View source for Mathematical fallacy

{{Redirect|Invalid proof|any type of invalid proof besides mathematics|Fallacy}}
{{Redirect|1=0 = 1|2=the algebraic structure where this equality holds|3=Null ring}}
{{short description|Certain type of mistaken proof}}
In [[mathematics]], certain kinds of mistaken [[Mathematical proof|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]] 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.<ref>{{harvnb|Maxwell|1959|p=9}}</ref> Therefore, these fallacies, for pedagogic reasons, usually take the form of spurious [[Mathematical proof|proofs]] of obvious [[contradiction]]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]] is here slightly different from the [[logical fallacy]]. 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]] of [[Euclidean geometry]],<ref name="Maxwell 1959">{{harvnb|Maxwell|1959}}</ref> the [[five color theorem|five colour theorem]] of [[graph theory]]). ''Pseudaria'', an ancient lost book of false proofs, is attributed to [[Euclid]].<ref>{{harvnb|Heath|Heiberg|1908|loc=Chapter II, §I}}</ref>

Mathematical fallacies exist in many branches of mathematics.  In [[elementary algebra]], typical examples may involve a step where [[division by zero]] is performed, where a [[root of a function|root]] 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]].<ref>{{Cite journal|last=Barbeau|first=Ed|date=1991|title=Fallacies, Flaws, and Flimflam|url=https://www.maa.org/sites/default/files/pdf/mathdl/CMJ/barbeau.pdf|journal=The College Mathematics Journal|volume=22|issue=5|issn=0746-8342}}</ref><ref>{{Cite web|url=https://math.stackexchange.com/q/348198 |title=soft question – Best Fake Proofs? (A M.SE April Fools Day collection)|website=Mathematics Stack Exchange|access-date=2019-10-24}}</ref>

==Howlers==
{{image frame|width=150|caption=Anomalous cancellation in calculus|border=no|content=<math>\begin{array}{l}
\;\;\; \dfrac     {d}      {dx} \dfrac{1}{x}
\\  =  \dfrac     {d}      {d}  \dfrac{1}{x^2}
\\  =  \dfrac{d\!\!\!\backslash}{d\!\!\!\backslash} \dfrac{1}{x^2}
\\  =  -                        \dfrac{1}{x^2}
\end{array}</math>}}
Examples exist of mathematically correct results derived by incorrect lines of reasoning. Such an argument, however true the conclusion appears to be, is mathematically [[Validity (logic)|invalid]] and is commonly known as a ''howler''. The following is an example of a howler involving [[anomalous cancellation]]:
<math display=block>\frac{16}{64} = \frac{16\!\!\!/}{6\!\!\!/4}=\frac{1}{4}.</math>

Here, although the conclusion {{sfrac|16|64}} = {{sfrac|1|4}} is correct, there is a fallacious, invalid cancellation in the middle step.<ref group="note">The same fallacy also applies to the following:

<math display=block>\begin{align}
\frac{19}{95} = \frac{19\!\!\!/}{9\!\!\!/5} &= \frac{1}{5} \\
\frac{26}{65} = \frac{26\!\!\!/}{6\!\!\!/5} &= \frac{2}{5} \\
\frac{49}{98} = \frac{49\!\!\!/}{9\!\!\!/8} &= \frac{4}{8} = \frac{1}{2}
\end{align}</math></ref> Another classical example of a howler is [[Cayley–Hamilton theorem#A bogus "proof": p(A) = det(AIn − A) = det(A − A) = 0|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 [[Edwin A. Maxwell|Maxwell]].<ref name="Maxwell 1959"/> Outside the field of mathematics the term ''howler'' has various meanings, generally less specific.

== Division by zero ==
The [[Division by zero|division-by-zero fallacy]] has many variants. The following example uses a disguised division by zero to "prove" that 2&nbsp;=&nbsp;1, but can be modified to prove that any number equals any other number.

