Talk:Mathematical fallacy

Clean up (Jan 2008)
I hate to say it, but this article is a complete mess. It contains a substantial amount of redundant content. Square roots produce 1=-1 because they can be multivalued. So do fourth roots, or 3/2 powers, or 3/2 powers of things that are equal to -1 but obfuscated. We don't need five examples of the exact same thing. That's, by my estimation, half or a third of the article. I believe with a little effort, this article can be substantially tidied. I would, however, raise my concerns now that this article isn't encyclopedic as it stands. Nothing in this article is verifiable, it's simply an anecdotal collection of all those impossible "proofs" that kids hear in math class. Like they say, Wikipedia is not for things made up in school one day, and judging by what I just archived, alot of the stuff in this article is exactly that. Surely, some of the fallacies are notable in their commonalities (dividing or multiplying by zero, using multivalued complex functions incorrectly), but these things should be discussed briefly in an encyclopedic fashion. They shouldn't be accumulation points for a giant collection of invalid deductions. This is on my mental to-do list, and I may start cutting redundant or unnecessary things from the article, so if anyone wants to help or contribute their thoughts, I'd encourage you to do so. I'd be more than happy if someone else did some of the work here or contributed their ideas about what should happen to this article. --Cheeser1 (talk) 10:00, 4 January 2008 (UTC)


 * You are right; none of these things appear to have encyclopedic value in themselves. A few of them might make sense as examples in an article about mathematical fallacies in general, but they would need to be subordinate to a general discussion of such fallacies. –Henning Makholm 19:43, 19 January 2008 (UTC)


 * I disagree, I find this encyclopaedic, albeit currently unreferenced. I strongly suspect it is quite referenceable to maths textbooks though, and I would hate to see it deleted.- (User) WolfKeeper (Talk) 20:51, 19 January 2008 (UTC)
 * This stuff does not appear in textbooks, I am fairly certain. --Cheeser1 (talk) 21:33, 19 January 2008 (UTC)
 * I'm a lot less certain. And I'm 100% certain that there are books with these kinds of things in that can be cited.- (User) WolfKeeper (Talk) 22:29, 19 January 2008 (UTC)
 * What, like a complex analysis textbook will explain how the square root is really "multi-valued"? Or an algebra book will explain that the 0 of a ring has no inverse? These joke proofs circulate on message boards and people show each other, but they have little value and I doubt they are laid out in such terms - perhaps as a "caution, do not do this" but we are not a textbook, and unless these are meaningful beyond instructional value (or to confuse/trick people on an internet forum) I don't see how it's encyclopedic. --Cheeser1 (talk) 23:38, 19 January 2008 (UTC)


 * Why would they need to be meaningful beyond instructional value? The whole point of an encyclopedia is to be instructional, and the wikipedia, being the biggest encyclopedia ever, has plenty of scope to include information that some will not find in any way important.- (User) WolfKeeper (Talk) 00:02, 20 January 2008 (UTC)


 * An encyclopedia is a reference text, not an instructional text. See WP:NOT. --Cheeser1 (talk) 00:23, 20 January 2008 (UTC)


 * There is a big difference between something that is instructional (as in educational) and a how-to. This is not a how-to.- (User) WolfKeeper (Talk) 01:20, 20 January 2008 (UTC)
 * WP:NOT mentions "instruction manuals" and "textbooks" in separate bulleted items. Wikipedia is supposed to be neither. "The purpose of Wikipedia is to present facts, not to teach subject matter." False proofs are not facts, if anything they are un-facts. –Henning Makholm 01:50, 20 January 2008 (UTC)

(unindent) My biggest concern is that this is essentially one big piece of original research, and we'd do best to simply cite a couple of well-known and commonly mistaken "proofs", rather than do the infinite number of ways of "proving" 1=-1. x42bn6 Talk Mess 01:26, 20 January 2008 (UTC)
 * Countably infinite, so at least we can put them in order. --Cheeser1 (talk) 01:30, 20 January 2008 (UTC)
 * I think you misread the meaning of facts in this context. WP writes about facts as in well known existing terms/notions/phenomenons/concepts/etc and not as in writing only about things, which are scientifically proven to exist. Those invalid proofs exists as philosophical concepts, religious terms, fanous mistakes or misnomers do and as such WP can of course write about them as long as it correctly (and factually) describes them as invalid proofs.--84.174.254.199 (talk) 13:37, 3 June 2008 (UTC)
 * Spelling errors exist, but an article on the topic should not consist of an indiscriminate collection of made-up examples of spelling errors, but instead of verifiable facts (or ascribable notable opinions) about spelling errors. --Lambiam 21:49, 3 June 2008 (UTC)
 * WRT 01:50, 20 January 2008 (UTC), Fact is anything that can be proven - these are factual, all right. Just not correct. —Preceding unsigned comment added by Raekuul (talk • contribs)

At the very least, if we keep this we should get rid of the "Proof that 2+2=5" style section headings. They convey no useful information about what happens in the sections; once one has "derived" a falsehood, it is trivial and arbitrary how one cleans up that falsehood to arrive at a more spectacular absurdity. –Henning Makholm 04:46, 27 January 2008 (UTC)

Ha! "Unverifiable material may be challenged and removed." I love it. OK, let's verify the invalid proofs. Seriously though, mathematical proofs (including, as here, proofs that certain proofs are invalid) should stand by themselves; it's useful to know who wrote about them first, or before, but the proof will still be valid if that information is missing 76.67.98.29 (talk) 21:54, 6 February 2008 (UTC).

