3192:
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92:. 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
1339:
3179:
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.
3182:
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
3178:
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
1374:; 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
3296:
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"
151:
72:
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
2676:
3096:
3686:{\displaystyle {\begin{aligned}{\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{aligned}}}
3283:
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.
1115:
4136:
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422:
81:. 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.
3274:
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
319:{\displaystyle {\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}}}
2320:
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
64:
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.
1969:
2547:
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2494:
3165:
1809:
1547:
2117:
3101:
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
1334:{\displaystyle \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}
2052:
639:
2720:
485:
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.
724:
2842:
2170:
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2254:
1477:
333:
Examples exist of mathematically correct results derived by incorrect lines of reasoning. Such an argument, however true the conclusion appears to be, is mathematically
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465:
Bogus proofs, calculations, or derivations constructed to produce a correct result in spite of incorrect logic or operations were termed "howlers" by
1574:
84:
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
4117:
4084:
4027:
3983:
3955:
3843:
3335: − 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.
4741:
1993:
Another example illustrating the danger of taking the square root of both sides of an equation involves the following fundamental identity
1344:
Since the difference between two values of a constant function vanishes, the same definite integral appears on both sides of the equation.
4398:
4240:
4076:
1101:
and shifting them by 1 or indeed any number is allowed. The error really comes to light when we introduce arbitrary integration limits
1909:
5265:
4552:
4167:
3897:
4213:
2671:{\displaystyle i={\sqrt {-1}}=\left(-1\right)^{\frac {2}{4}}=\left(\left(-1\right)^{2}\right)^{\frac {1}{4}}=1^{\frac {1}{4}}=1}
1093:
after which the antiderivatives may be cancelled yielding 0 = 1. The problem is that antiderivatives are only defined
5275:
3298:
3091:{\displaystyle {\begin{aligned}e^{2\pi i}&=1\\\left(e^{2\pi i}\right)^{i}&=1^{i}\\e^{-2\pi }&=1\\\end{aligned}}}
3380: − 1 = 0 horses in common, and thus are not necessarily the same colour as each other, so the group of
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2455:
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1762:
459:
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As a corollary, one can show that all triangles are equilateral, by showing that AB = BC and AC = BC in the same way.
1488:
458:
is correct, there is a fallacious, invalid cancellation in the middle step. Another classical example of a howler is
3320:
horses. By our previous assumption, all the horses are of the same colour in this new group, since it is a group of
4783:
4159:
2067:
941:
5191:
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132:
1999:
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3360: = 1 (i.e., one horse is a group where all the horses are the same colour). Thus, by induction,
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3384: + 1 = 2 horses is not necessarily all of the same colour. The implication "every
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The error in each of these examples fundamentally lies in the fact that any equation of the form
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Lewis
Carroll in numberland: his fantastical mathematical logical life: an agony in eight fits
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3316: − 1 horses of the same colour. If we add another horse, we have another group of
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963:
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482:
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by simply substituting the scalar variables of the characteristic polynomial by the matrix.
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3400: = 1. The basis case is correct, but the induction step has a fundamental flaw.
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89:
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The fallacy is in the second to last line, where the square root of both sides is taken:
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27:"Invalid proof" redirects here. For any type of invalid proof besides mathematics, see
2442:{\displaystyle 1={\sqrt {1}}={\sqrt {(-1)(-1)}}={\sqrt {-1}}{\sqrt {-1}}=i\cdot i=-1.}
5259:
4996:
4886:
4618:
4581:
4525:
4393:
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2962:). If this property is not recognized, then errors such as the following can result:
1362:. For instance, while squaring a number gives a unique value, there are two possible
901:
The fallacy is in line 5: the progression from line 4 to line 5 involves division by
645:
78:
3183:
always prove that quantities are being incorporated with their correct orientation.
2958:
When a number is raised to a complex power, the result is not uniquely defined (see
5008:
4427:
4202:
4196:
3872:
1746:{\displaystyle \left(5-{\frac {9}{2}}\right)^{2}=\left(4-{\frac {9}{2}}\right)^{2}}
105:
52:
are often exhibited, and sometimes collected, as illustrations of a concept called
34:"0 = 1" redirects here. For the algebraic structure where this equality holds, see
135:
are equated. Well-known fallacies also exist in elementary
Euclidean geometry and
4192:
4153:
4103:
4070:
4017:
4000:
3941:
3887:
1083:{\displaystyle \int {\frac {1}{x\,\log x}}\,dx=1+\int {\frac {1}{x\,\log x}}\,dx}
5201:
4978:
4915:
1387:
1363:
45:
1654:{\displaystyle 5^{2}-5\times 9+{\frac {81}{4}}=4^{2}-4\times 9+{\frac {81}{4}}}
417:{\displaystyle {\frac {16}{64}}={\frac {16\!\!\!/}{6\!\!\!/4}}={\frac {1}{4}}.}
5111:
5105:
5032:
4944:
2538:
is non-negative (when dealing with real numbers), which is not the case here.
1895:
have the same sign, which is not the case here. In this case, it implies that
4037:
3796:
3232:
Draw the perpendicular bisector of segment BC, which bisects BC at a point D.
5218:
5002:
4986:
4669:
3270:
The error in the proof is the assumption in the diagram that the point O is
3226:
35:
5148:
4058:. Mathematics and plausible reasoning. Vol. 1. Princeton. p. 120.
2341:
Invalid proofs utilizing powers and roots are often of the following kind:
4218:
4208:
3810:
5096:
3859:
Barbeau, Ed (1990). "Fallacies, Flaws and
Flimflam #19: Dolt's Theorem".
3203:
3175:
2681:
The error here lies in the incorrect usage of multiple-valued functions.
1375:
937:
156:
136:
131:
is incorrectly extracted or, more generally, where different values of a
4250:
3811:"soft question – Best Fake Proofs? (A M.SE April Fools Day collection)"
3432:
3255:, △ROB ≅ △QOC (∠BRO = ∠CQO = 90°; BO = OC (hypotenuse); RO = OQ (leg)).
85:
28:
3248:, △RAO ≅ △QAO (∠ORA = ∠OQA = 90°; ∠RAO = ∠QAO; AO = AO (common side)).
3976:
Fundamentals of
Complex Analysis Engineering, Science and Mathematics
1864:
894:
113:
17:
4022:(1st American ed.). New York: W.W. Norton. pp. 169–170.
3190:
1094:
119:
Mathematical fallacies exist in many branches of mathematics. In
68:
For example, the reason why validity fails may be attributed to a
2541:
Alternatively, imaginary roots are obfuscated in the following:
948:
can be used to give a false proof that 0 = 1. Letting
4222:
3218:
and may have been discovered by him. It was published in 1899.
2960:
Exponentiation § Failure of power and logarithm identities
1964:{\displaystyle 5-{\frac {9}{2}}=-\left(4-{\frac {9}{2}}\right)}
3238:
Draw line OR perpendicular to AB, line OQ perpendicular to AC.
3338:
Therefore, combining all the horses used, we have a group of
3392: + 1 horses are of the same colour" works for any
1394:. Failing to do so results in a "proof" of 5 = 4.
257:
246:
3892:(illustrated ed.). Courier Corporation. p. 207.
936:
can lead to mathematical fallacies — if the properties of
3999:
Carroll, Lewis (1899). Collingwood, Stuard Dogson (ed.).
112:, an ancient lost book of false proofs, is attributed to
4135:
Heath, Sir Thomas Little; Heiberg, Johan Ludvig (1908).
3312:
If we remove a horse from the group, we have a group of
3258:
Thus, AR = AQ, RB = QC, and AB = AR + RB = AQ + QC = AC.
