View source for Wiki143 talk:WikiProject Mathematics/Proofs/Archive 2

{{talkarchive}}
{|class="messagebox standard-talk"
|-
|align="center" style="font-size: x-large; font-weight: bold; line-height: 125%"|<div style="float:left;">[[Image:Nuvola apps edu mathematics-p.svg|50px]]</div>This is an old discussion page for <br/>[[Wikipedia:WikiProject Mathematics|WikiProject Mathematics]]
|-
| align="center" style="padding-top: 6pt; border-top: 1pt solid #c0c090;"|This page is devoted to discussions of when and how to include proofs in mathematics articles. The [[Wikipedia talk:WikiProject Mathematics|main discussion page]] should be used for other issues. 
|-
|align="center" style="font-size: small; padding-top: 6pt; border-top: 1pt solid #c0c090"|Please add new topics at the bottom of the page and [[WP:SIG|sign your posts]].
|}
{{archivebox|
[[/Archive 1]]: 8/16/2004 - 9/3/2005<br/>
[[/Archive 2]]: 7/13/2006 - 3/31/2008
}}

==Distinguishing between proofs in context and stand-alones==
This thread has produced many good ideas for proofs which exist mainly to support an article. In my opinion though, there are some proofs which are deserving of their own article. If the proof is '''only''' in existance to support a specific concept, one of the ideas outlined above would be appropriate, but some proofs have significant backgrounds of their own. Proofs with histories, versitile applications, and context in the development of math may well be worth their own article. I think most mathmaticians would agree that the classec 'proof' of the 4 color theorem, for example, is worth discussing on its own, seperately from the theorem itself because it has a long history, complex solution, and has had a noteworthy response from the mathematical community. Alternately though, we could likely come to a consensus that, for example, the double angle theorem proofs are not worthy of their own article but should be explained/referenced as a part of the theorems' article because the proofs are relatively trivial and did not mark a signifigant point in the development of math. [[User:48v|48v]] 05:36, 13 July 2006 (UTC)

:I believe that consensus already exists. Culturally/historically important proofs deserve their own stand-alone articles. The somewhat open question is what to do about "relatively trivial" proofs that "do not mark a significant point in the development of math.". At the moment, the latter are allowed, but shunted into the ghetto called [[:Category:Article proofs]]. NB. this latter category has been growing at the rate of one article proof per month; there is no particular demand. In important ways, [[Planetmath]] is a more suitable place for article proofs. [[User:Linas|linas]] 19:46, 10 December 2006 (UTC)

==Let's just create articles that gather related proofs==

I already started one, [[Proofs of trigonometric identities]]. --[[User:Kirbytime|<span style="color:red;">Ķĩřβȳ</span>]][[Islam|<span style="color:green;">♥</span>]]<font color="pink">;[[User_talk:Kirbytime|Ťįɱé]]</font>[[Special:Contributions/Kirbytime|<span style="color:blue;">Ø</span>]] 11:32, 20 October 2006 (UTC)

:I agree, Wikipedia should include 'naive' definitions etc for the general public, but there is no reason why we should try to exclude proofs. If we can somehow group proofs into sensible collections and then have links to the appropriate pages on an article then that would stop a proof/more rigour slowing the flow, whilst allowing the more 'expert' reader to delve into the subject deeper. --[[User:TM-77|TM-77]] 11:57, 28 October 2006 (UTC)

[[Proofs of trigonometric identities]] is, in its present form, a '''horrible mess'''.  Please help clean it up. [[User:Michael Hardy|Michael Hardy]] 19:11, 13 December 2006 (UTC)

How does it look now? --[[User:Kirbytime|<span style="color:red;">Ķĩřβȳ</span>]][[Islam|<span style="color:green;">♥</span>]][[User_talk:Kirbytime|<span style="color:pink;">Ťįɱé</span>]][[Special:Contributions/Kirbytime|<span style="color:blue;">Ø</span>]] 22:03, 18 December 2006 (UTC)

==/Proof subpages== <!-- Can we change the section heading without breaking existing links? -->
'''Proposition:''' Create a /Proofs subpage with mathematical articles, listing proofs of theorems and statements made in that article. For example, the page [[Linear_Algebra]] contains a section ''[[Linear_Algebra#Some_useful_theorems|Some useful theorems]]'', the proofs of which could be given on [[Linear_Algebra/Proofs]].

'''Arguments:''' (''Here is my argumentation, including many points made above'') As a mathematician I often read statements and theorems on Wikipedia pages of which the proof would interest me. Often, proofs are omitted though. One could argue that Wikipedia is a general encyclopedia and not a compilation of mathematical theorems and that therefore, proofs do not belong in it. Also, they would make pages unnecessarily cluttered and harder to read for the general public, and those just trying to grasp an idea without getting caught up in the details. On the other hand, one may as well argue that Wikipedia is (and is becoming more and more) a compilation of knowledge and should actually contain as much information as possible. 
Therefore, I'd propose to include proofs, but separated from the main article. To do this in a uniform way, a subpage of the article should be created: when one sees a statement on a mathematical page, one simply appends ''/Proofs'' to the page name to verify it. This would keep the page clean and readable, make the details available in an extremely easy (and uniform!) way for anyone interested and would have the advantage that proofs can be verified and corrected by anyone that can read them. A cite-like reference to the ''/Proofs'' page could be added to the statements of which the proof is available. 

For an example, see [[Addition_of_natural_numbers/proofs]]; I'm talking about generalizing this concept.

[[User:CompuChip|CompuChip]] 10:28, 12 November 2006 (UTC)

:This is a wonderful idea. I stopped editing and reading Wikipedia articles related to math when I saw that proofs are, by policy, not to be included in the Wikipedia. Reason being that I have no idea if a mathematical statement is correct without seeing the proof and have been burned too many times by taking a text book or author's word for something. If wikipedia is going to store real mathematical knowledge and be credible, it needs to store the proofs. This is the same as referencing a politics piece to a newspaper. None of the knowledge in Wikipedia regarding math is worthwhile without the proofs or references to them, and there is no reason not to include them on a separate page, allowing interested parties to see them and less interested parties to skip them. For recent mathematics knowledge, I see no reason a link to relevant articles can be used instead, but most of the stuff in Wikipedia is over 100 years old and I think including proofs is a worthwhile venture. I would also suggest a markup similar to <nowiki>{{fact}}</nowiki> that would indicate a proof was needed. [[User:Pdbailey|Pdbailey]] 17:43, 1 April 2007 (UTC)

::Although this discussiion seems to be a bit old, I find it very relevant. Proofs are such a vital part of mathematics that I think we MUST find a way to include them for the interested reader. This proposal seems to be a good solution. I'm new to editing Wikipedia but I'm already tired of the same things as the poster above. Is there any way to give this debate more attention among the mathematics editors as I think it is very vital to the mathematics articles. [[User:JKBlock|JKBlock]] 17:31, 18 June 2007 (UTC)

:A very good proposal! To support it and let us gain some experience with it, I've started a [[Moore-Penrose pseudoinverse/Proofs|new example]], which proves a relation in the main article that was questioned on the [[Talk:Pseudoinverse#Question_on_one_of_the_relations|Talk]] page.  I could not find a proof for this relation in the literature, so could not simply add a reference to the proof. Even if we did have a reference, it might be hard to come by and check (the reference). --[[User:RainerBlome|RainerBlome]] 10:29, 5 July 2007 (UTC)

: Interesting idea. However I think it is very important, that at least one reference is made in the main article to the proofs page if one such is made. Otherwise one would never guess that the article has a proofs subpage. Apart from people not being able to find what they are looking for, we risk duplication of material. Also the main article might be moved, with the proofs page staying behind. I'm not sure where best to put a reference to the proofs page (I think it should be placed prominently), but under See Also may be a start. [[User:Aenar|Aenar]] 03:46, 23 September 2007 (UTC)

== A lay perspective? ==
I do not know whether my comments are welcome here, as I am a very new contributor to WP. Anyhow, here goes. I am all for proofs, both complex and elementary. I agree that they cannot all appear in the main article on a subject, but I feel strongly that they should appear somewhere.

Is there any reason why WP cannot be a both an encyclopedia for lay people and an encyclopedia of mathematics. This would allow people who are interested in a subject to start out as a lay person and (assuming accurate articles meaningful interpretation) progress to higher levels of understanding. It is often mentioned in the long discussion about proofs, that they can be found in any mathematics textbook, yet how many lay people have mathematics textbooks in the bookcase at home? And how many people have more than one mathematics textbook? When i was studying, I had the vast library on campus to turn to for alternate proofs when the one my textbooks gave was above my understanding. I am no longer a student and without those reference works I am lost.

