Wikipedia talk:WikiProject Mathematics/Proofs

'Hidden' proofs
A lot of discussion went on at Wikipedia talk:WikiProject Mathematics/Proofs/Archive 2 about what we should do with regards to proofs, but the discussion generally petered out. I personally really like the style used in some articles I've seen where the proof is hidden i.e. Theorem 1: 0 ⋅ a = a ⋅ 0 = 0

is there any problem with trying to implement this style with some of the shorter proofs on WP, where the proof itself isn't of essential value? It seems to get over the issue of whether/when proofs disrupt the flow of the article... SetaLyas (talk) 11:06, 28 December 2008 (UTC)


 * Yes there is a big problem with doing that in WP. Currently when that is done the proof is hidden if javascript is not enabled and cannot be revealed. Before it could be done it would be necessary to fix the show-reveal handling so it reveals the text if javascript is disabled - I believe this can be done cleanly though it isn't altogether straightforward to stop it flashing on some browsers when loading the page if there is javascript. Dmcq (talk) 13:12, 28 December 2008 (UTC)


 * This seems like an excellent option to me. What is the general consensus about when a proof should be included in an article.  I was personally of the opinion that do not have much of a place, but I have a feeling I am in a minority. Thenub314 (talk) 15:48, 28 December 2008 (UTC)


 * There's been more talk about this at Wikipedia_talk:WikiProject_Mathematics/Archive_43. The best option I think is to put a proof into a footnotes section. There's an example of that as well as other options in User:Proteins/Math_sandbox. Footnotes don't involve any accessibility, javascript or printing problems. Dmcq (talk) 18:44, 28 December 2008 (UTC)

Is there consensus about when a proof is a valid article ingredient?
There is a debate on-going on WikiProject Mathematics about whether to add a detailed proof of the orthogonality of solutions to the Sturm-Liouville equation with distinct eigenvalues. The main issue under discussions appears to be whether proofs belong on Wikipedia. After reading some archives referenced by a contributor to that discussion, it seems no clear consensus exists. The contributor recommended discussing the issue on this page, rather than continuing to discuss it on the main page. So, can someone tell me whether any consensus has been reached on this issue? Dnessett (talk) 19:09, 16 April 2009 (UTC)


 * Last time we had a discussion about proofs, the consensus seemed to be that the proof had to be of encyclopedic interest. Ask yourself: If you were writing a survey article on this topic, would you include a proof? I brought up the example of Grothendieck universe before, and I still think it's a good one. There's a proof in the lead of the article. It's very short, only one line, and it illustrates very well how Grothendieck universes work: It's appropriate. But there's also a proof in the next section of the article which is terrible: It hides all the important concepts from the reader and the details of the proof are not very interesting. (The article even admits this!)  It's inappropriate.


 * What this means is that while certain proofs are appropriate, most proofs are inappropriate. For instance, for a well-known and difficult result such as Fermat's Last Theorem, a discussion of the proof is of interest, but a complete proof is far too long and contains far too many details to be of general interest. As another example, the isomorphism theorems are fundamental, to the point of often being used without acknowledgment, but they do not have interesting proofs. Detailed proofs are mostly a matter for textbooks and papers. On the other hand, double counting (proof technique) is about a method of proof, so it is entirely appropriate for it to have proofs illustrating that method. Ozob (talk) 02:25, 17 April 2009 (UTC)


 * I don't think that we reached an actual consensus and certainly that wasn't it. I recall several people saying that proofs are useful to them and should appear, though perhaps one step away from the main article. There was a long discussion on how this could be accomplished, e.g., hidden with CSS 'show' tags, or in a separate article such as Proof of Bertrand's postulate. I also recall several people saying proofs are unencyclopedic and should be abolished. So no, I don't think there has been an actual decision. --Zvika (talk) 05:22, 17 April 2009 (UTC)


 * Well, I'm going on the basis of our discussion at Wikipedia_talk:WikiProject_Mathematics/Archive_46. I (of course) am for this interpretation; people who believe something roughly similar to this include David Eppstein, Ryan Reich, and Charles Matthews. That's not the only discussion, of course; the next most recent one is at Wikipedia_talk:WikiProject_Mathematics/Archive_43. Notice, though, that the consensus there is that a proof must be "encyclopedic" to warrant inclusion, though the precise meaning of "encyclopedic" is left to the reader. People expressing this opinion include CBM, Michael Hardy, and Jakob.scholbach. Going to the archives of this talk page, we find that Jitse Niesen believes that proofs should have encyclopedic interest in order to be included. More negatively, we find Oleg Alexandrov saying that proofs are "not that welcome". Dysprosia doesn't seem to much like them either. Plclark has a long description of why he thinks they are inappropriate, mostly due to referencing and attribution issues.


 * This opinion is not universally shared. In the archives for this talk page, linas expressed the belief that proofs should generally be included (later he clarified this by saying that Wikipedia is not the right place for all proofs); people agreeing include Hv, Connelly, CompuChip, Pdbailey, JKBlock, payxystaxna, RainerBlome, and CRGreathouse.


 * Of course, some of these users may have changed their minds since the previous discussions.


 * It seems that there is at least consensus for the following statement: Some proofs are appropriate on Wikipedia, and some proofs are not. The question is really which proofs should be included. In my own opinion, it's a tautology to say that a proof should be included if and only if it's "encyclopedic", because after all we're an encyclopedia; the question is just what "encyclopedic" happens to mean. I gave some examples above of what I thought encyclopedic does and does not mean, and I take it that you disagree. I'd be curious to hear your thoughts on what is and isn't encyclopedic. Ozob (talk) 15:53, 17 April 2009 (UTC)

It seems to me that the discussion so far demonstrates there is no consensus on this issue. However, there does seem to be two camps (pro-proof and anti-proof (or at least "you had better make an irrefutable case for a proof in a Wikipedia article")). In practice that probably means it really depends who the active editors are for an article and which camp is in the majority. So, it seems to me you are going to have inconsistencies across the Wikipedia corpus of material. I suppose this is unavoidable in an open source project. Dnessett (talk) 00:01, 18 April 2009 (UTC)


