E-Book Content
beyond geometry
Math Chapter Title iii
the history of
beyond geometry a new mathematics of space and form
John Tabak, Ph.D.
BEYOND GEOMETRY: A New Mathematics of Space and Form Copyright © 2011 by John Tabak, Ph.D. All rights reserved. No part of this book may be reproduced or utilized in any form or by any means, electronic or mechanical, including photocopying, recording, or by any information storage or retrieval systems, without permission in writing from the publisher. For information contact: Facts On File, Inc. An imprint of Infobase Learning 132 West 31st Street New York NY 10001 Library of Congress Cataloging-in-Publication Data Tabak, John. Beyond geometry: a new mathematics of space and form / John Tabak. p. cm.—(The history of mathematics) Includes bibliographical references and index. ISBN 978-0-8160-7945-2 ISBN 978-1-4381-3623-3 (e-book) 1. Topology. 2. Set theory. 3. Geometry. I. Title. QA611.T33 2011 516—dc22 2010023887 Facts On File books are available at special discounts when purchased in bulk quantities for businesses, associations, institutions, or sales promotions. Please call our Special Sales Department in New York at (212) 967-8800 or (800) 322-8755. You can find Facts On File on the World Wide Web at http://www.infobaselearning.com Excerpts included herewith have been reprinted by permission of the copyright holders; the author has made every effort to contact copyright holders. The publishers will be glad to rectify, in future editions, any errors or omissions brought to their notice. Text design by David Strelecky Composition by Hermitage Publishing Services Illustrations by Dale Williams Photo research by Elizabeth H. Oakes Cover printed by Yurchak Printing, Inc., Landisville, Pa. Book printed and bound by Yurchak Printing, Inc., Landisville, Pa. Date printed: May 2011 Printed in the United States of America 10 9 8 7 6 5 4 3 2 1 This book is printed on acid-free paper.
For Stuart Dugowson. He was a generous spirit with an inquiring mind.
contents
Preface Acknowledgments Introduction
1 Topology: A Prehistory Euclid’s Axioms Euclidean Transformations The Beginnings of Calculus Counterexample 1: A Continuous Function That Is Not Everywhere Differentiable
2 A Failure of Intuition An Alternative to Euclid’s Axioms Bernhard Bolzano and Further Limitations on Geometric Reasoning Counterexample 2: A Continuous Nowhere Differentiable Function
3 A New Mathematical Landscape Richard Dedekind and the Continuum Georg Cantor and Set Theory Counterexample 3: Peano’s Space-Filling Curve Different-Looking Sets, Similar Properties: Part I Different-Looking Sets, Similar Properties: Part II
4 The First Topological Spaces Felix Hausdorff and the First Abstract Topological Spaces Topological Transformations
x xvi xvii
1 2 7 10 14
18 19 23 30
33 34 38 44 47 50
55 56 66
The Role of Examples and Counterexamples in Topology Counterexample 4: Sierpi´nski’s Gasket
74 76
5 The Standard Axioms and Three Topological Properties
80
The Standard Axioms Topological Property 1: Compactness Topological Property 2: Regularity The Role of Rigor in Mathematics Topological Property 3: Connectedness
81 84 91 92 98
6 Schools of Topology The Polish School Topology at the University of Texas The Moore Method Topology in Japan
7 Dimension Theory
102 103 108 112 114
120
Continuum Theory Inductive Definitions of Dimension A Noninductive Definition of Dimension and More Consequences of Dimension Theory Still Another Concept of Dimension: The Hausdorff Dimension
136
8 Topology and the Foundations of Modern Mathematics
143
Topology and the Language of Mathematic