I don't work in the field of mathematics, but mathematics is the human construct that I find most fascinating. I also love asking ''Why?''. I feel that if a person asks ''why?'' or thinks ''that cannot be right'' when reading an article, the answer or proof should be readily availible.

I could go on, but this page is quite long enough as it is ;-)

:I support the inclusion of proofs in WP, either as a subpage or as a separate article --[[User:Payxystaxna|payxystaxna]] 22:00, 27 November 2006 (UTC)

::Obviously (see my above post) so do I. The question is where to place this to reach consensus on it and get it implemented. I can start adding proofs to pages now (well, actually in a few days when I have time) using the ''/Proofs'' subpages I proposed, but I'd rather wait until it's made "official" and I'm sure everything is done correctly the first time. --[[User:CompuChip|CompuChip]] 15:34, 28 November 2006 (UTC)

:::The problem with proofs is this. To simply state every theorem in every textbook on mathematics, it would increase the number of WP math articles by a hundred-fold. I have, for example, entire shelves of math books which are currently summarized in WP by a handful of mostly small articles. So, just to recap the content would require an explosive growth. Now, as proofs are typically 3 to 100 times longer than the statement of a theorem, this would require another explosion in the quantity of content. Even if the explosion occurred, simply protecting it against vandalism would be a task.

:::Thus, I argue that the appropriate thing to do is to focus on providing missing content, rather than providing proofs. I would also like to suggest that a better repository for proofs might be [[Planetmath]], which does have a charter for this, and already has hundreds if not thousands of proofs. By contrast, we have only seventeen in [[:Category:Article proofs]] so far, and this cat is growing at the rate of one a month. [[User:Linas|linas]] 20:05, 10 December 2006 (UTC)

:::: I have long been taught that there are three kinds of proofs - those that establish a result, those that illuminate a result, and those that expand a result.  That is to say, there are proofs that are simply a way to get from point A to B - they don't really generalize, and there isn't much that can be learned from a close analysis of them.  The second class of proofs are proofs that, in doing them (or seeing them) one realizes something more fundamental, or important, about the thing being proved.  The last class is the type of proof where the method of proof, or construction used, is more important than the result found - like, say, Euclid's Method in a proof involving congruences.  It seems that the latter two are the most important for our purposes - and, especially the second, should be included in an article.  The first should be 'sourced', but probably not included.  Most mathematicians have an learned sense of what class a given proof falls into - and one will note that it is independent of the difficulty of the proof given.  In addition, length seems to be totally separate, as a consideration.  To relate to what you said - we should focus on adding content - however, we should not omit, or overlook, the fact that in many cases a proof 'is' content, and just as important for understanding.  [[User:Haemo|Haemo]] 09:01, 19 December 2006 (UTC)

::::Experience shows that explosive growth of Wikipedia happens anyway, proofs or not, so it will be dealt with anyway. Mathematics is just one area in Wikipedia, and adding proofs will not make a big difference in volume, I believe. The question to me is not "do we want growth?", but "what kind of growth do we want?"  You suggest to focus on missing content.  Proofs are content to me, and I and others miss them, and I and others want to add them.  PlanetMath has the disadvantage of not being a public Wiki, with all the associated advantages and disadvantages.  In particular, PlanetMath's potential growth speed is limited. My personal experience is  that Wikipedia's quality is high enough for my needs.  --[[User:RainerBlome|RainerBlome]] 10:18, 5 July 2007 (UTC)

==Suggestion==

Have you considered creating a Mathematical proof book in Wikibooks? That seems to be a natural place for a chunk load of proofs. or Wikitext. [[User:CommandoGuard|CommandoGuard]] 22:27, 19 December 2006 (UTC)

: See [[http://en.wikibooks.org/wiki/The_Book_of_Mathematical_Proofs]]. Cheers--[[User:Shahab|Shahab]] 18:11, 19 October 2007 (UTC)

== A student-mathematician's perspective ==

I am currently a student who loves Mathematics, especially the abstract kind. So of course I would like to have easy access to formal treatment in proofs of theorems. On the other hand there must be a sort of outline of the proof in layman terms for the general public. Thus I suggest that both be accomodated as far as possible.

Firstly, the main article should consist of the background and mathematical content relevant to the users who find and read it. This means that it should have in my opinion intuitive explanations rather than strict proofs, so that everyone can grasp the concepts involved in as little time as possible. Even for myself, I sometimes avoid reading long proofs or skip certain sections, and get confused further down, in the end realising that I had missed out something in the middle of the "chunk". So it would help everyone if there was a intuitive and preferably short explanation of the truth of the mathematical content discussed.

Secondly, those who are really interested in the rigorous mathematical establishment of claims and theorems would surely want an easily obtainable form. This could be placed apart from the main article either through the dynamic "Click to reveal hidden content" or a "Proof" link to a separate dedicated article. I do have an inclination for the former method, because I think that it might be a bit cumbersome if a separate webpage has to be accessed for every proper proof.

Lastly, I believe it would be neater if proofs were written out one mathematical statement per line, because it may otherwise be harder to see the development of the proof. One example is [[Approximation_theory/proofs]]. I do not intend to be critical, but it is at least to me quite hard to read as every statement is concatenated into one paragraph, which is then at the mercy of word-wrap. Also the formal proofs themselves should have a complete formal statement of the result to be proven. In this example it was and still is not clear to me what exactly the first sentence means in formal terms, especially "is optimal". When I referred to the main article in question, I found a circular reference, which should be avoided as well.

Thus I too recommend that those who author the proofs check for readability and minimize referencing necessary. In other words the "proof" page/section should be ideally stand-alone, so that readers from mathematical background could if they wish look at only those and nothing else, while other readers could if they wish look only at the intuitive interpretation and explanation of the mathematical results. And both groups of people should be satisfied with whatever they choose to read, without having to struggle with what they do not want to read.

[[User:Lim Wei Quan|Lim Wei Quan]] 10:19, 30 December 2006 (UTC)

==A guinea pig==
I have mainly been editing pages on [[sporadic groups|sporadic simple groups]]. In my study of them I work out proofs as I go, but have not tried to post any on Wikipedia. There are many statements about simple groups that few could hope to prove. Certainly it would be formidable (and voluminous) to include a proof of the [[classification of finite simple groups|classification theorem]].

I added a definition to the article on the [[Higman–Sims graph]]. Someone else then posted a colorful diagram of the graph. I have since worked out a proof that the graph is [[strongly regular graph|strongly regular]]. The proof is pretty simple, but there are some presuppositions based on properties of [[Mathieu groups]]. THe proof really ought to be accessible from another article, on the [[Higman–Sims group]] [[User:Scott Tillinghast, Houston TX|Scott Tillinghast, Houston TX]] 03:41, 10 April 2007 (UTC)

==Originality of proofs==
Some might consider an original proof to be original research. On the other hand, an unoriginal proof could violate a copyright. I would hope that a contributor would rework any existing proof.

I have not seen anyone's proof that the Himan-Sims graph is strongly regular. Nevertheless I did what almost any mathematican would probably do after reading this result on the Wikipedia page. [[User:Scott Tillinghast, Houston TX|Scott Tillinghast, Houston TX]] 04:05, 10 April 2007 (UTC)

:I don't really see the relevance whether a proof is original research or not. My main objection against original research is that it is in general hard to verify, an objection that vanishes here. I think in most cases copyright is not an issue, since proofs should be as straightforward as possible and will not involve creative work by one author. In most cases, there will be one or only a few ways to approach a proof - when any mathematician would logically take some path I think it's safe to consider the proof... what do you call it, common, standard, ...? 
:Only in rare cases a proof will involve sufficiently creative work that copyright may pose a problem, but then usually the proof will be exciting enough to provide a reference (and if not, it's not worth it anyway, is it).
:But I think most proofs -- at least I can't think of any proof I've seen up until now, and after three years of studying mathematics full-time, that's quite a lot -- can be considered general enough not to have copyright, are old enough for the copyright to have expired, and probably a lot of modern proofs will be too complicated / specialized to show anyway. --[[User:CompuChip|CompuChip]] 16:01, 10 April 2007 (UTC) <small>Note I used the words ''often'' and ''probably'' a lot in my reply, to allow for exceptions. Of course, there could be cases in which the copyright ''is'' relevant, but what I'm basically saying is: in general I think there's no problem, if it is an issue it can be viewed case by case.</small>

