Wikipedia talk:WikiProject Mathematics/Proofs/Archive 1

encyclopedia vs. text

This is an encyclopedia, not a collection of math texts; but we often want to include proofs, as a way of really exposing the meaning of some theorem, definition, etc. A downside of including proofs is that they may interrupt the flow of the article, whose goal is usually expository. Use your judgement; as a rule of thumb, include proofs when they are part of an explanation; don't include them when they are a justification whose conclusion is merely "... therefore, P is true".

Since many readers will want to skip proofs, it is a good idea to set them apart in some way, for instance by giving them a separate section.

It would be nice to have a recommended format for that. In many articles the :proofs are now part of text. Take for example this page: Wilson's theorem. :It contains two proofs. Also some articles (like this one) would be improved

by a clear statement in the form of a theorem. Especially when the complete

statement is made anyway, why not clearly mark it so.

Perhaps a format could look like this:

Theorem (Wilson) A positive integer $n$ is prime

if and only if

$(n-1)!\ \equiv \ -1\ ({\mbox{mod}}\ n)$

Proof 1

..group..

end of proof --user:sander123

---

Wouldn't the above be better if it started, eg. Theorem (Wilson 1770) A positive integer ...? ---- Charles Stewart 23:36, 16 Nov 2004 (UTC)

I think better still is to say "Wilson 1770" and include a page-internal link to the full-blown source for the theorem, like this:

Theorem (Wilson 1770) An positive integer $n$ is prime

if and only if

$(n-1)!\ \equiv \ -1\ ({\mbox{mod}}\ n)$

Proof 1

..group..

R. Séroul. "Wilson's Theorem." Programming for Mathematicians, §2.9. pp. 16-17. Berlin: Springer-Verlag. 2000.

Deco 23:45, 16 Nov 2004 (UTC)

Actually. I would say, just write proofs as normal prose in English. I'm not in favour of importing conventions from mathematics textbooks here as a rule. If that means WP has more descriptions of proofs than actual proofs, so much the better; textbooks exist, but mostly they are not so good at giving thumbnails of proofs to give access to non-specialists. So basically I disagree with the initial comment. I would certainly prefer to have the Wilson's theorem proof described loosely as 'by pairing residues with their modular inverses ...'. Also it is misleading to imply anything about the date of such a proof. Charles Matthews 08:35, 17 Nov 2004 (UTC)

Agree with Charles. Usually just the sketch of a proof is enough. A complete proof is useful only when it is short and clear enough that anyone with a basic grasp of the subject area will find it easy to understand. An easilly understood proof can serve a general reader; anything more is the realm of math textbooks. Isomorphic 15:41, 17 Nov 2004 (UTC)

I also think it should be a sketch of a proof possibly with a reference to some book. A detailed proof is not good, because we would also need to develop a whole tree-system of "you need to know [1], [5], and [75] before you can understand this." (Igny 17:09, 21 Apr 2005 (UTC))

the one just does not exclude the other, bah. We should have both informal and formal descriptions. MarSch 12:30, 22 Apr 2005 (UTC)

What we need on Wikipedia is less formal descriptions, not more. We should strive towards making the general public understand what math is, and not writing for the fellow mathematician who has better sources of learing about math than an encyclopedia. I would ecnourage anybody to think twice before making an article more rigurous, more mathematical, in particular, adding proofs. That greatly reduces its readership and comprehension with the general public. Oleg Alexandrov 15:24, 25 May 2005 (UTC)

Wikipedia is for everyone. Articles are for experts and the general public. An expert might want to look up the precise definition a group cocycle. A non-expert might want a general idea of what the Pythagorean theorem is about. There is no reason to exclude either. What proofs to include and in what format to include them depends entirely on the individual article and the judgement of the people writing the article. Wikipedia is much more than an encyclopedia--345Kai 02:10, 6 April 2006 (UTC)

Contrarian view

I'd like to present a contrarian view. As wikipedia develops, it will de-facto become more detailed and, as a side-effect, more authoritative. This is unavoidable; more readers will beget more authors, and it will grow. We don't yet know how far, or what will cause it to stop growing. As it becomes more authoritative (and this is truly unavoidable), the question will come up: How can I trust that what is written here is accurate? The current mechanism, involving watchlists and maintainers, is adequate for insuring the accuracy of WP, in most cases, for the most part. However, its worth asking what steps can be taken to take WP to the next level, and whether these should be taken.

