY

99

(100)

Introduction to Part Two

99

(2)

6. The Elusive Distinction between Mathematics and Natural Science

101

(11)

1. How Physics Blurs the Mathematical Physical Distinction

102

(5)

2. Some Other Attempts to Distinguish Mathematical from Physical Objects

107

(1)

3. Our Epistemic Access to Space-Time Points

108

(2)

4. Morals for the Epistemology of Mathematics

110

(2)

7. Holism: Evidence in Science and Mathematics

112

(25)

1. The Initial Case for Holism

114

(4)

2. Objections to Holism

118

(3)

3. Testing Scientific and Mathematical Models

121

(3)

4. Global and Local Theories

124

(6)

5. Revising Logic and Mathematics

130

(7)

8. The Local Conception of Mathematical Evidence: Proof, Computation, and Logic

137

(38)

1. Some Norms of Mathematical Practice

138

(10)

2. Computation and Mathematical Empiricism

148

(7)

3. Mathematical Proof, Logical Deduction and Apriority

155

(17)

4. Summary

172

(3)

9. Positing Mathematical Objects

175

(24)

1. Introduction

175

(2)

2. A Quasi-Historical Account

177

(5)

3. Mathematical Positing Naturalized?

182

(2)

4. Positing and Knowledge

184

(4)

5. Postulational Epistemologies and Realism

188

(11)

PART THREE: MATHEMATICS AS A SCIENCE OF PATTERNS

199

(76)

Introduction to Part Three

199

(2)

10. Mathematical Objects as Positions in Patterns

201

(23)

1. Introduction

201

(1)

2. Patterns and their Relationships

202

(7)

3. Patterns and Positions: Entity and Identity

209

(4)

4. Composite and Unified Mathematical Objects

213

(3)

5. Mathematical Reductions

216

(4)

6. Reference to Positions in Patterns

220

(2)

7. Concluding Remarks on Reference and Reduction

222

(2)

11. Patterns and Mathematical Knowledge

224

(19)

1. Introduction

224

(2)

2. From Templates to Patterns

226

(6)

3. From Proofs to Truth

232

(8)

4. From Old Patterns to New Patterns

240

(3)

12. What is Structuralism? And Other Questions

243

(32)

1. Introduction

243

(1)

2. On `Facts of the Matter'

243

(3)

3. Patterns as Mathematical Objects

246

(4)

4. Structural Relativity

250

(4)

5. Structuralist Formulations of Mathematical Theories?

254

(3)

6. The Status of Structuralism

257

(4)

7. Structuralism, Realism, and Disquotationalism

261

(4)

8. Epistemic vs. Ontic Structuralism: Structuralism All the Way Down

265

(5)

9. A Concluding Summary

270

(5)

Bibliography

275

(8)

Index

283

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Mathematics As a Science of Patterns

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