A Formal Background to Mathematics 2a A Critical Approach to Element
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R. E. Edwards
A Formal Background to Mathematics 2a A Critical Approach to Elementary Analysis
Springer-Verlag New York Heidelberg
Berlin
Dr. Robert Edwards Institute of Advanced Studies The Australian National University Canberra, Australia
AMS Subject Classifications: OOAOS, 00A2S, 03-C1, 26-01, 28-01,30-01
Library of Congress Cataloging In Publication Data Edwards, Robert E A formal background to mathematics. (Universitext) Includes bibliographies and indexes. CONTENTS: v.l. Logic, sets, and numbers. 2v.v.2. A critical approach to elementary analysis. 2v. 1. Mathematics-1961I. Title. QA37.2.E 38 510 79·15045
All rights reserved. No part of this book may be translated or reproduced in any form without written permission from Springer-Verlag.
© 1980 by Springer· Verlag New York Inc.
987654321 ISBN-13: 978-0-387-90513-6 DOl: 10.1007/978-1-4613-8096-2
e-ISBN-13: 978-1-4613-8096-2
Foreword to Volume 2
This volume is devoted to a treatment of some of the fundamentals of elementary analysis (including what is usually described as
"calculus") and a
critique of the expositions to be found in typical high school and other conventionally informal texts.
Attention is concentrated upon elementary analysis,
since it is the major portion of mathematics at the high school stage and, more importantly, is the portion fraught with more of the fundamentally difficult concepts (limits and convergence in various guises).
Other topics, such as
fragments of linear algebra, probability and statistics, sometimes treated in high school texts, have been ignored on tile grounds that they are not so heavily laden with basic mathematical difficulties (though some exhibit problems which are not basically mathematical).
Also omitted from the main text is all consideration of
the use and abuse of calculators as an alleged aid in
"getting across"
concepts of limit and calculus (but see the Epilogue (iv)).
the
Another omission is
the absence of any reference to t;le poss i bil it) (taken up by some writers) of adopting the concepts and techniques of non-standard analysis with the same end in view.
(This is a bold and interesting idea;
but its execution seems to me to
demand extreme caution, if it is not to back-fire and foster more misunderstanding than it eliminates.) Both the informal content and style of presentation is fairly routine, but there is some novelty in the references to formalities.
Instances of such
vi
references are to be found in VII.l.2, VII.l.4, VII.l.l3, VII.5.5, VII.5.6, VIII.4.4, VIII.6.9, IX.2.ll, X.2.9, X.3.4(iii), X.5, XI.7.5, XI.8.3, XII.2.l, XII.4.4(ii), XII.5.5(i), XII.6.3(i), XII.IO.lO(iii), XILl1.15, the Preliminaries to XIIL5, XIIL5.5, XV.7.6(vi), XV.7.l0, Remark(iv) terminating XVI.7.1, XVI.IO. and in some of the problem:, (Prob 1em XIII 85, for example).
These bri ef epi sodes
are intended to indicate to the reader what ought to be done on many other occasions, and to encourage him to undertake some such discussion for himself. In Volume I an unusually marked attention has been