Back in 2004 I observed that there was already a lot of redundancy, and I added this paragraph near the top of the article:


 * Most of these proofs depend on some variation of the same error. The error is to take a function f that is not one-to-one, to observe that f(x) = f(y) for some x and y, and to (erroneously) conclude that therefore x = y.  Division by zero is a special case of this; the function f is x → x &times; 0, and the erroneous step is to start with x &times; 0 = y &times; 0 and to conclude that therefore x = y.  Similarly, the argument below that purports to demonstrate 5=4 makes this same error with the function f(x) = x2.   The erroneous step starts with the correct assertion that for certain x and y, x2 = y2, and then makes the incorrect deduction that x = y.

It was removed in spring 2007, though. -- Dominus (talk) 14:47, 19 February 2008 (UTC)


 * It might be better to rename the article Mathematical fallacy (assuming there's too much detail to shoehorn into the existing fallacy article). Then, rather than laying out as a series of "proofs" followed by "the fallacy here is that ..." explanations, it could be categorised by types of fallacy, with an explanation of each followed by an illustrative example "proof". That way it might seem more like an encylopedia article. Matt 20:10, 20 February 2008 (UTC) (Actually, I've just noticed that "Mathematical fallacy" already redirects here...)


 * I agree: the encyclopedic content is the list of common fallacies, ideally with sources. What we can prove with them is interesting but secondary material to illustrate their use.  Of course, statements like 2=1, 3=0 etc. are equivalent because given any one of them as an axiom, the rest can be "proved" easily.  Debatably, most of the fallacies are also equivalent, but that is bordering on original research. Certes (talk) 21:29, 8 March 2008 (UTC)


 * Another vote for renaming this to 'mathematical fallacy'. In fact, I came across this article because I was looking for a reference on logic, not (non)algebra.  I also concur that the section headings should be re-named, and perhaps each fallacy could be done (only) twice: once in an 'obscured' form, and once in an obviously-false form.  not-just-yeti (talk) 15:02, 16 March 2008 (UTC)

I'm fine with renaming (and extending) the lemma to a more general article on mthematical fallacies. However i do consider the content (posssibly modified) of encyclopaedic value in the sense of listing typical (and well known) mathematical fallacies. And as mentioned above you can find at least some (or most?) examples in literature as well. One example would be Heuser's Lehrbuch der Analysis - Band I on page 51 (German). Additional literature can be found at http://www.cut-the-knot.org/proofs/index.shtml which lists a few invalid proofs with literature references.--Kmhkmh (talk) 16:14, 2 June 2008 (UTC)

References possible
WolfKeeper mentioned that he "strongly suspects it is quite referenceable to maths textbooks". I think this is likely to be somewhat true, especially if the article is merged to a mathematical fallacy article. For instance, I believe Stewart's calculus contains the "+C" indefinite/definite integral proof that 1=0 as an exercise. In my experience these sorts of things do make it as "interesting" exercises in basic texts on grade school algebra and introductory calculus. Of course, I agree with the previous comments that the current article is heavily redundant, and more or less not encyclopedic in its current form. I think a few good examples can be lifted and put into a more general article. JackSchmidt (talk) 07:31, 22 February 2008 (UTC)

Error
My edit doesn't always show up. Why is this? Oboeboy (talk) 19:19, 14 March 2008 (UTC)

Proof that the sum of all positive integers is negative
Isn't this much simpler:

Let S = 1 + 2 + 3 + 4 + ..., so 2S = 2 + 4 + 6 + 8 + .... Now subtract S from 2S: 2S =      2     + 4     + 6     ...  S =   1 + 2 + 3 + 4 + 5 + 6 + 7 ... - S = - 1     - 3     - 5     - 7 ... Given that the point of this example of an invalid proof is to show that you can't apply the usual rules to divergent series, an even simpler example of the perils is this proof of 0 is infinite: S = 1 + 2 + 3 + 4 + 5 + ... S =    1 + 2 + 3 + 4 + ... - 0 = 1 + 1 + 1 + 1 + 1 + ... --Lambiam 10:02, 16 March 2008 (UTC)


 * Yes, there is a fundamental tension between a "clean" invalid proof (where the fallacy is laid bare, like dividing by zero) and an invalid proof where it's not obvious what's gone wrong. (For example here, you could even use "S = 1+1+1+...".)  But anyway, I concur with what you're saying -- the version on the page should be replaced with the final version you give.  not-just-yeti (talk) 15:02, 16 March 2008 (UTC)

1+2+4+8...equals -1
Judging by the main page on this series this actually has some twisted mathematical merit to it. At any rate the current objection to the proof is "you don't use a convergence formula on something that doesn't converge", but the s=1+2s manipulation yields this without assuming anything about convergence, or at least not explicitly. This seems rather out of place in this page along with all the ha-ha-gotcha-we-divided-by-zero stuff and whatnot. --AceMyth (talk) 02:58, 31 October 2008 (UTC)


 * It is refreshing to see an example that isn't a simple division by zero. Raekuul 12:00, 17 December 2008 (UTC) —Preceding unsigned comment added by Raekuul (talk • contribs)


 * There isn't anything particularly twisted about the validity of the convergence of this series to -1 under the 2-adic metric. In fact it is a rather pedestrian result and a standard preliminary example in texts on the p-adic numbers (see Gouvea, "p-adic Numbers", 1997). It is fallacious for the standard metric on real numbers, since the formula sum_{k=0}^n a^n = (1-a^{n+1})/(1-a) only converges to 1/(1-a) for |a|<1 (type "geometric series" into wolfram alpha to verify that this condition on convergence is explicit for real numbers). See this article from MAA Online for a discussion of how Euler himself was misled into believing the result for the real numbers since mathematics at the time lacked a definition of convergence. Since the result is false for the standard metric, but true for the 2-adic metric, this example may be too subtle for a page on mathematical fallacies, and the best course of action may be to remove it. At the very least, this section should state that this is a fallacy under the standard metric since lim_{n->infinity} |(1-2^{n+1})/(1-2)| diverges to infinity because |2|>=1, and hence the series fails the convergence criterion. The validity of the result under the p-adic metric should be relegated to an afterthought. I'll see if I can find a reference which has free online access. pAddict--164.76.72.206 (talk) 18:25, 6 February 2012 (UTC)

Removal of unreferenced sections
I'm going to tag all the sections in this article. If they're not referenced within 3 months, then they're probably never going to be, and so they will be removed. I'm also not going to allow any new sections to be added unless they're referenced.- (User) Wolfkeeper (Talk) 05:24, 7 November 2008 (UTC)

Wait, What?

