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 &pi;, but also provides an irrationality measure for &pi;. Fredrik Johansson 17:02, 14 August 2007 (CDT)

I agree. Approximations to &pi; 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 &pi; 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)
 * 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)
 * 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 &pi;" 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 &pi; is irrational. I am going to add an article both there and here titled proof that &pi; 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)


 * 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)