# Talk:Pi (mathematical constant)/Proofs/An elementary proof that 22 over 7 exceeds π

## Title

Can we call this article "Approximations of Pi" and list all modern approximations; this seems awfully specific and targetted.--Robert W King 15:03, 14 August 2007 (CDT)

- Although specific, this result is quite famous and has inspired further research. Besides the paper by Lucas mentioned in the article, F. Beukers notably found a generalization that not only produces a sequence of rational approximations of π, but also provides an irrationality measure for π. Fredrik Johansson 17:02, 14 August 2007 (CDT)

I agree. Approximations to π is a far broader topic. I was so bold in the initial draft of article as to call this "startling and elegant" and I think that justifies it as a separate article. An article on approximations to π would be rather different in spirit. Michael Hardy 17:37, 14 August 2007 (CDT)

...I agree that it seems very specific, considering how many articles on broad and basic topics are not here yet. But it seems to be hoped that that last fact will not remain permanent. Michael Hardy 17:38, 14 August 2007 (CDT)

- My only objection to the title is that it
*seems*or*gives the impression*that it is an argument, or original research. I think the title should reflect something which is indicitive of mathematical correctness with respect to pi and not a point/counterpoint subject. --Robert W King 20:41, 14 August 2007 (CDT)

Certainly it is an argument. It is not "original research" in the sense of something published for the first time here, and that's clear and explicit in the fact that papers published in the 1940s are cited. I don't understand what it is in the article as it stands that gives you an impression that it's original research or that there's anything that could be called "point/counterpoint". And what's indicative of mathematical correctness is the fact that although few of us would ever think of writing down this integral and thereby discovering this argument, anyone who's had a freshman calculus course can easily check the correctness of the result. Since you mention the *title*, are you saying that is somehow not consonant with mathematical correctness? A major part of the undergraduate training of anyone who majors in mathematics is the writing of proofs and judging validity of proofs. The word "proof" is daily fare. Do you have some problem with that word? Michael Hardy 21:38, 14 August 2007 (CDT)

- There's nothing in the article that I'm concerned about; the
**title itself**is the only issue for me on the grounds that 1.) It's not very encyclopedic; 22/7 is always referred to as an "approximation" of pi. 2.) the title itself does not imply seriousness despite that it is very factual in mathematics. If we were to start using titles like "Proof that...", it would seriously inhibit our credibility; there is a certain position that proponents of this statement ("There is proof that", "I have proof".. etc) always take and I don't think we should cater. - I do not have issue with the word "Proof", I understand it's mathetical implications, but from a literary and encyclopedic standpoint I think it's a poor choice for an article title.--Robert W King 09:35, 15 August 2007 (CDT)
- For what it's worth, I agree 100% with Robert -- it just seems like a very strange *title* for an encyl. article. Maybe if it were a *famous* proof, known to any high-school student, say, the article could be called
**Peirce-King's Proof that 22 over 7 exceeds Pi**.... Right now it sounds as if it's a crankish bit of original research. Hayford Peirce 18:09, 15 August 2007 (CDT)

- For what it's worth, I agree 100% with Robert -- it just seems like a very strange *title* for an encyl. article. Maybe if it were a *famous* proof, known to any high-school student, say, the article could be called
- Well it looks as if we're bumping into the fact that non-mathematicians, as opposed to normal people, are weird---they don't know the standard usages and conventions. Michael Hardy 20:18, 15 August 2007 (CDT)

I don't understand why "Proof that..." would impair credibility. I think we may have many mathematical proof articles here eventually. On Wikipedia I created a page called proof that holomorphic functions are analytic; that's one of my favorite proofs in the theory of complex variables. What "position" are you talking about? The fact that *proofs* are a major component of mathematical practice, without which the field as we know it would not exist, must be reported in an encyclopedia. To do otherwise would be deceptive. Michael Hardy 13:15, 15 August 2007 (CDT)