 * To illustrate this, consider the following "proof" of $$1=0$$ that only uses convergent infinite sums, and only the law allowing to interchange two consecutive terms in such a sum, which is definitely valid:
 * $$\begin{align}1&=1+0+0+0+0+\cdots\\

&=0+1+0+0+0+\cdots\\ &=0+0+1+0+0+\cdots\\ &=0+0+0+1+0+\cdots\\ &=0+0+0+0+1+\cdots\\ &\vdots\\ &=0+0+0+0+0+\cdots\\ &=0\end{align}$$

I think that the fallacy in this "proof" also needs to be explained. Raekuul 12:12, 17 December 2008 (UTC) —Preceding unsigned comment added by Raekuul (talk • contribs)

This is simply trying to show that mathematical proofs must contain finite steps by showing that a proof with infinite steps lead to absurd things. —Preceding unsigned comment added by 99.38.249.252 (talk) 06:08, 3 May 2011 (UTC)

1 != -1
square(root(-a))=-a root(-a) = i root(a) square(i) = -1 square(root(a)) = a With me so far? you have to consider every step.

root(square(-a)) = a

square(-a) = square(a) x square(-1) square(-a) = a^2 x 1 root(a^2) = a squareroot on a number give a possitive awnser. The logic written in the "paradox" is used in equations because in equations you are "reverse-engeneering" —Preceding unsigned comment added by 83.176.244.236 (talk) 03:01, 26 December 2008 (UTC)

Title Change
How if we change the title to "Invalid Proofs". There are more than 1 invalid proof(s) on this page. —Preceding unsigned comment added by 125.161.199.246 (talk) 02:51, 23 January 2009 (UTC)

Suggestion
Could the fact that three mathematical thirds of one isnt in fact one but 9.9 recurring be included in the article, since it is very simplistic. 9.9 recurring is infinitely close, but as infinity is a destination and not a fixed point, it is not equal to one. If this logic applied correctly, it can be deduced that one third of one is not 3.3 recurring, but a very slightly different number, or that three thirds of one isnt one, a paradox. This cannot be applied to numbers divisible by three i.e. 9. One third of nine is three. Three times three is nine. However one third of 100 is in fact 33.3 recurring, and 33.3 recurring times three 99.9 recurrring. Phew! Maths. —Preceding unsigned comment added by 82.39.116.172 (talk) 22:08, 1 February 2009 (UTC)


 * Although we could include an invalid proof based on a failure to understand that 0.999... = 1, I'm not sure this is a good idea. We already have an in-depth article on 0.999..., so perhaps we should add a link to that article in the see-also section. --Zundark (talk)

About radicals
The text says: "Start with the identity
 * $-1 = -1 \,$

Convert both sides of the equation into the vulgar fractions
 * $\frac{1}{-1} = \frac{-1}{1}$

Apply square roots on both sides to yield
 * $\sqrt{\frac{1}{-1}} = \sqrt{\frac{-1}{1}}$
 * $\frac{\sqrt{1}}{\sqrt{-1}} = \frac{\sqrt{-1}}{\sqrt{1}}$

Multiply both sides by $\sqrt{1}\cdot\sqrt{-1}$ to obtain
 * $\sqrt{1}\cdot\sqrt{1} = \sqrt{-1}\cdot \sqrt{-1}$

Any number's square root squared gives the original number, so
 * $\displaystyle{1 = -1}$

Q.E.D.

The proof is invalid because it applies the following principle for square roots incorrectly:


 * $\sqrt{\frac{x}{y}} = \frac{\sqrt{x}}{\sqrt{y}}$

This is only true when x and y are positive real numbers, which is not the case in the proof above. Thus, the proof is invalid.
 * undefined"

I think instead that the mistake is one step before: the rule
 * $$a=b \rightarrow \sqrt{a}=\sqrt{b}$$

is applied for a and b which are complex numbers and this is not correct (not even meaningful).

Am I right?--pokipsy76 (talk) 16:33, 22 February 2009 (UTC)


 * The complex number i is obtained by applying square root over -1. How is this a problem? --Secretss (talk) 15:19, 2 March 2009 (UTC)

Both are right; you get to pick your poison. If you work with a square root function defined only on the non-negative reals the error is where Pokipsy points at. But it is not uncommon either to work with a square root that is defined for all complex numbers but discontinuous along the negative real axis (namely, for a nonzero argument choose the root that has positive real part or is pure imaginary with positive imaginary part). In this latter case $$\sqrt{1/-1} = \sqrt{-1/1}$$ is a valid conclusion, but then you're not allowed to distribute the roots over the divisions. –Henning Makholm (talk) 21:20, 2 March 2009 (UTC)

An obvious error
let the positive number be a

let x = -a

squaring both the sides:

x² = (-a)²

x² = -a.-a

x² = a²

x² = √a²

x = a

the a² becomes √a², but the x² doesn't become √x². Edited to:

let the positive number be a

let x = -a

squaring both the sides:

x² = (-a)²

x² = -a.-a

x² = a²

√x² = √a²

x = a

Not sure how such an obvious error was missed. If I've missed something out, or am horribly wrong, do say. —Preceding unsigned comment added by 88.107.76.20 (talk) 10:31, 30 August 2009 (UTC)

Proposed cleanup

 * Mathematical fallacy (which currently redirects here) is a better title for the article than invalid proof. The section titles "Proof that..." can then be renamed to the more correct and descriptive "Fallacy of..."
 * There are too many examples of essentially the same thing. I propose that some effort should be made to give sources for things, and ultimately when a source is found to be lacking, content should be removed.  (A better place for a lot of the article is Wikibooks.  Maybe someone could move it there?)
 * The article should have some prose indicating why the fallacies are notable, and in some sense what they are all about.
 * Focus should be placed on the notion of mathematical fallacy in general, rather than on some specific litany of non-notable examples. Wikipedia is not an indiscriminate collection of factoids.