3464:
3115:
3105:
only the principal value is chosen. When treated as
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on both sides, correctly reduces to 5 = 5.
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3435: – Statement that apparently contradicts itself
3364:
horses are the same colour for any positive integer
341:. The following is an example of a howler involving
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3109:, both sides produce the same set of values, being
2489:{\displaystyle {\sqrt {xy}}={\sqrt {x}}{\sqrt {y}}}
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4138:The thirteen books of Euclid's Elements, Volume 1
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3489:
3488:
3441: – Marking an argument as obvious or trivial
3160:{\displaystyle \{e^{2\pi n}|n\in \mathbb {Z} \}.}
2950:is equal to the left side of the first equality.
1804:{\displaystyle 5-{\frac {9}{2}}=4-{\frac {9}{2}}}
386:
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373:
372:
371:
256:
255:
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245:
244:
243:
3974:Saff, E. B.; Snider, Arthur David (2013-07-18).
3458:The same fallacy also applies to the following:
3372:The fallacy in this proof arises in line 3. For
3202:, Chapter II, § 1), purports to show that every
1542:{\displaystyle 5^{2}-5\times 9=4^{2}-4\times 9}
1370:. One value can be chosen by convention as the
473:has various meanings, generally less specific.
4359:Affirmative conclusion from a negative premise
3748:
3746:
3376: = 1, the two groups of horses have
4364:Negative conclusion from affirmative premises
4234:
3396: > 1, but fails to be true when
3210:, meaning that two sides of the triangle are
3198:The fallacy of the isosceles triangle, from (
2112:{\displaystyle \cos x={\sqrt {1-\sin ^{2}x}}}
8:
3763:
3151:
3116:
469:. Outside the field of mathematics the term
123:, typical examples may involve a step where
4072:Mathematical fallacies, flaws, and flimflam
3221:Given a triangle △ABC, prove that AB = AC:
2496:is generally valid only if at least one of
2333:, the second equation is rendered invalid.
5092:
4939:
4814:
4417:
4268:
4241:
4227:
4219:
3950:. Princeton University Press. p. 12.
3353: + 1 horses are the same colour.
944:are ignored. For instance, a naive use of
161:
160:
159:
56:. There is a distinction between a simple
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1366:of a positive number. The square root is
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3418: – Class of mathematical expression
3368:, and so all horses are the same colour.
2765:without a prior choice of branch, while
932:as the mathematical study of change and
3927:
3915:
3752:
3737:
3730:
3451:
3327:Thus we have constructed two groups of
3199:
2047:{\displaystyle \cos ^{2}x=1-\sin ^{2}x}
921:is undefined, the argument is invalid.
3235:Let these two lines meet at a point O.
652:, the right is factored by extracting
4214:Math jokes including an invalid proof
4209:More invalid proofs from AhaJokes.com
3838:(6th ed.). Teubner. p. 51.
7:
4105:Mathematical fallacies and paradoxes
4056:Induction and Analogy in Mathematics
3388:horses are of the same colour, then
3349:horses are all the same colour, any
3331:horses all of the same colour, with
2057:which holds as a consequence of the
1757:Take the square root of both sides:
1358:Many functions do not have a unique
634:{\displaystyle a^{2}-b^{2}=ab-b^{2}}
88:is here slightly different from the
4077:Mathematical Association of America
3978:. Harlow, Essex, England: Pearson.
3889:Mathematical Fun, Games and Puzzles
2715:{\displaystyle (-1)^{\frac {1}{2}}}
1386:Care must be taken when taking the
460:proving the Cayley–Hamilton theorem
3429: – Coincidence in mathematics
648:both sides: the left factors as a
328:Anomalous cancellation in calculus
25:
4199:(including literature references)
3943:An Imaginary Tale: The Story of "
3713:'s original "proof" was that any
3309:horses is all of the same colour.
3187:Fallacy of the isosceles triangle
2792:only denotes the principal value
2325:is positive. In particular, when
2061:. Then, by taking a square root,
719:{\displaystyle (a-b)(a+b)=b(a-b)}
5240:
5239:
3778:"Fallacies, Flaws, and Flimflam"
3717:girls have the same colour eyes.
3412: – Kind of arithmetic error
2837:{\displaystyle 1^{\frac {1}{4}}}
2337:Square roots of negative numbers
2165:{\displaystyle -1={\sqrt {1-0}}}
4005:. T.F. Unwin. pp. 190–191.
3861:The College Mathematics Journal
3785:The College Mathematics Journal
3292:There exist several fallacious
3174:Many mathematical fallacies in
821:Combine like terms on the left
4737:Correlation implies causation
4002:The Lewis Carroll Picture Book
3873:10.1080/07468342.1990.11973308
3836:Lehrbuch der Analysis – Teil 1
3299:all horses are the same colour
3136:
2698:
2688:
2390:
2381:
2378:
2369:
1903:, so the equation should read
1171:
713:
701:
692:
680:
677:
665:
426:Here, although the conclusion
41:Certain type of mistaken proof
1:
3305:Let us say that any group of
2452:The fallacy is that the rule
497:be equal, nonzero quantities
3886:Frohlichstein, Jack (1967).
3214:. This fallacy was known to
2785:{\displaystyle {\sqrt {-1}}}
48:, certain kinds of mistaken
4069:Barbeau, Edward J. (2000).
2249:{\displaystyle x^{2}=a^{2}}
1665:These are perfect squares:
1472:{\displaystyle 25-45=16-36}
1382:Positive and negative roots
337:and is commonly known as a
5292:
5161:I'm entitled to my opinion
4160:Cambridge University Press
3815:Mathematics Stack Exchange
2844:has four different values
1351:
33:
26:
5235:
5144:
5017:
4016:Wilson, Robin J. (2008).
3701:Hypotenuse–leg congruence
3422:List of incomplete proofs
3356:This is clearly true for
5266:Recreational mathematics
5187:Motte-and-bailey fallacy
4287:Affirming the consequent
4155:Fallacies in mathematics
3764:Heath & Heiberg 1908
3427:Mathematical coincidence
563:{\displaystyle a^{2}=ab}
483:division-by-zero fallacy
133:multiple valued function
5207:Two wrongs make a right
4538:Denying the correlative
4144:. The University Press.
3940:Nahin, Paul J. (2010).
2310:{\displaystyle x=\pm a}
2278:{\displaystyle a\neq 0}
1428:{\displaystyle -20=-20}
856:Divide by the non-zero
5276:Mathematical fallacies
5192:Psychologist's fallacy
5129:Argument to moderation
5119:Argument from anecdote
5069:Chronological snobbery
4693:Quoting out of context
4660:Overwhelming exception
4543:Suppressed correlative
4443:Quoting out of context
4318:quantificational logic
4292:Denying the antecedent
3834:Heuser, Harro (1989).
3687:
3410:Anomalous cancellation
3195:
3161:
3092:
2944:
2924:
2901:
2878:
2858:
2838:
2806:
2786:
2759:
2736:
2716:
2672:
2531:
2510:
2490:
2443:
2311:
2279:
2250:
2201:
2166:
2113:
2048:
1965:
1853:
1805:
1747:
1655:
1543:
1473:
1429:
1335:
1084:
909:, which is zero since
881:
846:
811:
766:
720:
635:
564:
519:
418:
343:anomalous cancellation
320:
127:is performed, where a
5155:The Four Great Errors
5135:Argumentum ad populum
5124:Argument from silence
4828:Argumentum ad baculum
4606:Faulty generalization
4297:Argument from fallacy
4102:Bunch, Bryan (1997).
3688:
3439:Proof by intimidation
3279:AB, while Q will lie
3241:Draw lines OB and OC.