:If a proof can be included in the wikipedia with some fruit (i.e. if anyone can understand it) it will be over 70 years old. About the most recent proof of interest I can think of would be that of the [[Banach–Tarski paradox]] (over 80 years old as of now). Also, I would argue that original proofs that other editors agree are correct are just fine. [[User:Pdbailey|Pdbailey]] 20:51, 11 April 2007 (UTC)
::I think this isn't true at all. Much of [[discrete mathematics]] is more recent: see [[Bellman-Ford algorithm]], for an example that is less than 50 years old. I agree that you probably can't copyright a proof though. -[[User:137.222.10.58|137.222.10.58]] 23:05, 20 May 2007 (UTC)

:::My understanding is: You cannot copyright the idea of a proof, that is, the general method used. But you can copyright a specific sequence of words or symbols which implement that idea, if the author made enough personal choices to make it something which someone else would not have created without copying him. So you should be OK, if you read a proof and then rewrite it in your own words, not quoting the text. That is, do not follow the original too closely, use different variables, et cetera. [[User:JRSpriggs|JRSpriggs]] 07:01, 24 May 2007 (UTC)

:Scott, a mathematical proof in itself cannot be copyrighted. A specific text describing such a proof can, as are text books, even if they describe old proofs. At best a proof could be patented -- if they weren't already explicitly prohibited from patentability (as it should be). [[User:Niczar|Niczar]] ([[User talk:Niczar|talk]]) 22:32, 3 July 2008 (UTC)

== What is Category:Proofs all about? ==

Does anybody know what [[:Category:Proofs]] is about? Most of the articles there seem to be just math articles which also happen to have a proof (e.g. [[König's lemma]], [[Pythagorean theorem]]). Others are articles that contain nothing but a proof of a theorem ([[Pythagorean theorem proof (rational trigonometry)]]). Others are topics about proofs ([[Q.E.D.]], [[Reductio ad absurdum]]). What is the true/original purpose of this category? --[[User:Zvika|Zvika]] 16:28, 23 May 2007 (UTC) <small>(Copied from [[:Category talk:Proofs|here]], hoping to get a response)</small>

:I believe the intention is not so much for "WP customers", but for us, the people on the proofs project, to have a place where we can find all the pages that would be affected.  That way, when someone makes a proposal ("proofs should be banned", "proofs should be footnotes to the main article", "proofs should avoid original research", ....) the proposal can be evaluated in the light of the existing body of WP proofs. [[User:William Ackerman|William Ackerman]] 20:00, 23 May 2007 (UTC)

::In other words, your understanding is that the category should contain pages with actual proofs? Rather than pages like [[Q.E.D.]]? In that case, shouldn't [[Q.E.D.]] and the like be removed? --[[User:Zvika|Zvika]] 21:04, 23 May 2007 (UTC)

:::I'd lean toward being inclusive -- "Anything that a person working in the general area of how proofs should be handled in WP ought to be aware of."  In that sense, Q.E.D. ought to be included.  Remember, it's for our own use. [[User:William Ackerman|William Ackerman]] 00:27, 25 May 2007 (UTC)

::::This still doesn't make sense to me. Sure, if someone is interested in ''any'' article related to proofs, they can find it here. But I think it would make more sense to split this up at least to two subcategories: one for articles ''about'' proofs and one for articles ''containing'' proofs. It seems to me that there are very different uses for these categories. For instance, someone might be interested in the philosophy of proof, but really doesn't care that an article with a proof of Pythagoras' theorem exists. The [[:Category:Article proofs]] is a step in this direction, I think. --[[User:Zvika|Zvika]] 06:22, 25 May 2007 (UTC)

Well. Since there seem to be no further comments, I am planning on organizing this category as follows:
* Articles containing only a proof or proofs will be moved to ArticleName/Proof or ArticleName/Proofs (where ArticleName is the article which contains the statements being proved) and given the template <nowiki>{{proof}}</nowiki>, which will place them in the (already existing) [[:Category:Article proofs]].
* Ordinary articles which also contain proofs will be moved to a new [[:Category:Articles containing proofs]], which will be a subcategory of [[:Category:Proofs]].
* Articles like [[Q.E.D.]] will remain in [[:Category:Proofs]].
Feel free to suggest, complain, or help out.. --[[User:Zvika|Zvika]] 19:58, 28 May 2007 (UTC)

:Since you ask: you just put the <nowiki>{{proof}}</nowiki> template in [[Proofs of Fermat's theorem on sums of two squares]]. I notice that after the preamble that indicates what this experiment is all about, the article immediately begins without starting a new line. Thus, the text of the article begins in the indented italic paragraph that is provided by the template. I think the template should be fixed so that this does not happen, and the body of the article begins in a new, unindented, line. [[User:Magidin|Magidin]] 18:21, 30 May 2007 (UTC)

::I changed the template to an infobox style. That should resolve the problem. [[User:CMummert|CMummert]] · <small>[[User talk:CMummert|talk]]</small> 22:39, 30 May 2007 (UTC)

:::It does. Thanks! [[User:Magidin|Magidin]] 18:48, 31 May 2007 (UTC)

== Template text ==
The {{tl2|proof}} template says, in part,

* This article is "experimental" in the sense that it is a test of one way [[Wikipedia:WikiProject Mathematics|we]] may be able to incorporate more detailed proofs in Wikipedia.

In my view the "experiment" has long since ended, in success. I see no reason to mention it in the template any further. Therefore I am revising the template, trimming it to the essentials. --[[User:KSmrq|KSmrq]]<sup>[[User talk:KSmrq|T]]</sup> 20:03, 2 August 2007 (UTC)

:Looks good to me. --[[User:Zvika|Zvika]] 07:34, 4 August 2007 (UTC)

== An argument against detailed proofs ==

I would in fact like to see fewer proofs on wikipedia.  Let me explain my reasoning:

(i) Every proof which appears on wikipedia is either original to the author or not.  To argue by cases:

(ia) If the proof is original to the author, there are issues about posting it on 
the internet for all to read.  Namely, in mathematics we have a refereeing process 
whereby a specific person or persons takes (some) responsbility for checking 
the correctness of the argument.  This is not the case on wikipedia, and moreover 
the wikipedist "everyone makes small modifications as they like" philosophy is not 
conducive to writing a correct proof.  

(ib) If the proof is not original to the author, then (obviously) the author is taking the proof from some other source.  I am not worried about copyright issues per se (I do not believe proofs are copyrightable), but there is a scholarship problem.  Namely, I think that an encylopedia of mathematics has more of a duty to getting the history, influences and attributions correct and explicit than a mathematics textbook (though I have come to wish that e.g. mid-level undergraduate texts were also more scholarly in this sense).  

(ii) It is more likely that an imitative attempt by a wikipedist (or a good faith 
effort by a student of the subject to gain understanding by working through the proof) will come out worse (pedagogically and/or logically) than the best of the treatments to be found in carefully refereed original sources.  It is also tedious to write proofs in html.  Because of this many proofs seem to come out with a rushed, breathless quality to them.   

(iii) Someone above wrote that they do not have access to the original sources in 
which the proofs appear.  I have all sympathy for this, but this is a swiftly 
changing situation.  I would submit that at the present time a large percentage of 
proofs of important theorems (and at any rate, the theorems discussed here!) are 
available SOMEWHERE online.  Rather than seeing a proof typed up on the fly, I would much rather see (a) a careful reference to the _first_ published proof (and a 
discussion of priority, where appropriate), and (b) references to one or more (but 
not too many; choose the best!) webpages which contain the proof in question.  (Sufficiently) many professional mathematicians put their cutting edge papers as well as their lecture notes available on the web as a matter of course.  There is no reason to rehash a proof if someone else has already done it better, and they 
usually have.

(iv) Writing proofs is a time-consuming effort which in many cases reduplicates 
efforts made by (more qualified) others.  In particular it takes away energy 
from the other things I expect to find in an encyclopedia article, namely historical information, synthesis, intellectual impacts and so forth.

(v) As an example of this, I rewrote the page on Tychonoff's theorem because the 
previous version spent too much time copying the proof of Munkres.  It was not 
said that Munkres' proof is a (nicely done) reworking of the proof of Cartan-Bourbaki.  It seemed clear that the author was more interested in demonstrating his 
own personal understanding of the proof of Tychonoff's theorem than in sorting 
through the history of the result and all the various proofs using different ideas.  My rewrite does not claim to prove the theorem even once, but it discusses the historical and intellectual interrelationships between five different proofs.  There is no shortage of proofs of Tychonoff's theorem 
available on the internet.  (However, my rewrite is not perfect or even complete: 
I have not yet had time to add enough references.  But I can find half a dozen 
online proofs of Tychonoff's theorem with no trouble at all.)  