 * Yes, that's probably unavoidable, considering there are so many editors involved in various math articles. For instance, I never heard of most of the discussions Ozob mentioned above. BTW, if anyone does come across such a discussion, I think the correct thing to do is to point them to this page where a centralized discussion on the topic can be carried out.
 * Ozob, thanks for that summary, which as I said points out some discussions I wasn't aware of. As for my personal opinion (since you asked): I tend to be an inclusionist when it comes to proofs. I think that if someone took the time to contribute a proof to Wikipedia, we shouldn't remove it unless there is good reason to, although I think that in most cases it should not be within the ordinary flow of the text. I don't accept the claim that just because a proof is not interesting, it is unencyclopedic. On the contrary, I find it difficult to imagine a more dreary topic than Lists of integrals, but that page is clearly useful and any respectable mathematics reference work would definitely have one. Proofs are similarly useful because they save the time to look up the original paper and because they are the best possible verification of the correctness of a result. Also, I often find proofs in WP to be clearer than those published by the original author. So I definitely don't think proofs should be removed as a matter of policy. Are they the most important things missing right now? Perhaps not, but if someone wants to add a proof to an article, then I'm all for it. --Zvika (talk) 18:59, 18 April 2009 (UTC)

An important issue in the debate whether to allow proofs in Wikipedia articles is whether all proofs are allowable or only some proofs. I think a Wikipedia community consensus on this issue does exist. Almost no one thinks any proof is a suitable candidate for inclusion in an article. There are limits. The canonical example is Fermat's Last Theorem. As far as I can tell, the inclusion of its proof has no support (including my own). So, a reasonable question is: when does a proof add significant value to an article? This is the issue on which there seems to be no consensus. C S in an entry in S-L Proposal dated (01:32, 19 April 2009 (UTC)), writes, "Thus there is a natural relectance to speak out too strongly against proofs (I know this is true for me and a couple others).  We don't want to give "them" too much fodder for arguments to delete proof articles." This is a bit of an eye-opener. The tone of this statement is that of someone who has been bullied. If I may take the liberty of paraphrasing this statement (and if I get it wrong, please C S correct me), "If we speak out too strongly against a particular proof, then that will provide resources to those who want no proofs on Wikipedia that will allow them to delete those proofs we have managed to save."

Given this instructive bit of information, I think I will modify my original titles for the two camps in the Wikipedia community. There is the "proof-rare" camp and the "proof-common" camp. An interesting question is what criteria do those in the "proof-rare" camp use to decide when a proof improves an article and what criteria do those in the "proof-common" camp use to decide when a proof would not improve an article. Dnessett (talk) 14:54, 19 April 2009 (UTC)


 * If I may play devil's advocate for a minute here, what is so evil about having a Proof of Fermat's last theorem on WP? I think that, in a historical perspective, such an article would be far more notable than, say, Bulbasaur. If someone wants to sit down and write it out, all 600 pages of it, would you stand up and object? --Zvika (talk) 17:22, 19 April 2009 (UTC)

OK, I'm wrong. There is at least 1 contributor (Zvika) who thinks a Wikipedia article that includes a Proof of Fermat's last theorem would be fine. While I still haven't changed my mind, I am open to arguments that such inclusion is a valuable addition to WP. Dnessett (talk) 01:03, 20 April 2009 (UTC)


 * I think you want me to supply an argument in favor of the existence of that page. I guess that's a matter of inclusionist vs. deletionist philosophy: at least regarding proofs, my point of view is that if an article doesn't violate any core WP principle, then there is no reason to remove it. I would turn the question around and ask: What reasons can you give for not including (i.e., deleting) an article on the proof of Fermat's last theorem? The only question as far as I see is one of notability, and in this case the proof has generated such a huge amount of buzz, even in the popular media, that I doubt anyone could claim it is non-notable. I would also argue that many people would have agreed to include the proof had it been a one-liner, but the distinction between a one-line proof and a 600-page proof on WP seems to me almost an anachronism; see WP:NOT. --Zvika (talk) 07:03, 20 April 2009 (UTC)

I can think of at least one reason not to include a proof of Fermat's last theorem on Wikipedia. However, I am in the proof-common camp, so I think it best if someone in the proof-rare camp provides their objections (if any are even reading, not to say participating in this discussion). Dnessett (talk) 15:10, 20 April 2009 (UTC)

I stand with Zvika on this one: picking around here and there when I wanted a quick lowdown on a math topic, often I found the inclusion of a proof to be useful to satisfy my curiosity or to get a better understanding, without reaching out for a book. Basically, I'm an "inclusionist": I don't think articles should be pruned from their information related to the subject. If a proof somewhere will disturb the flow of an article, that's an editing problem, not one of including the information: The proof could be placed elsewhere, in a bottom section for example, or even have its own article for reference (for those interested) if it is too big. That way math articles can serve everyone: those who often get a crash primer on a subject by it (many do!), those who want a quick reminder (eg math people), those who want a general idea (if possible), etc. As someone who is interested in them (depends, of course I won't read Fermat's last theorem's proof, but that's an extreme example, but even then, it won't hurt me to see a link to it that brings me to its own page), I don't see why should the articles become *just* the strict theorems (facts) and general idea without proofs. 70.81.15.136 (talk) 03:39, 6 June 2009 (UTC)

I don't think all that many articles have been published with the proof of Fermat's Last Theorem in them. I think what is notable is that it has been solved and the general methods used to solve it, that it took a few goes, that many people have tried to do it. A full proof of the Pythagorean theorem one the other hand is given in many places and in many different ways and is obviously notable in itself. I would have thought the proofs that are commonly included in multiple textbooks or articles are the ones that ought to be in Wikipedia. We should apply the usual WP:NOTE rules. Just because the Bible is mentioned all over the place doesn't mean its full text need be included. Dmcq (talk) 12:39, 6 June 2009 (UTC)


 * I think as for other topics we need to distinguish (1) notability, (2) noteworthiness, (3) whether a proof is appropriate to flesh out aspects of an article, and (4) proofs as verification. (1) The proofs of Fermat's Last Theorem and the four colour theorem are so notable that they may get their own articles. To treat the topic adequately, they will have to describe the proof in some (certainly not full) detail as well as explain history, significance, reception etc. (2) In an article on a theorem, its proof is generally noteworthy enough for one section that sketches it. If there are many proofs, there might be two or three such sections or perhaps just one listing various approaches with very brief descriptions. (3) Sometimes a little proof is the best possible explanation of a definition, but that should be used very sparingly. (4) Sometimes a piece of information obviously improves the presentation, and one has a simple proof but no reference. In this case I think it's OK, although not ideal, to put the proof in a footnote as I did here: . (When I later found out that Dedekind had proved it first, I added a reference to his paper.)--Hans Adler (talk) 13:29, 6 June 2009 (UTC)


