The image on the cover, “NeuralNet Quilt”, is a section of a planar repeating pattern of type pgg and was created by Mike Field, University of Houston. The pattern was generated by a determinstic, but chaotic, torus map, and the colors represent probabilities of visiting points on the pattern under the iteration. 2000 Mathematics Subject Classification. Primary 00A30, 00A35, 97C50. For additional information and updates on this book, visit Library of Congress Cataloging-in-Publication Data Borovik, Alexandre. Mathematics under the microscope : notes on cognitive aspects of mathematical practice / Alexandre V. Borovik. p. cm. Includes bibliographical references and index. ISBN 978-0-8218-4761-9 (alk. paper) 1. Mathematics—Psychological aspects. 2. Mathematical ability. I. Title. BF456.N7B67 2009 510.1 9—dc22 2009029174 Copying and reprinting. Individual readers of this publication, and nonprofit libraries acting for them, are permitted to make fair use of the material, such as to copy a chapter for use in teaching or research. Permission is granted to quote brief passages from this publication in reviews, provided the customary acknowledgment of the source is given. Republication, systematic copying, or multiple reproduction of any material in this publication is permitted only under license from the American Mathematical Society. Requests for such permission should be addressed to the Acquisitions Department, American Mathematical Society, 201 Charles Street, Providence, Rhode Island 02904-2294 USA. Requests can also be made by e-mail to c 2010 by the American Mathematical Society. All rights reserved. The American Mathematical Society retains all rights except those granted to the United States Government. Printed in the United States of America. The paper used in this book is acid-free and falls within the guidelines established to ensure permanence and durability. Visit the AMS home page at 10 9 8 7 6 5 4 3 2 1 15 14 13 12 11 10
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