- ...and now I've copied proof that holomorphic functions are analytic from Wikipedia to Citizendium.
- I think you're proposing in effect that Citizendium should eschew standard terminology used in the field. Michael Hardy 13:27, 15 August 2007 (CDT)
- Not at all, you misunderstand what I'm getting at. Despite whatever content the article has; the title which references the content is poorly or incorrectly worded. What I am suggesting is that from an encyclopedic standpoint, it would be better to rename them "Approximations for Pi" and "Homomorphic Functions" and index the content within the article in such a way that there are subheadings that illustrate these proofs in that context. I'm not suggesting anything about eschewing terminology, but I'm trying to illustrate the issue from a referencial point of view.--Robert W King 13:32, 15 August 2007 (CDT)

Your proposed titles are absurd. "Approximations to π" is a far broader topic than that of this article. This article is about one particularly striking and elegant elementary argument. "Holomorophic function" is also a far broader topic than the proof that is the content of that article. One can write a thick volume on holomorphic functions, and in fact many people have done so. This particular proof is found on one or two pages of such a volume. Michael Hardy 14:21, 15 August 2007 (CDT)

... Moreover, it is a frequent occurrence that an article in a scholarly journal that presents a proof is called "Proof that..." or "Proof of..." or "A New Proof of..." or the like. Michael Hardy 17:53, 15 August 2007 (CDT)

OK, by popular demand I've changed the title. Michael Hardy 20:19, 15 August 2007 (CDT)

Robert King, I'm afraid you are starting to try my patience. You say that people who use the word "proof" have some sort of agenda or attitude that causes you not to take them seriously. That is nonsense. Proof is an important concept in mathematics. You wrote: "there is a certain position that proponents of this statement ('There is proof that', 'I have proof'.. etc) always take and I don't think we should cater." I have no idea what "position" you are talking about. I do know that that statement is nonsense. If you will not be specific about what "position" you're referring to instead of writing about it in vague language like this, then you're just being abusive. I've been very patient with you and I'm starting to resent it. Michael Hardy 01:17, 16 August 2007 (CDT)

If I were to create an article titled "proofs of the Pythagorean theorem" or "proofs of the law of quadratic reciprocity" (those being two theorems for which the number of known proofs is very large) would I also be taking a "certain position" to which the Citizendium "should not cater"?

All high school students (except those who will never study mathematics excecpt to satisfy requirements so they can graduate) are taught what mathematical proofs are, and are told many hundreds of time "Write a proof that..." or "Prove that..." and are required to read many proofs of mathematical propositions and to learn how to judge their validity and otherwise to understand them. All undergraduates are required to read textbooks that present hundreds of theorem and accompanying proofs. All articles in scholarly journals (except in fields other than mathematics) present theorems and proofs. But we must not have an article about a proof that is identified in its title as an article about a proof, because "there is a certain position" to which "we should not cater" that is taken by everyone who says "there is a proof". And when anyone asks SPECIFICALLY what that alleged attitude is, we are told, IN FACT INCORRECTLY, the the title that calls a proof a proof is incorrectly worded. I suggest that I know what correct wording is in this field and Robert King does not. Robert King has told us that "there is a certain position" taken by everyone claiming to know of a proof, that "we should not cater to", and when asked SPECIFICALLY WHAT IT IS, declines to say, but only asserts that he has been misunderstood.

That is abuse. Michael Hardy 02:10, 16 August 2007 (CDT)

I am surprised to see that Wikipedia contains no proof that π is irrational. I am going to add an article both there and here titled **proof that π is irrational**. That fact was actually not proved until the 18th century. In the 20th century a short proof was found requiring no prerequisite knowledge beyond integral calculus. One usually sees this attributed to Ivan Niven, but a very similar proof was found, apparently earlier, by Mary Cartwright. Michael Hardy 03:02, 16 August 2007 (CDT)

- the title which references the content is poorly or incorrectly worded. What I am suggesting is that from an encyclopedic standpoint, it would be better to rename them "Approximations for Pi" and "Homomorphic Functions"

It is bizarre that someone who thinks "approximations to pi" would be a better title would preach about something being poorly or incorrectly worded. Michael Hardy 11:00, 16 August 2007 (CDT)

- Michael, for curiosity sakes I checked over at WP for this same article and discovered much of the same discussion, that from an encyclopedic standpoint, the title of the article is poorly worded. I notice that there are articles for "History of numerical approximations to Pi", "Software for Calculating Pi", "Computing Pi", which I think are all more appropriate articles that should include this particular article.
- As it stands, from an encyclopedic point of view (looking at how it might be indexed, sorted, referenced when researching these topics) it would be better placed in one of these broad articles as this article itself seems more "stub-like". Additionally, I am merely trying to convince you that it would be better to place this article as a subsection rather than as an independant entry--I am not trying to incur any kind of hostility or wrath. --Robert W King 12:37, 16 August 2007 (CDT)
- Additionally, including it in an article about Pi itself would also probably work well.

