Making and Breaking Mathematical Sense: Histories and Philosophies of Mathematical Practice

251 Pages·2017·1.721 MB·English

by  Wagner, Roi

Making and Breaking Mathematical Sense
■ ■ ■ ■ ■ ■ ■

Making and Breaking
Mathematical Sense
HISTORIES AND PHILOSOPHIES OF
MATHEMATICAL PRACTICE
■ ■ ■ ■ ■ ■ ■
Roi Wagner
PRINCETON UNIVERSITY PRESS
PRINCETON AND OXFORD
Copyright © 2017 by Princeton University Press
Published by Princeton University Press, 41 William Street, Princeton, New Jersey 08540
In the United Kingdom: Princeton University Press, 6 Oxford Street, Woodstock, Oxfordshire
OX20 1TR
press.princeton.edu
All Rights Reserved
Library of Congress Cataloging-in-Publication Data
Names: Wagner, Roi, 1973–
Title: Making and breaking mathematical sense : histories and philosophies of mathematical
practice / Roi Wagner.
Description: Princeton : Princeton University Press, [2017] | Includes bibliographical
references and index.
Identifiers: LCCN 2016022844 | ISBN 9780691171715 (hardcover : alk. paper)
Subjects: LCSH: Mathematics—Philosophy—History. | Mathematics—History
Classification: LCC QA8.4 .W334 2017 | DDC 510.1—dc23 LC record available at
https://lccn.loc.gov/2016022844
British Library Cataloging-in-Publication Data is available
This book has been composed in Linux Libertine O and Myriad Pro
Printed on acid-free paper. ∞
Printed in the United States of America
10 9 8 7 6 5 4 3 2 1
To Rhone, who’s too good for some cliché dedication

Contents
■ ■ ■ ■ ■ ■ ■ ■
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
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
viii • Contents
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
Contents • ix
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
Bibliography 219
Index 233