3194:
3162:
3107:multivalued functions
3093:
2945:
2925:
2902:
2879:
2859:
2839:
2807:
2787:
2760:
2737:
2717:
2673:
2532:
2511:
2491:
2444:
2312:
2285:, has two solutions:
2280:
2251:
2202:
2167:
2122:Evaluating this when
2114:
2049:
1966:
1854:
1806:
1748:
1656:
1544:
1474:
1430:
1348:Multivalued functions
1336:
1085:
930:Mathematical analysis
882:
847:
812:
810:{\displaystyle b+b=b}
767:
765:{\displaystyle a+b=b}
721:
650:difference of squares
636:
565:
520:
419:
321:
5173:Invincible ignorance
4979:Reductio ad Stalinum
4965:Reductio ad Hitlerum
4921:Wisdom of repugnance
4688:Moving the goalposts
4553:Illicit transference
4478:Begging the question
4399:Undistributed middle
4307:Mathematical fallacy
4282:Affirming a disjunct
4205:with some discussion
3776:Barbeau, Ed (1991).
3462:
3113:
2969:
2934:
2911:
2888:
2868:
2848:
2816:
2796:
2769:
2746:
2726:
2685:
2548:
2521:
2500:
2456:
2348:
2292:
2263:
2220:
2210:which is incorrect.
2200:{\displaystyle -1=1}
2182:
2137:
2068:
2000:
1910:
1837:
1763:
1671:
1575:
1489:
1445:
1407:
1390:of both sides of an
1354:Multivalued function
1116:
1004:
946:integration by parts
865:
845:{\displaystyle 2b=b}
827:
789:
744:
662:
583:
538:
503:
349:
152:
62:mathematical fallacy
54:mathematical fallacy
4906:Parade of horribles
4882:In-group favoritism
4708:Syntactic ambiguity
4351:Syllogistic fallacy
4274:propositional logic
3905:Extract of page 207
3294:proofs by induction
2059:Pythagorean theorem
1852:{\displaystyle 5=4}
1302:
1256:
1203:
1185:
1133:
880:{\displaystyle 2=1}
518:{\displaystyle a=b}
102:five colour theorem
4992:Poisoning the well
4809:Proof by assertion
4784:Texas sharpshooter
4718:Questionable cause
4655:Slothful induction
4614:Anecdotal evidence
4474:Circular reasoning
4369:Exclusive premises
4331:Illicit conversion
4110:Dover Publications
3963:Extract of page 12
3918:, Chapter VI, §I.1
3683:
3681:
3288:Proof by induction
3196:
3157:
3088:
3086:
2940:
2923:{\displaystyle -i}
2920:
2900:{\displaystyle -1}
2897:
2874:
2854:
2834:
2802:
2782:
2758:{\displaystyle -i}
2755:
2732:
2712:
2668:
2527:
2506:
2486:
2439:
2307:
2275:
2246:
2197:
2162:
2109:
2044:
1961:
1849:
1801:
1743:
1651:
1539:
1469:
1425:
1331:
1288:
1242:
1189:
1169:
1119:
1080:
877:
842:
807:
776:Use the fact that
762:
716:
631:
560:
515:
414:
316:
314:
310:
281:
262:
227:
208:
189:
177:
121:elementary algebra
98:Euclidean geometry
5253:
5252:
5231:
5230:
5227:
5226:
5167:Ignoratio elenchi
5079:
5078:
4929:
4928:
4891:Not invented here
4596:Converse accident
4518:Correlative-based
4495:Compound question
4438:False attribution
4433:False equivalence
4407:
4406:
4203:Classic fallacies
4119:978-0-486-29664-7
4086:978-0-88385-529-4
4029:978-0-393-06027-0
3985:978-1-292-02375-5
3957:978-1-4008-3029-9
3930:, Chapter VI, §II
3845:978-3-8351-0131-9
3677:
3664:
3647:
3611:
3597:
3580:
3544:
3530:
3513:
3477:
2954:Complex exponents
2943:{\displaystyle i}
2877:{\displaystyle i}
2857:{\displaystyle 1}
2831:
2805:{\displaystyle i}
2780:
2735:{\displaystyle i}
2709:
2659:
2641:
2595:
2565:
2530:{\displaystyle y}
2509:{\displaystyle x}
2484:
2477:
2467:
2416:
2406:
2393:
2362:
2160:
2107:
1974:which, by adding
1954:
1927:
1799:
1780:
1730:
1694:
1649:
1611:
1322:
1276:
1224:
1154:
1071:
1030:
409:
396:
360:
309:
280:
261:
226:
207:
188:
176:
16:(Redirected from
5283:
5243:
5242:
5214:Special pleading
5093:
4954:Appeal to motive
4940:
4916:Stirring symbols
4896:Island mentality
4834:Wishful thinking
4815:
4531:Perfect solution
4508:No true Scotsman
4503:Complex question
4488:Leading question
4467:Question-begging
4453:No true Scotsman
4418:
4341:Quantifier shift
4336:Proof by example
4269:
4243:
4236:
4229:
4220:
4181:
4145:
4143:
4131:
4098:
4075:. MAA Spectrum.
4060:
4059:
4048:
4042:
4041:
4013:
4007:
4006:
3996:
3990:
3989:
3971:
3965:
3961:
3937:
3931:
3925:
3919:
3913:
3907:
3903:
3883:
3877:
3876:
3856:
3850:
3849:
3831:
3825:
3824:
3822:
3821:
3807:
3801:
3800:
3782:
3773:
3767:
3766:, Chapter II, §I
3761:
3755:
3750:
3741:
3735:
3718:
3708:
3702:
3699:
3693:
3692:
3690:
3689:
3684:
3682:
3678:
3670:
3665:
3657:
3648:
3646:
3642:
3630:
3629:
3617:
3612:
3604:
3598:
3590:
3581:
3579:
3575:
3563:
3562:
3550:
3545:
3537:
3531:
3523:
3514:
3512:
3508:
3496:
3495:
3483:
3478:
3470:
3456:
3416:Division by zero
3166:
3164:
3163:
3158:
3150:
3139:
3134:
3133:
3097:
3095:
3094:
3089:
3087:
3073:
3072:
3053:
3052:
3036:
3035:
3030:
3026:
3025:
2991:
2990:
2949:
2947:
2946:
2941:
2930:, of which only
2929:
2927:
2926:
2921:
2906:
2904:
2903:
2898:
2883:
2881:
2880:
2875:
2863:
2861:
2860:
2855:
2843:
2841:
2840:
2835:
2833:
2832:
2824:
2811:
2809:
2808:
2803:
2791:
2789:
2788:
2783:
2781:
2773:
2764:
2762:
2761:
2756:
2741:
2739:
2738:
2733:
2721:
2719:
2718:
2713:
2711:
2710:
2702:
2677:
2675:
2674:
2669:
2661:
2660:
2652:
2643:
2642:
2634:
2632:
2628:
2627:
2622:
2618:
2597:
2596:
2588:
2586:
2582:
2566:
2558:
2536:
2534:
2533:
2528:
2515:
2513:
2512:
2507:
2495:
2493:
2492:
2487:
2485:
2480:
2478:
2473:
2468:
2460:
2448:
2446:
2445:
2440:
2417:
2409:
2407:
2399:
2394:
2368:
2363:
2358:
2332:
2316:
2314:
2313:
2308:
2284:
2282:
2281:
2276:
2255:
2253:
2252:
2247:
2245:
2244:
2232:
2231:
2206:
2204:
2203:
2198:
2171:
2169:
2168:
2163:
2161:
2150:
2129:
2118:
2116:
2115:
2110:
2108:
2100:
2099:
2084:
2053:
2051:
2050:
2045:
2037:
2036:
2012:
2011:
1989:
1987:
1986:
1983:
1980:
1970:
1968:
1967:
1962:
1960:
1956:
1955:
1947:
1928:
1920:
1858:
1856:
1855:
1850:
1830:
1828:
1827:
1824:
1821:
1810:
1808:
1807:
1802:
1800:
1792:
1781:
1773:
1752:
1750:
1749:
1744:
1742:
1741:
1736:
1732:
1731:
1723:
1706:
1705:
1700:
1696:
1695:
1687:
1660:
1658:
1657:
1652:
1650:
1642:
1625:
1624:
1612:
1604:
1587:
1586:
1568:
1566:
1565:
1562:
1559:
1548:
1546:
1545:
1540:
1526:
1525:
1501:
1500:
1478:
1476:
1475:
1470:
1434:
1432:
1431:
1426:
1340:
1338:
1337:
1332:
1323:
1321:
1304:
1301:
1296:
1277:
1275:
1258:
1255:
1250:
1225:
1223:
1205:
1202:
1197:
1184:
1179:
1174:
1155:
1153:
1135:
1132:
1127:
1089:
1087:
1086:
1081:
1072:
1070:
1052:
1031:
1029:
1011:
997:, we may write:
996:
994:
993:
988:
985:
972:
970:
969:
961:
958:
919:division by zero
886:
884:
883:
878:
851:
849:
848:
843:
816:
814:
813:
808:
771:
769:
768:
763:
725:
723:
722:
717:
656:from both terms
640:
638:
637:
632:
630:
629:
608:
607:
595:
594:
569:
567:
566:
561:
550:
549:
524:
522:
521:
516:
477:Division by zero
457:
455:
454:
451:
448:
441:
439:
438:
435:
432:
423:
421:
420:
415:
410:
402:
397:
395:
391:
379:
378:
366:
361:
353:
325:
323:
322:
317:
315:
311:
308:
307:
295:
282:
279:
278:
266:
263:
260:
249:
238:
228:
225:
224:
212:
209:
200:
190:
181:
178:
175:
164:
125:division by zero
70:division by zero
21:
5291:
5290:
5286:
5285:
5284:
5282:
5281:
5280:
5256:
5255:
5254:
5249:
5223:
5197:Rationalization
5140:
5087:
5075:
5013:
4935:Genetic fallacy
4925:
4838:
4813:
4788:
4712:
4703:Sorites paradox
4683:False precision
4664:
4645:Double counting
4600:
4575:
4547:
4512:
4499:Loaded question
4483:Loaded language
4462:
4403:
4345:
4311:
4258:
4247:
4189:
4184:
4170:
4148:
4141:
4134:
4120:
4101:
4087:
4068:
4064:
4063:
4050:
4049:
4045:
4030:
4015:
4014:
4010:
3998:
3997:
3993:
3986:
3973:
3972:
3968:
3958:
3939:
3938:
3934:
3926:
3922:
3914:
3910:
3900:
3885:
3884:
3880:
3858:
3857:
3853:
3846:
3833:
3832:
3828:
3819:
3817:
3809:
3808:
3804:
3780:
3775:
3774:
3770:
3762:
3758:
3751:
3744:
3736:
3732:
3727:
3722:
3721:
3709:
3705:
3700:
3696:
3680:
3679:
3649:
3631:
3618:
3600:
3599:
3582:
3564:
3551:
3533:
3532:
3515:
3497:
3484:
3460:
3459:
3457:
3453:
3448:
3406:
3290:
3189:
3172:
3119:
3111:
3110:
3085:
3084:
3074:
3058:
3055:
3054:
3044:
3037:
3011:
3007:
3006:
3003:
3002:
2992:
2976:
2967:
2966:
2956:
2932:
2931:
2909:
2908:
2886:
2885:
2866:
2865:
2846:
2845:
2819:
2814:
2813:
2794:
2793:
2767:
2766:
2744:
2743:
2724:
2723:
2722:has two values
2697:
2683:
2682:
2647:
2611:
2607:
2606:
2602:
2601:
2575:
2571:
2570:
2546:
2545:
2519:
2518:
2498:
2497:
2454:
2453:
2346:
2345:
2339:
2330:
2290:
2289:
2261:
2260:
2236:
2223:
2218:
2217:
2180:
2179:
2135:
2134:
2127:
2091:
2066:
2065:
2028:
2003:
1998:
1997:
1984:
1981:
1978:
1977:
1975:
1939:
1935:
1908:
1907:
1835:
1834:
1831:on both sides:
1825:
1822:
1819:
1818:
1816:
1761:
1760:
1715:
1711:
1710:
1679:
1675:
1674:
1669:
1668:
1616:
1578:
1573:
1572:
1569:on both sides:
1563:
1560:
1557:
1556:
1554:
1517:
1492:
1487:
1486:
1443:
1442:
1405:
1404:
1384:
1372:principal value
1356:
1350:
1308:
1262:
1209:
1139:
1114:
1113:
1056:
1015:
1002:
1001:
989:
986:
981:
980:
978:
962:
959:
956:
955:
953:
927:
863:
862:
825:
824:
787:
786:
742:
741:
660:
659:
621:
599:
586:
581:
580:
541:
536:
535:
501:
500:
479:
452:
449:
446:
445:
443:
436:
433:
430:
429:
427:
380:
367:
347:
346:
331:
330:
329:
326:
313:
312:
299:
284:
283:
270:
250:
239:
230:
229:
216:
192:
191:
168:
150:
149:
145:
90:logical fallacy
42:
39:
32:
23:
22:
15:
12:
11:
5:
5289:
5287:
5279:
5278:
5273:
5268:
5258:
5257:
5251:
5250:
5248:
5247:
5236:
5233:
5232:
5229:
5228:
5225:
5224:
5222:
5221:
5216:
5211:
5210:
5209:
5199:
5194:
5189:
5184:
5175:
5170:
5163:
5158:
5151:
5145:
5142:
5141:
5139:
5138:
5131:
5126:
5121:
5116:
5115:
5114:
5101:
5099:
5090:
5081:
5080:
5077:
5076:
5074:
5073:
5072:
5071:
5057:
5052:
5047:
5046:
5045:
5036:
5029:
5027:Accomplishment
5018:
5015:
5014:
5012:
5011:
5006:
4999:
4994:
4989:
4984:
4983:
4982:
4975:
4974:
4973:
4956:
4950:
4948:
4937:
4931:
4930:
4927:
4926:
4924:
4923:
4918:
4913:
4908:
4903:
4898:
4893:
4884:
4879:
4874:
4869:
4864:
4859:
4854:
4848:
4846:
4840:
4839:
4837:
4836:
4831:
4823:
4821:
4812:
4811:
4802:
4796:
4794:
4790:
4789:
4787:
4786:
4781:
4779:Slippery slope
4776:
4771:
4766:
4765:
4764:
4754:
4753:
4752:
4745:
4735:
4734:
4733:
4722:
4720:
4714:
4713:
4711:
4710:
4705:
4700:
4698:Slippery slope
4695:
4690:
4685:
4680:
4674:
4672:
4666:
4665:
4663:
4662:
4657:
4652:
4647:
4642:
4633:
4632:
4631:
4626:
4624:Cherry picking
4616:
4610:
4608:
4602:
4601:
4599:
4598:
4593:
4587:
4585:
4577:
4576:
4574:
4573:
4568:
4563:
4557:
4555:
4549:
4548:
4546:
4545:
4540:
4535:
4534:
4533:
4522:
4520:
4514:
4513:
4511:
4510:
4505:
4492:
4491:
4490:
4480:
4470:
4468:
4464:
4463:
4461:
4460:
4455:
4450:
4445:
4440:
4435:
4430:
4424:
4422:
4415:
4409:
4408:
4405:
4404:
4402:
4401:
4396:
4391:
4386:
4381:
4376:
4371:
4366:
4361:
4355:
4353:
4347:
4346:
4344:
4343:
4338:
4333:
4328:
4322:
4320:
4313:
4312:
4310:
4309:
4304:
4299:
4294:
4289:
4284:
4278:
4276:
4266:
4260:
4259:
4248:
4246:
4245:
4238:
4231:
4223:
4217:
4216:
4211:
4206:
4200:
4193:Invalid proofs
4188:
4187:External links
4185:
4183:
4182:
4168:
4150:Maxwell, E. A.