## Persistent misconceptions over the role of proofs in mathematics

Isn't there something we can do to avoid having this discussion over and over? Insisting that mathematics articles not say anything about proofs is akin to insisting that linguistics articles say nothing about words or phrases. Perhaps there would actually be more content under mathematics if we didn't waste so much time on arguments such as this one. Greg Woodhouse 12:47, 16 August 2007 (CDT)

- Greg, I'm not debating that articles shouldn't say anything about proofs. I'm just thinking that the title should be renamed or the article should be inclusive into a larger article so that it can be better referenced/indexed and does not appear to be a research paper or original research from the title alone. --Robert W King 12:55, 16 August 2007 (CDT)
- Greg and Michael, neither Robert nor I are saying that proofs don't belong in CZ articles. We're just saying that
**this particular article**should be renamed. We have given reasons for our thinking. Maybe we're right, maybe we're wrong, but we're talking about the**title**only, nothing else. Why can't we simply discuss the title without bringing in any other considerations? Hayford Peirce 13:10, 16 August 2007 (CDT)

- Greg and Michael, neither Robert nor I are saying that proofs don't belong in CZ articles. We're just saying that

- It certainly does not look like original research to me (at least not the mention of it here). For starters, anyone with a high school education will very likely have learned that 22/7 is a good rational approximation to pi. But where does it come from? It looks like a number pulled from the proverbial hat. In number theory, you learn that it is an approximant to the continued fraction expansion of pi. But is it possible to derive this same result using only elementary arguments (essentially calculus)? There is a long tradition in mathematics of attempting to establish results by elementary means that were previously only known be provable by more advanced methods. These types of results are the
*data*of mathematics, and this article reports one of those facts, much as a biology article might report that green plants are able to make use of light as an energy source. Greg Woodhouse 13:18, 16 August 2007 (CDT)

"Isn't there something we can do to avoid having this discussion over and over?" Yes. (1) If you're debating with a non-math editor, and you're a math editor, lay down the law. You're allowed. That's why we have editors. (2) If the issue is going to be controversial, even among editors, get as many math editors together as possible (from the math workgroup) and get them to sign off on a policy declaration. Both of these things take effort and gumption, but it's better than endless debating. --Larry Sanger 13:42, 16 August 2007 (CDT)

If the Citizendium ever advances beyond its present point where most links are red links and most topics that are must-haves are to be here, then, as with Wikipedia for some time now, there will be many articles on different aspects of π (look at Wikipedia's article titled "list of topics related to π"). The present topic is not one that would be obligatory in an article about π but it's also one that it would be unfortunate not to have here. The purpose of this article is *not* to answer the question of whether 22/7 is more than π, but rather to afford the insight that this particular method of proof gives. It is not about approximations to π in general, and there will probably eventually be another article about that. The title of this article is not something that anyone would search for, but potentially 10 or 20 other articles are likely to link to this one. It seems odd that someone who proposes titles as grossly inappropriate as some of the ones above would be the one to say that something looks like the work of a crackpot. I'm perfectly happy to hear of other titles, but if someone suggests that I should change the title to "space alien files paternity suit against Hillary Clinton" and then adds that the present title looks like the work of a crackpot, I'd hope that at least he'd be specific about his claims that "there is a certain position" taken by those who claim to have a proof and we "should not cater to" it. Inevitably, many encyclopedia articles will be devoted to presenting one particular mathematical proof, and their titles should be honest, descriptive, and straightforward. Where we stand, the two people who have said there's something unencyclopedic about this title still haven't even attempted to explain why. As for laying down the law, even if no such thing is done and everyone works in the same way as on Wikipedia, what will happen is that people who make claims of the kind above will get steamrollered the way they did when someone proposed deleting Wikipedia's counterpart of this article (in Janunary or February of 2006, I think?). Michael Hardy 16:19, 16 August 2007 (CDT)