 * Wikipedia is not a mirror of public domain or other source material. If a proof is published and available elsewhere, then unless it's really short and unobtrusive, I think it should just be summarized and referenced. If it isn't already published and available elsewhere, Wikipedia is not for original research applies. —Largo Plazo (talk) 14:07, 1 March 2010 (UTC)

Proposal: Start writing the guideline
I've read most of this discussion, including the two archives, and it seems to me that the time has come to put what has been agreed upon so far into a proposed guideline. So I propose the following outline with details to be filled in. Most of this is result of trying to organize the previous discussion into a coherent list of topics, but I've taken the liberty of filling in some gaps.
 * Introductory section
 * Pros and cons of including proofs in Wikipedia
 * Differences between textbook proofs and Wikipedia proofs
 * Scope of the guideline
 * List of applicable general guidelines
 * Proofs as articles (e.g. Proof of Bertrand's postulate, Furstenberg's proof of the infinitude of primes)
 * Notability criteria
 * Naming convention
 * Articles about proofs (e.g. Feit–Thompson theorem, Wiles' proof of Fermat's Last Theorem)
 * Notability criteria
 * Outlining the proof
 * Proofs within articles (e.g. Wilson's theorem, Mathematical induction)
 * Rationale for including proofs
 * When to include, when to avoid
 * Proof sketches
 * Style guideline
 * Style guidelines inherited from MOSMATH
 * Level of detail and accessibility issues
 * Article flow
 * Links to example articles

With near certitude of oversimplification, my interpretation of much of the discussion is that WP:NOTTEXTBOOK implies that Wikipedia should not be a repository for the proofs of mathematical theorems, so there needs to be stringent criteria applied as to when a proof should be included. However, some proofs do aid in the understanding of a subject, or are suitable subjects for articles themselves, so an outright ban is not appropriate either (see WP:NOTPAPER). So it seems to me that this dichotomy of interests should be the basis for the criteria for the inclusion of proofs: If the proof meets standards of notability in its own right then it is appropriate to create an article for it, and if a proof aids in the understanding of a subject then it is appropriate to include it within an article, but proofs that do not meet these criteria should not be included.--RDBury (talk) 11:11, 14 December 2009 (UTC)


 * I think this is a reasonable idea overall, and I'm glad you decided to poke people about it. I have seen and been involved with various proposals for new guidelines, and there are some hidden pitfalls that need to be avoided. But overall, I think that an essay that describes the math project's best practices for proofs would be a helpful thing to have around. &mdash; Carl (CBM · talk) 12:52, 14 December 2009 (UTC)


 * One pitfall to avoid: it's bad to mark the page as a "proposed guideline", "proposed policy", or anything like that for quite a while. For several reasons (that I will not go into here), people like to scuttle proposed guidelines. It's better to just mark it as a description of the math project's opinion about best practice, and emphasize it is not binding on other projects. That makes it seem much less threatening to others. &mdash; Carl (CBM · talk) 12:44, 14 December 2009 (UTC)


 * About "notability criteria": it's better to just say "such articles must meet all the existing notability guidelines" rather than attempting to define new notability guidelines. By avoiding contentious areas such as that, we can focus on getting a more reasonable description of actual practice. &mdash; Carl (CBM · talk) 12:46, 14 December 2009 (UTC)
 * Thanks for the input. I'll look around for alternatives to 'proposed guideline'; 'essay' sounds good. In keeping with that, I wasn't really thinking of defining new notability criteria but more how we think they should be interpreted in this context.--RDBury (talk) 14:11, 14 December 2009 (UTC)


 * There is an essay template that can be used. This has the advantage that there is no claim that an essay represents a general consensus across all areas of the wiki. Explaining how to interpret the notability guidelines is a good idea; we just want to make sure we present it as "here is our suggestion" rather than "this is how it must be interpreted". One difficulty that has arisen in the past with notability guidelines is that people try to make them too specific, which results in some more-or-less arbitrary list of overly specific criteria; then people squabble over the fine points instead of the big picture. I think Notability (sports) is an example of what to avoid. But if we stick to more general considerations and try to avoid laundry lists, it can make it easier to find agreement. &mdash; Carl (CBM · talk) 14:51, 14 December 2009 (UTC)

I don't quite agree that WP:NOTTEXTBOOK means there cannot be regular proofs. However personally I'm not that keen for normal proofs in WP as well, since they are potential maintenance problem and there is a good chance for authors constantly quarreling about whether or not to add a proof to an article, i.e. 2 good reasons to avoid proofs (for the most part). The guideline otherwise looks good to me, but it should however provide a pointer to the concerned wikibook projects, so that authors know where they could provide or contribute proofs (within Wikimedia). Also I'd like to emphasize again, that the new guideline should be sanction by larger number of math editors before it becomes a definite guideline, since this is a guideline about a potentially contentious issue.--Kmhkmh (talk) 14:58, 14 December 2009 (UTC)


 * I do not think that stringently applying WP:N to proofs is the right thing to do. Notability guidelines are intended to dissuade people from writing articles about trivial details which don't interest anybody else, or about topics they want to promote. In these cases, removing trivial information helps WP by improving the signal to noise ratio, without harming anyone except those who wish to use WP for their own purposes. By contrast, anyone writing a proof of a theorem is not doing this for self-promotion but rather because they believe this is interesting, necessary, or helpful to the topic. If delegated to a separate article, I don't see how a proof could bother anyone, and I do think it could be of use. Deleting such articles strikes me as unkind and unnecessary. At the very least, if there is a consensus not to have some kinds of proofs on WP, then there should be a documented procedure for transferring them to some other Wikimedia project like Wikibooks, with a pointer such as Wikibooks. --Zvika (talk) 06:37, 15 December 2009 (UTC)
 * You have a point, but if you go back through this discussion you'll find people who are opposed to almost any proof in Wikipedia. One of the tricky aspects of doing this is going to be coming down in the middle and making everyone equally unhappy. I like your phrase "interesting, necessary, or helpful to the topic" and it might make a useful starting point; I hope no one would argue that uninteresting, unnecessary and unhelpful material should be in Wikipedia. There are a couple of Wikibooks cleanup tags, but they don't seem quite appropriate to these articles, maybe a new one should be created or an existing one can be modified. In any case, the people at Wikibooks may have something to say before a bunch of articles get copied to their domain.--RDBury (talk) 09:45, 15 December 2009 (UTC)


