The authors begin by focusing on exercises in refuting erroneous mathematical arguments and their classification. The remaining chapters discuss examples of false arguments in arithmetic, algebra, geometry, trigonometry, and approximate computations. Ideally, students will come to the correct insights and conclusions on their own; however, each argument is followed by a detailed analysis of the false reasoning. Stimulating and unique, this book is an intriguing and enjoyable way to teach students critical mathematical reasoning skills.
The authors begin by focusing on exercises in refuting erroneous mathematical arguments and their classification. The remaining chapters discuss examples of false arguments in arithmetic, algebra, geometry, trigonometry, and approximate computations. Ideally, students will come to the correct insights and conclusions on their own; however, each argument is followed by a detailed analysis of the false reasoning. Stimulating and unique, this book is an intriguing and enjoyable way to teach students critical mathematical reasoning skills.
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The authors begin by focusing on exercises in refuting erroneous mathematical arguments and their classification. The remaining chapters discuss examples of false arguments in arithmetic, algebra, geometry, trigonometry, and approximate computations. Ideally, students will come to the correct insights and conclusions on their own; however, each argument is followed by a detailed analysis of the false reasoning. Stimulating and unique, this book is an intriguing and enjoyable way to teach students critical mathematical reasoning skills.
Product Details
| ISBN-13: | 9780486816579 |
|---|---|
| Publisher: | Dover Publications |
| Publication date: | 10/28/2016 |
| Series: | Dover Books on Mathematics |
| Sold by: | Barnes & Noble |
| Format: | eBook |
| Pages: | 224 |
| File size: | 23 MB |
| Note: | This product may take a few minutes to download. |
Read an Excerpt
Lapses in Mathematical Reasoning
By V. M. Bradis, V. L. Minkovskii, A. K. Kharcheva, E. A. Maxwell, J. J. Schorr-Kon
Dover Publications, Inc.
Copyright © 2016 V. M. BradisAll rights reserved.
ISBN: 978-0-486-81657-9
CHAPTER 1
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 which — in a way that, on the face of it, is strictly logical — it 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."
In the methodological circular of the Ministry of Education RSFSR "On the Teaching of Mathematics in the grades V–X" (1952, p. 41) it is indicated that "a very useful help for the development of the logical abilities of pupils is given by all kinds of sophisms."
I. Mathematical Sophisms and their Pedagogical Value
Sophism is a word of Greek origin, meaning in translation a wily fabrication, trick, or puzzle. The matter concerns a "proof," aimed at a formally logical establishment of an absurd premise.
Mathematical sophisms involve lapses in mathematical arguments where, even though the result is patently false, the errors leading to them are more or less well concealed. To uncover a sophism means to show the error in the argument by means of which the outer appearance of proof has been created. The demonstration of the error is usually arrived at by counterposing the correct argument against the false one.
In the main, mathematical sophisms are constructed on the basis of incorrect usage of words, on the inaccuracy of formulations, very often on neglecting the conditions of applicability of theorems, on a hidden execution of impossible operations, on invalid generalizations, particularly in passing from a finite number of objects to an infinite one, and on masking of erroneous arguments or assumptions by means of geometrical "obviousness". V. I. Lenin gives a general formulation characterizing sophistry as "... the grasping of the outer coincidence of cases outside the relation of events...."
The intricacy of a mathematical sophism increases with the subtlety of the error concealed in it, with the lack of advance warning in ordinary school instruction on the subject, and with the artfulness of its concealment by inaccuracies of verbal expression. For purposes of concealment one usually complicates the plot of the sophism, i.e. one formulates a situation, for the proving of which one must use some true mathematical propositions, bringing about a distraction which makes the reader look for the error along a false path. In some sophisms this kind of distraction is successfully aided by an optical illusion.
The basic aim of introducing sophisms in school, lies in accustoming the students to critical thinking, to knowing not only how to carry out definite logical schemes and definite processes of thought, but also how to examine critically every stage of an argument in accordance with established principles of mathematical thought and computational practice.
