Making and breaking mathematical sense : histories and philosophies of mathematical practice / Roi Wagner.

Format:

Book

Author/Creator:

Wagner, Roi, 1973- author.

Language:

English

Subjects (All):

Mathematics--Philosophy--History.

Mathematics.

Mathematics--History.

Mathematics--Philosophy.

History.

Genre:

History.

Physical Description:

ix, 236 pages ; 25 cm

Place of Publication:

Princeton : Princeton University Press, [2017]

Contents:

Chapter 1 Histories of Philosophies of Mathematics 13

History 1 On What There Is, Which Is a Tension between Natural Order and Conceptual Freedom 14

History 2 The Kantian Matrix, Which Grants Mathematics a Constitutive Intermediary Epistemological Position 22

History 3 Monster Barring, Monster Taming, and Living with Mathematical Monsters 28

History 4 Authority, or Who Gets to Decide What Mathematics is About 33

The "Yes, Please!" Philosophy of Mathematics 37

Chapter 2 The New Entities of Abbacus and Renaissance Algebra 39

Abbacus and Renaissance Algebraists 39

The Emergence of the Sign of the Unknown 40

First Intermediary Reflection 45

The Arithmetic of Debited Values 46

Second Intermediary Reflection 51

False and Sophistic Entities 53

Final Reflection and Conclusion 56

Chapter 3 A Constraints-Based Philosophy of Mathematical Practice 59

Dismotivation 59

The Analytic A Posteriori 63

Consensus 67

Interpretation 72

Reality 81

Constraints 84

Relevance 90

Conclusion 97

Chapter 4 Two Case Studies of Semiosis in Mathematics 100

Ambiguous Variables in Generating Functions 101

Between Formal Interpretations 101

Models and Applications 107

Openness to Interpretation 109

Gendered Signs in a Combinatorial Problem 112

The Problem 112

Gender Role Stereotypes and Mathematical Results 116

Mathematical Language and Its Reality 120

The Forking Paths of Mathematical Language 122

Chapter 5 Mathematics and Cognition 128

The Number Sense 129

Mathematical Metaphors 137

Some Challenges to the Theory of Mathematical Metaphors 142

Best Fit for Whom? 143

What Is a Conceptual Domain? 146

In Which Direction Does the Theory Go? 150

So How Should We Think about Mathematical Metaphors? 154

An Alternative Neural Picture 156

Another Vision of Mathematical Cognition 163

From Diagrams to Haptic Vision 164

Haptic Vision in Practice 171

Chapter 6 Mathematical Metaphors Gone Wild 177

What Passes between Algebra and Geometry 177

Piero della Francesca (Italy, Fifteenth Century) 178

Omar Khayyam (Central Asia, Eleventh Century) 179

René Descartes (France, Seventeenth Century) 181

Rafael Bombelli (Italy, Sixteenth Century) 183

Conclusion 187

A Garden of Infinities 188

Limits 189

Infinitesimals and Actual Infinities 194

Chapter 7 Making a World, Mathematically 199

Fichte 201

Schelling 206

Hermann Cohen 209

The Unreasonable(?) Applicability of Mathematics 213.

Notes:

Includes bibliographical references and index.

ISBN:

9780691171715

0691171718

OCLC:

948560962