HOME

TheInfoList



OR:

In
mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, certain kinds of mistaken proof are often exhibited, and sometimes collected, as illustrations of a concept called mathematical fallacy. There is a distinction between a simple ''mistake'' and a ''mathematical fallacy'' in a proof, in that a mistake in a proof leads to an invalid proof while in the best-known examples of mathematical fallacies there is some element of concealment or deception in the presentation of the proof. For example, the reason why validity fails may be attributed to a
division by zero In mathematics, division by zero is division where the divisor (denominator) is zero. Such a division can be formally expressed as \tfrac, where is the dividend (numerator). In ordinary arithmetic, the expression has no meaning, as there is ...
that is hidden by algebraic notation. There is a certain quality of the mathematical fallacy: as typically presented, it leads not only to an absurd result, but does so in a crafty or clever way. Therefore, these fallacies, for pedagogic reasons, usually take the form of spurious
proofs Proof most often refers to: * Proof (truth), argument or sufficient evidence for the truth of a proposition * Alcohol proof, a measure of an alcoholic drink's strength Proof may also refer to: Mathematics and formal logic * Formal proof, a co ...
of obvious
contradiction In traditional logic, a contradiction occurs when a proposition conflicts either with itself or established fact. It is often used as a tool to detect disingenuous beliefs and bias. Illustrating a general tendency in applied logic, Aristotle's ...
s. Although the proofs are flawed, the errors, usually by design, are comparatively subtle, or designed to show that certain steps are conditional, and are not applicable in the cases that are the exceptions to the rules. The traditional way of presenting a mathematical fallacy is to give an invalid step of deduction mixed in with valid steps, so that the meaning of
fallacy A fallacy is the use of invalid or otherwise faulty reasoning, or "wrong moves," in the construction of an argument which may appear stronger than it really is if the fallacy is not spotted. The term in the Western intellectual tradition was intr ...
is here slightly different from the
logical fallacy In philosophy, a formal fallacy, deductive fallacy, logical fallacy or non sequitur (; Latin for " tdoes not follow") is a pattern of reasoning rendered invalid by a flaw in its logical structure that can neatly be expressed in a standard logic syst ...
. The latter usually applies to a form of argument that does not comply with the valid inference rules of logic, whereas the problematic mathematical step is typically a correct rule applied with a tacit wrong assumption. Beyond pedagogy, the resolution of a fallacy can lead to deeper insights into a subject (e.g., the introduction of
Pasch's axiom In geometry, Pasch's axiom is a statement in plane geometry, used implicitly by Euclid, which cannot be derived from the postulates as Euclid gave them. Its essential role was discovered by Moritz Pasch in 1882. Statement The axiom states tha ...
of
Euclidean geometry Euclidean geometry is a mathematical system attributed to ancient Greek mathematician Euclid, which he described in his textbook on geometry: the '' Elements''. Euclid's approach consists in assuming a small set of intuitively appealing axioms ...
, the five colour theorem of
graph theory In mathematics, graph theory is the study of '' graphs'', which are mathematical structures used to model pairwise relations between objects. A graph in this context is made up of '' vertices'' (also called ''nodes'' or ''points'') which are conn ...
). ''Pseudaria'', an ancient lost book of false proofs, is attributed to
Euclid Euclid (; grc-gre, Εὐκλείδης; BC) was an ancient Greek mathematician active as a geometer and logician. Considered the "father of geometry", he is chiefly known for the '' Elements'' treatise, which established the foundations of ...
. Mathematical fallacies exist in many branches of mathematics. In
elementary algebra Elementary algebra encompasses the basic concepts of algebra. It is often contrasted with arithmetic: arithmetic deals with specified numbers, whilst algebra introduces variables (quantities without fixed values). This use of variables entail ...
, typical examples may involve a step where
division by zero In mathematics, division by zero is division where the divisor (denominator) is zero. Such a division can be formally expressed as \tfrac, where is the dividend (numerator). In ordinary arithmetic, the expression has no meaning, as there is ...
is performed, where a
root In vascular plants, the roots are the organs of a plant that are modified to provide anchorage for the plant and take in water and nutrients into the plant body, which allows plants to grow taller and faster. They are most often below the su ...
is incorrectly extracted or, more generally, where different values of a multiple valued function are equated. Well-known fallacies also exist in elementary Euclidean geometry and
calculus Calculus, originally called infinitesimal calculus or "the calculus of infinitesimals", is the mathematics, mathematical study of continuous change, in the same way that geometry is the study of shape, and algebra is the study of generalizati ...
.