# Let ''a'' and ''b'' be equal, nonzero quantities
#:<math>a = b</math>
# Multiply by ''a''
#:<math>a^2 = ab</math>
# Subtract ''b''<sup>2</sup>
#:<math>a^2 - b^2 = ab - b^2</math>
# [[Factorization|Factor]] both sides: the left factors as a [[difference of squares]], the right is factored by extracting ''b'' from both terms
#:<math>(a - b)(a + b) = b(a - b)</math>
# Divide out (''a'' − ''b'')
#:<math>a + b = b</math>
# Use the fact that ''a'' = ''b''
#:<math>b + b = b</math>
# Combine like terms on the left
#:<math>2b = b</math>
# Divide by the non-zero ''b''
#:<math>2 = 1</math>
:''[[Q.E.D.]]''<ref>{{citation|first=Harro|last=Heuser|title=Lehrbuch der Analysis – Teil 1|edition=6th|publisher=Teubner|year=1989|isbn=978-3-8351-0131-9|page=51}}</ref>

The fallacy is in line 5: the progression from line 4 to line 5 involves division by ''a''&nbsp;−&nbsp;''b'', which is zero since ''a''&nbsp;=&nbsp;''b''. Since [[division by zero]] is undefined, the argument is invalid.

==Analysis==
[[Mathematical analysis]] as the mathematical study of change and [[limit of a function|limits]] can lead to mathematical fallacies — if the properties of [[integral]]s and [[differential (mathematics)|differentials]] are ignored. For instance, a naive use of [[integration by parts]] can be used to give a false proof that 0&nbsp;=&nbsp;1.<ref>{{citation|first=Ed|last=Barbeau|journal=The College Mathematics Journal|volume=21|number=3|year=1990|pages=216–218|title=Fallacies, Flaws and Flimflam #19: Dolt's Theorem|doi=10.1080/07468342.1990.11973308}}</ref> Letting ''u''&nbsp;=&nbsp;{{sfrac|1|[[common logarithm|log]] ''x''}} and ''dv''&nbsp;=&nbsp;{{sfrac|''dx''|''x''}}, we may write:

: <math>\int \frac{1}{x \, \log x} \, dx = 1 + \int \frac{1}{x \, \log x} \, dx</math>

after which the antiderivatives may be cancelled yielding 0&nbsp;=&nbsp;1. The problem is that antiderivatives are only defined [[up to]] a [[Constant function|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''.

: <math>\int_a^b \frac{1}{x \, \log x} \, dx = 1 |_a^b + \int_a^b \frac{1}{x \, \log x} \, dx = 0 + \int_a^b \frac{1}{x \log x} \, dx = \int_a^b \frac{1}{x \log x} \, dx</math>

Since the difference between two values of a constant function vanishes, the same definite integral appears on both sides of the equation.

== Multivalued functions ==
{{Main article |Multivalued function}}
Many functions do not have a unique [[inverse function|inverse]]. For instance, while squaring a number gives a unique value, there are two possible [[square root]]s of a positive number. The square root is [[multivalued function|multivalued]]. One value can be chosen by convention as the [[principal value]]; 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]]s.

=== Positive and negative roots ===

Care must be taken when taking the [[square root]] of both sides of an [[Equality (mathematics)|equality]]. Failing to do so results in a "proof" of<ref>{{cite book |title=Mathematical Fun, Games and Puzzles |edition=illustrated |first1=Jack |last1=Frohlichstein |publisher=Courier Corporation |year=1967 |isbn=0-486-20789-7 |page=207 |url=https://books.google.com/books?id=w7CVzMosF-kC}} [https://books.google.com/books?id=w7CVzMosF-kC&pg=PA207 Extract of page 207]</ref> 5&nbsp;=&nbsp;4.

Proof:
:Start from
::<math>-20 = -20</math>
:Write this as
::<math>25-45 = 16-36</math>
:Rewrite as
::<math>5^2-5\times9 = 4^2-4\times9</math>
:Add {{sfrac|81|4}} on both sides:
::<math>5^2-5\times9+\frac{81}{4} = 4^2-4\times9+\frac{81}{4}</math>
:These are perfect squares:
::<math>\left(5-\frac{9}{2}\right)^2 = \left(4-\frac{9}{2}\right)^2</math>
:Take the square root of both sides:
::<math>5-\frac{9}{2} = 4-\frac{9}{2}</math>
:Add {{sfrac|9|2}} on both sides:
::<math>5 = 4</math>
:''[[Q.E.D.]]''

The fallacy is in the second to last line, where the square root of both sides is taken: ''a''<sup>2</sup>&nbsp;=&nbsp;''b''<sup>2</sup> only implies ''a''&nbsp;=&nbsp;''b'' if ''a'' and ''b'' have the same sign, which is not the case here. In this case, it implies that ''a''&nbsp;=&nbsp;–''b'', so the equation should read
:<math>5-\frac{9}{2} = -\left(4-\frac{9}{2}\right)</math>
which, by adding {{sfrac|9|2}} on both sides, correctly reduces to 5&nbsp;=&nbsp;5.