## Why do we have Mathematics Editors?

I stumbled upon this page by chance, mainly because I was curious about the large number of edits on the Talk page. It is elementary mathematical knoweldge that proofs are an integral component of maths. When there are two editors here, it is absurd that the debate continues: I have had to lay down the law as the only active Economics Editor in rather more complex cases. Subject Editors decide the academic content of CZ: full stop [point, period etc]. --Martin Baldwin-Edwards 16:40, 16 August 2007 (CDT)

## Proof -- The E.B.

Just as an aside, I have the 24-volume Encyclopaedia Brit., 1941 edition. In Vol. 18, Plants to Raym, on pages 580-581 it has entries for Pronunciation, Proof ("that which establishes the truth of a fact or the belief in the truth", Proof-reading, Proof Spirit, and Propaganda. Nothing else. Throughout the volumes there are many articles with graphs, tables, charts, and complex mathematical formulae. Hayford Peirce 19:23, 16 August 2007 (CDT)

- What is your point? Greg Woodhouse 20:38, 16 August 2007 (CDT)

- I imagine it is that EB is a little weak in the field of mathematics:-)--Martin Baldwin-Edwards 21:43, 16 August 2007 (CDT)

## Separate articles about proofs?

It seems everybody agrees that there is a place in Citizendium for the information in the article, so let's not discuss that. The question is merely where to put it.

Looking at the titles that Robert and Hayford propose, they want it not in a separate article but integrated in a bigger article like "Approximations to π". As Michael said, the problem with that is that this proof will take a disproportionate amount of space in the article if we put it there (assuming that the rest of the article will be written). Nevertheless, I have sympathy for Robert and Hayford's statement that "Proof that ..." is a strange title and topic for an *encyclopaedia* article, which is a different beast than a *journal* article.

I quite like the idea proposed by Mark W Donoghoe at the forum of putting the proofs in a subpage. That's also similar to what PlanetMath does. What do you think of that? -- Jitse Niesen 22:57, 16 August 2007 (CDT)

- That is what subpages are for: good idea. --Martin Baldwin-Edwards 23:06, 16 August 2007 (CDT)

I am suspicious of "subpages". I just created the page titled proof that π is irrational. I *could* be a subpage of π. It could be a subpage of irrational number, along with various other proofs of irrationality of various numbers. Maybe it could be a subpage of reductio ad absurdum along with various other examples. We probably cannot anticipate all of the possibilities. Michael Hardy 00:26, 17 August 2007 (CDT)

This is actually important. Let us be very clear about what the function of subpages is: it is to provide a home for reference information of any sort that does not fit into an introductory encyclopedic narrative on a topic. It is up to math editors whether *in fact* proofs do not fit into such narratives. What is clear to me, in any case, is that such proofs are not *themselves* encyclopedic narratives; proofs are different items than encyclopedic narratives. So, if a proof is not itself included in other articles--such as an article *about* the proof, and which then contains the proof--then it really would belong on a subpage.

At the root of the issue here are precisely the notions that we are (1) attempting to make clear distinctions between different types of reference information, and (2) subordinating all *other* types to what I called "introductory encyclopedic narratives." Item (2) is what gives rise to Mike's objection, and it is one that I anticipated when planning out the subpage scheme. Not every subpage has a single obvious root page. But we can easily finesse this problem simply by choosing one, such as pi/Proofs, and then creating links from the other pages, such as irrational number/Proofs. Note that there is a class of subpages that are in fact sub-sub-pages. Signed Articles are an example. One creates a subpage to list all the subpages of a given type. So from the user perspective, it is very similar to presented with An elementary proof that 22 over 7 exceeds π and An elementary proof that 22 over 7 exceeds π (notice the address). --Larry Sanger 01:04, 17 August 2007 (CDT)