Howlers

Examples exist of mathematically correct results derived by incorrect lines of reasoning. Such an argument, however true the conclusion appears to be, is mathematically
invalid Invalid may refer to: * Patient, a sick person * one who is confined to home or bed because of illness, disability or injury (sometimes considered a politically incorrect term) * .invalid, a top-level Internet domain not intended for real use As ...
and is commonly known as a ''howler''. The following is an example of a howler involving anomalous cancellation: \frac = \frac=\frac. Here, although the conclusion = is correct, there is a fallacious, invalid cancellation in the middle step.The same fallacy also applies to the following: \begin \frac = \frac &= \frac \\ \frac = \frac &= \frac \\ \frac = \frac &= \frac = \frac \end Another classical example of a howler is proving the Cayley–Hamilton theorem by simply substituting the scalar variables of the characteristic polynomial by the matrix. Bogus proofs, calculations, or derivations constructed to produce a correct result in spite of incorrect logic or operations were termed "howlers" by Maxwell. Outside the field of mathematics the term ''howler'' has various meanings, generally less specific.


Division by zero

The division-by-zero fallacy has many variants. The following example uses a disguised division by zero to "prove" that 2 = 1, but can be modified to prove that any number equals any other number. # Let ''a'' and ''b'' be equal, nonzero quantities #:a = b # Multiply by ''a'' #:a^2 = ab # Subtract ''b''2 #:a^2 - b^2 = ab - b^2 #
Factor Factor, a Latin word meaning "who/which acts", may refer to: Commerce * Factor (agent), a person who acts for, notably a mercantile and colonial agent * Factor (Scotland), a person or firm managing a Scottish estate * Factors of production, suc ...
both sides: the left factors as a
difference of squares In mathematics, the difference of two squares is a squared (multiplied by itself) number subtracted from another squared number. Every difference of squares may be factored according to the identity :a^2-b^2 = (a+b)(a-b) in elementary algebra. P ...
, the right is factored by extracting ''b'' from both terms #:(a - b)(a + b) = b(a - b) # Divide out (''a'' − ''b'') #:a + b = b # Use the fact that ''a'' = ''b'' #:b + b = b # Combine like terms on the left #:2b = b # Divide by the non-zero ''b'' #:2 = 1 :''
Q.E.D. Q.E.D. or QED is an initialism of the Latin phrase , meaning "which was to be demonstrated". Literally it states "what was to be shown". Traditionally, the abbreviation is placed at the end of mathematical proofs and philosophical arguments in pri ...
'' The fallacy is in line 5: the progression from line 4 to line 5 involves division by ''a'' − ''b'', which is zero since ''a'' = ''b''. Since
division by zero In mathematics, division by zero is division where the divisor (denominator) is zero. Such a division can be formally expressed as \tfrac, where is the dividend (numerator). In ordinary arithmetic, the expression has no meaning, as there is ...
is undefined, the argument is invalid.