Another example illustrating the danger of taking the square root of both sides of an equation involves the following fundamental identity<ref>{{harvnb|Maxwell|1959|loc=Chapter VI, §I.1}}</ref>
:<math>\cos^2x=1-\sin^2x</math>
which holds as a consequence of the [[Pythagorean theorem]]. Then, by taking a square root,
:<math>\cos x = \sqrt{1-\sin^2x}</math>
Evaluating this when ''x''&nbsp;=&nbsp;{{pi}} , we get that
:<math>-1 = \sqrt{1-0}</math>
or
:<math>-1 = 1</math>
which is incorrect.

The error in each of these examples fundamentally lies in the fact that any equation of the form
:<math>x^2 = a^2</math>

where <math>a \ne 0</math>, has two solutions:
:<math>x=\pm a</math>
and it is essential to check which of these solutions is relevant to the problem at hand.<ref>{{harvnb|Maxwell|1959|loc=Chapter VI, §II}}</ref>  In the above fallacy, the square root that allowed the second equation to be deduced from the first is valid only when cos&nbsp;''x'' is positive. In particular, when ''x'' is set to {{pi}}, the second equation is rendered invalid.

=== Square roots of negative numbers ===

Invalid proofs utilizing powers and roots are often of the following kind:
:<math>1 = \sqrt{1} = \sqrt{(-1)(-1)} = \sqrt{-1}\sqrt{-1}=i \cdot i = -1.</math>

The fallacy is that the rule <math>\sqrt{xy} = \sqrt{x}\sqrt{y}</math> is generally valid only if at least one of <math>x</math> and ''<math>y</math>'' is non-negative (when dealing with real numbers), which is not the case here.<ref>{{cite book |title=An Imaginary Tale: The Story of "''i''" |first1=Paul J. |last1=Nahin |publisher=Princeton University Press |year=2010 |isbn=978-1-4008-3029-9 |page=12 |url=https://books.google.com/books?id=PflwJdPhBlEC}} [https://books.google.com/books?id=PflwJdPhBlEC&pg=PA12 Extract of page 12]</ref>

Alternatively, imaginary roots are obfuscated in the following:

:<math>i=\sqrt{-1} = \left(-1\right)^\frac{2}{4} = \left(\left(-1\right)^2\right)^\frac{1}{4} = 1^\frac{1}{4} = 1</math>

The error here lies in the incorrect usage of multiple-valued functions. <math>(-1)^\frac{1}{2}</math> has two values <math>i</math> and <math>-i</math> without a prior choice of branch, while <math>\sqrt{-1}</math> only denotes the principal value <math>i</math>. <ref>{{cite book | last=Saff | first=E. B. | last2=Snider | first2=Arthur David | title=Fundamentals of Complex Analysis Engineering, Science and Mathematics | publisher=Pearson | publication-place=Harlow, Essex, England | date=2013-07-18 | isbn=978-1-292-02375-5}}</ref> Similarly, <math>1^\frac{1}{4}</math> has four different values <math>1</math>, <math>i</math>, <math>-1</math>, and <math>-i</math>, of which only <math>i</math> is equal to the left side of the first equality.

=== Complex exponents ===
When a number is raised to a complex power, the result is not uniquely defined (see {{slink|Exponentiation|Failure of power and logarithm identities}}). If this property is not recognized, then errors such as the following can result:

:<math>
\begin{align}
e^{2 \pi i} &= 1 \\
\left(e^{2 \pi i}\right)^{i} &= 1^{i} \\
e^{-2 \pi} &= 1 \\
\end{align}
</math>

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]]s, both sides produce the same set of values, being <math>\{e^{2 \pi n} | n \isin \mathbb{Z} \}.</math>

== Geometry ==
Many mathematical fallacies in [[geometry]] 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 ===
[[File:Fallacy of the isosceles triangle2.svg|thumb]]
The fallacy of the isosceles triangle, from {{harv|Maxwell|1959|loc=Chapter II, § 1}}, purports to show that every [[triangle]] is [[isosceles triangle|isosceles]], meaning that two sides of the triangle are [[congruence (geometry)|congruent]]. This fallacy was known to [[Lewis Carroll]] and may have been discovered by him. It was published in 1899.<ref>{{citation |  title=The Lewis Carroll Picture Book|editor=S.D.Collingwood| pages=190–191| publisher=Collins| year=1899}}</ref><ref>{{citation| title=Lewis Carroll in Numberland| author=Robin Wilson| pages=169–170| publisher=Penguin Books| isbn=978-0-14-101610-8| year=2008}}</ref>

Given a triangle △ABC, prove that AB = AC:
# Draw a line [[bisection|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 [[solution of triangles|AAS]], △RAO ≅ △QAO (∠ORA = ∠OQA = 90°; ∠RAO = ∠QAO; AO = AO (common side)).
# By [[congruence (geometry)|RHS]],<ref group="note">Hypotenuse–leg congruence</ref> △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&nbsp;+&nbsp;RB, but AC is actually AQ&nbsp;−&nbsp;QC; and thus the lengths are not necessarily the same.