4146:
4132:
4118:
4099:
4085:
4065:
4062:
4061:
4043:
4028:
4008:
3991:
3984:
3966:
3956:
3932:
3920:
3908:
3898:
3878:
3867:(3): 216–218.
3851:
3844:
3826:
3802:
3768:
3756:
3742:
3729:
3728:
3726:
3723:
3720:
3719:
3703:
3694:
3676:
3673:
3668:
3663:
3660:
3655:
3652:
3650:
3645:
3641:
3634:
3628:
3621:
3615:
3610:
3607:
3602:
3601:
3596:
3593:
3588:
3585:
3583:
3578:
3574:
3567:
3561:
3554:
3548:
3543:
3540:
3535:
3534:
3529:
3526:
3521:
3518:
3516:
3511:
3507:
3500:
3494:
3487:
3481:
3476:
3473:
3468:
3467:
3450:
3449:
3447:
3444:
3443:
3442:
3436:
3430:
3424:
3419:
3413:
3405:
3402:
3370:
3369:
3354:
3343:
3336:
3325:
3310:
3289:
3286:
3260:
3259:
3256:
3249:
3242:
3239:
3236:
3233:
3230:
3188:
3185:
3171:
3168:
3156:
3153:
3149:
3145:
3142:
3138:
3132:
3129:
3126:
3122:
3118:
3099:
3098:
3083:
3080:
3077:
3075:
3071:
3068:
3065:
3061:
3057:
3056:
3051:
3047:
3043:
3040:
3038:
3034:
3029:
3024:
3021:
3018:
3014:
3010:
3005:
3004:
3001:
2998:
2995:
2993:
2989:
2986:
2983:
2979:
2975:
2974:
2955:
2952:
2939:
2919:
2916:
2896:
2893:
2873:
2853:
2830:
2827:
2822:
2812:. Similarly,
2801:
2779:
2776:
2754:
2751:
2731:
2708:
2705:
2700:
2696:
2693:
2690:
2679:
2678:
2667:
2664:
2658:
2655:
2650:
2646:
2640:
2637:
2631:
2626:
2621:
2617:
2614:
2610:
2605:
2600:
2594:
2591:
2585:
2581:
2578:
2574:
2569:
2564:
2561:
2556:
2553:
2526:
2505:
2483:
2476:
2471:
2466:
2463:
2450:
2449:
2438:
2435:
2432:
2429:
2426:
2423:
2420:
2415:
2412:
2405:
2402:
2397:
2392:
2389:
2386:
2383:
2380:
2377:
2374:
2371:
2366:
2361:
2356:
2353:
2338:
2335:
2318:
2317:
2306:
2303:
2300:
2297:
2274:
2271:
2268:
2257:
2256:
2243:
2239:
2235:
2230:
2226:
2208:
2207:
2196:
2193:
2190:
2187:
2173:
2172:
2159:
2156:
2153:
2148:
2145:
2142:
2130:, we get that
2120:
2119:
2106:
2103:
2098:
2094:
2090:
2087:
2082:
2079:
2076:
2073:
2055:
2054:
2043:
2040:
2035:
2031:
2027:
2024:
2021:
2018:
2015:
2010:
2006:
1972:
1971:
1959:
1953:
1950:
1945:
1942:
1938:
1934:
1931:
1926:
1923:
1918:
1915:
1899: = –
1869:
1868:
1861:
1860:
1859:
1848:
1845:
1842:
1813:
1812:
1811:
1798:
1795:
1790:
1787:
1784:
1779:
1776:
1771:
1768:
1755:
1754:
1753:
1740:
1735:
1729:
1726:
1721:
1718:
1714:
1709:
1704:
1699:
1693:
1690:
1685:
1682:
1678:
1663:
1662:
1661:
1648:
1645:
1640:
1637:
1634:
1631:
1628:
1623:
1619:
1615:
1610:
1607:
1602:
1599:
1596:
1593:
1590:
1585:
1581:
1551:
1550:
1549:
1538:
1535:
1532:
1529:
1524:
1520:
1516:
1513:
1510:
1507:
1504:
1499:
1495:
1481:
1480:
1479:
1468:
1465:
1462:
1459:
1456:
1453:
1450:
1439:Write this as
1437:
1436:
1435:
1424:
1421:
1418:
1415:
1412:
1383:
1380:
1352:Main article:
1349:
1346:
1342:
1341:
1330:
1327:
1320:
1317:
1314:
1311:
1307:
1300:
1295:
1291:
1287:
1284:
1281:
1274:
1271:
1268:
1265:
1261:
1254:
1249:
1245:
1241:
1238:
1235:
1232:
1229:
1222:
1219:
1216:
1212:
1208:
1201:
1196:
1192:
1188:
1183:
1178:
1173:
1168:
1165:
1162:
1159:
1152:
1149:
1146:
1142:
1138:
1131:
1126:
1122:
1091:
1090:
1079:
1076:
1069:
1066:
1063:
1059:
1055:
1050:
1047:
1044:
1041:
1038:
1035:
1028:
1025:
1022:
1018:
1014:
1009:
926:
923:
899:
898:
890:
889:
888:
887:
876:
873:
870:
854:
853:
852:
841:
838:
835:
832:
819:
818:
817:
806:
803:
800:
797:
794:
774:
773:
772:
761:
758:
755:
752:
749:
728:
727:
726:
715:
712:
709:
706:
703:
700:
697:
694:
691:
688:
685:
682:
679:
676:
673:
670:
667:
643:
642:
641:
628:
624:
620:
617:
614:
611:
606:
602:
598:
593:
589:
572:
571:
570:
559:
556:
553:
548:
544:
527:
526:
525:
514:
511:
508:
478:
475:
413:
408:
405:
400:
394:
390:
383:
377:
370:
364:
359:
356:
327:
306:
302:
298:
292:
289:
286:
285:
277:
273:
269:
259:
253:
248:
242:
235:
232:
231:
223:
219:
215:
206:
203:
197:
194:
193:
187:
184:
174:
171:
167:
158:
157:
148:
147:
146:
144:
141:
79:contradictions
40:
24:
14:
13:
10:
9:
6:
4:
3:
2:
5288:
5277:
5274:
5272:
5269:
5267:
5264:
5263:
5261:
5246:
5238:
5237:
5234:
5220:
5217:
5215:
5212:
5208:
5205:
5204:
5203:
5200:
5198:
5195:
5193:
5190:
5188:
5185:
5183:
5179:
5176:
5174:
5171:
5169:
5168:
5164:
5162:
5159:
5157:
5156:
5152:
5150:
5147:
5146:
5143:
5137:
5136:
5132:
5130:
5127:
5125:
5122:
5120:
5117:
5113:
5110:
5109:
5108:
5107:
5103:
5102:
5100:
5098:
5094:
5091:
5089:
5082:
5070:
5067:
5066:
5065:
5061:
5058:
5056:
5053:
5051:
5048:
5044:
5040:
5037:
5035:
5034:
5030:
5028:
5025:
5024:
5023:
5020:
5019:
5016:
5010:
5007:
5005:
5004:
5000:
4998:
4995:
4993:
4990:
4988:
4985:
4981:
4980:
4976:
4972:
4969:
4968:
4967:
4966:
4962:
4961:
4960:
4957:
4955:
4952:
4951:
4949:
4947:
4946:
4941:
4938:
4936:
4932:
4922:
4919:
4917:
4914:
4912:
4909:
4907:
4904:
4902:
4899:
4897:
4894:
4892:
4888:
4887:Invented here
4885:
