Making and Breaking Mathematical Sense

Acknowledgments xi
Introduction 1
What Philosophy of Mathematics Is Today 1
What Else Philosophy of Mathematics Can Be 3
A Vignette: Option Pricing and the Black-Scholes Formula 6
Outline of This Book 10
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
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
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
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
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
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
7: Making a World, Mathematically 199
Fichte 201
Schelling 206
Hermann Cohen 209
The Unreasonable(?) Applicability of Mathematics 213
Bibliography 219
Index 233