== Proof by induction ==
There exist several fallacious [[Proof by induction|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]].<ref>{{cite book|title=Induction and Analogy in Mathematics | series=Mathematics and plausible reasoning |volume=1 |first = George | last=Pólya | author-link=George Pólya| year=1954 |page=120 | publisher=Princeton}}</ref><ref group="note">[[George Pólya]]'s original "proof" was that any ''n'' girls have the same colour eyes.</ref>

# 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''&nbsp;−&nbsp;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''&nbsp;−&nbsp;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''&nbsp;+&nbsp;1 horses of the same colour.
# Thus if any ''N'' horses are all the same colour, any ''N''&nbsp;+&nbsp;1 horses are the same colour.
# This is clearly true for ''N''&nbsp;=&nbsp;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'', and so all horses are the same colour.

The fallacy in this proof arises in line 3. For ''N''&nbsp;=&nbsp;1, the two groups of horses have ''N''&nbsp;−&nbsp;1&nbsp;=&nbsp;0 horses in common, and thus are not necessarily the same colour as each other, so the group of ''N''&nbsp;+&nbsp;1&nbsp;=&nbsp;2 horses is not necessarily all of the same colour. The implication "every ''N'' horses are of the same colour, then ''N''&nbsp;+&nbsp;1 horses are of the same colour" works for any ''N''&nbsp;>&nbsp;1, but fails to be true when ''N''&nbsp;=&nbsp;1. The basis case is correct, but the induction step has a fundamental flaw.

== See also ==
*{{annotated link|Anomalous cancellation}}
*{{annotated link|Division by zero}}
*{{annotated link|List of incomplete proofs}}
* {{annotated link|Mathematical coincidence}}
* {{annotated link|Paradox}}
* {{annotated link|Proof by intimidation}}

==Notes==
{{Noteslist|group=note}}

==References==
{{Reflist}}
{{refbegin}}
* {{Citation | last1=Barbeau | first1=Edward J. | title=Mathematical fallacies, flaws, and flimflam | publisher=[[Mathematical Association of America]] | series=MAA Spectrum | isbn=978-0-88385-529-4 |mr=1725831 | year=2000 |url=https://books.google.com/books?id=ugrXqDEyPU8C}}.
* {{Citation | last1=Bunch | first1=Bryan | title=Mathematical fallacies and paradoxes | publisher=[[Dover Publications]] | location=New York | isbn=978-0-486-29664-7 |mr=1461270 | year=1997 |url=https://books.google.com/books?id=M6DvzoKlcicC}}.
* {{citation | first1=Sir Thomas Little|last1=Heath|first2=Johan Ludvig|last2=Heiberg|title=The thirteen books of Euclid's Elements, Volume 1|year=1908|publisher=The University Press}}.
* {{Citation | last1=Maxwell | first1=E. A. | author-link= Edwin A. Maxwell | title=Fallacies in mathematics | publisher=[[Cambridge University Press]] |mr=0099907 | year=1959 |url=https://books.google.com/books?id=zNvvoFEzP8IC |isbn=0-521-05700-0}}.
{{refend}}

==External links==
{{Commons category|Invalid proofs}}
*[http://www.cut-the-knot.org/proofs/index.shtml Invalid proofs] at [[Cut-the-knot]] (including literature references)
*[http://www.math.toronto.edu/mathnet/falseProofs/ Classic fallacies] with some discussion
*[http://www.ahajokes.com/math_jokes.html More invalid proofs from AhaJokes.com]
*[https://web.archive.org/web/20071101040235/http://www.jokes-funblog.com/categories/49-Math-Jokes Math jokes including an invalid proof]

{{Formal fallacies}}

{{DEFAULTSORT:Mathematical fallacy}}
[[Category:Recreational mathematics]]
[[Category:Proof theory]]
[[Category:Mathematical fallacies|*]]