Analysis

Mathematical analysis Analysis is the branch of mathematics dealing with continuous functions, limits, and related theories, such as differentiation, integration, measure, infinite sequences, series, and analytic functions. These theories are usually studied ...
as the mathematical study of change and
limits Limit or Limits may refer to: Arts and media * ''Limit'' (manga), a manga by Keiko Suenobu * ''Limit'' (film), a South Korean film * Limit (music), a way to characterize harmony * "Limit" (song), a 2016 single by Luna Sea * "Limits", a 2019 ...
can lead to mathematical fallacies — if the properties of
integral In mathematics, an integral assigns numbers to functions in a way that describes displacement, area, volume, and other concepts that arise by combining infinitesimal data. The process of finding integrals is called integration. Along with ...
s and differentials are ignored. For instance, a naive use of
integration by parts In calculus, and more generally in mathematical analysis, integration by parts or partial integration is a process that finds the integral of a product of functions in terms of the integral of the product of their derivative and antiderivat ...
can be used to give a false proof that 0 = 1. Letting ''u'' =  and ''dv'' = , we may write: : \int \frac \, dx = 1 + \int \frac \, dx after which the antiderivatives may be cancelled yielding 0 = 1. The problem is that antiderivatives are only defined
up to Two mathematical objects ''a'' and ''b'' are called equal up to an equivalence relation ''R'' * if ''a'' and ''b'' are related by ''R'', that is, * if ''aRb'' holds, that is, * if the equivalence classes of ''a'' and ''b'' with respect to ''R'' ...
a constant and shifting them by 1 or indeed any number is allowed. The error really comes to light when we introduce arbitrary integration limits ''a'' and ''b''. : \int_a^b \frac \, dx = 1 , _a^b + \int_a^b \frac \, dx = 0 + \int_a^b \frac \, dx = \int_a^b \frac \, dx Since the difference between two values of a constant function vanishes, the same definite integral appears on both sides of the equation.


Multivalued functions

Many functions do not have a unique
inverse Inverse or invert may refer to: Science and mathematics * Inverse (logic), a type of conditional sentence which is an immediate inference made from another conditional sentence * Additive inverse (negation), the inverse of a number that, when a ...
. For instance, while squaring a number gives a unique value, there are two possible
square root In mathematics, a square root of a number is a number such that ; in other words, a number whose '' square'' (the result of multiplying the number by itself, or  ⋅ ) is . For example, 4 and −4 are square roots of 16, because . ...
s of a positive number. The square root is multivalued. One value can be chosen by convention as the
principal value In mathematics, specifically complex analysis, the principal values of a multivalued function are the values along one chosen branch of that function, so that it is single-valued. The simplest case arises in taking the square root of a posit ...
; in the case of the square root the non-negative value is the principal value, but there is no guarantee that the square root given as the principal value of the square of a number will be equal to the original number (e.g. the principal square root of the square of −2 is 2). This remains true for
nth root In mathematics, a radicand, also known as an nth root, of a number ''x'' is a number ''r'' which, when raised to the power ''n'', yields ''x'': :r^n = x, where ''n'' is a positive integer, sometimes called the ''degree'' of the root. A root ...
s.


Positive and negative roots

Care must be taken when taking the
square root In mathematics, a square root of a number is a number such that ; in other words, a number whose '' square'' (the result of multiplying the number by itself, or  ⋅ ) is . For example, 4 and −4 are square roots of 16, because . ...
of both sides of an equality. Failing to do so results in a "proof" of 5 = 4. Proof: :Start from ::-20 = -20 :Write this as ::25-45 = 16-36 :Rewrite as ::5^2-5\times9 = 4^2-4\times9 :Add on both sides: ::5^2-5\times9+\frac = 4^2-4\times9+\frac :These are perfect squares: ::\left(5-\frac\right)^2 = \left(4-\frac\right)^2 :Take the square root of both sides: ::5-\frac = 4-\frac :Add on both sides: ::5 = 4 :''
Q.E.D. Q.E.D. or QED is an initialism of the Latin phrase , meaning "which was to be demonstrated". Literally it states "what was to be shown". Traditionally, the abbreviation is placed at the end of mathematical proofs and philosophical arguments in pri ...
'' The fallacy is in the second to last line, where the square root of both sides is taken: ''a''2 = ''b''2 only implies ''a'' = ''b'' if ''a'' and ''b'' have the same sign, which is not the case here. In this case, it implies that ''a'' = –''b'', so the equation should read :5-\frac = -\left(4-\frac\right) which, by adding on both sides, correctly reduces to 5 = 5. Another example illustrating the danger of taking the square root of both sides of an equation involves the following fundamental identity :\cos^2x=1-\sin^2x which holds as a consequence of the
Pythagorean theorem In mathematics, the Pythagorean theorem or Pythagoras' theorem is a fundamental relation in Euclidean geometry between the three sides of a right triangle. It states that the area of the square whose side is the hypotenuse (the side opposit ...
. Then, by taking a square root, :\cos x = \sqrt Evaluating this when ''x'' =  , we get that :-1 = \sqrt or :-1 = 1 which is incorrect. The error in each of these examples fundamentally lies in the fact that any equation of the form :x^2 = a^2 where a \ne 0, has two solutions: :x=\pm a and it is essential to check which of these solutions is relevant to the problem at hand. In the above fallacy, the square root that allowed the second equation to be deduced from the first is valid only when cos ''x'' is positive. In particular, when ''x'' is set to , the second equation is rendered invalid.