[[User:Plclark|Plclark]] 01:52, 18 September 2007 (UTC)Plclark


:Plclark. Also I am new here at Wikipedia (trying to figure out whether I belong here or not …). I have read the contributions above on this page, and I essentially agree with them all. I agree mostly with your opinion, too. But there is one catch in it I would like you to comment on. If you would allow me.

:You say<br />
:”If the proof is not original to the author, then (obviously) the author is taking the proof from some other source”<br />

:I have a concrete example, that MIGHT have slipped your attention as a principle example of ”proof without any other source than the author”: In prevalent literature, nuclear physics, an already well known approximation formula for nuclear radiuses is partly used and partly experimentally proven ”approximately OK” as the so called (my name) cube graph r = r0(A^1/3). But, as far as I know (despite some Internet research and other), there is no corresponding ”mathematical deduction” to this seemingly simple and direct expression. However, there is, nevertheless, a simple way to EXPLAIN a WHY and HOW, a simple mathematical deduction with the same basic preferences as those used by the water drop model presumptions of ”homogeneous substance density” and which leads to the actual expression<br /> 
:r = r0(A^1/3). Just a single line, really. No extensive work. (See example at talk page on Nuclear size).

:The author (in the exemplified talk page) do not know whether (such) an already existent (published) proof already exists. So, the author MIGHT be a unique source here to that particular proof (although it is not likely he is — with respect to the relatively long history of nuclear physics). IF that is the case, it would be an example that slipped out of your observation. Meaning: we cannot categorize (exactly) according to your explained paragraphs. There are (categories of) exceptions to take account of. 

:Meaning in general: Clarifying descriptive explanations (proofs) should be added to, or incorporated with, or given as appendix or footnote or other available context to any main article where it so has come to a consensus between the collaborators (additional proofs MUST have consensus, not by vote but by argumentation in accord with already deducible mathematical laws, in order to exclude hazard or ”MathWacko-communities”). I think this is (approximately) what (nearly all of) the above contributors already have proposed. Question is only: how organize it? Or am I missing something, Plclark? --[[User:85.89.80.140|85.89.80.140]] 17:20, 20 September 2007 (UTC)

By way of response: first, your example has something to do with physics, whereas I was thinking only about mathematical articles (and the heading of this page explicitly restricts 
to mathematics).  More than that, I'm afraid I didn't quite understand what you said.  But if 
you are saying that it is possible for an author to independently come up with a proof which is 
not in fact "original" in that it someone else, unbeknownst to the author, already came up with 
it, then I agree.  I don't think this has much effect on what I was saying about writing wikipedia articles though, except that if you find your own proof because you thought there wasn't one already in the literature you are committing a(n understandable, and forgivable) scholarly error. [[User:Plclark|Plclark]] 06:08, 23 September 2007 (UTC)Plclark
<br />
<br />
:Plclark. Thanks for the response. I was trying to exemplify what this project page exposes,<br />
:”This page is devoted to discussions of when and how to include proofs in mathematics articles.”,<br />
:in the light of an obvious conflict with Wikipedia’s already stated No Original Research<br />
:in trying to give a concrete example that ”mathematics articles”, really, not easily can be distinguished from articles in general containing mathematical descriptions: I was trying to convince you, perhaps by an odd example, that descriptive and explanatory novels in Wikipedia inevitably appears spontaneously IF the Wikipedia aim is to let also (type) not high school educated people get a glimpse, an idea, of a specific subject. Meaning: original research in Wikipedia already exists, it is already established and launched — through mathematically connected articles. We cannot stop it. It is already here. Mathematical novelty is already working at full in Wikipedia, and we find it in pure mathematical articles as well as in mathematical physics articles. <br />

:'''Example''' (the best at present, my opinion): [[Planck law|Planck radiation law]]. Without the contributed novel in the example, explaining the details, only readers with a (type) high school education in mathematics would be able to read the stuff. Or as it is stated by the actual contributor:<br />
:”I gave a self-contained derivation suitable for wikipedia. You won't find this derivation anywhere in the literature, because textbooks usually assume that the student knows the basic stuff and they'll proceed from there. Giving the standard derivation that you can find in textbooks serves no purpose, because the people who can follow that are precisely the ones who already know it. What I did here was to imagine that the reader knows nothing except what he/she can find on wikipedia. He/she must, of course, have some basic mathematical skills. So, perhaps a sentence needs to be included at the start of the derivation saying that we give a self contained derivation and that other derivations can be found in books such as.... Count Iblis 16:29, 3 October 2006 (UTC) ”<br />
:@INTERNET Wikipedia, Talk:Planck’s law, references<br />

:All mathematics. Excellent generalized example of the project core quest on this page, my opinion.<br />
:My quest to you Plclark was just: do you object to such a stand alone mathematical novelty in Wikipedia, by principle? It shouldn’t be there (you say?). Remove it (you say?). Is that what you say? <br />
:Please, give your opinion to clarify on the Planck math example as a core example, if you approve.--[[User:85.89.80.140|85.89.80.140]] 16:50, 23 September 2007 (UTC)

As I tried to say above: Planck's law is a physics topic, and the page "Planck's law" is a physics page.  Everything 
I said above applies only to mathematics articles.  When I say that the page is not a math page, I don't mean it in a 
valuative sense.  What I am really claiming is that it is outside of my area of technical and cultural expertise: I'm 
sure that physicists have their own standards about mathematical proofs, and I know nothing about the physical 
literature.  So whether or not my remarks are appropriate to this example, I don't know -- I don't have an "expert" 
opinion on it, and I suppose you wouldn't want an uninformed opinion.  But in case you do: before I looked at the page I would have suspected that in fact the derivation of something as (presumably; again, I don't know about such things) 
basic as Planck's law would have already appeared in the literature in some form equivalent to the way it was presented in the article.  When I looked at the article I found that there was indeed a reference to [4], which 
seems to undercut Count Iblis' claim that "you won't find this derivation anywhere in the literature".  (In my 
experience, people who say things like this haven't looked hard enough.)  In that the current article includes 
print references (good!) but does not include a reference to any other online source that gives a derivation, I 
approve of the relatively short proof: if it were not there, then, were I interested in the proof, I wouldn't know 
how to find it without leaving my chair and going to a science library (which I would have to be awfully interested 
in order to do).  I hope this response is satisfactory to you.  I won't speak on it further because I feel I am not 
really qualified. [[User:Plclark|Plclark]] 05:54, 24 September 2007 (UTC)Plclark

''Every proof which appears on wikipedia is either original to the author or not.'' Ok. What about the proof of [[quadratic equation]]?
Is it OR? Surely not! Is anybody able to give the reference of the original proof? I really don't think so! Is it a well known proof that is sufficiently important to be included in any encyclopedia having an entry for quadratic equation? Of course! What source will you give for it? It depends on your country, your personal preferences, etc. as there are exactly no objective reason to prefer some book for this proof. [[User:Taxipom|pom]] 08:21, 24 September 2007 (UTC)

If you read the article on the quadratic equation, you will see that it does give a historical tracking of various 
special cases through various civilizations.  (It is a very nice article.)  I hope you will agree that this part of 
the article is more interesting and more useful than the derivation of the quadratic formula (which I don't object to, by the way).  So I don't understand your point.  Moreover, this is certainly an atypical example -- most of the proofs included in wikipedia articles are of theorems proved by a particular person at a particular time.  
[[User:Plclark|Plclark]] 17:08, 24 September 2007 (UTC)Plclark

:: I don't think this is an atypical example. Mathematics are full of "folklore results" with uncertain origins and elementary well known unatributed proofs. Yet another example, called the "Isometry Extension Theorem" in undegraduate textbooks: If you have two famillies of points (a_i) and (b_i) of an Euclidean space such that dist(a_i,a_j)=dist(b_i,b_j) then there exists an isometry mapping the a_i's to the b_i's. The well known proof is for sure much more recent than the statement, and both have an uncertain origin. [[User:Taxipom|pom]] 18:50, 24 September 2007 (UTC)

I am a bit baffled that in response to my statement "most of the proofs included in wikipedia articles....at a particular time", you give an example of a theorem whose proof is not included in a wikipedia article.  For what it's 
worth, my claim is quite verifiable.  For instance, I am now looking at the 
list "Articles devoted to theorems of which a (sketch of a) proof is given", 
and in a list of about 50 theorems, the ones which are not easily attributable 
to someone (so far as I know!) are the Five Lemma, the Intermediate 
Value Theorem, the Mean Value Theorem, Quadratic Equation, splitting 
lemma, squeeze theorem.  Two that I am unsure about are the ultraparallel 
theorem and the ultrafilter theorem (probably the latter is due to 
Henri Cartan and the former to Bolyai, Lobachevksii and/or Gauss).  This 
seems to bear out my claim, n'est-ce pas?  