I peronally don't know if WP will become "authoritative" but I don't think it should. WP is good as an entrypoint and should give objective informations on how to get further informations on topics which are presented. To be authoritative using the current editing mechanism which neither ensures a proper peer-review, nor an authorship responsability, nor a global coherence (such as the one of Bourbaki, for instance), nor the perenity of the entries seems dangerous to me. pom 19:58, 14 November 2005 (UTC)

I see proofs and citations as a valid and effective mechanism that can be used to maintain and improve accuracy. Now, let me make clear one critically important point: I think almost all such proofs should be out-of-line, in their own separate articles. The references to such proofs should almost always bee teeny, tiny foot-note like bullets. Also, let me make clear: proof is maybe too strong a word. A demonstration might be a better term. A bit of twiddling to show how a formula in an article was derived. Let me not be vague: lets take a look at some examples -- these are from PlanetMath --

• [1] and [2] These two show the derivation of some formulas, and have little utility other than to confirm the consistency of formulas given in a master article.
• [3] and [4] These are two different proofs (one rather long) of a theorem that is important enough to have an actual name, the Bohr-Mollerup theorem.

The first two might have some marginal utility in WP. If there had been something like this for the Fourier transform, maybe some of the recent controversy about possible, misinterpreted, well-meaning vandalism might have been avoided. See this talk page. But these kinds of demonstrations should not be prominent in the article in any way.

The second two could arguably be merged with the article on Bohr-Mollerup, but might also sit more elegantly on their own.

If you accept this contrarian viewpoint, then what I want to know is: what is the manual of style, how should we put a ticky-mark/footnote on a formula to indicate that there is a proof/demonstration on some other page? May as well establish a convention for this now. One a related topic, I'd like to know how to mark up a single formula with a specific book (or online) reference for that one formula.

To conclude, WP will grow. Its unavoidable. It won't be an encyclopedia much longer; it will get much bigger and denser than that. It will subsume the contents of entire textbooks, it will become far more comprehensive than MathWorld. It already has done so in several areas. Neither you nor I can stop this evolution from "encyclopedia" to something vaster. I suggest only that the groundwork be laid so that there will be some uniformity of presentation. linas 03:20, 3 Apr 2005 (UTC)

I disagree that including proofs will help improve accuracy. Proofs in most areas of mathematics will usually only be comprehensible to someone who already has expertise in that area. And someone who has enough expertise to understand a proof will generally also know the correct statement of a theorem, and therefore be able to fix the error regardless of whether a proof is provided or not. Even if one doesn't know the statement, it is almost always easier to look it up in a textbook rather than wade through a difficult proof. References and citations are useful to maintain accuracy, proofs are not. Redquark 01:46, 14 Apr 2005 (UTC)

Since this seems to have turned into a talk page :(

Do you see what you are saying? Yes if you already know what you are reading you will not learn new things. If you do not already know it you had better read a book. As if they are without errors! Thus i am forced to conclude that you are recommending everyone to not read WP. Yes for some it might be too hard and for some ancient history, but there are also people who are _in between_ and who learn a lot from proofs. The proofs do not hurt anybody and they make it possible for WP to be a reference itself, like a good encyclopedia should be. This is not possible without the proofs. MarSch 13:39, 14 Apr 2005 (UTC)

Proofs can certainly be valuable information in their own right; all I said was that Linas' claim that they would improve accuracy is wrong. Redquark 00:17, 17 Apr 2005 (UTC)

Proofs are, in my experience, vital to the growth and understanding of mathehmatics. Some people may not be able to afford so many books to look up mathematical proofs(like me ;). Additionally, if I see a radical mathematical statement without proof, I would not accept it completely until a proof is found. I believe Linas is right. We can include proofs as links so that the mathematical article about it is credible, yet the article isn't full of proofs, which may be confusing to ameteurs(like me ;). JLJ 3:13, 20 Apr 2005 (UTC)

