Wikipedia:WikiProject Mathematics/Proofs

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This is a discussion page for WikiProject Mathematics

This page is devoted to discussions of when and how to include proofs in mathematics articles. The main discussion page should be used for other issues.

Please add new topics at the bottom of the page and sign your posts.

Archives /Archive 1: 8/16/2004 - 9/3/2005

Contents 1 Distinguishing between proofs in context and stand-alones 2 Let's just create articles that gather related proofs 3 /Proof subpages 4 A lay perspective? 5 Suggestion 6 A student-mathematician's perspective

[edit] 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 signifigant 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/refrenced 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. 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. linas 19:46, 10 December 2006 (UTC)

[edit] Let's just create articles that gather related proofs

I already started one, Proofs of trigonometric identities. --Ķĩřβȳ♥;ŤįɱéØ 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. --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. Michael Hardy 19:11, 13 December 2006 (UTC)

How does it look now? --Ķĩřβȳ♥ŤįɱéØ 22:03, 18 December 2006 (UTC)

[edit] /Proof subpages

Proposition: Create a /Proof subpage with mathematical articles, listing proofs of theorems and statements made in that article. For example, the page Linear_Algebra contains a section 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.

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 {{fact}} that would indicate a proof was needed. Pdbailey 17:43, 1 April 2007 (UTC)

[edit] 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 --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. --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. 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. Haemo 09:01, 19 December 2006 (UTC)

[edit] 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. CommandoGuard 22:27, 19 December 2006 (UTC)

[edit] 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.

Lim Wei Quan 10:19, 30 December 2006 (UTC)