4883:
4880:
4878:
4875:
4873:
4870:
4868:
4865:
4863:
4860:
4858:
4855:
4853:
4850:
4849:
4847:
4845:
4841:
4835:
4832:
4830:
4829:
4825:
4824:
4822:
4820:
4816:
4810:
4806:
4803:
4801:
4798:
4797:
4795:
4791:
4785:
4782:
4780:
4777:
4775:
4772:
4770:
4767:
4763:
4760:
4759:
4758:
4755:
4751:
4750:
4746:
4744:
4743:
4739:
4738:
4736:
4732:
4729:
4728:
4727:
4724:
4723:
4721:
4719:
4715:
4709:
4706:
4704:
4701:
4699:
4696:
4694:
4691:
4689:
4686:
4684:
4681:
4679:
4676:
4675:
4673:
4671:
4667:
4661:
4658:
4656:
4653:
4651:
4650:False analogy
4648:
4646:
4643:
4641:
4637:
4634:
4630:
4627:
4625:
4622:
4621:
4620:
4619:Sampling bias
4617:
4615:
4612:
4611:
4609:
4607:
4603:
4597:
4594:
4592:
4589:
4588:
4586:
4584:
4583:
4582:Secundum quid
4578:
4572:
4569:
4567:
4564:
4562:
4559:
4558:
4556:
4554:
4550:
4544:
4541:
4539:
4536:
4532:
4529:
4528:
4527:
4526:False dilemma
4524:
4523:
4521:
4519:
4515:
4509:
4506:
4504:
4500:
4496:
4493:
4489:
4486:
4485:
4484:
4481:
4479:
4475:
4472:
4471:
4469:
4465:
4459:
4456:
4454:
4451:
4449:
4446:
4444:
4441:
4439:
4436:
4434:
4431:
4429:
4426:
4425:
4423:
4419:
4416:
4414:
4410:
4400:
4397:
4395:
4394:Illicit minor
4392:
4390:
4389:Illicit major
4387:
4385:
4382:
4380:
4377:
4375:
4372:
4370:
4367:
4365:
4362:
4360:
4357:
4356:
4354:
4352:
4348:
4342:
4339:
4337:
4334:
4332:
4329:
4327:
4324:
4323:
4321:
4319:
4314:
4308:
4305:
4303:
4300:
4298:
4295:
4293:
4290:
4288:
4285:
4283:
4280:
4279:
4277:
4275:
4270:
4267:
4265:
4261:
4256:
4252:
4244:
4239:
4237:
4232:
4230:
4225:
4224:
4221:
4215:
4212:
4210:
4207:
4204:
4201:
4198:
4194:
4191:
4190:
4186:
4179:
4175:
4171:
4169:0-521-05700-0
4165:
4161:
4157:
4156:
4151:
4147:
4140:
4139:
4133:
4129:
4125:
4121:
4115:
4111:
4107:
4106:
4100:
4096:
4092:
4088:
4082:
4078:
4074:
4073:
4067:
4066:
4057:
4053:
4052:Pólya, George
4047:
4044:
4039:
4035:
4031:
4025:
4021:
4020:
4012:
4009:
4004:
4003:
3995:
3992:
3987:
3981:
3977:
3970:
3967:
3964:
3959:
3953:
3949:
3948:
3944:
3936:
3933:
3929:
3924:
3921:
3917:
3912:
3909:
3906:
3901:
3899:0-486-20789-7
3895:
3891:
3890:
3882:
3879:
3874:
3870:
3866:
3862:
3855:
3852:
3847:
3841:
3837:
3830:
3827:
3816:
3812:
3806:
3803:
3798:
3794:
3790:
3786:
3779:
3772:
3769:
3765:
3760:
3757:
3754:
3749:
3747:
3743:
3739:
3734:
3731:
3724:
3716:
3712:
3707:
3704:
3698:
3695:
3674:
3671:
3666:
3661:
3658:
3653:
3651:
3643:
3639:
3632:
3626:
3619:
3613:
3608:
3605:
3594:
3591:
3586:
3584:
3576:
3572:
3565:
3559:
3552:
3546:
3541:
3538:
3527:
3524:
3519:
3517:
3509:
3505:
3498:
3492:
3485:
3479:
3474:
3471:
3455:
3452:
3445:
3440:
3437:
3434:
3431:
3428:
3425:
3423:
3420:
3417:
3414:
3411:
3408:
3407:
3403:
3401:
3399:
3395:
3391:
3387:
3383:
3379:
3375:
3367:
3363:
3359:
3355:
3352:
3348:
3344:
3341:
3337:
3334:
3330:
3326:
3323:
3319:
3315:
3311:
3308:
3304:
3303:
3302:
3300:
3295:
3287:
3285:
3282:
3278:
3273:
3268:
3265:
3264:
3257:
3254:
3250:
3247:
3243:
3240:
3237:
3234:
3231:
3228:
3224:
3223:
3222:
3219:
3217:
3216:Lewis Carroll
3213:
3209:
3205:
3201:
3193:
3186:
3184:
3180:
3177:
3169:
3167:
3154:
3143:
3140:
3130:
3127:
3124:
3120:
3108:
3104:
3081:
3078:
3076:
3069:
3066:
3063:
3059:
3049:
3045:
3041:
3039:
3032:
3027:
3022:
3019:
3016:
3012:
3008:
2999:
2996:
2994:
2987:
2984:
2981:
2977:
2965:
2964:
2963:
2961:
2953:
2951:
2937:
2917:
2914:
2894:
2891:
2871:
2851:
2828:
2825:
2820:
2799:
2777:
2774:
2752:
2749:
2729:
2706:
2703:
2694:
2691:
2665:
2662:
2656:
2653:
2648:
2644:
2638:
2635:
2629:
2624:
2619:
2615:
2612:
2608:
2603:
2598:
2592:
2589:
2583:
2579:
2576:
2572:
2567:
2562:
2559:
2554:
2551:
2544:
2543:
2542:
2539:
2537:
2524:
2503:
2481:
2474:
2469:
2464:
2461:
2436:
2433:
2430:
2427:
2424:
2421:
2418:
2413:
2410:
2403:
2400:
2395:
2387:
2384:
2375:
2372:
2364:
2359:
2354:
2351:
2344:
2343:
2342:
2336:
2334:
2328:
2324:
2304:
2301:
2298:
2295:
2288:
2287:
2286:
2272:
2269:
2266:
2241:
2237:
2233:
2228:
2224:
2216:
2215:
2214:
2211:
2194:
2191:
2188:
2185:
2178:
2177:
2176:
2157:
2154:
2151:
2146:
2143:
2140:
2133:
2132:
2131:
2126: =
2125:
2104:
2101:
2096:
2092:
2088:
2085:
2080:
2077:
2074:
2071:
2064:
2063:
2062:
2060:
2041:
2038:
2033:
2029:
2025:
2022:
2019:
2016:
2013:
2008:
2004:
1996:
1995:
1994:
1991:
1957:
1951:
1948:
1943:
1940:
1936:
1932:
1929:
1924:
1921:
1916:
1913:
1906:
1905:
1904:
1902:
1898:
1894:
1890:
1886:
1883: =
1882:
1879:only implies
1878:
1875: =
1874:
1867:
1866:
1862:
1846:
1843:
1840:
1833:
1832:
1814:
1796:
1793:
1788:
1785:
1782:
1777:
1774:
1769:
1766:
1759:
1758:
1756:
1738:
1733:
1727:
1724:
1719:
1716:
1712:
1707:
1702:
1697:
1691:
1688:
1683:
1680:
1676:
1667:
1666:
1664:
1646:
1643:
1638:
1635:
1632:
1629:
1626:
1621:
1617:
1613:
1608:
1605:
1600:
1597:
1594:
1591:
1588:
1583:
1579:
1571:
1570:
1552:
1536:
1533:
1530:
1527:
1522:
1518:
1514:
1511:
1508:
1505:
1502:
1497:
1493:
1485:
1484:
1482:
1466:
1463:
1460:
1457:
1454:
1451:
1448:
1441:
1440:
1438:
1422:
1419:
1416:
1413:
1410:
1403:
1402:
1400:
1399:
1398:
1395:
1393:
1389:
1381:
1379:
1377:
1373:
1369:
1365:
1361:
1355:
1347:
1345:
1328:
1325:
1318:
1315:
1312:
1309:
1305:
1298:
1293:
1289:
1285:
1282:
1279:
1272:
1269:
1266:
1263:
1259:
1252:
1247:
1243:
1239:
1236:
1233:
1230:
1227:
1220:
1217:
1214:
1210:
1206:
1199:
1194:
1190:
1186:
1181:
1176:
1166:
1163:
1160:
1157:
1150:
1147:
1144:
1140:
1136:
1129:
1124:
1120:
1112:
1111:
1110:
1108:
1104:
1100:
1096:
1077:
1074:
1067:
1064:
1061:
1057:
1053:
1048:
1045:
1042:
1039:
1036:
1033:
1026:
1023:
1020:
1016:
1012:
1007:
1000:
999:
998:
992:
984:
977: =
976:
968:
965:
952: =
951:
947:
943:
942:differentials
939:
935:
931:
924:
922:
920:
916:
913: =
912:
908:
905: −
904:
897:
896:
892:
891:
874:
871:
868:
861:
860:
859:
855:
839:
836:
833:
830:
823:
822:
820:
804:
801:
798:
795:
792:
785:
784:
783:
779:
775:
759:
756:
753:
750:
747:
740:
739:
737:
733:
729:
710:
707:
704:
698:
695:
689:
686:
683:
674:
671:
668:
658:
657:
655:
651:
647:
644:
626:
622:
618:
615:
612:
609:
604:
600:
596:
591:
587:
579:
578:
577:
573:
557:
554:
551:
546:
542:
534:
533:
532:
528:
512:
509:
506:
499:
498:
496:
492:
488:
487:
486:
484:
476:
474:
472:
468:
467:Edwin Maxwell
463:
461:
424:
411:
406:
403:
398:
392:
388:
381:
375:
368:
362:
357:
354:
344:
340:
336:
304:
300:
296:
290:
287:
275:
271:
267:
251:
240:
233:
221:
217:
213:
204:
201:
195:
185:
182:
172:
169:
165:
142:
140:
138:
134:
130:
126:
122:
117:
115:
111:
107:
103:
99:
95:
94:Pasch's axiom
91:
87:
82:
80:
76:
71:
66:
63:
59:
55:
51:
47:
37:
30:
19:
5271:Proof theory
5182:Naturalistic
5165:
5153:
5133:
5104:
5088:of relevance
5031:
5009:Whataboutism
5001:
4977:
4971:Godwin's law
4963:
4943:
4826:
4819:Consequences
4800:Law/Legality
4774:Single cause
4747:
4740:
4580:
4448:Loki's Wager
4428:Equivocation
4421:Equivocation
4306:
4197:Cut-the-knot
4154:
4137:
4108:. New York:
4104:
4071:
4055:
4046:
4018:
4011:
4001:
3994:
3975:
3969:
3946:
3942:
3935:
3928:Maxwell 1959
3923:
3916:Maxwell 1959
3911:
3888:
3881:
3864:
3860:
3854:
3835:
3829:
3818:. Retrieved
3814:
3805:
3788:
3784:
3771:
3759:
3753:Maxwell 1959
3738:Maxwell 1959
3733:
3714:
3711:George Pólya
3706:
3697:
3454:
3397:
3393:
3389:
3385:
3381:
3377:
3373:
3371:
3365:
3361:
3357:
3350:
3346:
3345:Thus if any
3339:
3332:
3328:
3321:
3317:
3313:
3306:
3291:
3280:
3276:
3271:
3269:
3266:
3262:
3261:
3225:Draw a line
3220:
3200:Maxwell 1959
3197:
3181:
3173:
3102:
3100:
2957:
2680:
2540:
2517:
2451:
2340:
2326:
2322:
2319:
2258:
2212:
2209:
2174:
2123:
2121:
2056:
1992:
1973:
1900:
1896:
1892:
1888:
1884:
1880:
1876:
1872:
1870:
1863:
1396:
1385:
1364:square roots
1357:
1343:
1106:
1102:
1092:
990:
982:
974:
966:
949:
928:
914:
910:
906:
902:
900:
893:
857:
781:
777:
735:
731:
730:Divide out (
653:
575:
530:
529:Multiply by
494:
490:
480:
470:
464:
425:
338:
332:
118:
109:
106:graph theory
83:
67:
61:
57:
53:
43:
5202:Red herring
4959:Association
4640:Conjunction
4561:Composition
4458:Reification
4374:Existential
4326:Existential
3740:, p. 9
3297:shows that
1483:Rewrite as
1401:Start from
1388:square root
1368:multivalued
77:of obvious
46:mathematics
5260:Categories
5178:Moralistic
5112:Sealioning
5106:Ad nauseam
5033:Ipse dixit
4945:Ad hominem
4769:Regression
4571:Ecological
4384:Four terms
4302:Masked man
3820:2019-10-24
3725:References
2329:is set to
5219:Straw man
5097:Arguments
5086:fallacies
5060:Tradition
5050:Etymology
5022:Authority
5003:Tu quoque
4987:Bulverism
4757:Gambler's
4726:Animistic
4670:Ambiguity
4636:Base rate
4379:Necessity
4251:fallacies
4038:227016411
3797:0746-8342
3227:bisecting
3212:congruent
3208:isosceles
3144:∈
3128:π
3070:π
3064:−
3020:π
2985:π
2915:−
2892:−
2775:−
2750:−
2692:−
2613:−
2577:−
2560:−
2434:−
2425:⋅
2411:−
2401:−
2385:−
2373:−
2321:cos
2302:±
2270:≠
2186:−
2155:−
2141:−
2102:
2089:−
2075:
2039:
2026:−
2014:
1944:−
1933:−
1917:−
1789:−
1770:−
1720:−
1684:−
1633:×
1627:−
1595:×
1589:−
1534:×
1528:−
1509:×
1503:−
1464:−
1452:−
1420:−
1411:−
1376:nth roots
1316:
1290:∫
1270:
1244:∫
1218:
1191:∫
1148:
1121:∫
1065:
1049:∫
1024:
1008:∫
938:integrals
708:−
672:−
619:−
597:−
574:Subtract
291:−
258:∖
247:∖
110:Pseudaria
36:Null ring
5245:Category
4877:Ridicule
4862:Flattery
4852:Children
4749:Post hoc
4629:McNamara
4591:Accident
4566:Division
4413:Informal
4152:(1959).
4054:(1954).