 * My point was that someone might find a certain proof helpful/necessary and would then sit down and write. Others—perhaps most of us—might later disagree, but still we should not completely delete that person's hard (and useful) work. At most there should be a procedure for moving it to a different Wikimedia project. Certainly we need to think about exactly how this procedure would work. This is separate from the issue of what, according to the community consensus, constitutes a useful proof. --Zvika (talk) 11:35, 15 December 2009 (UTC)
 * That is exactly why we should get that sanctioned by a larger number of math editors, because not all but just most or maybe almost all agree with that. Otherwise we wouldn't have had all the discussions in the past or feel the need for an explicit guideline to begin with. The Wikibook projects are well suited to avoid to avoid the deletion of useful work or rather guide potential authors to a more appropriate place for regular proofs to begin with. Of course we cannot simply dump stuff into Wikibook project without the consent of their authors, however some Wikibook project are already specifically designed for that purpose, that is collected all sorts of proofs for standard or well known theorems. The important point here is anyhow to let any Wikipedia math editor know, where he could contribute a proof that would be accessible to Wikipedia readers (via link) without conflicting with our guidelines.--Kmhkmh (talk) 14:51, 15 December 2009 (UTC)

I may have found an appropriate place for proofs that we want to remove from WP: a Wikibook called Famous Theorems of Mathematics. It was previously called The Book of Mathematical Proofs. I left a comment on the talk page there to see if someone is still maintaining that page, and if they would be willing to accept our proofs. --Zvika (talk) 08:26, 17 December 2009 (UTC)


 * I don't believe that anyone on wikibooks other then me is monitoring the famous theorems of mathematics page. It's likely that your additions will go unnoticed for some time by other contributors there. But as the original author of the book I certainly have a few reservations about adding proofs in that book without any categorization and in an unorganized fashion. Remember that wikibooks is about books, not about repositories. A book is supposed to have a compactness which your efforts might destroy(no disrespect intended). Also if other wikipedians don't take an interest in adding proofs there, which I seriously believe is going to be the case, then the resultant module will be neither here nor there.(Back when I started the book with the name The Book of Mathematical proofs, I posted a suggestion on wikipedia regarding posting of proofs there, but many wikipedians quite rightly pointed out the practical problem of the giganticity of such a task and the resultant mess the book was likely to become.) I believe that the best option would be to start a proof wiki. If no one is sufficiently interested in taking the lead then people can add proofs on an existing proofwiki here, and a template can be added in the relevant articles linking to that proof. Even if its not part of wikimedia, the purpose to the reader will be served, and that's what is really important. Another option, though not a better one, would be to add proofs on wikiversity, which is at least supposed to contain all educational resources and not books. Regards-Shahab (talk) 08:15, 22 December 2009 (UTC)

Some flesh on the bones
I have covered what seem to me to be the main points of an outline, on the page. Charles Matthews (talk) 08:39, 15 December 2009 (UTC)
 * It looks great to me, except for the last paragraph; I couldn't quite figure out what you are referring to there. --Zvika (talk) 08:49, 15 December 2009 (UTC)
 * I think there could be a bit more expansion in some areas but overall it looks like a good start. This discussion goes back 5 years and I want to make sure that the viewpoints are fairly represented. I'm not sure I agree with the section on scope; I'm thinking there is a lot material that looks like a proof that isn't actually labeled a proof. An example that immediately comes to mind (probably because I had a lot to do with writing it) is Catenary. The top half is very non-mathie and has lots of pretty pictures, but the bottom half is full of derivations and formulas. I think this is entirely appropriate (see previous parenthetical remark) but I think the material is sufficiently proof-like to come under the scope of this page.--RDBury (talk) 12:38, 15 December 2009 (UTC)
 * I wrote Work is needed in this area just to suggest that this is the "guideline" part, and what we need to discuss some more. For the "necessary conditions" I think we need to get away from talking about "notability", because that is at best a sketchy concept, and at worst leads to citation counting and other hopeless kinds of argument. And as it says, it is not too controversial that most proofs aren't worth an article. For the "sufficient condition" side we need to remember the basic arguments on summary style: if the proof of Theorem X isn't included in the article Theorem X, then it should not stand on a "subpage", but as a topic in its own right. So the discussion needs to be had, how do we extract some general ideas of what should be acceptable, from the kind of indications given before? Roughly, if we can say "The proof of Theorem X has Attribute A" implies the proof is of interest, then we should add Attribute A to a list of indications to authors. An example not yet mentioned would be A = "leads to a named number being defined", example Skewes' number. I don't think it would be that easy to get a list defined, but it does seem worthwhile going into more detail there, rather than on the negative side. Charles Matthews (talk) 12:50, 15 December 2009 (UTC)
 * I agree that we shouldn't be getting too detailed about notability at the moment. Right now I'm like that old supreme court ruling; I can't define it but I know what it is when I see it. I think the best we can do now list good examples and not so good examples, my selection of good examples being the links I put into the outline.--RDBury (talk) 13:31, 15 December 2009 (UTC)
 * I'd still like to see a pointer to Wikibooks in the final version (see above).--Kmhkmh (talk) 14:52, 15 December 2009 (UTC):
 * I added a paragraph to start. &mdash; Carl (CBM · talk) 14:57, 15 December 2009 (UTC)


 * I agree that a list of examples – both good and bad – is more helpful than a list of vague criteria. Also, we may find consensus more easily that a particular article is good, or is bad, if we don't also have to find consensus on exactly why it is good or bad. &mdash; Carl (CBM · talk) 15:00, 15 December 2009 (UTC)