But in truth the claim is not vitally important: as I said above, 
when there is not a clear attribution to be made, it becomes more important 
that an encyclopedia article discuss the history of the result, because the 
history is complicated.  With regard to your example on extension of isometries 
from subspaces, I would be very interested to know in which works of geometry 
this fact does / does not appear.  Even the five lemma must have an interesting 
history -- it is not less interesting because most of us don't know about it! 
[[User:70.155.76.162|70.155.76.162]] 01:08, 26 September 2007 (UTC)Plclark  


<br />
:OK Plclark. Thanks again for your patience, taking your time with my elusive examples. (They really don’t throw the sufficient light upon the quest, poor bastards). So, let’s cut the crap and get to the point. In this project page header [[Wikipedia:WikiProject Mathematics]] is stated<br />

:”In its broadest terms, mathematics includes its overlaps with physics, computer science, operations research, and other areas. However, there are now separate WikiProjects in many of these areas, and so the focus of WikiProject Mathematics has shifted towards articles which are primarily mathematical (although it still includes many aspects of theoretical physics, for example).”<br />

:Meaning: we cannot, contrary your opinion Plclark as above, really separate between mathematics and physics in any sense as to state that THIS project page is RESERVED for pure mathematical articles only. OK? I respect your mathematical part, and I will not argue on it further.<br />

:It would be nice with a concrete resolution with achieved results on this project page (due to its almost 100% consensus), so that the article creativity on mathematics and physics could proceed with good guiding. The consensus of this project page, then as I see it, needs clarification from you, Plclark (see your heading): Are you, or not, against mathematical clarifications (”proofs”, detailed or not), even to the price of violating the Wikipedia policy of No Original Research? I don’t know, Plclark, but it seems you are the only one active individual on this page to be, sort of, against. Am I wrong? Please clarify.--[[User:85.89.80.203|85.89.80.203]] 21:09, 24 September 2007 (UTC)

I'm not sure what you mean by cut the crap, but I don't find it appropriate 
in what for my part is a frank, considered and well-intentioned discussion.  You seem disappointed by my unwillingness to have the scope of my opinion 
extend into the realms of physics.  Regarding what you quote, I can only say 
that the statement about the intended scope of WikiProject Mathematics into 
related fields is a very measured one; e.g. I am (honestly) unsure about whether the Planck's law page falls under the scope of theoretical physics.  
But although there are some borderline cases, one can certainly point to many wikipedia articles which are focused only on mathematics and not physics or 
other sciences, and it is to these articles and these articles only that my 
opinion applies.  Maybe you are advocating for an official wikipedia policy 
on proofs that covers mathematics and all of its adjacent fields.  That is (of course) a perfectly good opinion; for my part, it seems that the more "local" 
the policy, the more nuanced and hence the more useful.

As to the fact that on this webpage I am the only one expressing my opinion: 
yes, that's true.  What I will say is that my opinion on mathematical (but not 
physical!) matters is an expert one: I have described a set of priorities 
for writing math articles on wikipedia which, if followed, would (according 
to me, of course) contribute the most to the larger mathematical community.  
I think my views have a large currency among mathematicians.  Most young mathematicians I know are involved in wikipedia to the extent of being willing 
to look up an article here and see whether what comes up is helpful or not.  I 
think relatively few mathematicians are active wikipedists (and at any 
rate most math articles seem not to be written by research mathematicians).  Consensus on wikipedia is a kind of nebulous thing -- how many people who regularly look up math articles on wikipedia are even aware that this page exists?  (It took me a lot of random searching to amble onto it.)  As defense 
for my position that there are, very often, more important things to include 
in wikipedia articles than theorem proofs, I can refer you to the one page 
that I completely rewrote: Tychonoff's Theorem.  The old version, which I 
didn't like so much, had a sketch of the proof.  I changed it to a version 
(which again, I like okay, not tremendously; really good writing is difficult 
and time-consuming) which provides much broader context but no proof of 
Tychonoff's theorem.  I did this more than a month ago and no one except me has since made any but the most cosmetic changes; in particular, no one has clamored "what happened to the proof sketch?"  

I am surprised that you regard my position as being unclear, since I have 
tried to describe it carefully and illustrated it with examples.  If anyone else out there is reading this page and doesn't understand what I am advocating, please let me know.  [[User:70.155.76.162|70.155.76.162]] 01:08, 26 September 2007 (UTC)Plclark

:Sorry, Plclark. Just a short interjection:<br />
:Do you mean by ”what I am advocating” the (i-v) paragraphs on top of this block?--[[User:85.89.80.203|85.89.80.203]] 05:07, 26 September 2007 (UTC)

::People, you've spent so much time arguing your opinions here, and it still sounds like you disagree on very little if anything. It seems like you're just talking about different meanings of the word "proof". May I propose that you use your spare time to improve articles as you see fit — rather than discussing how other people should improve articles as you see fit? If somebody doesn't like a change you've made, they will let you know, and then you can start a discussion. From my experience, discussions are a lot more focused when a concrete example is being considered. --[[User:Zvika|Zvika]] 06:11, 26 September 2007 (UTC)

With respect, this is not a very insightful comment.  As far 
as I can tell, the meaning of the word "proof" is not in 
dispute whatsoever.  I also, I did of course include many 
concrete examples in my discussion.  Part of the problem is 
that we do not seem much interested in each other's concrete 
examples.

And of course it is more efficient to try to influence the 
wikipedic behavior of dozens or hundreds of other people than to act on my own opinions personally.  I realize that 
other people (whatever other people are following this page 
at this time) don't seem to agree with my opinions.  That's my point -- I want to get an opinion which I think many, many mathematicians share on record and make sure everyone who is busily setting up already extant proofs in html at 
least understands why I think this is a relatively poor 
use of their time.  

You do know, I guess, that there is already a webpage which is devoted to giving very careful definitions and proofs of 
an ever-expanding list of mathematical terms and theorems: 
it is called planetmath.org.  In that more of the articles 
on planetmath are written by math grad students or mathematicians, a larger percentage of the articles look 
satisfactory to me.  Moreover, each article on planetmath 
is "owned" by a particular person, so if I spend half an hour perusing planetmath and find four small mistakes, I can send four quick emails to these authors, and they will usually write back and say, "Thank you; I fixed it."  With wikipedia, if I try to suggest minor changes on the talk page people say (if anything) "Be bold; do it yourself", and 
then I have to decide whether I have the time and the inclination to fix it myself (always wondering whether someone else will change it back).  It would be easier for 
me to write my own article from scratch (using latex instead of html) and put it on "plclark's world of mathematics", i.e., my own webpage.  In fact I do so often.

The real issue here is how will you handle input from experts?  The way it is going right now, I predict wikipedia will be displaced by planetmath, scholarpedia, and other sites in which experts can contribute in an efficient way.  [[User:72.152.92.18|72.152.92.18]] 07:26, 26 September 2007 (UTC)Plclark


:Hello Zvika. Nice hearing from you.<br />
:When I came to this page, I realized, directly the situation: You have, yourself been a pioneering force to bring the mathematics proof articles to an organized deposit, supported and promoted by practically ALL familiar with this page, included me. I saw, at once, there was only one individual, the User Plclark, who, sort of, did not approve, and deviated in a strange, but fairly declared way. So, I thought it would be worth a try to ”win him on our side”, bringing a 100% consensus, for further. So, User Zvika, I have nothing to argue or protest on to the people having worked successfully with this project (since the end of 2004). I am all in concord. But now, suddenly, it seems you disagree on points I thought already were settled, making the discussion with Plclark even more differentiated and diverse. I would agree to Plclark’s comment on your comment, Zvika; you say we do not agree on the meaning of ”proof”. You are wrong there. Neither is our disagreement trivial. If you see it that way, you have lost the point, and I am afraid my presence on this page has contributed to a more, not less, chaotic situation. Ask me to leave, Zvika, and I’ll do it. I’ll drop it all. I won’t say any more.--[[User:85.89.80.203|85.89.80.203]] 08:18, 26 September 2007 (UTC)


