Mathematical Knowledge and the Interplay of Practices Unauthenticated Download Date | 6/27/16 3:50 PM This page intentionally left blank Unauthenticated Download Date | 6/27/16 3:50 PM Mathematical Knowledge and the Interplay of Practices J O S É F E R R E I R Ó S PRINCETON UNIVERSITY PRESS PRINCETON AND OXFORD Unauthenticated Download Date | 6/27/16 3:50 PM Copyright © 2016 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 1TW press.princeton.edu Jacket art: Courtesy of Shutterstock All Rights Reserved ISBN 978- 0- 691- 16751- 0 Library of Congress Control Number: 2015935302 British Library Cataloging- in- Publication Data is available This book has been composed in ITC New Baskerville Printed on acid- free paper  Printed in the United States of America 10 9 8 7 6 5 4 3 2 1 Unauthenticated Download Date | 6/27/16 3:50 PM Contents List of Illustrations ix Foreword xi 1 On Knowledge and Practices: A Manifesto 1 2 The Web of Practices 17 2.1. Historical Work on Practices 18 2.2. Philosophers Working on Practices 22 2.3. What Is Mathematical Practice, Then? 28 2.4. The Multiplicity of Practices 34 2.5. The Interplay of Practices and Its Basis 39 3 Agents and Frameworks 44 3.1. Frameworks and Related Matters 45 3.2. Interlude on Examplars 55 3.3. On Agents 59 3.4. Counting Practices and Cognitive Abilities 65 3.5. Further Remarks on Mathematics and Cognition 74 3.6. Agents and “Metamathematical” Views 79 3.7. On Systematic Links 83 4 Complementarity in Mathematics 89 4.1. Formula and Meaning 89 4.2. Formal Systems and Intended Models 94 4.3. Meaning in Mathematics: A Tentative Approach 99 4.4. The Case of Complex Numbers 104 Unauthenticated Download Date | 6/27/16 3:52 PM vi Contents 5 Ancient Greek Mathematics: A Role for Diagrams 112 5.1. From the Technical to the Mathematical 113 5.2. The Elements: Getting Started 117 5.3. On the Euclidean Postulates: Ruling Diagrams (and Their Reading) 127 5.4. Diagram- Based Mathematics and Proofs 131 5.5. Agents, Idealization, and Abstractness 137 5.6. A Look at the Future—Our Past 147 6 Advanced Math: The Hypothetical Conception 153 6.1. The Hypothetical Conception: An Introduction 154 6.2. On Certainty and Objectivity 159 6.3. Elementary vs. Advanced: Geometry and the Continuum 163 6.4. Talking about Objects 170 6.5. Working with Hypotheses: AC and the Riemann Conjecture 176 7 Arithmetic Certainty 182 7.1. Basic Arithmetic 182 7.2. Counting Practices, Again 184 7.3. The Certainty of Basic Arithmetic 189 7.4. Further Clarifications 195 7.5. Model Theory of Arithmetic 198 7.6. Logical Issues: Classical or Intuitionistic Math? 200 8 Mathematics Developed: The Case of the Reals 206 8.1. Inventing the Reals 207 8.2. “Tenths” to the Infinite: Lambert and Newton 215 8.3. The Number Continuum 221 8.4. The Reinvention of the Reals 227 8.5. Simple Infinity and Arbitrary Infinity 231 8.6. Developing Mathematics 236 8.7. Mathematical Hypotheses and Scientific Practices 241 Unauthenticated Download Date | 6/27/16 3:52 PM Contents vii 9 Objectivity in Mathematical Knowledge 247 9.1. Objectivity and Mathematical Hypotheses: A Simple Case 249 9.2. Cantor’s “Purely Arithmetical” Proofs 253 9.3. Objectivity and Hypotheses, II: The Case of (N) 257 9.4. Arbitrary Sets and Choice 261 9.5. What about Cantor’s Ordinal Numbers? 265 9.6. Objectivity and the Continuum Problem 273 10 The Problem of Conceptual Understanding 281 10.1. The Universe of Sets 283 10.2. A “Web- of- Practices” Look at the Cumulative Picture 290 10.3. Conceptual Understanding 296 10.4. Justifying Set Theory: Arguments Based on the Real- Number Continuum 305 10.5. By Way of Conclusion 310 References 315 Index 331 Unauthenticated Download Date | 6/27/16 3:52 PM This page intentionally left blank Unauthenticated Download Date | 6/27/16 3:52 PM List of Illustrations Figure 1 A schematic picture of interrelated practices. 38 Figure 2 Two representations of a theory, idealized vs. concretely given in practice. 53 Figure 3 Diagram from a late edition of the Zhou Bi (an early text in astronomy), related to third- century commentaries of this work and of the Jiu Zhang or Nine Chapters. 121 Figure 4 Diagrams corresponding to the Elements Book I. Prop. 5, Book I. Prop. 32, and Book III. Prop. 2. 136 Figure 5 First page of the Paganini- Pacioli edition of the Elements (Venice, 1509), with diagrams of line, point, angle, circle, etc. 145 Figure 6 The cone representation of the world of sets 292 Unauthenticated Download Date | 6/27/16 3:53 PM This page intentionally left blank Unauthenticated Download Date | 6/27/16 3:53 PM