I have a feeling that a good way to approach cleanup overall is to attempt to rewrite the article from sources. There is no shortage of books on mathematical fallacies. 71.182.236.206 (talk) 13:04, 6 November 2009 (UTC)


 * I have put citation needed on all the proofs without any citation. I think asking for a citation is a good way of cutting down the proofs to those which people though were worth printing and might help to remove errors. I will have a look at providing citations for some which I know I've seen in books but otherwise I think the remainder should be removed after a week or so. Dmcq (talk) 17:17, 7 November 2009 (UTC)


 * I seem to have mislaid the old book I was going to look in "Riddles in Mathematiics' by Eugene P Northop. If itr is the one I was thinking of it has a number of interesting fallacies in it plus puzzles like the one about the probable length of a chord of a circle. Dmcq (talk) 23:49, 7 November 2009 (UTC)

Explanation of revert
I am sure that you are editing with good faith, but your two most recent edits were not appropriate. One of the (false) proofs your deleted was an attempt to correct another similar proof, and should be included in the article. This example is a particularly basic instance of how proofs can have subtle errors, and how subtle alterations may be necessary (and in this case, the proof cannot be fixed). If you defend your revert here, it is certainly possible to restore your edit. However, conforming to WP guidelines, once an editor has challenged an edit (by reverting), the person behind the challenged edit must defend the edit (and reach consensus) before adding it back. Generally in Wikipedia, we add rather than delete content and I think that this should be at the forefront of your mind when editing. You may certainly rewrite the article, but please be careful in terms of deleting content which was maintained for so many years by hard-working editors. -- PS T  01:59, 7 November 2009 (UTC)


 * If your attitude is simply to revert any good faith effort to reform the article, then I see no way that progress can possibly be made. There is a clear and obvious consensus on this very discussion page that there are too many examples, and that many of the examples are almost exactly like one another.  Moreover, the templates at the top of the page indicate that (1) much of the content is not encyclopedic (and so should logically be removed), and (2) that the article requires cleanup.  I assume that you disagree with these, but you seem to be in the minority.  My purpose is to completely rewrite the article from sources: in particular, this will establish appropriate context for a few choice examples, rather than be an endless litany of almost identical ones.  Wikipedia is not an indiscriminate collection of information and, as I suggest above, a better place for the article (as it currently stands) is WikiBooks. 74.98.44.216 (talk) 14:08, 7 November 2009 (UTC)

Error in Proof that 2 =1 in section Calculus
"no integer is an interior point of the real line" That is just plain false, every integer is an interior point of the real line. The mistake lies in applying the differentiation rule for finite sums in a case where the bounds of summation are not independent of the differentiation variable. I do not know how the current section meshes with the texts it cites, and whether these sources are really relevant here, but this needs to be addressed. Regards, Paradoctor (talk) 10:54, 7 November 2009 (UTC)


 * I have fixed the issue you point out. The cited source isn't clear on where the error lies.  I disagree with your proposed analysis of the fallacy, however.  Each side of the equation is a well-defined function of the integer x.  However, the derivative of a function of integer argument is not defined (in the naive calculus sense, at least) because the integers are not interior points (of the set of integers).  So we don't even get to the point where it is possible to apply the differentiation rule for sums.  As PST points out, there are generalizations of the derivative that do make sense.  An alternative analysis might run something like this: "equality of two functions evaluated at integer points does not imply equality of the derivatives."  But I think that this is less clear, since the function is a priori only defined for integer arguments: there is no canonical way to "interpolate" it.  74.98.44.216 (talk) 14:36, 7 November 2009 (UTC)


 * A few comments, but let's take of care business first: Thanks for the quick reaction, this gives me the warm and fuzzy feeling that keeps me hooked to my keyboard. ;)
 * If the source isn't clear on where the error lies, providing our own analysis constitutes OR. Routine calculations does not apply here, as evidenced by our current disagreement.
 * "original proof": Is there another proof discussed in the section I'm not aware of?
 * Presuming that "proof" in this article refers to mathematical proof, the phrase "proof is false" is at best confusing. I suggest something along the lines of "proof is faulty", "demonstration fails" (to convince), "argument is invalid", i. e. concentrate on showing that there is a flaw in the argument, rather than pointing to the conclusion's truth value. Proof is generally understood to imply validity, an invalid argument loses its proofness.
 * "line 3 only holds when x on each side is an integer": (as do lines 2&4, BTW) It is true that this notation presupposes integer x in the case of "naive" calculus (is that a term of the trade?). But, analogous to fractional calculus, nothing prevents us from defining a fractional summation compatible with the traditional definition, in which the bounds may range over the reals, a simple case being rounding of the bounds. This concept can be generalized to functions, leading to the possiblity of applying a function &pi; times to an argument. A instructive case is the Gamma function, which extends to the factorial to complex arguments, permitting definition of a special case of fractional multiplication:
 * $$\Gamma(\pi+1) = \pi! = \Pi_{i=1}^{\pi}i$$.
 * The pertinent question is whether the statement about line 3 is in the literature. If it is, an inline citation is essential, as this is a central point in the analyis of the argument.
 * As long as an argument is not fully formal, you're always faced with the task of interpreting the text, which practically always introduces ambiguities. The Gretchenfrage here is what is meant by "derivative with respect to x", further implying the question of what x's domain is intended to be. I daresay that the majority of readers will imply "naive" calculus and real numbers, respectively, but without some serious evaluation, that is just an educated guess.
 * "Each side of the equation is a well-defined function of the integer x": Sorry, but first and foremost, they are expressions. You may use these expressions to define functions, of course, but doing so constitutes a choice not implicit in the text of the argument. Of course, you could avail yourself to the argument directly above. ;)
 * "don't even get to the point where it is possible to apply the differentiation rule for sums": I see what you're getting at, but as I think I've shown above, that would only be true under assumptions not implicit in the text. Would "in a case not applicable in standard calculus" slake your bloodthirst? ;)
 * Regards, Paradoctor (talk) 18:21, 7 November 2009 (UTC)