::Well, I'm glad I got you two to agree on something at least :)
::First let me say that I had no intention to ask either of you to leave and I apologize if I was understood that way. Everyone is entitled and invited to try to change other people's opinions. I only humbly suggested that you might find it more productive to concentrate on editing specific cases which you find relevant, rather than theoretically discussing possible changes. I made this suggestion particularly since it seems that the two of you have quite different interests, and it is possible that a general rule appropriate for one field of study is inappropriate in another.
::Plclark: I like PlanetMath and use it a lot. (Incidentally, a lot of material in WP is taken from PM and vice versa; see [[Wikipedia:WikiProject Mathematics/PlanetMath Exchange]].) Apart from its more limited scope (I am interested in math articles only insomuch as they relate to my main field, which is electrical engineering), the main difference between PM and WP is Wikipedia's philosophy that anyone can edit a page. Of course this has its ups and downs, but the main advantage is that even casual readers can fix a mistake or add content, rather than having to write an email to someone explaining what the problem is. To me, fixing a small mistake in WP seems much simpler than writing an email explaining where that mistake is and how it should be corrected. Editing WP articles is quick and easy and [[WP:MATH|you can use LaTeX]]. This is why I suggested previously that you "be bold" and make the changes yourself — that is just the Wiki philosophy. Unless you delete others' work without explanation, it is unlikely that your changes will be reverted without discussion.
::I recognize your goal to convince others to spend their time improving articles, rather than writing up proofs. Point taken. However, it seems to me that in many concrete examples you agree with most of us ([[quadratic equation]], [[Euclidean algorithm]]), which is why I made my suggestion.
::One final technical comment: Plclark, please login so that your signatures display your username and not your IP.
::[[User:Zvika|Zvika]] 09:13, 26 September 2007 (UTC)

== Addendum ==

To sum up the previous long post: when I look up "Theorem X" on wikipedia, I would 
like to find:

(i) a careful, reliable statement of the theorem in question (reliability ensured 
by references both online and otherwise).

(ii) information which complements, rather than duplicates, the complete 
proof I would find by looking in one of these references.  [[User:Plclark|Plclark]] 01:57, 18 September 2007 (UTC)Plclark

:These are legitimate things to look for in an article on "Theorem X". It also seems to me, in many cases, legitimate to look for a proof (if it is short) or a commented proof skeleton. In many cases this helps understanding or exemplifies a general proof method. In the end, I guess it depends on the situation: I think the article on the [[Euclidean algorithm]] would be deficient without the proof, whereas obviously we aren't going to include the proof of [[Fermat's last theorem]].
:As for Wikipedians not being the right kind of people for writing proofs, five years ago I would have thought the same about writing any serious math article, yet fortunately I was proven wrong. Why not give us a chance? --[[User:Zvika|Zvika]] 18:32, 20 September 2007 (UTC)

I agree that some discussion of the proof is often helpful or integral.  If the proof is really 
easy enough, then yes, I agree that giving it in the context of the article probably makes 
for easier reading than referring the reader to an external source.  The Euclidean algorithm is a good example of this.  What I am saying is that if it is much harder than that I would prefer 
not to see a proof.  

Regarding your second paragraph: when I look through wikipedia math articles I see some lovely 
ones (that I, as a mathematician in a different field, can use as my first reference) and some that are deficient in various distressing ways.  (The article on number theory is an embarrassment to me, to the extent that I nominated it for deletion, but unsurprisingly that didn't work.)  I should admit that I have not seen a proof on wikipedia that I thought was 
wrong, only proofs that I thought were not as good as the ones found in standard, easily accessible sources.  Again though, my point is this: an encyclopedia article is a work of scholarship.  It is not the same kind of scholarship as a technical book or article in mathematics, but it is not lesser or less important for this.  Most math books and articles 
provide too little in the way of "encyclopedic scholarship", so there is a lot of work to do 
in most encyclopedia articles in mathematics.  When there is so much work to be done, why try to reinvent the wheel?  [[User:Plclark|Plclark]] 06:08, 23 September 2007 (UTC)Plclark

:It sounds like we can pretty much agree that proofs or proof skeletons are sometimes useful in an article, and sometimes unnecessary. It also sounds like you know a lot about many of the subjects here and have a clear vision of how they could be improved. Certainly you know more than me in these matters; I am not even a mathematician. That being the case, might I suggest that you [[WP:BOLD|be bold]] and improve articles as you see fit -- not by nominating core articles for deletion, but by adding what knowledge you have to them. Please keep in mind that Wikipedia is a work in progress, so some articles -- even important ones -- are bound to be deficient. The thinking in our community should not be "delete anything that contains serious flaws" but rather "fix the flaws, one by one." That is how we got to where we are today. Hopefully, with the help of people like you and me, we will improve articles like [[Number theory]] and get them to the level of articles like [[Game theory]]. --[[User:Zvika|Zvika]] 07:12, 23 September 2007 (UTC)

== Comments on some articles in "Category: Article proofs" ==

Perhaps it will be helpful if I comment specifically on some of the proof pages.

Addition of natural numbers: There are no references given.  Presumably these facts were 
proven in Peano's original paper, which should be cited, along with a more contemporary reference.

Approximation theory: a mess.  Here we have a proof without a precisely stated theorem.  (The 
parent page is not very good either.)

Bertrand's postulate: this is nicely done.  My only complaint is that it gives a proof and says 
"the gist" of it is due to Erdos.  This isn't very clear.  Also, the reference to Erdos'' paper 
should be given.  

Boy's surface: There is no reference to Bryant's paper.  Also, what is the significance of all this?  In what sense do the results of Morin and Bryant improve upon Boy's original construction?  Did Boy want an explicit parameterization but not have one?  It is an instance 
where the (simple enough) proof is being given instead of other more interesting information.

Cardiod: again, too little context is provided.  Why do we care about this curve?  Do we, in 
fact, still care about it?  [[User:Plclark|Plclark]] 06:37, 23 September 2007 (UTC)Plclark

:Exactly. That is why "article proofs" were considered to be "second class citizens", in a kind of purgatory. They do not live up to higher standards, but then, aren't requested to live up to higher standards. That is why we don't want them in the main article space. ''If'' they were good, notable proofs, then, yes, we could put them into article space, add citations and references, etc. But they're not.

:Personally, I looked upon all this as a mechanism to placate the (younger? less sophisiticated?) authors who felt a need to write up proofs, avoid the resultant pollution to article space, and avoid the confrontation and alientation that usually results from deletions: the "I'm taking it personally" feeling when you read that "your proof of xyz is substandard (aka "it sucks") and we're going to negate your time and effort (your work is meaningless and not wanted here)". So, for me, it was a live-and-let-live  stance. [[User:Linas|linas]] ([[User talk:Linas|talk]]) 04:50, 26 November 2007 (UTC)

==An alternative method for handling proofs for articles==

I have just finished writing a template {{tl|seemore}}, which may be useful for handling some of the sorts of proofs which /Proof pseudo-subpages were originally proposed for. I have illustrated the alternative method below using one of the identities from the [[Totient function]] page, and its proof from the [[Totient function/Proofs]] pseudo-subpage.

===An identity for Euler's function===

:<math>\sum_{k=1}^n\frac{\varphi(k)}{k} = \sum_{k=1}^n\frac{\mu(k)}{k}\left\lfloor\frac{n}{k}\right\rfloor</math>

{{hat|Click on "show" to see a proof of this identity.}}
Click on "hide" to the right of the line above to hide the proof.

'''Proof:''' The proof of the identity

:<math>\sum_{k=1}^n\frac{\varphi(k)}{k} = \sum_{k=1}^n\frac{\mu(k)}{k}\left\lfloor\frac{n}{k}\right\rfloor</math>

is by [[Mathematical_induction|mathematical induction]] on <math>n</math>. 
The base case is <math>n=1</math> and we see that the claim holds:

:<math>\varphi(1)/1 = 1 = \frac{\mu(1)}{1} \left\lfloor 1\right\rfloor.</math>

For the induction step we need to prove that

:<math>\frac{\varphi(n+1)}{n+1} = 

\sum_{k=1}^n\frac{\mu(k)}{k}
\left(\left\lfloor\frac{n+1}{k}\right\rfloor - \left\lfloor\frac{n}{k}\right\rfloor\right)
+ \frac{\mu(n+1)}{n+1}.</math>

The key observation is that

:<math>
\left\lfloor\frac{n+1}{k}\right\rfloor - \left\lfloor\frac{n}{k}\right\rfloor
\; = \; \begin{cases} 1, & \mbox{if }k|(n+1) \\ 0, & \mbox{otherwise, }
\end{cases}</math>

so that the sum is
:<math>
\sum_{k|n+1,\; k<n+1} \frac{\mu(k)}{k} + \frac{\mu(n+1)}{n+1}
= \sum_{k|n+1} \frac{\mu(k)}{k}.</math>