In the opinion of experienced teachers the possibilities of effective applications of mathematical sophisms increase as the students advance through school and their interest in the logical structure of science grows. This work may be proposed in a particularly profound and useful form before a mathematical club of students of the higher grades, where a heightened interest in the logical bases of the methods of logical proof usually shows itself.
Mathematical sophisms require that their texts be read with particular attention and great caution. In studying them one has to seek carefully the proper accuracy of statement and notation, the observance of all the conditions governing theorems, the absence of inadmissible generalizations, forbidden operations, reliance on apparent properties of figures in auxiliary constructions. All these points are methodologically valuable, since they are aimed at a complete mastery of the subject in contradistinction to a formal one, which "is characterized by undue domination in the consciousness and memory of the students of the accustomed outer (verbal, symbolic, or pictorial) expression of the mathematical fact over the content of that fact" (A. Ya. Khinchin).
The degree of mastery of a mathematical fact is considerably strengthened when its perception is stimulated by the absurd allegation contained in the formulation of the sophism.
Exercises in uncovering sophisms do not guarantee the absence of similar errors in the students' own arguments, but they do give the possibility of uncovering and understanding more quickly any error that may appear. This thought, as applied to pedagogy, consists in the fact that mathematical sophisms proposed to students should, as a rule, be used not only to prevent errors, but to check the degree of familiarity and firmness of grasp of the given material. Upon this proposition is based the working practice of our better teachers of mathematics who, to some extent, use sophisms in the concluding stages of Exercises to a given Chapter and in revision.
The pedagogue prevents students' mistakes by a thorough analysis of the concepts studied in class. The teachers' own familiarity with typical students' errors, their origin, and the material of mathematical sophisms helps towards a better attainment of this goal. The degree of the teacher's preparation in this direction is usually shown in the choice of examples and in the clarification of existing variations of a given type, with the aim of preventing the appearance of one-sided associations and incorrect generalizations.
Most teachers agree that in explaining new material, one should, as a rule, avoid fixing the attention of the students upon errors about to appear, in order not to create false intuitive impressions.
Pedagogically warranted use of mathematical sophisms does not exclude the formulation of problems in misleading form, but, on the contrary, often uses it as a preliminary stage of the work as a source of instructive errors. For these problems the student finds no ready-made answers in the teacher's textbook. What is here required of the student is an understanding of the essence of the theoretical material studied, independent thought, and deliberate operation of the known stock of mathematical facts. Some such problems are:
1. When is a/b equal to unity?
2. From the fact that a >b, is it possible to conclude that |a| > |b|?
3. From the equality (a - b)2 = (m - n)2, may one draw the conclusion that a - b = m - n?
4. Does the formula
√(x) × √(y) = √(xy)
hold for all values of x and y?
5. Is the identity
log x2 = 2 log x
valid for all positive values of x?
6. Define the meaning of the symbol [disjunction] in the notation 2a [disnjunction] a, putting a equal to: (a) log 1/2 (b) log cos α, where 0 ≤ α < 90°, and generalize.
7. For what values of x do the following expressions lose their meaning?
x3 - 1/x - 1, 1/x2 - 1, 3/cos x, x/log x.
8. Establish the error and correct the statement of a theorem given by a student of geometry: "A straight line parallel to one of the sides of a triangle cuts from it a triangle similar to the given one."
9. In a right-angled triangle can a median dropped to an arm coincide with the bisectrix?
It is necessary, when applying any mathematical sophism, to instruct a pupil that there should be at his disposal the requisites for uncovering that sophism. The nonobservance of this necessary condition not only completely invalidates the use of sophisms, but also makes them harmful: the student, not being able to find his way in the essence of the problem, helplessly catches hold of external methods, reducing his work to simple guesswork, loses his equilibrium, and develops streaks of indecision. All this, of course, has nothing in common with the problem of gradual and persistent testing of caution in assertions, or with the acknowledged necessity to understand the conditions of a problem and the means for its effective solution. At every point of the course, too, the teacher should be completely candid with the pupil, openly pointing out to him those logical gaps in his exposition which are the result of deliberate pedagogical adjustment.