3404:See also
3204:triangle
3176:geometry
3170:Geometry
1392:equality
1099:constant
925:Analysis
917:. Since
137:calculus
5064:Novelty
5039:Poverty
4901:Loyalty
4867:Novelty
4844:Emotion
4793:Appeals
4762:Inverse
4742:Cum hoc
4731:Furtive
4249:Common
4178:0099907
4128:1461270
4095:1725831
3433:Paradox
3324:horses.
3281:outside
1988:
1976:
1829:
1817:
1567:
1555:
1397:Proof:
1360:inverse
995:
979:
971:
954:
456:
444:
440:
428:
335:invalid
143:Howlers
86:fallacy
58:mistake
29:Fallacy
5149:Cliché
5084:Other
5055:Nature
5043:Wealth
4678:Accent
4264:Formal
4176:
4166:
4126:
4116:
4093:
4083:
4036:
4026:
3982:
3954:
3896:
3842:
3795:
3277:within
3272:inside
3263:Q.E.D.
2907:, and
2259:where
1865:Q.E.D.
934:limits
895:Q.E.D.
646:Factor
471:howler
339:howler
114:Euclid
100:, the
75:proofs
60:and a
4911:Spite
4805:Stone
4142:(PDF)
3791:(5).
3781:(PDF)
3446:Notes
1095:up to
50:proof
4997:Tone
4872:Pity
4857:Fear
4255:list
4164:ISBN
4114:ISBN
4081:ISBN
4034:OCLC
4024:ISBN
3980:ISBN
3952:ISBN
3894:ISBN
3840:ISBN
3793:ISSN
2742:and
2516:and
1891:and
1815:Add
1553:Add
1105:and
973:and
940:and
493:and
489:Let
481:The
129:root
4316:In
4272:In
4195:at
3869:doi
3253:RHS
3251:By
3246:AAS
3244:By
3229:∠A.
3206:is
2175:or
2093:sin
2072:cos
2030:sin
2005:cos
1887:if
1313:log
1267:log
1215:log
1145:log
1062:log
1021:log
964:log
108:).
104:of
96:of
44:In
18:1=0
5262::
5180:/
5062:/
5041:/
4889:/
4807:/
4638:/
4501:/
4497:/
4476:/
4174:MR
4172:.
4162:.
4158:.
4124:MR
4122:.
4112:.
4091:MR
4089:.
4079:.
4032:.
3865:21
3863:.
3813:.
3789:22
3787:.
3783:.
3745:^
3620:49
3609:98
3606:49
3553:26
3542:65
3539:26
3486:19
3475:95
3472:19
3301:.
2884:,
2864:,
2437:1.
1644:81
1606:81
1558:81
1467:36
1461:16
1455:45
1449:25
1423:20
1414:20
1378:.
1109:.
1097:a
983:dx
975:dv
780:=
738:)
734:−
442:=
437:64
431:16
369:16
358:64
355:16
345::
139:.
116:.
4257:)
4253:(
4242:e
4235:t
4228:v
4180:.
4130:.
4097:.
4040:.
3988:.
3960:.
3947:"
3945:i
3902:.
3875:.
3871::
3848:.
3823:.
3799:.
3715:n
3675:2
3672:1
3667:=
3662:8
3659:4
3654:=
3644:8
3640:/
3633:9
3627:/
3614:=
3595:5
3592:2
3587:=
3577:5
3573:/
3566:6
3560:/
3547:=
3528:5
3525:1
3520:=
3510:5
3506:/
3499:9
3493:/
3480:=
3398:N
3394:N
3390:N
3386:N
3382:N
3378:N
3374:N
3366:N
3362:N
3358:N
3351:N
3347:N
3340:N
3333:N
3329:N
3322:N
3318:N
3314:N
3307:N
3155:.
3152:}
3148:Z
3141:n
3137:|
3131:n
3125:2
3121:e
3117:{
3103:i
3082:1
3079:=
3067:2
3060:e
3050:i
3046:1
3042:=
3033:i
3028:)
3023:i
3017:2
3013:e
3009:(
3000:1
2997:=
2988:i
2982:2
2978:e
2938:i
2918:i
2895:1
2872:i
2852:1
2829:4
2826:1
2821:1
2800:i
2778:1
2753:i
2730:i
2707:2
2704:1
2699:)
2695:1
2689:(
2666:1
2663:=
2657:4
2654:1
2649:1
2645:=
2639:4
2636:1
2630:)
2625:2
2620:)
2616:1
2609:(
2604:(
2599:=
2593:4
2590:2
2584:)
2580:1
2573:(
2568:=
2563:1
2555:=
2552:i
2525:y
2504:x
2482:y
2475:x
2470:=
2465:y
2462:x
2431:=
2428:i
2422:i
2419:=
2414:1
2404:1
2396:=
2391:)
2388:1
2382:(
2379:)
2376:1
2370:(
2365:=
2360:1
2355:=
2352:1
2331:π
2327:x
2323:x
2305:a
2299:=
2296:x
2273:0
2267:a
2242:2
2238:a
2234:=
2229:2
2225:x
2195:1
2192:=
2189:1
2158:0
2152:1
2147:=
2144:1
2128:π
2124:x
2105:x
2097:2
2086:1
2081:=
2078:x
2042:x
2034:2
2023:1
2020:=
2017:x
2009:2
1985:2
1982:/
1979:9
1958:)
1952:2
1949:9
1941:4
1937:(
1930:=
1925:2
1922:9
1914:5
1901:b
1897:a
1893:b
1889:a
1885:b
1881:a
1877:b
1873:a
1847:4
1844:=
1841:5
1826:2
1823:/
1820:9
1797:2
1794:9
1786:4
1783:=
1778:2
1775:9
1767:5
1739:2
1734:)
1728:2
1725:9
1717:4
1713:(
1708:=
1703:2
1698:)
1692:2
1689:9
1681:5
1677:(
1647:4
1639:+
1636:9
1630:4
1622:2
1618:4
1614:=
1609:4
1601:+
1598:9
1592:5
1584:2
1580:5
1564:4
1561:/
1537:9
1531:4
1523:2
1519:4
1515:=
1512:9
1506:5
1498:2
1494:5
1458:=
1417:=
1329:x
1326:d
1319:x
1310:x
1306:1
1299:b
1294:a
1286:=
1283:x
1280:d
1273:x
1264:x
1260:1
1253:b
1248:a
1240:+
1237:0
1234:=
1231:x
1228:d
1221:x
1211:x
1207:1
1200:b
1195:a
1187:+
1182:b
1177:a
1172:|
1167:1
1164:=
1161:x
1158:d
1151:x
1141:x
1137:1
1130:b
1125:a
1107:b
1103:a
1078:x
1075:d
1068:x
1058:x
1054:1
1046:+
1043:1
1040:=
1037:x
1034:d
1027:x
1017:x
1013:1
991:x
987:/
967:x
960:/
957:1
950:u
915:b
911:a
907:b
903:a
875:1
872:=
869:2
858:b
840:b
837:=
834:b
831:2
805:b
802:=
799:b
796:+
793:b
782:b
778:a
760:b
757:=
754:b
751:+
748:a
736:b
732:a
714:)
711:b
705:a
702:(
699:b
696:=
693:)
690:b
687:+
684:a
681:(
678:)
675:b
669:a
666:(
654:b
627:2
623:b
616:b
613:a
610:=
605:2
601:b
592:2
588:a
576:b
558:b
555:a
552:=
547:2
543:a
531:a
513:b
510:=
507:a
495:b
491:a
453:4
450:/
447:1
434:/
412:.
407:4
404:1
399:=
393:4
389:/
382:6
376:/
363:=
305:2
301:x
297:1
288:=
276:2
272:x
268:1
252:d
241:d
234:=
222:2
218:x
214:1
205:d
202:d
196:=
186:x
183:1
173:x
170:d
166:d
38:.
31:.
20:)
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