- I like the idea of proofs on subpages. I'm afraid that rarely the proofs are simple enough to fit in our basic article, as it is described in CZ:Article mechanics. And in general I'm afraid that not too much of math staff concerning given notion fits in the basic article *about* that notion. The encyclopaedic narrative article, as far as I can tell, basically lives on meta-level: describes/ defines briefly the notion itself, tries to explain it in layman's terms (or with the minimum prerequisites), describes its connections/interactions with other notions, applications, history, wide perspective (if applicable). It is
*about*. In no case it's meant as a training to use the notion in practice. - For example an encyclopaedic article titled "22/7 exceeds pi" (not only "A proof that..") could begin with "This is a well known/celebrated fact, discovered by... in... there are elementary proofs;" etc. I'm afraid there is nothing much to add *about* it -- and so this is hardly an encyclopaedic article. Consequently, there is no place for a proof "that 22/7 exceeds Pi" in the basic formula; proof would better go to a subpage (with the present title, though). Doesn't really matter what subpage since readers will find it via wikilinks and keywords.
- If, however, there is an interesting story about the fact and the proofs, then "22/7 exceeds Pi" article pays for its own page; a the proof might be included as its part. Basically, it's an editorial decision (but the present form of the article and arguments presented in the talk indicate that a subpage could be more appropriate). Maybe if some ideas indicated in the article were developed --what are "larger patters" suggested in the text; it is connected to what notions? this is intriguing and may be elaborated on; what are other proofs of the fact and how can they be related one to another ? Why the fact is so important/characteristic? Did it inspire some research/development?-- this article could stand on its own. However in this case I'd suggest the title "22/7 exceeds pi", since there is more to write about that fact than just about one proof.
- It was long, sorry. There is another story around problem of proofs,
~~yet to come~~. Aleksander Stos 04:21, 17 August 2007 (CDT)

- So let me continue on a more general level.
- I said I'm afraid that not much math stuff fits in the basic CZ formula. And since Britannica was mentioned... I'd venture to say that EB is not so weak in math -- from encyclopaedic point of view it just contains the important topics and treats them as encyclopaedia is expected to: narratively, not so technically. In fact, EB tries to follow our basic formula ;-) More seriously, this gives us an image where IMHO our Article Mechanics may lead to.
- My point is that the basic formula, maybe correct, well chosen and justified, is just not sufficient! It results in great general prose... which for many purposes might be next to useless. And since CZ aims to be not only a classical encyclopaedia but also a general ultimate reference source, a few days ago I started to play with an idea of "advanced presentation" *subpage*. It could be best defined by the audience it should address: the math *users* as opposed to laymans, math-educated public, perhaps young students who know already something about it and want to find a formula, more advanced students (graduate, PhD level) and, why not, professionals. Maybe these people are potentially most frequent users of our math content. An unfinished example can be seen on User:Aleksander Stos/ComplexNumberAdvanced. In such a subpage there could be much more place for proofs and computational examples (still some specific or long proofs could be best presented on a separate subpage, e.g. a proof for fundamental theorem of algebra). Although to some extent my example is redundant with the basic article, it's really shorter/compact and can/will be extended with more formal/"advanced" stuff. It addresses different public. Consider also that if we have an "advanced" presentation then we can get rid of some formal stuff from the basic article. For example, there were serious doubts by a respected editor whether the formal definition fits in Complex number. Now, no problem -- we can really have the basic article without it if it's moved to the subpage. This way we explicitly
*layer*information and everyone would know where to go. I'd appreciate very much anyone's input regarding the idea which is still under construction (if interested let's discuss it on the talk of the example, where you'll find some more background -- or on the forum). Aleksander Stos 07:23, 17 August 2007 (CDT)

OK, take a look at Wikipedia's article titled "Pythagorean theorem". I think it would be unfortunate if all of the proofs appeared only in subpages or only on separate pages. In *some* cases, it makes sense to put proofs on separate pages; that is clearly not one of those cases. In some cases (as in elementary proof that 22 over 7 exceeds π) the proof is of more interest than the fact being proved. Michael Hardy 09:46, 17 August 2007 (CDT)

- Exactly. Greg Woodhouse 10:10, 17 August 2007 (CDT)