I agree with your conclusion, but from a different angle. I come to WP mostly to learn, and one of the areas in which I want to learn more is mathematics. I expect a WP article to contain enough information (either inline or via crossreferences) to allow me to learn a subject from nothing, and in mathematics the exposition of proofs is an essential aspect of that. I agree that they should be separated from the main article for a theorem if they would detract from the flow, but the proof should be understandable to someone with a basic knowledge of English and rudimentary maths, along with the text and linked (online, preferably WP) references. Hv 00:13, 17 August 2005 (UTC)

Regarding proofs and accuracy: please note that reference works can contain errors as well. For example on 11 April 2005, a young college student made changes to the article Bessel function to bring it into line with a well-known and respected work, Numerical Recipies in C: The Art of Scientific Computing from Cornell. Unfortunately, this book is in error (a tiny error) in this particular case (it must have slipped past the proof-readers). I reverted the change, and provided discussion on the talk page. I believe that having a "proof page", as described below, could provide a more permanent and reliable repository for this correction, than the talk page could.linas 00:50, 28 Apr 2005 (UTC)

I tend to agree with linas's contrarian view.

1. I don't like reading textbooks (heavy papyrus tomes, not universally accessible on the Web? c'mon that's pathetic...this is the 21st century.)
2. I don't like proofs to be forced on the reader. They interrupt the flow of the text, and though I am a math major, often I don't care how something is proved -- it's a low level detail to be apprehended as needed.
3. I would like non-trivial proofs to be separate articles in Wikipedia. Why? Let me explain. On 7 Sept 2005 I saw a formula on the Rank (linear algebra) page which I had never seen before. This formula would be useful in proving a conjecture of my own, but (a) I needed to double-check the correctness of the Wikipedia statement, to make sure it wasn't mathematical flim flam; and (b) I should've liked to have an idea of how the formula is proved, so I could generalize that idea to my own situation. The proof was not in my linear algebra book or two other linear algebra books online, and my university library didn't have the reference that User:Jitse Niesen provided for the proof. In the end, Jitse Niesen provided the proof in the Talk page. Was this proof useful for me? Yes. Required for my learning about this subject? Yes!
4. I don't think my scenario is unique. Many people don't care about proofs, but might occasionally want to check a proof for better understanding or to dispute or double-check the correctness of the Wikipedia article. As User:JLJ said, this knowledge is essential for understanding mathematics. While textbook references are fine for some people, I see no reason why Wikipedia can't be more complete and be "self hosted" in the sense that many nontrivial mathematical results are proved. Thus, I really can't see any harm in this, and it would be of great benefit to people like me. - Connelly 06:09, 9 September 2005 (UTC)

OK. The whole wiki thing says supply and demand are two aspects of what the community discussion propels. The article request page can be used to request proofs - why not? Charles Matthews 10:24, 9 September 2005 (UTC)

Prototype for proving article claims

After some discussion about whether certain formulas in an article were correct or not, and the proper means of deriving them, I was movitivated to construct a prototype experimental page demonstrating a "proof" for a claim made in an article. The goal of this page is as discussed above: to provide support for claims, without cluttering the main article itself. To see this in action, please visit the article Laplace operator (as of april 27 2005). About 2/3rds of the way down will be a formula, and to the right of the formula, in some smaller letters, a link (proof) which will take you to an article Laplace operator/Proofs which will provide the detail for the claim.

The current standing proposal is to put all article proofs into a /Proofs subdirectory, so that any such secondary, supporting material always ends up in the same location for any article, and is thus easy to find. However, this is appearently a controversial area within WP, see Wikipedia:Subpages.