Questionable examples
Carl asked for bad examples as well as good. I won't say these are bad necessarily, but maybe questionable or at least in need of improvement. So there are six questionable (imo) ones to match the six good ones I gave earlier. I think they show the kind of diversity of issues we're dealing with. There are some articles there just too far over my head technically for me to make any kind of judgment, but overall I'd say there is more good than bad out there.--RDBury (talk) 10:02, 16 December 2009 (UTC)
 * Nondeterministic finite state machine/Proofs -- This page is almost all symbols with no prose. The basic idea is given in the article Powerset construction and I have it tagged for a merge.
 * Linear/Proof -- Too much detail and questionable notability.
 * Parabola/Proofs -- The high school geometry, two-column format is entirely inappropriate for an encyclopedia article. The statement being proved does not need to be justified in a survey article.
 * Moore–Penrose pseudoinverse/Proofs -- This is worthwhile material but it's in textbook style rather than encyclopedia style. It seems to be a good candidate to move to Wikibooks but the question is where? The nearest candidate I can find is Linear Algebra, but this seems like an undergraduate level text and the Moore–Penrose pseudoinverse is more of a graduate level topic. On the other hand, it doesn't seem a suitable for a book on it's own.
 * Boy's surface/Proofs -- The article actually was copied to Wikibooks where it is now residing uncomfortably in Famous Theorems of Mathematics. Same issue as the last one; it's easy to say move it to Wikibooks but where exactly should it go?
 * -- This is an old version of an article I've been trying to clean up. To me it's an example of the kind of thing that can happen when people fill in their own proofs instead of using a reliable source; I'm sure the proofs are mathematically valid, but pages are used to show what can be proved in a few lines. For example see this edit.


 * Thanks. I added two of these; maybe we could add a couple more. Four sounds like the limit that anyone reading the page here would actually have the patience to look at, and we should be able to cover the "most common" problems with four.


 * I appreciate the Wikibooks issue; I alluded to it on the main page with "Occasionally, it is suggested that proofs that are removed from Wikipedia articles might be good source material for Wikibooks, although it is not obvious how to accomplish this easily." Personally, I have no idea how Wikibooks works or how to move a proof from here to there.


 * It seems to me that there are two different issues with many of the bad proofs: th choice of which thing to prove, and the proof itself. I need to think about this more, but probably we could add a section to this page describing how proofs are usually written (for example, using prose instead of a two-column format). &mdash; Carl (CBM · talk) 12:04, 16 December 2009 (UTC)


 * The actual mechanics of moving the article isn't that hard; you post a request and an admin will copy the article. I conjecture that copy/paste is a no-no because of GFDL issues.--RDBury (talk) 19:14, 17 December 2009 (UTC)


 * I added a Style section to indicate prose style is preferred. This is really just inherited from MOSMATH so I don't think it's controversial.--RDBury (talk) 20:26, 17 December 2009 (UTC)


 * There are about 30 or so article proofs, and this number has not changed in about 5 years. I fail to understand the deletionist urge ... why not leave well-enough alone? linas (talk) 16:48, 15 January 2010 (UTC)

Further edits by Ozob
Not that happy with these. My views on using "notability" are above: it is an unhelpful concept here, I think. Those who think that notability discussions of the normal kind are going to get us the right results in this area are just wrong, I believe. Please discuss here. Charles Matthews (talk) 21:05, 22 December 2009 (UTC)
 * I think overall I like the changes. It's good to have a summary of the purpose that proofs can serve in Wikipedia, and how proofs here might differ from those in those in textbooks. To me the notability section is just summarizing the WP:GNG. Maybe we should be just pointing to the GNG instead of summarizing here, but the GNG is going to apply either way. Otherwise we're saying that proof articles have a lower standard and that would be a policy change. The interpretation of the GNG is still wide open, anything from "if it's in a textbook then it can be put in Wikipedia," to "Wikipedia should just be about the facts so proofs shouldn't be here." I think you're right in that we don't want to get specific now, but I don't think the changes really do.--RDBury (talk) 23:17, 22 December 2009 (UTC)


 * Yes, the reason why I wrote it with so much attention to the general notability guideline was because I didn't think we could get away with anything else. But as I was doing it, I convinced myself that the GNG actually serves our purposes pretty well if we give instructions on how to interpret it correctly. Remember that the GNG says, "If a topic has received significant coverage in reliable sources that are independent of the subject, it is presumed to satisfy the inclusion criteria for a stand-alone article." A "reliable source" for a proof should mean a refereed paper, monograph, or book. "Independent of the subject" means that it's been discussed outside of the original paper in which it was proved, and should probably also mean that it's been discussed by someone other than the original author. (Note that this means the proof, not just the result.) "Significant coverage" should mean that the proof actually receives some discussion, as in, "This proof is interesting because ..." or "This proof contains the following clever trick ..." or something like that. Discussion of the theorem that the proof demonstrates isn't relevant. And "presumption" means the same as it does in the GNG.


 * What my text doesn't address is the issue of proofs inside articles; for example, how do we justify discussing Euclid's proof of the infinitude of primes in the middle of prime number? Wikipedia doesn't have a general content guideline in the same way that it has a general notability guideline. In fact, I think proofs in articles is going to end up as the harder part of this page. I'm interested to hear ideas (for this and the other parts of this page). Ozob (talk) 06:42, 23 December 2009 (UTC)
 * To some degree (like the prime number case) it may simply have to remain at the discretian of the (main) editor of the article.--Kmhkmh (talk) 07:10, 23 December 2009 (UTC)


 * Still not happy. General notability still affects proofs, but "reliably sourcing" a proof is a dead-end for any serious discussion. Proofs here will appear as some summary or variant of proofs elsewhere, and we should not be worrying about whether a concise or thumbnail version has been seen in exactly that form. If general considerations were enough, we needn't have bothered with trying for an more explicit and friendly guideline, really. And reducing notability to issues of sourcing is basically fallacious. If we are back to asking "how do we justify discussing Euclid's proof of the infinitude of primes?" the whole thing has fairly clearly gone in the wrong direction: we have some truisms trickling down from general policy, but a failure to treat proofs as the kind of content that we clearly do want to include and encourage. Sorry, it just seems wrong-headed to me, and certainly misses the point of previous discussion. Charles Matthews (talk) 08:47, 23 December 2009 (UTC)


 * Any proof we can't reliably source is in violation of policy, and there's no way we're going to get around that. The only thing I think we can do is interpret the terms of GNG in our context: We can specify, for example, that a reliable source means a paper or book that follows the same rough outline of the proof (but not every little detail, because of course that would be plagiarism). I don't see how that' s objectionable, because if there are no sources for the proof, then it's OR and we're not supposed to include it. Now, in practice, if I'm putting an argument into an article, then I don't look up the argument and base my text on what I find (unless I'm writing the article to teach myself); but I consider that a weakness in my own writing. Not citing prior work is bad practice in an encyclopedia. It's bad practice in a paper, too: after all I can't publish if I haven't given appropriate credit.