Square roots of negative numbers

Invalid proofs utilizing powers and roots are often of the following kind: :1 = \sqrt = \sqrt = \sqrt\sqrt=i \cdot i = -1. The fallacy is that the rule \sqrt = \sqrt\sqrt is generally valid only if at least one of x and ''y'' is non-negative (when dealing with real numbers), which is not the case here. Alternatively, imaginary roots are obfuscated in the following: :i=\sqrt = \left(-1\right)^\frac = \left(\left(-1\right)^2\right)^\frac = 1^\frac = 1 The error here lies in the third equality, as the rule a^ = (a^b)^c only holds for positive real ''a'' and real ''b'', ''c''.


Complex exponents

When a number is raised to a complex power, the result is not uniquely defined (see ). If this property is not recognized, then errors such as the following can result: : \begin e^ &= 1 \\ \left(e^\right)^ &= 1^ \\ e^ &= 1 \\ \end The error here is that the rule of multiplying exponents as when going to the third line does not apply unmodified with complex exponents, even if when putting both sides to the power ''i'' only the principal value is chosen. When treated as
multivalued function In mathematics, a multivalued function, also called multifunction, many-valued function, set-valued function, is similar to a function, but may associate several values to each input. More precisely, a multivalued function from a domain to a ...
s, both sides produce the same set of values, being .


Geometry

Many mathematical fallacies in
geometry Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is c ...
arise from using an additive equality involving oriented quantities (such as adding vectors along a given line or adding oriented angles in the plane) to a valid identity, but which fixes only the absolute value of (one of) these quantities. This quantity is then incorporated into the equation with the wrong orientation, so as to produce an absurd conclusion. This wrong orientation is usually suggested implicitly by supplying an imprecise diagram of the situation, where relative positions of points or lines are chosen in a way that is actually impossible under the hypotheses of the argument, but non-obviously so. In general, such a fallacy is easy to expose by drawing a precise picture of the situation, in which some relative positions will be different from those in the provided diagram. In order to avoid such fallacies, a correct geometric argument using addition or subtraction of distances or angles should always prove that quantities are being incorporated with their correct orientation.


