V. M. Bradis - Lapses in Mathematical Reasoning

LAPSES IN MATHEMATICAL REASONING

V. M. BRADIS, V. L. MINKOVSKII and A. K. KHARCHEVA

Translated by

J. J. SCHORR-KON

English Translation Edited by

E. A. MAXWELL

DOVER PUBLICATIONS, INC.

Mineola, New York

Bibliographical Note

This Dover edition, first published in 1999 and reissued in 2016, is an unabridged and unaltered republication of the English translation of the second Russian edition (1959), as published in The Commonwealth and International Library of Science, Technology, Engineering and Liberal Studies by Pergamon Press, Oxford and London, and the Macmillan Company, New York, in 1963.

Library of Congress Cataloging-in-Publication Data

Bradis, V. M. (Vladimir Modestovich), 1890

[Oshibki v matematicheskikh rassuzhdeniiakh. English]

Lapses in mathematical reasoning / V.M. Bradis, V.L. Minkovskii, and A.K. Kharcheva; translated by J.J. Schorr-Kon ; English translation edited by E.A. Maxwell.Dover ed.

p. cm.

Originally published: Oxford : Pergamon Press ; New York : Macmillan, 1963.

Includes bibliographical references.

ISBN-13: 978-0-486-40918-4

ISBN-10: 0-486-40918-X (pbk.)

1. MathematicsProblems, exercises, etc. 2. Problem solving. I. Minkovskii, V.L. (Vladimir Lvovich) II. Kharcheva, A.K. (Avgusta Konstaninovna) III. Title.

QA43 .B6953 1999

510'.76dc21

99-045828

Manufactured in the United States by RR Donnelley

40918X03 2016

www.doverpublications.com

Contents

From the Preface to the First Edition

I NFINITELY varied lapses occur and keep occurring in the course of mathematical arguments. It is worthwhile to discuss with high-school students some of these errors for two reasons: first, by thoroughly familiarizing ourselves with a mistake we protect ourselves from repeating it in the future; second, it is easy to make the very process of looking for the mistake fascinating for the student, and the study of lapses becomes a means of stimulating interest in the study of mathematics.

In the majority of cases it is easy to lead an argument, in which a given lapse is introduced towards a clearly false conclusion. This gives the semblance of a proof of some obvious absurdity, or a so-called sophism. To analyse a sophism means to point to the lapse which has been introduced in the argument and because of which the absurd conclusion has been made.

Many such erroneous arguments from various branches of mathematics are known and there exist several compilations of them. The present collection is meant for high-school students and contains material of varying difficulty which may be proposed by teachers at most levels. It is valuable to make use of this material in the activities of school mathematics clubs, but some of the problems may be profitably analysed also in the course of ordinary classes, particularly while revising.

We note that in the course of analysing lapses it is absolutely necessary to press for complete clarity: the students should establish clearly wherein consists the lapse contained in the argument and how it is to be corrected. Taking this into account, the .) Such inaccuracies are met not only in the answers of students but also in commonly accepted formulations.

The work of A. K. Kharcheva Mathematical Sophisms and Their Application in the School, presented by her as her thesis for the diploma at the Kalinin Pedagogical Institute, forms the basis of the present collection. The final editing of the book and some additions to the original text are by V. M. Bradis.

.

At that time, before the introduction of state examinations, the graduates of pedagogical institutes had to produce and defend a diploma thesis.

Preface to the Second Edition

T HE book by V. M. Bradis and A. K. Kharcheva Lapses in Mathematical Reasoning, published in 1938 and long out of print, enjoyed in its time, a considerable success among teachers. According to an agreement with the authors I have undertaken its revision for publication. In preparing the new edition I have made use of my paper An Attempt at Classification of Exercises on Refutation of False Mathematical Arguments, printed in 1956 in the Academic Records of the Chairs of the Faculty of Physics and Mathematics of the Orlov State Pedagogical Institute (vol. XI, no. II, ). Also some of the less successful chapters of the first edition of the book are omitted, a few new erroneous arguments are added and the explanations are carried out in separate parts of the respective chapters.

In the present book the false proofs are distributed according to a scheme of classification by subject-matter. This means, that the traditional division of the material into arithmetic, algebra, geometry and trigonometry is retained, but within parts of the school mathematical curriculum, the division of the examples is carried out in accordance with the classification set forth in the first chapter.

In putting together the present compilation the authors have made use of various manuals, among them:

Obreymov, V. 1., Mathematical Sophisms (Matematicheskiye sofizmy), 3rd ed. Petersburg (1898).

Goryachev, D. N., and A. M. Voronetz, Problems, Questions and Sophisms for Mathematics Lovers (Zadachi, voprosy i sofizmy dlya lyubitelei matematiki). Moscow (1903).

Lyamin, A. A., Mathematical Paradoxes and InterestingProblems (Matematicheskiye paradoksy i interesnyye zadachi). Moscow (1911).

Lyanchenkov, M. S., Mathematical Anthology (Matematicheskaya Khrestomatiya). Petersburg (19111912).

I hope that those who have familiarized themselves with the book and have comments thereon will direct them to the mathematical editors of the Uchpedgiz (State Publishing House for Teaching and Pedagogical Literature) at the following address: Moscow, 118, 3ii proezd Marinoi roshchi, dom 41.

V. L. M INKOVSKII

CHAPTER I

Exercises in Refuting Erroneous Mathematical Arguments and their Classification

Introduction

In science every positive or negative assertion may be called a thesis. For example, in proving some theorem, we have a thesis the text of that theorem.

To prove a thesis means to establish its truth. To refute a thesis means to demonstrate its falsity.

The verification of a thesis consists in its proof or refutation.

The refutation of a proof does not necessarily imply the refutation of the thesis. If the thesis is true, then the refutation of the proof testifies only to the fact that incorrect arguments are brought in its defence or else that an error in the argument is introduced. However, the truth of the thesis remains in question as long as the necessary arguments are not presented together with a logically faultless statement of proof.

In checking a proof supporting a true or apparently true thesis, it is by no means easy always to note the presence of an error. The problem becomes much easier if, knowing that an error is actually present in the proof, we start with the particular aim of exhibiting it.

If the thesis expresses a false opinion, then any proof of that thesis will always be false. The ability to refute the proof of a thesis in the case of falsity is just as necessary as the ability to prove a thesis in the case of accuracy.

In the course of political, scientific and everyday disputes, in the process of a court investigation and analysis, in attempts to solve various problems, one must learn not only to prove, but also to refute.

V. I. Lenin, analysing conscious and subconscious errors in the domain of logical thought of his political adversaries, used to recall those arguments which are called mathematical sophisms by mathematicians and in whichin a way that, on the face of it, is strictly logicalit is proven that twice two makes five, that a part is greater than the whole, etc.; and he used to point out that there exist collections of such mathematical sophisms, and they bring profit to schoolchildren.