 * Also, I don't think it's unacceptable to require an argument for including Euclid's proof of the infinitude of primes. If some editor demanded one and we chose to pity him, then we could easily find a reference that explained that the proof was important; it is, after all, an important proof! But Wikipedia, being an unthinking collection of electrons, does not know that it is an important proof unless it has citations and references for that.


 * Let me turn the question around and ask: If the GNG is not appropriate in our context, what would be? How would you like to determine whether or not we should include a proof? Ozob (talk) 05:57, 24 December 2009 (UTC)


 * When you say "The only thing I think we can do is interpret the terms of GNG in our context", you are being entirely "reductionist", and it is so unhelpful. This is an essay trying to explain what more there is to say; or at least was. And when you say "Wikipedia ... does not know that it is an important proof unless it has citations and references for that", you fall into the trap of equating "notability" with enough verifiability, which is a fallacy. You have cut out the bit saying At the same time, our best articles provide a useful reference for readers familiar with a field who wish to look up particular facts. The role of proofs, which may be short but correct arguments, or sketches of longer arguments serving more as a map of complete proofs, is to support the "survey" and "reference" ambitions. Frankly, with that, and switching round the "proofs as article topics" and "proofs in articles", thus confusing one kind of inclusion with another, you have subverted what was being said completely. Editors can perfectly well include material about proofs to support the content of an article, just because the material is sensible, as long as they steer clear of "original research". And the occurrence of a proof in half-a-dozen textbooks does not confer notability. I'm reverting since I feel you have missed the point of the page as was. Charles Matthews (talk) 08:45, 24 December 2009 (UTC)


 * I think the best solution is to try to merge the two versions somehow; I'll try to look at it later to come up with a proposal. I'm thinking that we're reaching the point where there should be input from other editors before major changes are made.--RDBury (talk) 09:30, 24 December 2009 (UTC)


 * I have the feeling that I'm not understanding your perspective here. I didn't think our versions were very different, but you find them to be opposites. For the moment I'm going to hold back and wait for the input of other editors here, since I don't feel comfortable editing a page that I don't comprehend. Ozob (talk) 19:35, 24 December 2009 (UTC)

I'm with Charles Matthews on this. Currently, there are about 30 article proofs, and this number has not changed in 5 years. If there was a lot of pent up energy to create new article proofs,then more discussion would be merited. Conversely, a lack of interest is not tantamount to a need to delete the existing proofs. I can't help but view Ozob's changes as an invitation to apply some of WP's poorer policies in order to delete the existing proofs. (For the record, I believe that that many WP policies that are treated as "holy" by certain groups of "enforcers" are, in fact, rather poor policies for WP as a whole. Many others fail dramatically when applied to certain subtopics, such as mathematics or the sciences -- yet the "enforcers" persist in trying to apply them, driven by some urge, while demonstrating a lack of experience and understanding. Knowledge is a complex thing, and the idea that one set of policies can rule over all of WP is dangerous and mis-guided.) linas (talk) 16:45, 15 January 2010 (UTC)

Style: Pronouns, imperative, Q.E.D.
Proofs normally use the imperative heavily (e.g. "Let ABC be a triangle...") and sometimes use first person ("We find that..."). This seems to be confusing for non-mathematicians who aren't used to reading proofs and may raise style flags. I seem to recall there was a recent thread on Wikipedia talk:WikiProject Mathematics about articles being tagged with essay-like because of something like this. The use of the imperative might make some readers confuse a proof with a How-To section and raise a flag because of it. For the use of the imperative I'd like to add to the style section a sentence or two to approve the use of the imperative in proofs, just to have something to point to if the issue comes up. I'm not so sure what to do with the first person issue; it's generally discouraged (though frequently used anyway) and I can't think of reason that proofs should be made an exception. As an example, Area of a disk uses first person several times. This article has an A rating but it failed a GA review and one of the main reasons given was the use of 'we', though it's used all over the article and not just in that proof. I'd like to add a sentence to discourage first person in proofs in keeping with general style guidelines, but I'd like to think about possible alternatives first. Finally, I'd like to add a sentence to discourage the use of Q.E.D. at the end of proofs. First, it's jargon and many readers will need to look up the meaning. Second, it seems to be out of fashion now; most modern texts either use nothing or a tombstone. Third, the end of a proof in Wikipeda is almost always marked by a section heading or the end of the article itself, so it doesn't really serve any purpose here. Fourth, the correct usage is debatable, for example I don't think should be used at the end of an indirect proof (Q.E.A instead). These changes could be controversial so I wanted to get some feedback before putting them in the article.--RDBury (talk) 10:48, 24 December 2009 (UTC)


 * It should be enough to note that proofs are expected to be discussed and described in complete sentences, in ordinary English. If you like, append some comments on formulation, removal of first-person, and markers for the end of proofs. Charles Matthews (talk) 11:47, 24 December 2009 (UTC)

Progress on /Proofs articles.
I'm in the process of dealing with (i.e. merging, renaming, changing to redirects) articles with the suffix '/Proofs' since they don't meet Wikipedia naming guidelines. There have been several previous discussions on this. This current status can be found here. I've merged several of the articles and found redirects for others, plus many of the articles were already done by other people. There are seven articles outstanding: --RDBury (talk) 16:12, 16 January 2010 (UTC)
 * Boy's surface/Proofs: This was copied to Wikibooks so I PRODed it, the PROD was removed however. I'll try AfD next.
 * Ellipse/Proofs: I'm planning to add a summary of this proof to the main article. But this article is currently in a state of flux and it is unclear at the moment what the definitions will be.
 * Laplace–Beltrami operator/Proofs, Moore–Penrose pseudoinverse/Proofs, Christoffel symbols/Proofs, Addition of natural numbers/Proofs, Ordinary least squares/Proofs: These seem to contain a number of more or less straightforward proofs of technical facts which don't appear (at least to me) to have encyclopedic content. I am encountering some resistance in changing these to redirects and I haven't decided on a next step yet.