I request the community to play a bit with the concept and the layout: How does it feel? Can we make this work? Can we get (proof) to sit to the far-right of the formula somehow? Replace it with a cute icon or a macro of some kind? What about the proof page itself? What should the "house style" be? Will we be able to keep proof pages from morphing into full-blown articles? Or will they morph into some other ugly beast? Give it a shot. linas 00:15, 28 Apr 2005 (UTC)

I don't like the title Laplace operator/Proofs. It it subordinates the article to another article. Certainly they should link to each other, but a reader may come to this article without having read the one whose title precedes the slash. Michael Hardy 21:21, 26 July 2005 (UTC)

Another option would be to put them in a namespace, like Proof:Laplace operator. Hv 11:06, 9 September 2005 (UTC)

A list of all such "proof" pages can now be found at Category:Article proofs. There is a template, {{proof|Article name}}, that adds boilerplate to such pages. linas 01:29, 4 August 2005 (UTC)

Cardioid

I can't say I agree with the concept of cardioid article proofs. The kind of proofs that don't belong in a main article are potentially those we don't need ...

Charles Matthews 16:32, 18 May 2005 (UTC)

Yes, well, that's the issue, isn't it? Did this article need that proof, or didn't it? Someone went through a lot of trouble to add that proof. That someone is User:AugPi; a review of his contributions shows that he works mostly in articles covering more elementary mathematics. Why did he think the proof was good for the article? What about User:MarSch and the proofs in Laplacian? Personally, I went to cardioid because I was looking for some formulas that might give isometries, and sadly found nothing. While the article gave the superficial appearance of being weighty and authoritative, on further examination, the "weight" was a proof that was rather elementary. It seemed to be a big clutter-up of the article. It occured to me that this "weight" might be discouraging other contributors from expanding the cardioid article in needed ways. So I moved the proof out of the way. FWIW, I have anti-deletionist leanings. linas 17:15, 18 May 2005 (UTC)

You both and I agree that the cardioid proof does not belong on cardioid, so let us move on to the next issue: do they belong on Wikipedia at all? I say no, proofs do not belong in an encyclopaedia, they belong in textbooks or monographs. On the other hand, I cannot get excited about it, since they do little harm if they are spun off in a different article. -- Jitse Niesen 11:33, 19 May 2005 (UTC)

Several quick remarks: 1) a separate 'proof page' can in some cases be treated as the distilled wisdom of a discussion on a talk page. 2) Some of the math on WP is highly non-trivial. Look at the proof page as a way to provide additional help for the reader, without 'cluttering' up the main article. linas 05:23, 20 May 2005 (UTC)

If the proofs prove too elementary, it may be feasible for us to state the results without proof. I just don't want this to be a precedent where possibly a large chunk of content of an article is moved elsewhere simply because of the fact that the content consists of proofs; or that "technical" detail is moved elsewhere in order to make the article more accessible to the mainstream. I have to disagree with Jitse in saying whether proofs belong at all -- I think that for the right results, and worded in a lucid fashion, proofs can contribute greatly to an article. Dysprosia 11:41, 19 May 2005 (UTC)

I did not have the intention to be as general as my message above came out. Indeed proofs can contribute to an article. As an example, the 'proof' on Maximum principle is well worth having. However, I wanted to say that the detailed proof in cardioid article proofs (this refers to the old proof which is a computation of 20-odd lines) does not belong on Wikipedia. It may be possible to summarize my view as "sketch of proof is okay, details are not," though there is certainly a grey area which calls for case-by-case decisions. -- Jitse Niesen 12:49, 19 May 2005 (UTC)

Here's another point: math topics that are transparent to Charles Matthews are often opaque to me. Similarly, I found the Cardiod article rather elementary, but I strongly suspect the person who first typeset that proof, User:AugPi, did not find it trivial. I suppose that a high school sophomore, reading the article on the Cardioid, might actually find that proof useful and insightful. Should we make it so that WP math articles are accessible only to PhD's? (No). So how can we write articles so that they are accessible to a wide audience, without cluttering them up? The proposal: move the 'clutter' to a 'proof' page. linas 05:23, 20 May 2005 (UTC)

My apologies for the misinterpretation. Dysprosia 13:57, 20 May 2005 (UTC)

very enlightening to see a second example of your proofs-in-a-separate-page-thingy. I must say that I agree that the proof for cardioid is horrible, but I would really like a not-horrible proof in the article itself. I will see if I can clean it up. -MarSch 11:55, 19 May 2005 (UTC)

have cleaned it up to 3 lines, simply by inverting the coordinate change. -MarSch 11:55, 19 May 2005 (UTC)

