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The Number Sense: How the Mind Creates

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The Number Sense: How the Mind Creates The Number Sense This page intentionally left blank The Number Sense How the Mind Creates Mathematics Stanislas Dehaene OXFORD UNIVERSITY PRESS OXFORD UNIVERSITY PRESS Oxford New York Athens Auckland Bangkok Bogota Buenos Aires Calcutta Cape Town Chennai Dar es Salaam Delhi Florence Hong Kong Istanbul Karachi Kuala Lumpur Madrid Melbourne Mexico City Mumbai Nairobi Paris Sao Paulo Singapore Taipei Tokyo Toronto Warsaw and associated companies in Berlin Ibadan Copyright © 1997 by Stanislas Dehaene First published by Oxford University Press, Inc., 1997 First issued in 1999 as an Oxford University Press paperback 198 Madison Avenue, New York, New York 10016 Oxford is a registered trademark of Oxford University Press All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording, or otherwise, without the prior permission of Oxford University Press. Library of Congress Cataloging-in-Publication Data Dehaene, Stanislas. The number sense : how the mind creates mathematics / by Stanislas Dehaene. p. cm. Includes bibliographical references and index. ISBN 0-19-511004-8 ISBN 0-19-513240-8 (Pbk.) 1. Number concept. 2. Mathematics—Study and teaching— Psychological aspects. 3. Mathematical ability. I. Title. QA141.D44 1997 510'.1'9—dc20 96-53840 13579 10 8642 Printed in the United States of America Contents Preface ix Introduction 3 Part I Our Numerical Heritage Chapter 1 Talented and Gifted Animals 13 A Horse Named Hans 14 Rat Accountants 17 How Abstract Are Animal Calculations? 23 The Accumulator Metaphor 28 Number-Detecting Neurons? 31 Fuzzy Counting 34 The Limits of Animal Mathematics 38 From Animal to Human 39 Chapter 2 Babies Who Count 41 Baby Building: Piaget's Theory 41 Piaget's Errors 44 Younger and Younger 47 Babies' Power of Abstraction 5° How Much Is 1 plus 1 ? 52 The Limits of Infant Arithmetic 56 Nature, Nurture, and Number 61 Chapter 3 The Adult Number Line 64 1,2,3, and Beyond 66 Approximating Large Numbers 70 v i Contents The Quantity Behind the Symbols 73 The Mental Compression of Large Numbers 76 Reflexive Access to Number Meaning 78 A Sense of Space 80 Do Numbers Have Colors? 83 Intuitions of Number 86 Part II Beyond Approximation Chapter 4 The Language of Numbers 91 A Short History of Number 92 Keeping a Permanent Trace of Numerals 95 The Place-Value Principle 98 An Exuberant Diversity of Number Languages 100 The Cost of Speaking English 102. Learning to Label Quantities 106 Round Numbers, Sharp Numbers 108 Why Are Some Numerals More Frequent Than Others? no Cerebral Constraints on Cultural Evolution 115 Chapter 5 Small Heads for Big Calculations 118 Counting: The ABC of Calculation 119 Preschoolers as Algorithm Designers 122 Memory Appears on the Scene 124 The Multiplication Table: An Unnatural Practice? 126 Verbal Memory to the Rescue 130 Mental Bugs 131 Pros and Cons of the Electronic Calculator 134 Innumeracy: Clear and Present Danger? I36 Teaching Number Sense 139 Chapter 6 Geniuses and Prodigies 144 A Numerical Bestiary 147 The Landscape of Numbers 149 Phrenology and the Search for Biological Bases of Genius 152 Is Mathematical Talent a Biological Gift? 157 When Passion Produces Talent 162 Ordinary Parameters for Extraordinary Calculators 164 Contents v i i Recipes for Lightning Calculation 167 Talent and Mathematical Invention 170 Part III Of Neurons and Numbers Chapter 7 Losing Number Sense 175 Mr. N, the Approximate Man 177 A Clear-Cut Deficit 181 A Champion in Numerical Non-sense 185 Inferior Parietal Cortex and the Number Sense 189 Seizures Induced by Mathematics 191 The Multiple Meanings of Numbers 192 The Brain's Numerical Information Highways 194 Who Orchestrates the Brain's Computations? 199 At the Origins of Cerebral Specialization 202 Chapter 8 The Computing Brain 207 Does Mental Calculation Increase Brain Metabolism? 208 The Principle of Positron Emission Tomography 211 Can One Localize Mathematical Thought? 213 When the Brain Multiplies or Compares 218 The Limits of Positron Emission Tomography 220 The Brain Electric 221 The Time Line of the Number Line 223 Understanding the Word "Eighteen" 227 Numerate Neurons 228 Chapter 9 What Is a Number? 231 Is the Brain a Logical Machine? 