II. Classification of Exercises in Refuting False Mathematical Arguments
In the history of the development of science an essential role was played by mathematical sophisms (once called paradoxes). They demanded increased attention to the requirements of pithy analysis and of strict proof and have led early to a prolonged prohibition (at least, official) of the use of those concepts and methods which were still not accessible to strict logical elaboration. This makes it easy to understand the early interest in the study, systematization, and pedagogical application of patently false proofs.
The recognition of the pedagogical role of mathematical exercises refuting false proofs suggests an effort to find and characterize their basic forms as a necessary condition if there is to be a rational choice and application of this material in school.
The first attempt to set up a compilation of geometrical sophisms had been understood by the author of the Principles, Euclid of Alexandria. Regrettably, this work of Euclid's, bearing the name of Pseudaria, is considered as irretrievably lost. Proclus (410–485) tells us of its purpose and contents. From his words it is apparent that the work was meant for beginners in geometry. Its aim was to teach students to recognize false conclusions and thus be able to avoid them.
For recognizing errors Euclid sets forth ingenious methods, which he enumerates in a definite order, accompanying them by corresponding exercises. To a false proof Euclid contrasts the correct one and shows that sometimes intuition may serve as the source of error.
The eminent Russian mathematician and pedagogue V. I. Obreimov (1843–1910) has proposed his "attempt at grouping" exercises of such types and has enumerated these groups.
The first three groups of Obreimov's classification (equality of unequals, inequality of equals, and smaller exceeding the greater) embrace those false proofs whose theses contradict the application of the criteria of comparison of magnitudes, i.e. of the concepts greater than, less than, and equal to.
A fourth group consists in geometric absurdities. In it are included deductions in which an absurd conclusion follows from an error in the diagram in the presence of an irreproachable execution of all remaining logical arguments.
A fifth group is formed by "imaginary is real." Here are grouped false proofs connected with the incorrect treatment of the concept of complex numbers.
Obreimov's classification is not free of imperfections. First, in enumerating the forms of the concept being classified, the principle of uniqueness of the basis for classification of the errors is not sustained. Instead there is, on the one hand, the criteria of comparison, and, on the other, the inclusion in geometry or in some portion of the algebra course (complex numbers). Secondly, as the basis of the first three groups of Obreimov's classification, a purely external, rather general criterion of classification has been chosen, which is quite inessential as far as the characterization of false proofs is concerned. Because of this, material referring to the clarification of a given error is distributed over different divisions. Misunderstandings in connexion with division by zero are expounded in the first and second divisions, whereas misunderstandings connected with failing to change the sign in multiplying both parts of an inequality by a negative number are set forth in the second and third chapters, and so on.
A positive feature of Obreimov's classification is the separation into a distinct group of false proofs based on errors in construction.
We now proceed to study the proposals for a classification of false proofs by the German scientist Herman Schubert (1848–1911). He calls for a distinction of four forms of false proofs, based on division by zero, on the ambiguity of the square root, on geometrical deception (error in construction), and on ascribing an infinitely great value to the sum of an infinite set of numbers.
What is important in Schubert's classification is the division of false proofs according to the character of the errors which lead to them. However, classification according to the principles chosen has remained undeveloped. The four forms of false proofs enumerated by Schubert do not exhaust even the minimal content of the concept under consideration. In particular, nothing was said about false proofs constructed on the basis of a mistaken trust in geometrical intuition in those cases when there is no direct geometrical deception.