Ah, well, I haven't looked at your edits, but ... are you sure that was a good idea? The original proof was adequate for average 13-14 year old math students. Is your new/shorter proof still accessible to this audience? linas 05:23, 20 May 2005 (UTC)

I just looked, perfect! Now that fits the bill! linas 05:26, 20 May 2005 (UTC)

Wow! I think MarSch's edits to the cardiod proof makes it clear that User:AugPi actually struggled to find the proof, and his struggle/battle thus made him think that it was worthy of inclusion in WP. I can only conclude that a short, simple, clear proof such as MarSch's is exactly the sort of thing that can open doors for a certain class of readers. linas 05:33, 20 May 2005 (UTC)

Thank you. Yes, simple proofs is what we need. And simple implies short in my book. Linas (and others), do you still want to keep the proof in a separate page or... MarSch 16:16, 20 May 2005 (UTC)

Surely we can use talk pages for this kind of discussion and upgrading of proofs. Charles Matthews 16:22, 20 May 2005 (UTC)

Maximum principle seems to be a theorem on elliptic operators. For articles that are theorems I think we need to give a complete proof.--MarSch 16:38, 20 May 2005 (UTC)

The Feit-Thompson theorem? I don't think we need abstract criteria for adding proofs — I think any proof, like any other content, has to justify being here. Completeness is for textbook treatments. Charles Matthews 16:46, 20 May 2005 (UTC)

I am a frequent user of the mathematical articles on Wikipedia, and I believe that at least a link to a proof adds to the usefulness and authority of wikipedia. After all, articles in other areas commonly provide links or references to their sources, so why not for mathematics?

Where to place the proof is a problem, the options seem to be:

1. On the article pages themselves: this is only really practical for small elegant proofs which add to the understanding of the topic.
2. On a subpage, such as at Cardioid/Proofs.
3. On a page in a sister project such as Wikibooks or Wikisource
4. A link to an external site such as PlanetMath.

Linking to proofs on external sites obviously has its problems (such as lack of control), but i reckon options 2 and 3 could be useful, in particular I believe wikisource would be an appropriate place for proofs. -3mta3 03:26, 20 July 2005 (UTC)

Short elegant proofs surely make sence in many circumstances, if they illuminate the subject. Subpages are out of question, by Wikipeia policy. Wikibooks or Wikisource and Wikisource probably won't work, they are not meant for that. External link to planetmath would be fine too.

So I agree with items 1 and 4, and think that items 2 and 3 won't work. Oleg Alexandrov 03:30, 20 July 2005 (UTC)

Re subpages, I've asked for "special dispensation" for this special case, at Wikipedia:Subpages. My personal view is that proof pages are like glorified talk pages, or should be thought of as "sister pages" to talk pages. So I go with 1,2 or 4. Problem with 4 is that there is not enough cross-activity between PM and WP, and I don't know how to encourage such activity. Having to learn only one system is nicer than having to learn two. linas 01:37, 4 August 2005 (UTC)

Magic unrolling boxes

The French Wikipedia has an elegant answer for this problem, in the form of a box which you can click to unroll/close up an optional section. The template's called fr:Modèle:Boîte déroulante and you can see it in action at such pages as fr:À la croisée des mondes (click "dérouler") on the right-hand side). I believe it's sufficiently magical that we can't just copy the template code—it needs some kind of stylesheet upgrade for it to work—but it does seem like an ideal solution. —Blotwell 08:54, 29 August 2005 (UTC)

Don't know. I stay by my opinion that proofs are not that welcome. If using the trick above, the proof should be by default hidden, to be opened only if the user choses so. Oleg Alexandrov 15:22, 29 August 2005 (UTC)

Very nice. The more I think about it the more I like it. --MarSch 13:07, 1 September 2005 (UTC)

We will need a community wide discussion before considering on whether to adopt this. Oleg Alexandrov 16:16, 1 September 2005 (UTC)