Now the fact that
:<math>
\sum_{k|n+1} \frac{\mu(k)}{k} = \frac{\varphi(n+1)}{n+1}
</math>

is a basic totient identity. To see that it holds, let 
<math>p_1^{v_1} p_2^{v_2} \ldots p_q^{v_q}</math>
be the prime factorization of n+1. Then

:<math>\frac{\varphi(n+1)}{n+1} = \prod_{l=1}^q \left( 1 - \frac{1}{p_l} \right)
= \sum_{k|n+1} \frac{\mu(k)}{k} </math>

by definition of <math>\mu(k).</math> This concludes the proof.<br/>
'''[[QED]]'''
{{hab}}
&mdash;[[User:David_J_Wilson|David&nbsp;Wilson]]&nbsp;<small>([[User talk:David_J_Wilson|talk]]&nbsp;·&nbsp;[[Special:Contributions/David_J_Wilson|cont]])</small> 18:24, 20 December 2007 (UTC)

:The use of CSS to hide parts of articles is not well liked. I know it's used in some navboxes, but I think that people would frown upon its use in article text. What would be more reasonable would be the use of the {{tl|further}} template, or a similar template made just for proofs, that would link to the right place in a proofs article. &mdash;&nbsp;Carl <small>([[User:CBM|CBM]]&nbsp;·&nbsp;[[User talk:CBM|talk]])</small> 18:43, 20 December 2007 (UTC)

:::"The use of CSS to hide parts of articles is not well liked."
::I'm sorry to hear that. Are there any technical reasons why the use of this mechanism would be inadvisable?  Are there any ''specific'' policies or guidelines which can be used to argue against it?
::I agree that a {{tl|further}} template linking to a ''Proofs'' article would generally be preferable, ''provided'' such an article can be defended against calls for it to be deleted or merged with its "parent".  Unfortunately, discussions on the AfD nominations for the articles [[WP:Articles for deletion/Totient function/Proofs|Totient function/Proofs]] and [[WP:Articles for deletion/Boy's surface/Proofs|Boy's surface/Proofs]] seem to indicate that ''Proofs'' articles, especially when they have slashes in their tiles, don't seem to be well liked either. The problems with both the above-mentioned articles is that they don't satisfy the Wikipedia [[WP:NAME#Do not use an article name that suggests a hierarchy of articles|naming convention policy]] and are widely perceived as not satisfying the [[WP:N|notability guideline]] either. The first of these issues has already led to the first of those two articles being moved to [[Proofs_involving_the_totient_function|Proofs involving the totient function]], even though it survived the AfD nomination, and this in turn has led to [[Talk:Proofs involving the totient function#Requested move|further comment]] in favour of its being transwikified (and presumably then deleted). If [[WP:IDONTLIKEIT|IDONTLIKEIT]] is the only objection that can be raised against the mechanism I have suggested, then it should be easier for editors who find it suits their needs to defend when a separate ''Proofs'' article might be vulnerable to the same sorts of objections that were raised against the two above-mentioned articles.
::Another option, of course, is to put the proofs in a separate section at the end of the main article, and link to them from appropriate places in the article's body.  I wouldn't argue against that solution if it was what an article's editors chose to adopt.  However, I can well see why they might not want to do so, and the mechanism I have suggested simply provides them with another option to consider.
::&mdash;[[User:David_J_Wilson|David&nbsp;Wilson]]&nbsp;<small>([[User talk:David_J_Wilson|talk]]&nbsp;·&nbsp;[[Special:Contributions/David_J_Wilson|cont]])</small> 02:35, 21 December 2007 (UTC)

:::There is some older discussion about this at [[Wikipedia:HiddenStructure]]. It would need to be discussed on one of the village pumps before being implemented; it's not clear to me that there would be general support for introducing show/hide boxes for article prose. Some people (fortunately a minority) don't like to see any proofs. Convincing them that we should have proofs that are so tangential to the content that the proofs can be hidden would be difficult. &mdash;&nbsp;Carl <small>([[User:CBM|CBM]]&nbsp;·&nbsp;[[User talk:CBM|talk]])</small> 02:54, 21 December 2007 (UTC)

::::It seems to me that a serious drawback of CSS-hidden prose in the article text is that you lose it when the page is printed. It also creates difficulties for people wanting to port Wikipedia content to other platforms. In short, this mechanism is based on the assumption of a user interactively reading a Wikipedia page using a web browser, and this isn't always the case. --[[User:Zvika|Zvika]] ([[User talk:Zvika|talk]]) 13:36, 21 December 2007 (UTC)

On reading back through the [[/Archive_1 | archive]] I see that this issue [[/Archive_1#Magic unrolling boxes| has already been discussed]]. Although no consensus seems to have been reached, opposition to the idea certainly seems to have outweighed the support for it.  Given the technical problems you have raised I reluctantly agree that the idea is probably not worth pursuing.  Nevertheless, I think that's a pity.
&mdash;[[User:David_J_Wilson|David&nbsp;Wilson]]&nbsp;<small>([[User talk:David_J_Wilson|talk]]&nbsp;·&nbsp;[[Special:Contributions/David_J_Wilson|cont]])</small> 16:09, 22 December 2007 (UTC)

==Proofs in Wikipedia - another approach?==
I am looking at [[Union of two regular languages]] right now, which is an expired PROD that I am keeping open for a few more minutes with <nowiki>{{inuse}}</nowiki>, but which I will likely deprod as not an uncontroversial deletion.  There is a second type of solution to the introduction of proofs into Wikipedia other than subpages or moving to Wikibooks - I hope you've not discussed this previously - and that is to treat the proof as 'supporting information' and upload it as a PDF where it would be stored in the 'image' namespace.  As far as I know, there is no bar to taking that route, which is sorely overutilized by people uploading copyrighted images and underutilized by people uploading illustrative figures, data analyses (not bordering on [[WP:OR]] but little different from creating an image yourself for upload) and supporting information.  Thoughts? --User:Ceyockey (<small>''[[User talk:Ceyockey|talk to me]]''</small>) 19:20, 27 December 2007 (UTC)
:What would be the advantage of this over having the proof in mainspace? I can think of several disadvantages (in terms of portability, for example). --[[User:Zvika|Zvika]] ([[User talk:Zvika|talk]]) 19:35, 27 December 2007 (UTC)
::The lack of opposition - the social factor. --User:Ceyockey (<small>''[[User talk:Ceyockey|talk to me]]''</small>) 19:41, 27 December 2007 (UTC)
::The other thing is that PDF need not be the format.  For instance, one could use wikimarkup and save the page into the image namespace ... I think ... or use some other form of portable mathematical markup. --User:Ceyockey (<small>''[[User talk:Ceyockey|talk to me]]''</small>) 20:34, 27 December 2007 (UTC)
:::Would that make the text editable like an ordinary wiki page? --[[User:Zvika|Zvika]] ([[User talk:Zvika|talk]]) 08:24, 28 December 2007 (UTC)
::::Not sure - needs to be verified. --User:Ceyockey (<small>''[[User talk:Ceyockey|talk to me]]''</small>) 15:18, 28 December 2007 (UTC)
==Incorrect use of subpages==
Many of the "proof subpages" directly violate the wikipedia guideline [[WP:SP#Articles do not have sub-pages (main namespace)]]. Every article in the encyclopedia should be self-standing (in form only, of course; not content wise. i.e. they should have a lead, headings, proper english, and other MoS type things) and titles shouldn't use slashes to indicate belonging to some parent page.  [[Special:Contributions/128.139.226.37|128.139.226.37]] ([[User talk:128.139.226.37|talk]]) 11:33, 21 February 2008 (UTC)
:That is probably true. If you see a page such as "X/Proof", please move it to something like "Proof of X". --[[User:Zvika|Zvika]] ([[User talk:Zvika|talk]]) 13:40, 21 February 2008 (UTC)

:The list below are the ones I know about. Please add if you see more, or remove if no longer there. &nbsp;--[[User talk:Lambiam|Lambiam]] 16:55, 29 March 2008 (UTC)

===List of violating "proof subpages"===
*[[Addition of natural numbers/Proofs]]
*[[Approximation theory/Proofs]]
*[[Boy's surface/Proofs]]
*[[Cardioid/Proofs]]
*[[Chebyshev polynomials/Proofs]]
*[[Christoffel symbols/Proofs]]
*[[Cone (geometry)/Proofs]]
*[[Conic section/Proofs]]
*[[Conservative force/Proofs]]
*[[Distributive lattice/Proofs]]
*[[Ellipse/Proofs]]
*[[Geodesic (general relativity)/Proofs]]
*[[Kronecker's lemma/Proof]]
*[[Laplace-Beltrami operator/Proofs]]
*[[Law of large numbers/Proof]]
*[[Moore-Penrose pseudoinverse/Proofs]]
*[[Nondeterministic finite state machine/Proofs]]
*[[Orthogonal polynomials/Proofs]]
*[[Parabola/Proofs]]
*[[Torque/Proofs]]
*[[Vector Laplacian/Proofs]]