OK, let's see WP:Pythagorean theorem. Agree that *some* proofs would fit in the main text in this case. They are just pretty simple. Above I implicitly claimed that simple proofs can make part of the main exposition (just was afraid there are not too many examples). Pythagorean theorem, as seen on WP, would IMHO look better if *some* more proofs were delegated to subpages -- one actually *is*. I'd choose maybe one or two simplest for the main text. According to subpages' model, the section about proofs would better shortly *comment* on each: its role, status, methods, complication level, field where it belongs, history, whatever. Look at some interesting comments about the "perverse" proof by differential equations (and check the author ;-)). And if e.g. Greg (below) emphasizes the pedagogical role of the project, then he could find a good occasion to add in the *main* text that the status of such proof in maths is dramatically different that other presented in this section (this is what is lacking in the WP, BTW). This would be a short *story about* proofs -- the proofs themselves would live just in neighbourhood. I'd venture to say that the main article would be even more readable like this. And with subpages we can insert definitely *more* proofs and *talk about* them. This should be satisfying those who defend them (me too, I'd defend insertion of proofs, the question is *how* do we do it).

Encyclopaedia is a *tertiary* source, while proofs are part of primary and secondary ones. We can shortly cite them or *summarise* in the main text --that's a very good and important idea!-- but putting them *in extenso* is IMHO somewhat strange to encyclopaedia. Of course, proofs are welcome, just build homes for them. BTW, above I'm just proposing a place where proofs would be more welcome directly in the "main" presentation.

To your point "*In some cases the proof is of more interest than the fact being proved*". Right. But if there is nothing much to say *about* the proof, why not put it as a properly linked subpage *anywhere* and describe the proof itself yet in more details, yet more to your liking, under the title of your choice (e.g. the initial one of this article)? This is just organisational question -- how do we think our reference source should look like at the structural level. IMHO the subpage model subordinated to a classical encyclopaedia is quite powerful. Aleksander Stos 12:19, 17 August 2007 (CDT)

## A home for the article?

So, there seems to be a sense that the article doesn't have a natural home - at least yet. Whether or not everyone agrees with this point, it doesn't seem unreasonable to say that if, at some future time, there is a more natural home for the article, it can be moved there. I'm sympathetic with both points of view here, but I'm a little surprised at all the controversy here. If you think mathematics articles exist solely to provide a handy list of formulas for reference, then relegating all proofs to subpages might make sense. But if you think of encyclopedias as pedagogical tools, then it clearly does not. If you think of James Burke's series *Connections* and compare it to this article, perhaps you'll see the point. It may be of interest that modern computers use integrated circuits, but the series doesn't merely try to inform you of this fact, but show ICs fit into the web of ideas and historical development. The whole point of this article is to draw a **connection**. Greg Woodhouse 10:27, 17 August 2007 (CDT)

- Ekhem, does anybody suggest that "
*mathematics articles exist solely to provide a handy list of formulas for reference*"? Relegating AFAIK is suggested for narrative style reason. Never mind. Some answers to your points may be found in my post above. Aleksander Stos 13:01, 17 August 2007 (CDT)

I doubt that anyone would really suggest relegating *all* proofs to subpages. And I continue to be very wary of subpages even in cases in which proofs are put on *separate* pages from the theorems that they prove. Michael Hardy 14:24, 17 August 2007 (CDT)

## Maybe this is useless, but I'm really puzzled

Jitse Niesen, who, like me, is a Citizendium mathematics editor, is telling me at CZ talk:Mathematics Workgroup that since Robert W. King and Hayford Pierce are not willing to clarify what they meant after I've asked them to do so, I should just ignore them. Robert W. King said "there is a certain position" that is "always" taken by people who say "I have a proof" or "There is a proof" and that Citizendium should not cater to that position. That of course is nonsense. Mathematicians are expected to teach standard proofs to publish novel proofs, to teach students how to go about finding novel proofs, to teach students how to judge the validity of proofs. It is commonplace to see articles in journals with titles like "Proof of a Conjecture of J. Smith" or "A New Proof of X's Theorem" or the like. It is as if he proposed that articles on architecture should not use the word "building" when an article is about a particular building, since "there is a certain position" that people who use that word "always" take and we should not cater to it. Similarly, the title of an article on music should not mention the word "symphony" when it's about a particular symphony, since that would impair the encyclopedia's credibility, since there is a certain position that people who use the word "symphony" always take, that we should not cater to, and it makes the article look like the work of a crank. Et cetera, et cetera.