I suggest taking another one of User:AugPi's pages (he's our main proof-provider) and mocking that up so that we can see how it works. I was unable to find "dérouler" anywhere on fr:À la croisée des mondes. Maybe this is a browser dependent thing? linas 04:48, 2 September 2005 (UTC)

While feeling quite irritable over the matter, I determined that my browser displays the lists perma-unrolled. I then tried out mozilla, and mozilla indeed rolls and unrolls these tabs; however, the rendering is totally broken. I presume you are using Opera, or MSIE? I'd prefer not to have MSIE-only markup on WP. In fact, I'd scream in agony. linas 04:57, 2 September 2005 (UTC)

FWIW, they roll and unroll nicely for me with Mozilla on Linux. Hv 12:18, 2 September 2005 (UTC)

Both Netscape 7.0 and Firefox 1.0.6 work perfectly for me. Given that there doesn't appear to be any outcry over the existing use of the template on fr:, I assume it at least works with MSIE too. So I'm curious (i) what linas's default browser is that is causing problems, and (ii) what linas means by "the rendering is totally broken [on Mozilla]". Also, it appears that User:Aoineko (mostly active on fr:) knows something about this template, so I've left a note on his talk page. —Blotwell 06:29, 2 September 2005 (UTC)

You don't, perchance, have JavaScript or CSS disabled? That would certainly kill it. —Blotwell 06:40, 2 September 2005 (UTC)

I have javascript disabled for security reasons. There is, after all, an old advisory stronlgy recommending this :-) (I work in the computer industry). When using Mozilla 1.7.8 - Mozilla/5.0 (X11; U; Linux i686; en-US; rv:1.7.8) Gecko/20050513 Debian/1.7.8-1, the text of the titles of the boxes got rendered on top of each other, causing it to be unreadable. The rollup buttons worked. My default browser is Konqueror 3.3.2. I stick to Konq mostly because its extremely fast and has a very well-designed user interface. I found Firefox to to slow and bloated, and the back button on Firefox is insanely mis-designed. (I depend heavily on the back button when editng WP). linas 14:22, 2 September 2005 (UTC)

Uh, no. Wikipedia articles should not have dancing text like this featured so prominently. We aim to be a reference work. Reference works include proofs sometimes -- if one wants to avoid a proof, one can simply scroll past it. If there are "too many proofs", then its best to judge what can be removed and what can stay. If the proofs are too complex to warrant removal, then perhaps the proof should be explained in more detail so it becomes less complex. Dysprosia 06:46, 2 September 2005 (UTC)

Well, I think the appearance of (long) proofs in WP over time is inevitable, so the issue is how to technically accomodate proofs. We have three alternatives:

• Inline proofs (must be kept short)
• Out-of-line proof pages (for longer proofs or proofs that disrupt article flow)
• Dancing text

One reason to segregate proof text from main articles is that I don't want to patrol proof text; minor edits to proofs will show up as minor edits to the article, and that may be more than I can take. Another reason to avoid dancing text is that articles will rapidly hit the 65KB limit. linas 14:22, 2 September 2005 (UTC)

It may be appropriate in some circumstances to put extremely long proofs in a seperate article. IMO, inline proofs should not be always artificially kept short, however; they should be elucidated enough so the logical outline is clearer than if the proof was kept as concise as possible. Dysprosia 02:43, 3 September 2005 (UTC)

Seems to me that linas' objections here are purely technical and both could be elegantly circumvented by transclusion of long proofs so that they are stored as separate articles but appear inline. (In case you're wondering, this would not require moving the proofs to the template namespace). This would apply equally to dancing text and to normal inline proofs, except that we are unlikely to want long inline proofs for reasons already discussed. — Blotwell 06:41, 3 September 2005 (UTC)

Transcluding proofs serves only to keep the load times down on editing the page. That's about it, though. Dysprosia 10:33, 3 September 2005 (UTC)

Well, I wonder who it is who is gagging to include these long proofs. Basically we have quite enough to do with stating the content of contemporary mathematics. Charles Matthews 13:32, 3 September 2005 (UTC)