I'll make some adjustments to the text to attempt to address some of these points. Let me respond to the ones that I disagree with here: --71.182.216.82 (talk) 19:41, 7 November 2009 (UTC)
 * The WP:OR point is one that could be made of pretty much the entire article, by the way. So far I think I am the only editor who has made any systematic effort to source the claims of the article.  At any rate, the OR issue in this particular case is not serious, since the fallacy itself appeared in a WP:RS: anyone familiar with the rigorous foundations of calculus can spot the error.
 * While we are certainly free to redefine things in any way that suits us, as long as we stick to conventional mathematics, the meaning of a finite sum is fairly unambiguous. In fact, most professional mathematicians are probably scarcely aware of the existence of the fractional calculus.  So, incidentally, the WP:OR point would probably be doubly an issue for using an unconventional notion of "summation", as opposed to the usual notion of "differentiation".  It is, at any rate, an utterly unsupported interpretation.
 * The distinction between "expression" and "function" in your post is somewhat of a red herring. The derivative (in what amounts to a "naive" calculus interpretation) is defined for functions, not expressions.  Is the line between expression and function really so sharp that anyone is likely to be confused by this?


 * I think that's my cue for going AWOL, too many bosses in here. Later. Paradoctor (talk) 13:56, 8 November 2009 (UTC)

Cleanup
The page has been moved, and I have rewritten the lead section. I have also cut six of the examples for various reasons. I'm not too worried about "original research" in the article, since most of the ideas involved are at least a century old, and the generation of a fresh example from a traditional idea for a fallacy is not in a serious sense research at all. I felt justified in taking down two of the tags. Charles Matthews (talk) 10:52, 8 November 2009 (UTC)


 * Excellent job on the lead. The title of the article should probably be "mathematical fallacy" in the singular, rather than the plural of "mathematical fallacies", per Wikipedia's typical naming conventions, but otherwise I strongly support the new title.  I agree with most (if not all) of the remaining cleanup.  "Taking in hand" indeed! 71.182.244.158 (talk) 13:14, 8 November 2009 (UTC)

Time for another move
The title of this page is "Mathematical fallaciy". Paradoctor (talk) 14:13, 8 November 2009 (UTC)

Please someone explain to me what is the falasy in the following proof
1 = 1 / 3 + 1 / 3 + 1 / 3

=(0.333...) +(0.333...) + 0.333...)

=(.0999......)

Lmt k --> &#8734; 1 ^ k = Lmt k -->&#8734;(0.999......) ^ k

Therefore,

1 = 0

Can someone explain here what was the wrong step?

If this proof were right can we conclude that

1 <> .09999999999999..........

Thanks —Preceding unsigned comment added by 208.120.156.114 (talk) 20:14, 6 December 2009 (UTC)


 * This page is for improving the article. Questions like this should be put on The maths reference dssk. And please put in a terminating pre like I stuck in in above if you put it there. Dmcq (talk) 20:33, 6 December 2009 (UTC)

Fallacy of the isosceles triangle - Someone clean the segment up, doesn't make sense.
"Furthermore, it can be further shown that, if AB is shorter than AC, then E will lie outside of AB, while F will lie within AC (and vice versa)."

This doesn't make any sense, because there is no mention of points "E" and "F" on the text, neither in the picture accompanying the text. —Preceding unsigned comment added by 87.94.140.223 (talk) 04:23, 7 July 2010 (UTC)
 * They should be Q and R, fixed Dmcq (talk) 07:05, 7 July 2010 (UTC)


 * WOW! 12 years later and this section is in gross error! Specifically: Line 7 in the "proof" claims that the two triangles BOR and COQ are congruent! LMFAO!! Simple inspection of the accompanying diagram shows they are not. (Not that that means much, but still your first clue ought to be a visual 'consistency check'.) Yes, both share a 90 degree angle, but clearly the angles RBO and QCO are NOT equal! (nor, of course are angles ROB and QOC.) I guess the "hidden fallacy" is assuming that BOQ and COR are (straight) lines (line segments). They're not. (Depending on how the fallacy is set up, you could also propose/pretend that the perpendicular line from AC to B (and also from AB to C) intersects at O, which it doesn't (generally).) I find it hard to believe that this has been allowed to exist for 12 (plus?) years! I keep going over the thing, and ...nope, Line #7 does NOT follow from any preceding line, nor is it correct. Segments BR and CQ aren't (necessarily) equal, nor are angles BOR and COQ. Segments BO and CO are equal, since O is on the perp. bisector. So we have two triangles with one (90°) angle and one (opposite) side equal. So what? What follows line 7 is utter rubbish, by the way. This needs a rewrite, and this time I politely suggest it be coherently written by someone who understands middle-school geometry.98.21.219.152 (talk) 17:23, 31 August 2022 (UTC)


