Mathematical Fallacies and Paradoxes

245 Pages·1997·2.68 MB·English

by  Bryan Bunch

Copyright Copyright © 1982 by Bryan H. Bunch All rights reserved.

Bibliographical Note This Dover edition, first published in 1997, is an
unabridged and corrected republication of the work first published by The Van
Nostrand Reinhold Company, New York, in 1982.

Library of Congress Cataloging-in-Publication Data Bunch, Bryan H.
Mathematical fallacies and paradoxes / Bryan Bunch.
p. cm.
Originally published: New York : Van Nostrand Reinhold Co., c1982.
9780486137933
1. Logic, Symbolic and mathematical. I. Title.
QA9.B847 1997
97-15386
511.3—dc21
CIP

Manufactured in the United States by Courier Corporation
29664406
www.doverpublications.com
To
Arline H. Bunch
who always expected me to
write a book
Mathematics is often erroneously referred to as the science of common sense.
Edward Kasner and James R. Newman

Mathematics may be defined as the subject in which we never know what we are
talking about nor whether what we are saying is true.
Bertrand Russell

There is no permanent place in the world for ugly mathematics.
G. H. Hardy

Cantor was aware of this problem but took it in his stride, which I think was very
perceptive of him. Russell also found out about it, and he worried about it.
J. N. Crossley

; ; ;this was sometime a paradox, but now time gives it proof.
William Shakespeare
Preface
This book is a collection and analysis of the most interesting paradoxes and
fallacies from mathematics, logic, physics, and language. It also treats important
results in mathematics that are based in paradox, notably Gödel’s theorem of
1931 and decision problems in general.
The material is arranged so that the rather tenuous relationship between
mathematical reality and physical reality becomes the subject of the book, while
the paradoxes and fallacies are tools for exploring this relationship. As a result,
although the material contains a number of topics that are often presented in an
anthology format, there is a definite progression from the first chapter to the
eighth. It is possible, however, to read most of the individual paradoxes or
fallacies in whatever order takes one’s fancy.
The first three chapters are largely concerned with examples that today are
generally classed as fallacies. As such, they have specific defects in the
mathematics, defects upon which all mathematicians are in agreement.
It seems appropriate to encourage the reader to try finding those defects.
Therefore, I stop at a point where the presentation of the fallacy is complete and
ask: Can You Find the Flaw? I also provide a hint. The remaining chapters deal
with topics for which there is no single, accepted explanation, so this feature is
dropped in Chapters 4 through 8.
I assume throughout that the reader has some experience with the content of
first-year high-school algebra. A few of the results also draw upon parts of high-
school geometry. Any mathematics that is needed beyond these levels is
developed as a topic in the body of the book. This includes a brief introduction
to the basic ideas of complex numbers in Chapter 1; mathematical induction, the
notion of the limit of a series, and some ideas from probability in Chapter 2;
indirect proof in Chapter 3; and elementary set theory in Chapter 5. These are all
necessary to a complete understanding of many of the paradoxes. Even so, there
are some mathematical complexities that I have omitted deliberately. For
example, the discussion of Gödel’s incompleteness theorem is necessarily
simplified (even the type of incompleteness to which it applies is omitted), as is
the analysis of special relativity. These simplifications in no way affect the
the analysis of special relativity. These simplifications in no way affect the
results stated. In a few cases, it has seemed better to state results without proof,
rather than to become bogged down in lengthy and difficult mathematical
development.
I wish to express my gratitude to Dr. Phillip S. Jones of the University of
Michigan for his thoughtful comments on much of the material in the
manuscript; and to my wife, Mary, for retyping the whole manuscript, as well as
for her help and support in so many other ways.

Briarcliff Manor, NY
B.H.B.
Table of Contents
Title Page
Copyright Page
Dedication
Preface
1 - Thinking Wrong about Easy Ideas
2 - Thinking Wrong about Infinity
3 - Using a Wrong Idea to Find Truth
4 - Speaking with Forked Tongue
5 - Paradoxes That Count
6 - The Limits of Thought?
7 - Misunderstanding Space and Time
8 - Moving Against Infinity
Selected Further Reading
Index