So I asked Robert W. King just what that "position" is. And he won't answer!

He complains that I misunderstand; that it's the *title* that he object to (in fact, I never thought it was anything else), and he says he's not against using the standard nomenclature of the field. His two statement contradict each other. And then he proposes alternative titles so grotesquely inappropriate that one doubts he ever looked at the content of the article. I've taught mathematics at five different universities; I've attended many hundreds of seminars in which mathematicians present recent research findings; I've seen many thousands of articles in mathematics research journals and in expository journals; I have a good idea of what terminology is standard in titles and in articles.

And I treated Robert W. King respectfully by asking him to clarify his position. And he wouldn't do so.

Is it *really* possible that Robert W. King and Hayford Peirce just came to this page in order to laugh at people so gullible as to treat them with respect? I hesitate to believe that, but I don't know how else to explain what they have done. People who seem more sensible than I am tell me to ignore them. Larry Sanger tells me to lay down the law to them. Sensible, maybe, but still I'm puzzled and I'd like to understand. Michael Hardy 20:47, 17 August 2007 (CDT)

- Jitse has already summed up the position that Hayford and myself exclusively wanted to make:

*"It seems everybody agrees that there is a place in Citizendium for the information in the article, so let's not discuss that. The question is merely where to put it.*

*Looking at the titles that Robert and Hayford propose, they want it not in a separate article but integrated in a bigger article like "Approximations to π". As Michael said, the problem with that is that this proof will take a disproportionate amount of space in the article if we put it there (assuming that the rest of the article will be written). Nevertheless, I have sympathy for Robert and Hayford's statement that "Proof that ..." is a strange title and topic for an encyclopaedia article, which is a different beast than a journal article.*

*I quite like the idea proposed by Mark W Donoghoe at the forum of putting the proofs in a subpage. That's also similar to what PlanetMath does. What do you think of that? -- Jitse Niesen 22:57, 16 August 2007 (CDT)"*

*Text here was removed by the Constabulary on grounds that it is needlessly inflammatory. (The author may replace this template with an edited version of the original remarks.)*

- Additionally, Aleksander also touched on the point we were making.--Robert W King 21:06, 17 August 2007 (CDT)

For some things, putting proofs in subpages may make sense (although I would still prefer to use separate pages rather than subpages, especially since there may be several pages on quite different topcis that the proof could be a subpage of), but:

- Foregoing any proofs of the Pythagorean theorem within an article about it would be weird, and there are other such cases;
- In cases like the present one, there isn't anything for it to be a subpage of, since the technique of proof is of interest but the fact being proved is probably not worthy of its own article;
- In cases like proof that holomorphic functions are analytic, there's probably no reason to have an article devoted to the theorem itself, that the proof would be a subpage of.

*Text here was removed by the Constabulary on grounds that it is needlessly inflammatory. (The author may replace this template with an edited version of the original remarks.)* Michael Hardy 22:48, 17 August 2007 (CDT)

Let's be clear:

- "Foregoing any proofs of the Pythagorean theorem within an article about it would be weird, and there are other such cases"
- Reply: I'm not sure anyone is saying there shouldn't be a proof of the Pythagorean Theorem within the Pythagorean Theorem article (i.e., an article
*about*the theorem). Has anyone said that all proofs*must*be on proof subpages?

- Reply: I'm not sure anyone is saying there shouldn't be a proof of the Pythagorean Theorem within the Pythagorean Theorem article (i.e., an article
- "In cases like the present one, there isn't anything for it to be a subpage of, since the technique of proof is of interest but the fact being proved is probably not worthy of its own article"
- Reply: well if indeed the proof illustrates an interesting technique of proof, shouldn't there be an article about that technique? Couldn't the proof be a subpage of that page?