 * The reason why BOR and COQ are congruent is neatly explained in the text, and supported by relevant sources. The point is that O cannot be where it is the drawing, so you better don't do a "simple inspection of the accompanying diagram". You have been tricked into trying to reason by diagram inspection, which is the actual point of this fallacy puzzle. You've been had - DVdm (talk) 20:43, 31 August 2022 (UTC)
 * I think that's backward. The text does indeed explain that BOR and COQ are congruent by hypotenuse-leg equality; but it asserts without proof that the hypotenuses are equal, and they aren't. This is actually quite clear from the diagram. The real error is trusting the text.
 * O actually can be where the drawing shows it. The error is not in assuming that O is inside the triangle. If you limit the claim to "a triangle that contains its circumcenter is isoceles" then it's still wrong. 2600:1700:975F:F010:F43D:87EE:EEA7:CFDE (talk) 17:29, 12 May 2023 (UTC)
 * The hypotenuses of BOR and COQ are resp. BO and OC, and by design, of course they are equal. That is immediately inferred by looking at the triangles BOD and DOC of which they are the hypotenuses too. - DVdm (talk) 18:52, 12 May 2023 (UTC)

Q.E.D.?
I think it would be nice if somebody could clean up the inappropriate "Q.E.D."s that have been used all over the place. For example by making them appropriate by stating up front what is going "to be demonstrated". It would make things a lot more readable and as well as not look so illiterate. AlexFekken (talk) 10:09, 16 September 2010 (UTC)


 * I have just made a number of minor edits to deal with this. In doing so I found that the entries with inappropriate QEDs were invariably lacking a citation. For this and other reasons there is more scope for cleaning up for somebody who has the time. It is a nice collection of examples. AlexFekken (talk) 03:16, 17 October 2010 (UTC)

Errors in multivalued fuction section
The first sentence: "Functions that are multivalued have poorly defined inverse functions over their entire range." is false. For example a vertical line is a multivalued function because for the given value of x there are infinitely many value of y. The inverse of a vertical line is a horizontal line which is perfectly well defined. The sentence should read: Functions that are not injective have poorly defined inverse functions (multivalued functions) over their range.

In the explanation of the "proof" that 2\pi = 0 the text states that the arcsin is a infinitely mulitvalued function. This is also false. The arcsin function is a single valued function. Given any value of x, arcsin(x) is a number between -\pi/2 and \pi/2. This means that arcsin(sin(x)) only equals x if x is between -\pi/2 and \pi/2. This is the fallacy in the given proof since 2\pi is not in the requisite range. 67.142.166.22 (talk) 01:11, 6 March 2012 (UTC)

What's the point of this article?
This article primarily consists of (unreferenced) examples of false proofs (for which there is no hope of ever supplying references). There is no shortage of published accounts of individual mathematical fallacies from which to draw, if necessary. But the article simply cannot be an endless account of published false proofs. What, then? It seems to me that the article should discuss the notion of a mathematical fallacy (as it has been discussed in the peer-reviewed literature: e.g., mathematics, philosophy, psychology). This discussion can be informed by examples, but it does not seem like a worthy subject of an encyclopedia article for it to be about examples in their own right, which is the way things currently stand. I suggest stubbing the article, and trying to compile some sources on this talk page. Perhaps some help can be conscripted from users at WT:WPM. This is, after all, something of a "fun" area that might be a nice collaborative project. Sławomir Biały (talk) 00:28, 11 March 2012 (UTC)


 * There's been a number of books on mathematical fallacies and I think it should be straightforward enough to provide citations for most of the article. The references at the end aren't just published false proofs, they are books on fallacies. Dmcq (talk) 02:11, 11 March 2012 (UTC)


 * There's no shortage of cool fallacies out there, many of them in published books. That's kind of my point.  This article isn't a book on the subject of mathematical fallacies, and our purpose isn't to emulate them.  Instead, it's an encyclopedia article about mathematical fallacies.  So I think the question we need to address is: what do we put in it?  It's clear (to me at least) that it shouldn't just be a laundry list of fallacies sorted by topic.  Otherwise we would look like the already excellent cut-the-not link.  It has to do something other than this.  Our pillars (NOR, V, NOT) might help to produce a more encyclopedic article.   Sławomir Biały  (talk) 02:42, 11 March 2012 (UTC)


 * I think you're taking this article too seriously. We should have a category 'recreational mathematics' on Wikipedia where stuff like this can go perhaps. What you are talking about sounds more like maths pedagogy and correcting students' misconceptions which is nothing at all like what this article is about. There is a popular topic of amusing fallacies which is exploited by Cut the Knot along with some books. The article just summarizes a notable topic just like most other articles in Wikipedia.Dmcq (talk) 11:00, 11 March 2012 (UTC)

I just want to say that I agree that this is not a kind of article I'm proud of having in wikipedia. On the other hand, I don't see what can be done, either. Reading the thread above, this seems like a long-time problem, and no one figured out how to solve it. Maybe someone does! (I would concentrate on correct stuff.) -- Taku (talk) 15:28, 11 March 2012 (UTC)

Uncited content
I originally removed the following as failing verification. It did not appear in the originally cited source. It has since been restored without a reference, which makes the entry original research, and thus forbidden by policy. I have moved it here in case there is need to discuss the matter further. Sławomir Biały (talk) 19:47, 11 February 2013 (UTC)

Extraneous solutions 3 = 0 The following illustrates a subtle misuse of extraneous solution.