== Where oh where has my little proof gone? ==

=== Comments from [[WT:WPM]] discussion ===
This is a copy of the last <strike>&nbsp;4&nbsp;</strike> several messages from the most recent conversation on "to include or not to include":

Zaxxonal and silly rabbit: It isn't a question of whether the topic is notable enough - we already have an article on the topic. The question is whether including this derivation runs afoul of our mission of being an encyclopedia rather than a textbook. There is no clear agreement at the moment about "proof subpages" or "derivation subpages"; in any case, the existence of some such subpages can't be used as an argument for including more of them. My personal opinion is that lengthy derivations are not in line with our mission, but may be in line with the mission of our sister project [[Wikibooks]]. &mdash;&nbsp;Carl <small>([[User:CBM|CBM]]&nbsp;·&nbsp;[[User talk:CBM|talk]])</small> 19:12, 27 March 2008 (UTC)

: I feel rather strongly that proofs do belong in Wikipedia.  The question to ask here is, does this proof deserve an article?  In this case I'm not familiar with the area so I don't feel qualified to answer. [[User:CRGreathouse|CRGreathouse]]<small> ([[User talk:CRGreathouse|t]] | [[Special:Contributions/CRGreathouse|c]])</small> 03:05, 28 March 2008 (UTC)

: I respect Carl's position, but as he himself pointed out, there is no clear agreement on the matter.  I personally see no harm in having proofs in Wikipedia.  I have no opinion on whether they should be subpages or something else; I do agree they should not clutter up main articles.  (I also express no opinion about this particular page since it is far from my specialty.)  There is no reason why a proof is inherently unencyclopedic.  Has this idea been revisited much lately?  I know there have been some isolated discussions flaring up over the last few years, but maybe we need to work out a consensus once and for all about this proof business.  Otherwise, we will see fragmented discussions just like this pop up every time someone challenges the existence of some proof page.  [[User:VectorPosse|VectorPosse]] ([[User talk:VectorPosse|talk]]) 07:36, 28 March 2008 (UTC)

::I am aware of the lack of agreement here, and was only offering a personal opinion. I have a lot of respect for the viewpoint that we should include a very complete collection of proofs, and I think it's mainly a question of project scope. I agree that ''some'' proofs belong in our articles, and I've put some in myself. I was trying to say above that lengthy derivations are what I'm not convinced about. In many cases I think it's better for us to discuss the ideas in the proof and the insights it provides. That sort of meta-analysis is what Wikipedia is good at. &mdash;&nbsp;Carl <small>([[User:CBM|CBM]]&nbsp;·&nbsp;[[User talk:CBM|talk]])</small> 12:56, 28 March 2008 (UTC)

::: I see now.  The distinction between derivations and proofs is an important one and I would tend to agree with you on that count.  [[User:VectorPosse|VectorPosse]] ([[User talk:VectorPosse|talk]]) 23:55, 28 March 2008 (UTC)

::::The issue here IMO is whether we're dumping this proof because people with experience in the subject don't think it's important, or because it's boring to that hypothetical bright 14-year-old that many people take to be the target audience around here.  24-hour news channels suck, in part, because they follow an iron rule that anything that isn't entertaining gets dumped instantly.  Wikipedia does better, and must do better.  It just seems inherently wrong to me to treat whether "boring" content stays as a fight, with winners and losers.  Surely there's a way for everyone to win.  What occurs to me, and I don't know if this has been tried, is to make more use of the the Wikibook icon, and it might say "Wikibooks has a proof of this at" with a link to the section giving the proof:
{{wikibooks|Article#specific section}}
::::That is, if a proof is deemed too boring for Wikipedia, insert the wikibook link at a specific section in the Wikipedia text, and insert some kind of graphic in the Wikibooks text too with a link back to Wikipedia, so that people will know not to delete the text from Wikibooks without making a change back at Wikipedia.  Is this a good idea?  Is there something better? - Dan [[User:Dank55|Dank55]] ([[User talk:Dank55|talk]]) 00:30, 29 March 2008 (UTC)

I concur with Carl. As the author has admitted, this is his first piece of writing on wikipedia. I feel that we should provide some guidance to him about what wikipedia is and what it is not. It may be tempting to view wikipedia as a universal depository of all knowledge, whether it be done for altruistic or selfish reasons, but there are inherent dangers in such inclusiveness. Hence, we have policies such as [[WP:NOT#TEXT|"Wikipedia is not a textbook"]] which restrict the scope of the project. The piece under discussion (I will not go so far as to call it an article) is a ''textbook case'' of violating this policy. There are many wikis out there that collect technical information of various sorts, which usually have a narrow topical focus and may aim at supplanting or even replacing the monographs on their subject, e.g. [http://tosio.math.toronto.edu/wiki/index.php/Main_Page Dispersive PDE wiki]. But wikipedia's primary aim is to be an ''encyclopaedia''. In this sense, I think that distiction between derivations and proofs is an important one, and we should be careful not to open the floodgates to all sorts of technical writing and data storage (the criterion here is not whether it is boring, but whether it is encyclopaedic). [[User:Arcfrk|Arcfrk]] ([[User talk:Arcfrk|talk]]) 02:17, 29 March 2008 (UTC)

: http://en.wikibooks.org/wiki/The_Book_of_Mathematical_Proofs is the link at Wikibooks for a module specifically intended to hold proofs of theorems stated outside of Wikibooks.  Wikipedia is not a textbook, but Wikibooks is.  I don't have an opinion on which proofs should be in Wikipedia, but for any algorithm or theorem in Wikipedia, is there any good reason to ''exclude'' a cited, important proof from Wikibooks, or not to have links going both ways, from the section with the theorem to the section on Wikibooks with the proof and vice versa? - Dan [[User:Dank55|Dank55]] ([[User talk:Dank55|talk]]) 02:49, 29 March 2008 (UTC)

::I can't see why we would avoid linking to wikibooks when the content there is relevant. It's a Wikimedia Foundation project, after all. &mdash;&nbsp;Carl <small>([[User:CBM|CBM]]&nbsp;·&nbsp;[[User talk:CBM|talk]])</small> 21:23, 29 March 2008 (UTC)

::Michael - The changes your have indicated as needed have been made.  Your feedback has been ''constructive'' and ''exceptional''.  Thank you. --[[User:Zaxxonal|Zaxxonal]] ([[User talk:Zaxxonal|talk]]) 16:28, 30 March 2008 (UTC)

::: But, with the exception of the first paragraph, it still fails "Wikipedia is not a textbook". [[User:Arcfrk|Arcfrk]] ([[User talk:Arcfrk|talk]]) 04:32, 31 March 2008 (UTC)

=== New discussion ===
I see in previous conversations on this /Proofs page above that there was a suggestion that hidden text or images might solve the problem, but neither seemed to have support.  Carl said that {{tl|further}} might be useful.

I don't think anyone who's experienced in a subject, including academics, should self-censor in an attempt to be "encyclopedic", as long as what they want to say is important in their field.  Yes, Wikipedia has had phenomenal growth, I think we're up to 7th in the U.S. in number of hits, and yes, that suggests that we should have a certain respect for the principles that got us here, and some of those principles might be described as rules to keep the information from appearing too specialized or boring to a general audience.  But general arguments about keeping Wikipedia interesting must not be allowed to slide into non-scientific bias against any group or subject, including math.  The subject came up in the current [[WP:GAN]] for the Monty Hall problem that some of Marilyn Vos Savant's columns on math problems have attracted phenomenal response from readers of ''Parade'', which is as general an audience as you can get.  Many people study graduate math, and proofs are almost always a large part of the curriculum.  The fact that proofs seem boring or irrelevant to some people is immaterial.  Articles should be as accessible as possible, given the material, but there's no rule that Wikipedia has to limit itself to articles that every 14-year-old can understand.  On the other hand, when, for whatever reason, material that we believe to be important and relevant doesn't survive in Wikipedia, I don't think we should miss the opportunity to keep the material in Wikibooks (if it's typical textbook material) and link it, from the right place in the Wikipedia text (and not just with a "more information is available in Wikibooks" graphic at the end, which sounds more like a disclaimer than like a true extension of the text). - Dan [[User:Dank55|Dank55]] ([[User talk:Dank55|talk]]) 15:30, 31 March 2008 (UTC)