 * I looked at the addition of natural numbers one. I don't see any activity on those pages recently, so I went ahead and redirected it. I also put a merge tag on the main article, since I think addition could be covered perfectly well in Natural number. &mdash; Carl (CBM · talk) 00:02, 21 January 2010 (UTC)


 * I'm sorry, but as I see it, what Carl did is delete a page without due process. The fact that he called this a "redirect" is just a euphemism. To say the least, there is no consensus that proofs should be deleted (see the comments above for examples). Even if we agree that proofs are not appropriate on WP (and we haven't yet), there is no argument that writing a proof is a lot of work which might be useful for someone. Therefore, deleting the content instead of moving it to any of the several alternative homes we discussed amounts to trashing someone else's good work. I really do not see how this helps in any way. Please finish the discussion and get a consensus for the guideline before implementing it. --Zvika (talk) 07:07, 21 January 2010 (UTC)


 * I don't know if you've read everything above, but to clarify: (1) this is not a proposal for a guideline, it's an essay, and will never become a guideline; and (2) we are not talking about "whether proofs are appropriate on Wikipedia" - everyone agrees they are sometimes appropriate. We are talking about subpages dedicated only to proofs. &mdash; Carl (CBM · talk) 13:01, 21 January 2010 (UTC)

Re RDBury: do you have a list of all the unresolved /Proofs pages? I see the ones above, but I don't know how many others there are. It might help organize things if there is a master list somewhere. &mdash; Carl (CBM · talk) 13:01, 21 January 2010 (UTC)
 * The whole list is here. I'm going to wrap this up in the next day or so with the exception of Ellipse and Boy's surface. Several of the articles still have notability issues but those can be raised later in AfD's.--RDBury (talk) 16:37, 21 January 2010 (UTC)
 * Thanks. I have proposed addition of natural numbers for a merge as well. &mdash; Carl (CBM · talk) 16:49, 21 January 2010 (UTC)

All the articles are now done. The only article remaining with the /Proofs suffix is Boy's surface/Proofs and I AfD'd that one.--RDBury (talk) 19:32, 21 January 2010 (UTC)


 * That's really great; I appreciate all the work to handle those. I am contemplating putting the natural numbers one on AFD. &mdash; Carl (CBM · talk) 19:46, 21 January 2010 (UTC)

Carl, I too am talking about articles dedicated only to proofs. Reading the above discussion again, I still see nothing close to an agreement that such articles should be deleted, and I still don't understand why you "redirected" (i.e. effectively deleted) the page Proofs involving the addition of natural numbers. At the very least it should be transwikied and then PRODed, not removed without discussion. --Zvika (talk) 20:13, 21 January 2010 (UTC)


 * I removed it because as far as I can tell there is general agreement that the proof subpages were an experiment that did not work out. For example, several editors have contributed to the essay on the other side of this page, which says
 * It is widely accepted, however, that if a proof is made a topic of its own dedicated Wikipedia article, the proof must be significant as a proof, not merely "routine".
 * The proofs in the addition page are completely routine. In general, I thought that simply redirecting the page was the best solution for content that (1) doesn't really belong on WP and (2) is completely standard, so it could be easily replaced elsewhere. &mdash; Carl (CBM · talk) 20:23, 21 January 2010 (UTC)
 * I agree that the proofs in the addition page are routine but I'd suggest reviewing the AfD's for similar articles (see links above) before doing this one. I think the best argument is WP:NOTTEXTBOOK and there is little hope that a rewrite will fix the issue. It might be cool to write a wikibook that covers the construction of real numbers starting from the Peano axioms, where the addition article might be page 3, but I don't think that will happen soon.--RDBury (talk) 01:21, 22 January 2010 (UTC)
 * PS. The Proofs section in the WP:MSM guideline may also be relevant.--RDBury (talk) 02:26, 22 January 2010 (UTC)
 * Guys, I understand you believe you are doing this for the best, but I urge you to stop a moment and consider the other side because I think you are doing this wrong. The other side is a person who contributed time, effort and knowledge in writing a proof. This proof might not have encyclopedic value; it might even seem trivial to you. I am willing to grant that it might not be appropriate on WP. But that does not mean that the person's work has no value or use. People can and will be interested in seeing that proof. For example, Proofs involving the Moore–Penrose pseudoinverse is the first hit when googling for pseudoinverse proof. And indeed that page consistently receives hundreds of page views a month. Rather than delete the material forever, we have had several constructive suggestions above: I proposed transferring to Wikibooks and User:Shahab suggested transferring to proofwiki. This will allow the information to still be accessible without interfering with our encyclopedic standards. I thought that this point was agreed on by everyone, which is why I was a bit surprised to see Addition of natural numbers/Proofs suddenly disappear. Are there additional proofs pages which have been removed? If so, please let me know which. I would be glad to transwiki them myself, and I think the redirect should be changed to an interwiki soft redirect. --Zvika (talk) 09:50, 22 January 2010 (UTC)
 * I realize that no one likes to see their work lost and transferring to Wikibooks has been discussed as an option before. For articles that are more textbook style than encyclopedia style I have nothing against this except for the practical issue that there isn't always a book for the article to be placed into. The problem with writing an article 'Proof of fact about X' which should really be a section in a book called 'The Theory of X' is that you're writing a snippet and not the entire textbook. As for the fate of the remaining other proofs articles, keep in mind that there is an entire spectrum of quality from well written and accessible to unbelievable cruft. I've spent many hours summarizing and merging the articles that I felt had worthwhile material into the main article. The articles I judged to be pure cruft I replaced with a redirect rather than going though the hassle of AfD's, but I suppose now that the immediate issue of fixing the article names has been resolved these could be resurrected and AfD'd properly. My overall response is, just as with any other type of article, if you don't want it to be deleted then improve the article. Find references, add context, make it understandable or expand it to an actual book if it should be in a book, so it meets Wikipedia or Wikibooks standards and won't be deleted.--RDBury (talk) 15:55, 22 January 2010 (UTC)

Really I don't want to get into a deletionist/inclusionist argument here, but I don't think it is my responsibility to improve all these articles within the 7 days of the AfD just so they aren't permanently deleted. Keep in mind that WP:WIP. Also, as we discussed above, there is both an appropriate book in wikibooks, and an entire wiki (proofwiki) dedicated to holding these "snippets". So why the urge to delete content? Again, please, provide a list of proofs which you have redirected and/or nominated for deletion. I think that is only fair so that myself and others can follow up on this. --Zvika (talk) 08:29, 24 January 2010 (UTC)