Assume the following equation for $$ x $$:
 * $$ x^2+x+1=0\,$$

Then:
 * $$ x^2=-x-1 \,$$

Divide by x (assume x is not 0)
 * $$ x=-1 -1/x \,$$

Substituting the last expression for x in the original equation we get:
 * $$ x^2+(x)+1=0\,$$
 * $$ x^2 + (-1-1/x) +1 = 0 \,$$
 * $$ x^2 + (-1/x) = 0 \,$$
 * $$ x^2=1/x \,$$
 * $$ x^3=1 \,$$
 * $$ x=1 \,$$

Substituting x=1 in the original equation yields:
 * $$ 1^2+1+1 = 0 \,$$
 * $$ 3 = 0 \,$$

Q.E.D.

The fallacy is in the last part of the "proof" before "Substituting...", which is to conclude that because
 * $$ x^3=1$$

then
 * $$ x=1. $$

In fact, equation
 * $$ x^3=1 $$

has three solutions, and
 * $$ x=1$$

is just one of them (the other two are solutions to the original equation).

The first part of proof is correct, which is to show that a solution to the original equation
 * $$ x^2+x+1=0 $$

is a solution to
 * $$ x^3=1. $$

It is also correct that
 * $$ x=1$$

is a solution to
 * $$ x^3=1.$$

However, this does not imply that
 * $$ x=1$$

is also a solution to the original equation. No where in the "proof" it is shown that a solution to
 * $$ x^3=1$$

is also a solution to the original equation. In fact, $$x=1$$ is not a solution to
 * $$ x^2+x+1=0.$$

Therefore, it is incorrect to substitute $$ x=1 $$ into the original equation.


 * Quite right I think remove the stuff. This article in particular needs citations to stop people's brainwaves over breakfast being stuck in. I also stuck a query at User talk:Tuntable about the citation that was there which didn't actually pan out - that is a bit worrying. Dmcq (talk) 21:09, 11 February 2013 (UTC)


 * The fallacy as illustrated by the original example is commonly discussed with extraneous solutions. I agree this article should not be an endless list of fallacies, but rather it should address the notion, and types of common fallacies. Extraneous solution is such an important type that should not be excluded. Furthermore, the whole section on Multivalued functions of this article doesn't have a reference either.Hguy (talk) 22:04, 11 February 2013 (UTC)


 * Indeed, it doesn't. There was a bunch of stuff there that lacked references.  I removed these, and other uncited items, and left the stub section header in, in the hopes that it would encourage someone with sources to add those.  We don't need more uncited stuff in this article.  What we need are more reliable sources on mathematical fallacies.   Sławomir Biały  (talk) 23:56, 11 February 2013 (UTC)

One removed was pretty and I couldn't remember it 109.152.222.39 (talk) 23:53, 1 August 2013 (UTC)

Any angle is zero

Construct a rectangle ABCD. Now identify a point E such that CD = CE and the angle DCE is a non-zero angle. Take the perpendicular bisector of AD, crossing at F, and the perpendicular bisector of AE, crossing at G. Label where the two perpendicular bisectors intersect as H and join this point to A, B, C, D, and E.

Now, AH=DH because FH is a perpendicular bisector; similarly BH = CH. AH=EH because GH is a perpendicular bisector, so DH = EH. And by construction BA = CD = CE. So the triangles ABH, DCH and ECH are congruent, and so the angles ABH, DCH and ECH are equal.

But if the angles DCH and ECH are equal then the angle DCE must be zero, which is a contradiction.

The error in the proof comes in the diagram and the final point. An accurate diagram would show that the triangle ECH is a reflection of the triangle DCH in the line CH rather than being on the same side, and so while the angles DCH and ECH are equal in magnitude, there is no justification for subtracting one from the other; to find the angle DCE you need to subtract the angles DCH and ECH from the angle of a full circle (2π or 360°).

Parting words
After a few days of trying to contribute and make this article a little better (it is already great), I'm leaving this article and will stop modifying it again simply because I don't have enough time to deal with the ongoing events.

I came across this article while searching for a "sophisticated proof" of 1=0, and found the original section "x cubed =1" perfect in that regard. Having realized that the original statement didn't convey what the fallacy intends to convey, I contributed a modification. However, my contribution was repeatedly reverted and removed with reasons as "the original version was correct", and then "it was in a wrong section" after explanation was given why the original version was not appropriate, and then "there was no reference" after the modification was moved a separate section, and then "it was not a reference" after a reference was provided.

I understand the need to police the page. However, by looking at the history of the article, it is not the case that the article had a history of being vandalized previously. After all, this is a small article compared with most Wikipedia articles. Even if the section is not perfect, would the Wikipedia community be better served with this particular example or without? I'd definitely say the answer is the former.

Now this begs the question why Sławomir kept removing such a common example of fallacy from this article, given there has been other section in the article without a reference. I don't have the answer, but I hope it is not to try to avoid the embarrassment of, as a regular of this article, not spotting the issue with the example previously, and subsequently insisting that the original version was correct after it was pointed out that the original example was not appropriately stated.