- "In cases like proof that holomorphic functions are analytic, there's probably no reason to have an article devoted to the theorem itself, that the proof would be a subpage of."
- Reply: OK, but does it follow from that that there is therefore no obvious place where the proof might be filed?
- replay by AS:"Proof that holomorphic functions are analytic" is a natural subpage to "holomorphic function". Proof of just one property. I see no problem with organising things like this. In fact, I see some advantages. Aleksander Stos 02:16, 18 August 2007 (CDT)

A little more of the milk of human kindness, please! --Larry Sanger 01:32, 18 August 2007 (CDT)

### Reply to one of Larry's points

- Reply: well if indeed the proof illustrates an interesting technique of proof, shouldn't there be an article about that technique? Couldn't the proof be a subpage of that page?

In this case, no. The proof naturally leads the reader to suspect that it's merely one instance of a more generally applicable pattern without actually saying what that is. And quite possibly it can be generalized in more than one way. Some work on this is done in at least one of the cited papers, but it hasn't been taken very far. Michael Hardy 20:10, 21 August 2007 (CDT)

### And another

Making this a subpage of the π page doesn't really seem to follow from the proposed rationale for subpages. Proof that π is irrational might, since it answers a question that would naturally be in the reader's mind when reading about π. This situation where the argument rather than the result is what the article is about is quite different. Michael Hardy 20:13, 21 August 2007 (CDT)

- Michael, upon looking as a math author into the article and its potential development I can agree with your points here -- exactly as I suggested before. Let's make it clear. CZ is organised so that the mainspace is filled with narratives, just like a more classical encyclopaedia. Any "attachments" go to subpages. The only question is whether this article stands on its own ("is narrative enough") or should be go to an attachment.
- As it stands the answer is not very clear but... The article *is* narrative (although IMHO just starting), and it contains not the proof *itself* but a *description* of the meaning of the proof. The proof itself is shortened to one-line math formula. I agree that the article could stand on its own, especially if someone could *elaborate* on that *really* *intriguing* pattern. Furthermore, there are some other possibilities to expand the *description* of the meaning of this proof. Therefore, this goes to the *mainspace* and should be named as it stands (notice that neither Larry nor my humble self, we never claimed otherwise). I could even suggest that upon expanding the narrative side of the article we could also expand the --elementary, BTW-- computation of the integral. The more, the merrier ;-) Depending on relative proportions of the occupied space this computation could be in the main article or go to an attachment -- so that the flow of the main text be natural. Aleksander Stos 03:29, 22 August 2007 (CDT)

I've been asked here and elsewhere why I persist in beating a dead horse. It's as if the reason I raised the topic of what I was puzzled about was to get compliance of some kind. My actual purpose was to get information---I was trying to find out just what Robert W. King meant by his comments. I'm not going to pursue that further at this point because my attempts don't seem to be working. Once upon a time I was startled to find out that some people are deeply offended by the word *counterexample*; now I wonder if the same is true of the word *proof*. Michael Hardy 21:23, 18 August 2007 (CDT)

- It appears so. Greg Woodhouse 23:04, 18 August 2007 (CDT)

I agree that not pursuing the "initial" discussion is a good solution. In fact, starting from Jitse's excellent remark on subpages, we are discussing more substantial matter (the role of proofs, organisation of math articles, subpages, general style...) and how to apply it in this particular case. And maybe even this more interesting discussion comes slowly to an end. Aleksander Stos 03:54, 22 August 2007 (CDT)

## Reviving a dead horse

I stumbled upon this article and am wondering why this is called a proof. I say this because the page contains a single statement (equation), not a proof. The single equation states that zero is less than the integral which equals 22/7 - pi.

Nowhere in the article is a proof given, just a statement of fact. Where does the Pi in the equation come from? To be a proof page, shouldn't more detail be given, leading the reader from say equation A to equation B to ... the final ah-ha moment of proof? David E. Volk 00:14, 20 March 2009 (UTC)

- I don't know whether it's a proof or not, but I do know that the very short article generated a lot of discussion, as you can see. User:Hans Adler, a math Editor, seems to be fairly active recently -- why don't you ask him what *he* thinks of it? Hayford Peirce 02:01, 20 March 2009 (UTC)

- What you were looking for, David, is given at the Student level subpage, as indicated in the article right below the equation. Remains to be discussed whether this is the best place to put it... --Daniel Mietchen 11:30, 8 July 2009 (UTC)