Visualization, Explanation and Reasoning Styles in Mathematics
This book contains groundbreaking contributions to the philosophical analysis of mathematical practice. Several philosophers of mathematics have recently called for an approach to philosophy of mathematics that pays more attention to mathematical practice
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SYNTHESE LIBRARY / VOLUME 327
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PAOLO MANCOSU,
VISUALIZATION, EXPLANATION VISUALIZATION, EXPLANATION AND STYLESININ AND REASONING REASONING STYLES MATHEMATICS MATHEMATICS
VISUALIZATION, EXPLANATION AND
Edited by Paolo Mancosu, Klaus Frovin Jørgensen and Stig Andur Pedersen
VISUALIZATION, EXPLANATION AND REASONING STYLES IN MATHEMATICS
SYNTHESE LIBRARY STUDIES IN EPISTEMOLOGY, LOGIC, METHODOLOGY, AND PHILOSOPHY OF SCIENCE
Editor-in-Chief:
VINCENT F. HENDRICKS, Roskilde University, Roskilde, Denmark JOHN SYMONS, University of Texas at El Paso, U.S.A.
Honorary Editor: JAAKKO HINTIKKA, Boston University, U.S.A.
Editors: DIRK VAN DALEN, University of Utrecht, The Netherlands THEO A.F. KUIPERS, University of Groningen, The Netherlands TEDDY SEIDENFELD, Carnegie Mellon University, U.S.A. PATRICK SUPPES, Stanford University, California, U.S.A. JAN WOLEN´SKI, Jagiellonian University, Kraków, Poland
VOLUME 327
VISUALIZATION, EXPLANATION AND REASONING STYLES IN MATHEMATICS Edited by
PAOLO MANCOSU University of California, Berkeley, U.S.A.
KLAUS FROVIN JØRGENSEN Roskilde University, Denmark and
STIG ANDUR PEDERSEN Roskilde University, Denmark
A C.I.P. Catalogue record for this book is available from the Library of Congress.
ISBN 1-4020-3334-6 (HB) ISBN 1-4020-3335-4 (e-book)
Published by Springer, P.O. Box 17, 3300 AA Dordrecht, The Netherlands. Sold and distributed in North, Central and South America by Springer, 101 Philip Drive, Norwell, MA 02061, U.S.A. In all other countries, sold and distributed by Springer, P.O. Box 322, 3300 AH Dordrecht, The Netherlands.
Printed on acid-free paper
All Rights Reserved © 2005 Springer No part of this work may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, microfilming, recording or otherwise, without written permission from the Publisher, with the exception of any material supplied specifically for the purpose of being entered and executed on a computer system, for exclusive use by the purchaser of the work. Printed in the Netherlands.
CONTENTS Contributing Authors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix P. MANCOSU, K.F. JØRGENSEN AND S.A. PEDERSEN / Introduction . 1 PART I. MATHEMATICAL REASONING AND VISUALIZATION P. MANCOSU / Visualization in Logic and Mathematics . . . . . . . . . . . . . . 13 1. Diagrams and Images in the Late Nineteenth Century . . . . . . . . . . . 2. The Return of the Visual as a Change in Mathematical Style . . . . 3. New Directions of Research and Foundations of Mathematics . . . Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
13 17 21 26 27 28
M. GIAQUINTO / From Symmetry Perception to Basic Geometry . . . . . 31 Int
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