As I'm leaving, I challenge the readers of this article to find a "proper" reference for the original example of "3=0" and have this example included back into this page, because the Wikipedia community is better served with this example on this page than without. I'd be happy to see this example be included in this article, even if it is in the original version. Hguy (talk) 01:25, 12 February 2013 (UTC)
 * I have removed the addition again. It referenced another Wikipedia article - one without any references. That other problems exist is no reason to start sticking in more when other people disagree. This article really has had too much uncited stuff stuck into it and needed to be cleaned up. What one has to do is find a book or paper or newspaper article or something that says something and reference that if other people disagree with something being included. Verifiability does not mean everything needs a citation in the article - but it does mean that if anyone disagrees and thinks something needs a citation they can remove the stuff if a citation can't be found in a reasonable time see WP:V. In particular verifiability is not provided by someone working something out for themselves and being sure it is right, see WP:OR about that. Dmcq (talk) 08:49, 12 February 2013 (UTC)
 * I don't really appreciate having ulterior motives suggested for my actions. This suggests that I wasn't acting in good faith.  I'm certainly not embarrassed about anything.  I think the fallacy is a better illustration of the fallacy of extracting the root of $$x^3=1$$, which makes it belong in the "Power and root" section under the heading of $$x^3=1$$.  We can quibble over the semantics of the best way to lay out the fallacy then, but I still firmly believe that the original version was better than your proposed one.  Your proposed version makes no mention whatsoever of complex numbers.  Nowhere does it say that the two non-real roots of $$x^3=1$$ are precisely the roots of the equation $$x^2+x+1$$.  I don't see how you can claim to offer a correct resolution of the fallacy without saying these these things that lie at the heart of the matter.   Sławomir Biały  (talk) 12:55, 12 February 2013 (UTC)

By HL
ClueBot NG rightfully reverted the question "WHAT THE HECK IS HL???" added by IP 192.16.204.218 below the line:
 * 9. By HL, △ROB ≅ △QOC (RO = QO since △RAO ≅ △QAO; BO = CO since △ODB ≅ △ODC; ∠ORB and ∠OQC are right angles)

I must say, I have no idea "what the heck" HL might be either. I went to to the wikilinked article solution of triangles and couldn't find any clue as to what HL might stand for. Can someone clarify? - DVdm (talk) 08:17, 21 January 2014 (UTC)


 * ✅, thanks Google. Made a change to make it clear for future readers. - DVdm (talk) 09:32, 21 January 2014 (UTC)

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Requested move 17 April 2018

 * The following is a closed discussion of a requested move. Please do not modify it. Subsequent comments should be made in a new section on the talk page. Editors desiring to contest the closing decision should consider a move review. No further edits should be made to this section. 

The result of the move request was: no consensus to move the page at this time, per the discussion below. Dekimasu よ! 02:45, 24 April 2018 (UTC)

Mathematical fallacy → Invalid proof – As stated in the lead section, mathematical fallacy is a special case of invalid proof. And because they are both covered by this article, I suggest to use the broader title, which was used until 2009. Petr Matas 15:42, 17 April 2018 (UTC)


 * People write books with 'mathematical fallacy' in the title. They don't write ones about 'invalid proof'. The better common name for the topic is mathematical fallacy. As described in the lead there's a lot more to being a mathematical fallacy than it simply being an invalid proof. Dmcq (talk) 16:57, 17 April 2018 (UTC)


 * The above discussion is preserved as an archive of a requested move. Please do not modify it. Subsequent comments should be made in a new section on this talk page or in a move review. No further edits should be made to this section.

Line 3 = zero
user:DVdm In line 3 of section Division by zero, show me any number where the expression does not come back to 0 = 0, there is none, zero then populates line 4 to factor zero and line 5 to reduce/divide by zero, if we are going to use this example it needs to be correct and shown so that non-math people can understand it. Please revert your revert and check the math first and show me a number, any number121.99.108.78 (talk) 23:12, 17 November 2019 (UTC)
 * Yes, the equation (not the "expression") comes back to 0 = 0. But you wrote that "Line 3 equals zero, line 4 factors zero and the fallacy is completed in line 5". Your wording was sloppy: lines don't have values, lines don't factor values. You could have said that "both terms in line 3 equal zero, a factor zero is factored out in line 4, and the fallacy occurs in line 5". Saying that "the fallacy is completed in line 5", suggests that the fallacy started before line 5. It didn't. I think that non-math people deserve precision, so I reverted your edit because of this sloppiness, and because the original wording was correct and sufficient. There were no errors or fallacies in lines 3 and 4, and your explanation sort of drew the attention away from the real fallacy. Hope this helps. - DVdm (talk) 08:20, 18 November 2019 (UTC)
 * DVdm Thanks for the reply, however I still cannot see the distinction, the lead reads...
 * 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
 * ... and the divide by zero is invoked, so to speak, by having a disguised zero = zero in line 3, i.e. false or irrelevant logic e.g. a fallacy, this is then on-sold by factoring ((needlessly in reality) before dividing. Surely getting a bad expression past the reader (the student, fallacy target etc. not wiki reader here) is the actual invoking of the fallacy condition. Happy to learn, hence re-visiting, cheers121.99.108.78 (talk) 10:31, 25 November 2019 (UTC)


 * Yes, there is a "disguised" 0=0 in lines 3 and 4, but there is no fallacy because indeed 0=0. The action of factoring out (a-b) is not irrelevant. On the contrary, it is necessary as a preparation for the actual fallacy of dividing by zero, which happens in line 5. And of course, also note that what you call "irrelevant logic", is still valid logic, and certainly no fallacy. - DVdm (talk) 11:14, 25 November 2019 (UTC)

Northrop
Just been reading an book by Eugene P Northrop which covers practicaly all the ground of this article and a lot more in detail. It dates from 1944 but Dover have reprinted it

NadVolum (talk) 10:14, 1 November 2021 (UTC)

"X=y" listed at Redirects for discussion
A discussion is taking place to address the redirect X&. The discussion will occur at Redirects for discussion/Log/2021 December 4 until a consensus is reached, and readers of this page are welcome to contribute to the discussion. User:1234qwer1234qwer4 (talk) 17:56, 4 December 2021 (UTC)

clarification needed
I consider that there is a need to clarify https://en.wikipedia.org/wiki/Mathematical_fallacy#Complex_exponents in light of https://en.wikipedia.org/wiki/Imaginary_unit#i_raised_to_the_power_of_i 2001:6B0:E:2B18:0:0:0:71 (talk) 11:00, 14 August 2022 (UTC)