232 Analog Computations in the Brain 235 When Intuition Outruns Axioms 238 Platonists, Formalists, and Intuitionists 242 The Construction and Selection of Mathematics 245 The Unreasonable Effectiveness of Mathematics 249 Appendix 253 Notes and References 255 Index 267 This page intentionally left blank Preface e are surrounded by numbers. Etched on credit cards or engraved on coins, printed on pay checks or aligned on computerized spread Wsheets, numbers rule our lives. Indeed, they lie at the heart of our technology. Without numbers, we could not send rockets roaming the solar sys- tem, nor could we build bridges, exchange goods, or pay our bills. In some sense, then, numbers are cultural inventions only comparable in importance to agricul- ture or to the wheel. But they might have even deeper roots. Thousands of years before Christ, Babylonian scientists used clever numerical notations to compute astronomical tables of amazing accuracy. Tens of thousands of years prior to them, Neolithic men recorded the first written numerals by engraving bones or by painting dots on cave walls. And as I shall try to convince you later on, millions of years earlier still, long before the dawn of humankind, animals of all species were already registering numbers and entering them into simple mental compu- tations. Might numbers, then be almost as old as life itself? Might they be engraved in the very architecture of our brains? Do we all possess a "number sense," a special intuition that helps us make sense of numbers and mathematics? Fifteen years ago, as I was training to become a mathematician, I became fas- x Preface cinated by the abstract objects I was taught to manipulate, and above all, by the simplest of them—numbers. Where did they come from? How was it possible for my brain to understand them? Why did it seem so difficult for most people to master them? Historians of science and philosophers of mathematics had provid- ed some tentative answers, but to a scientifically oriented mind their speculative and contingent character was unsatisfactory. Furthermore, scores of intriguing facts about numbers and mathematics were left unanswered in the books I knew of. Why did all languages have at least some number names? Why did everybody seem to find multiplications by seven, eight, or nine particularly hard to learn? Why couldn't I seem to recognize more than four objects at a glance? Why were there ten boys for one girl in the high-level mathematics classes I was attending? What tricks allowed lightning calculators to multiply two three-digit numbers in a few seconds? As I learned increasingly more about psychology, neurophysiology, and com- puter science, it became obvious that the answers had to be looked for, not in his- tory books, but in the very structure of our brains—the organ that enables us to create mathematics. It was an exciting time for a mathematician to turn to cogni- tive neuroscience. New experimental techniques and amazing results seemed to appear every month. Some revealed that animals could do simple arithmetic. Others asked whether babies had any notion of 1 plus 1. Functional imaging tools were also becoming available that could visualize the active circuits of the human brain as it calculates and solves arithmetical problems. Suddenly, the psy- chological and cerebral bases of our number sense were open to experimenta- tion. A new field of science was emerging: mathematical cognition, or the scientific inquiry into how the human brain gives rise to mathematics. I was lucky enough to become an active participant in this quest. This book provides a first glance at this new field of research that my colleagues in Paris and several research teams throughout the world are still busy developing. I am indebted to many people for helping me complete the transition from mathematics to neuropsychology. First and foremost, my research program on arithmetic and the brain could never have developed without the generous assis- tance of three outstanding teachers, colleagues, and friends who deserve very special thanks: Jean-Pierre Changeux in neurobiology, Laurent Cohen in neu- ropsychology, and Jacques Mehler in cognitive psychology. Their support, advice, and often direct contribution to the work described here have been of invaluable help. Through the years, I have also benefited from the advice of many other emi- nent scientists. Mike Posner, Don Tucker, Michael Murias, Denis Le Bihan, Andre Syrota, and Bernard Mazoyer shared with me their in-depth knowledge of brain imaging. Emmanuel Dupoux, Anne Christophe, and Christophe Pallier advised Preface x i me in psycholinguistics. I am also grateful for ground-shaking debates with Rochel Gelman and Randy Gallistel and for judicious remarks by Karen Wynn, Sue Carey, and Josiane Bertoncini on child development. The late professor Jean-Louis Signoret had introduced me to the fascinating domain of neuropsychology. Subsequently, numerous discussions with Alfonso Caramazza, Michael McCloskey, Brian Butterworth, and Xavier Seron greatly enhanced my understanding of this discipline. Xavier Jeannin and Michel Dutat, finally, assisted me in programming my experiments. I also gratefully acknowledge the crucial contribution of several students to my research: Rokny Akhavein, Serge Bossini, Florence Chochon, Pascal Giraux, Markus Kiefer, Etienne Koechlin, Lionel Naccache, and Gerard Rozsavolgyi. The preparation of this book greatly benefited from the close scrutiny of Brian Butterworth, Robbie Case, Markus Giaquinto, and Susana Franck for the English edition, and of Jean-Pierre Changeux, Laurent Cohen, and Ghislaine Dehaene-Lambertz for the French edition. Warm thanks go also to Joan Bossert, my editor at Oxford University Press, John Brockman, my agent, and Odile Jacob, my French editor.
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