Fallacy of the isosceles triangle

The fallacy of the isosceles triangle, from , purports to show that every
triangle A triangle is a polygon with three edges and three vertices. It is one of the basic shapes in geometry. A triangle with vertices ''A'', ''B'', and ''C'' is denoted \triangle ABC. In Euclidean geometry, any three points, when non- colline ...
is
isosceles In geometry, an isosceles triangle () is a triangle that has two sides of equal length. Sometimes it is specified as having ''exactly'' two sides of equal length, and sometimes as having ''at least'' two sides of equal length, the latter versio ...
, meaning that two sides of the triangle are
congruent Congruence may refer to: Mathematics * Congruence (geometry), being the same size and shape * Congruence or congruence relation, in abstract algebra, an equivalence relation on an algebraic structure that is compatible with the structure * In mod ...
. This fallacy was known to
Lewis Carroll Charles Lutwidge Dodgson (; 27 January 1832 – 14 January 1898), better known by his pen name Lewis Carroll, was an English author, poet and mathematician. His most notable works are '' Alice's Adventures in Wonderland'' (1865) and its sequ ...
and may have been discovered by him. It was published in 1899. Given a triangle △ABC, prove that AB = AC: # Draw a line bisecting ∠A. # Draw the perpendicular bisector of segment BC, which bisects BC at a point D. # Let these two lines meet at a point O. # Draw line OR perpendicular to AB, line OQ perpendicular to AC. # Draw lines OB and OC. # By AAS, △RAO ≅ △QAO (∠ORA = ∠OQA = 90°; ∠RAO = ∠QAO; AO = AO (common side)). # By RHS,Hypotenuse–leg congruence △ROB ≅ △QOC (∠BRO = ∠CQO = 90°; BO = OC (hypotenuse); RO = OQ (leg)). # Thus, AR = AQ, RB = QC, and AB = AR + RB = AQ + QC = AC. ''Q.E.D.'' As a corollary, one can show that all triangles are equilateral, by showing that AB = BC and AC = BC in the same way. The error in the proof is the assumption in the diagram that the point O is ''inside'' the triangle. In fact, O always lies on the circumcircle of the △ABC (except for isosceles and equilateral triangles where AO and OD coincide). Furthermore, it can be shown that, if AB is longer than AC, then R will lie ''within'' AB, while Q will lie ''outside'' of AC, and vice versa (in fact, any diagram drawn with sufficiently accurate instruments will verify the above two facts). Because of this, AB is still AR + RB, but AC is actually AQ − QC; and thus the lengths are not necessarily the same.


Proof by induction

There exist several fallacious proofs by induction in which one of the components, basis case or inductive step, is incorrect. Intuitively, proofs by induction work by arguing that if a statement is true in one case, it is true in the next case, and hence by repeatedly applying this, it can be shown to be true for all cases. The following "proof" shows that all horses are the same colour.
George Pólya George Pólya (; hu, Pólya György, ; December 13, 1887 – September 7, 1985) was a Hungarian mathematician. He was a professor of mathematics from 1914 to 1940 at ETH Zürich and from 1940 to 1953 at Stanford University. He made fundamenta ...
's original "proof" was that any ''n'' girls have the same colour eyes.
# Let us say that any group of ''N'' horses is all of the same colour. # If we remove a horse from the group, we have a group of ''N'' − 1 horses of the same colour. If we add another horse, we have another group of ''N'' horses. By our previous assumption, all the horses are of the same colour in this new group, since it is a group of ''N'' horses. # Thus we have constructed two groups of ''N'' horses all of the same colour, with ''N'' − 1 horses in common. Since these two groups have some horses in common, the two groups must be of the same colour as each other. # Therefore, combining all the horses used, we have a group of ''N'' + 1 horses of the same colour. # Thus if any ''N'' horses are all the same colour, any ''N'' + 1 horses are the same colour. # This is clearly true for ''N'' = 1 (i.e. one horse is a group where all the horses are the same colour). Thus, by induction, ''N'' horses are the same colour for any positive integer ''N''. i.e. all horses are the same colour. The fallacy in this proof arises in line 3. For ''N'' = 1, the two groups of horses have ''N'' − 1 = 0 horses in common, and thus are not necessarily the same colour as each other, so the group of ''N'' + 1 = 2 horses is not necessarily all of the same colour. The implication "every ''N'' horses are of the same colour, then ''N'' + 1 horses are of the same colour" works for any ''N'' > 1, but fails to be true when ''N'' = 1. The basis case is correct, but the induction step has a fundamental flaw.


See also

* * * * * *


Notes


References

* . * . * . * .


External links


Invalid proofs
at
Cut-the-knot Alexander Bogomolny (January 4, 1948 July 7, 2018) was a Soviet-born Israeli-American mathematician. He was Professor Emeritus of Mathematics at the University of Iowa, and formerly research fellow at the Moscow Institute of Electronics and Math ...
(including literature references)
Classic fallacies
with some discussion
Math jokes including an invalid proof
{{DEFAULTSORT:Mathematical fallacy Recreational mathematics Proof theory *