Response to comments made in main article
Some comments were made in the main article that I would like to respond to. First, to the statement that the practice of using the /Proofs 'subpage' names was found to be in violation of naming guidelines the comment "When? This project applied for an exception, it had been granted; has this been reversed?" was added. Second, to the statement that the issue is currently being addressed the comment "In what way?" was added. The initial comment that led to this was made User:Miym in Wikipedia talk:WikiProject Mathematics/Archive 56 who pointed out the guideline WP:SP and was not contradicted. User:CBM suggested handling these articles on a case by case basis which is what has been done. If there was an exception granted then this is the first I'm hearing of it. It wasn't mentioned in that discussion or the subsequent discussion at Wikipedia talk:WikiProject Mathematics/Archive 57 so if an exception was made then it doesn't seem to be generally known. In any case, comments about the main page should be made here and not on the page itself so I'm removing them.--RDBury (talk) 05:03, 22 January 2010 (UTC)

Pons Asinorum
Was there a reason for cutting out this example? It is, after all, one of the few examples one could give of a proof a generally-educated person could be expected to have heard of, as a topic (whether on not they could cite the mathematics involved). In other words it has notability of the proverbial kind, which you couldn't say is shared with the other examples. Charles Matthews (talk) 11:14, 24 January 2010 (UTC)
 * The proof moved to a different article, apparently (Isosceles triangle theorem). I added that article as an example. &mdash; Carl (CBM · talk) 13:13, 24 January 2010 (UTC)


 * I added a comment to Talk:Pons asinorum that I'd like to see a diagram there. Thde problem with the isosceles triangle theorem article is that it does not deal with the classical pons asinorum but with later proofs. Pons asinorum is Euclid's original proof and it is that which is most notable as a fully worked out proof. Dmcq (talk) 13:48, 24 January 2010 (UTC)


 * Could you double-check the wording in the proofs at Isosceles triangle theorem for historical accuracy about what Euclid did? I am not overly familiar with Eudlid. &mdash; Carl (CBM · talk) 14:09, 24 January 2010 (UTC)


 * I've found it at gutenberg, the proof starts on the pdf page 21 which is the books page 11 and the diagram is on the next page. It looks simple enough, I'll tackle it later if no one beats me to it. Dmcq (talk) 14:45, 24 January 2010 (UTC)
 * In fact there#s a parallel Greek and English version in the pdf accessible from . I think I'd go for the Greek letters for the bridge. Dmcq (talk) 14:54, 24 January 2010 (UTC)


 * I'm still working on the proofs section, there are a number of notable ones besides Euclid's, which is just as difficult as its reputation would lead you to believe. I didn't add Euclid's original proof but Proclus' variation which is slightly simpler while maintaining the general method. There is probably enough information out there on the Elements to write a couple dozen articles; a good place to start is Heath's translation and commentary. Getting off-topic a bit, doing the research I was surprised by how big a part the Elements played in Western culture until about a century ago. If you mentioned "Euclid I 47" to a 19th century mathie they would immediately know you were talking about the Pythagorean theorem.--RDBury (talk) 20:28, 24 January 2010 (UTC)


 * The proofs section is done as far as I'm concerned. Heath notes a pre-Euclidean proof which may be notable but it assumes so much in the way of other theorems that it's not really a proof.--RDBury (talk) 18:02, 27 January 2010 (UTC)

derivative of antilog x
Shrenujparekh (talk) 13:17, 8 April 2011 (UTC) related to calculus,derivatives

i really did no where to post this......so posted here as an appropriate place(maths proofs).... what is the derivative of y=antilog x(the base of antilog is e)

WAIT BEFORE YOU READ IT FURTHER!! 1st try it yourself on a paper(i know i'm a bit childish,but still 1st try on your own)

calculation:

y=antilogx...................(1)

logy=x{antilog when goes to other side becomes log and vice-versa}

diffrentiate both sides w.r.t x

1/y (dy/dx)=1.............................(via implicit functions)

(dy/dx)=y

therefore ans is y but y=antilogx..........from(1) therefore the derivative of antilogx is...antilogx itself........and similarly integral of antilogx...........is the same

so can anyone put this in a wiki article??
 * The antilogarithm for base e is the inverse function of the logarithm to base e, which is the natural logarithm. The inverse of the natural logarithm is the exponential function. It is well known that this function is its own derivative; the fact is mentioned in that article. Should we also give a proof of this well-known fact? One problem is that there are many ways to define the exponential function, and what is a good proof of this fact depends on which specific definition you have chosen. If you define it as the inverse of the natural logarithm (which is indeed a definition some authors have used), then your proof is fine. Our article gives another definition, for which it is also easy to give a proof, except that it requires some more machinery if you want to be rigorously precise. --Lambiam 19:31, 17 April 2011 (UTC)

"Imperative"
I don't really see the point of the "imperative" section, which anyway seems a bit confused &mdash; this let is not an imperative in the ordinary sense, but more of a subjunctive ("third-person imperative" if you like). Admittedly it looks formally like an imperative, but it's hard to believe many native speakers think they're being commanded to let x be a real number.

Whether you agree with that analysis or not, the main point is, what is the section doing here? The essay seems to be about when proofs are appropriate. Stylistic suggestions for proofs don't seem to me to be a good fit. --Trovatore (talk) 20:45, 13 September 2012 (UTC)

Adding a Proof template?
The guidelines page on proofs mentions the possibility to use collapse boxes to include non-essential proofs. I was wondering whether it would make sense to have a specific template for this, as (for example) the french wikipedia has: fr:Mod%C3%A8le:D%C3%A9monstration ? Are there reasons for this not to be used also in the English wikipedia? Note I am asking about this, although I would not be able to implement it myself. EtudiantEco (talk) 05:24, 11 January 2015 (UTC)


 * Here lies the end... Of your search: https://en.wikipedia.org/wiki/Template:Collapse
 * You are welcome. Temnothorax (talk) 23:21, 3 November 2023 (UTC)
 * More about it here: https://simple.wikipedia.org/wiki/Template:Collapse_bottom Temnothorax (talk) 23:23, 3 November 2023 (UTC)

Right aligned tombstone in Wikipedia?
Hi. I was wondering if Wikipedia had a template that produced a right-aligned tombstone symbol, similar to how a right-aligned tombstone is added at the end of a \begin{proof}...\end{proof} environment in LaTeX. The Tombstone template appears to only produce a tombstone symbol without any alignment. Although I did search for it, I was unable to find a template that produces a right-aligned tombstone. Thank you! Mgkrupa (talk) 14:56, 3 August 2020 (UTC)