The French pedagogue and historian of mathematics E. Fourier relegates to geometrical sophisms all those sophisms whose formulations relate to geometrical objects. Thus, he includes here also geometrical embodiments of algebraic sophisms, based on the presence of purely algebraic errors. Of course, these errors may be masked not only by various geometrical assumptions but also by assumptions referring to other branches of mathematics. It is clear that a classification based on the outward form of the erroneous argument is a purely external classification.
The geometrical sophisms in the broad sense indicated are divided by Fourier into two forms: those based on errors in construction and those based on errors in argument. In the errors of argument Fourier distinguishes between errors connected with deviation from precise definitions and those connected with the carrying out of inadmissible operations on numbers.
We now give our attempt to classify exercises about refuting false mathematical arguments into the more important classes of errors in speech and thought which, usually without even naming them, the teacher stubbornly fights in his daily work both in the teaching of theory and in carrying out the exercises.
The proposed classification is aimed, to begin with, at the teacher. With this goal in view, the very name emphasizes the specific pedagogical intention of a given exercise for each form of error, thus creating the possibility of fast orientation in the material and preventing its unsystematic use.
Our teaching experience allows us to assert that this classification is useful also to students of the higher grades, whose arguments begin to be executed not only in correspondence with definite principles but also on the basis of consciousness of these principles. During this transition period the students feel a continual need to verify the accuracy of their knowledge, their logical development, the degree of their comprehension, and the adequacy of their verbal expression.
We are far from the thought that the proposed classification is free from imperfections which are characteristic, for example, of the classification of arithmetical problems according to the essential peculiarities of the methods of their solution. The use of a wider working experience and criticism will help to suggest the necessary corrections. However we take the liberty of thinking that, even in its present form, it represents a certain step forward in comparison with existing classifications pursuing, just as this one, purely pedagogical aims.
We now go on to the actual consideration of the problem.
1. Incorrectness of speech
A systematic analysis of sophisms was first given by Aristotle (384–322 B.C.) in a special treatise devoted to the refutation of sophistries in which all the errors are divided into two classes: "incorrectnesses of speech" and errors "outside speech," i.e. in thinking.
There is no need to prove that every lesson correctly planned and executed in the subject of mathematics is at the same time also a lesson in developing the powers of exposition of the students. On the pages of methodological literature one can often see the emphasis of the favourable influence of mathematics upon the improvement of the student's gift of speech, in the sense of its precision and consistency. However, these aims are not attained automatically. To attain them, daily work is necessary on the part of the mathematics teacher as he watches the student's choice of words, the form of expression of his thoughts, both in verbal answers and in written tasks. The intensive elimination of faults encountered in the students' speech involves interesting the students themselves in correcting the answers of their fellows. It should be distinctly brought to the attention of the students that irregularities of speech not only make the study of mathematics more difficult but are also responsible for numerous other fallacies.
(Continues...)
Excerpted from Lapses in Mathematical Reasoning by V. M. Bradis, V. L. Minkovskii, A. K. Kharcheva, E. A. Maxwell, J. J. Schorr-Kon. Copyright © 2016 V. M. Bradis. Excerpted by permission of Dover Publications, Inc..
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Table of Contents
Designed as a method for teaching correct mathematical thinking to high school students, this book contains a brilliantly constructed series of what the authors call "lapses," erroneous statements that are part of a larger mathematical argument. These lapses lead to sophism or mathematical absurdities. The ingenious idea behind this technique is to lead the student deliberately toward a clearly false conclusion. The teacher and student then go back and analyze the lapse as a way to correct the problem.
The authors begin by focusing on exercises in refuting erroneous mathematical arguments and their classification. The remaining chapters discuss examples of false arguments in arithmetic, algebra, geometry, trigonometry, and approximate computations. Ideally, students will come to the correct insights and conclusions on their own; however, each argument is followed by a detailed analysis of the false reasoning. Stimulating and unique, this book is an intriguing and enjoyable way to teach students critical mathematical reasoning skills.
Dover (2016) republication of the edition originally published by the Pergamon Press, Oxford and London, and the Macmillan Company, New York, 1963.
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