Talk:Invalid proof
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[edit] 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)
- 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)
- This stuff does not appear in textbooks, I am fairly certain. --Cheeser1 (talk) 21:33, 19 January 2008 (UTC)
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- 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)
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- An encyclopedia is a reference text, not an instructional text. See WP:NOT#TEXT. --Cheeser1 (talk) 00:23, 20 January 2008 (UTC)
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- 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#TEXT 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)
- 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)
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(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)
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- 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)
- 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)
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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 × 0, and the erroneous step is to start with x × 0 = y × 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...)
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- 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)
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- 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)
[edit] 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)
[edit] Error
My edit doesn't always show up. Why is this? Oboeboy (talk) 19:19, 14 March 2008 (UTC)
[edit] 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)
