Talk:Pasch's theorem

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 Field:  Geometry

Cut and Paste[edit]

This stub is a cut and paste copy of the given reference. Worse, it tells nothing about the subject; it provides no context. I don't argue for deletion of this obvious copyvio; rather, an article must be written.

Why is this useful or important and in what way? — Xiongtalk* 23:38, 29 January 2008 (UTC)

This theorem by Pasch is used in Neutral Geometry ( Geometry without Euclid's Parallel Postulate ). It is a fundamental axiom for proving certain properties of line segments, and is also useful in other applications of Between-ness. There is more guidance in Greenberg's Geometry text. I will need to do a proper citation later. Kuttaka (talk) 17:49, 14 July 2009 (UTC)

I was searching for the proof, but i only found two articles: [1] and [2]. These things proove Hilbert's axioms II. 5 And not the one related to II. 4. Which one is the real Pasch theorem? This page [[3]] is also affected. --Gabor8888 (talk) 22:08, 13 February 2010 (UTC)

References[edit]

I added Coxeter (1969) as a reference, but left a {{refimprove}} tag because that source does not actually support anything in the article except the statement of the theorem. Deltahedron (talk) 21:40, 3 February 2013 (UTC)

The only other sentence in the article matches the info in the external link to mathworld. The article needs to be expended with more content, and that content will need more sources, but I can't see why the two sentences that the article currently includes ought to be specifically tagged for lacking references. — Carl (CBM · talk) 21:51, 3 February 2013 (UTC)
Fair enough. Deltahedron (talk) 21:56, 3 February 2013 (UTC)
I clarified the Coxeter reference–he only lists this result amongst several that are attributed to Pasch (via Veblen). Greenberg does not mention this result and calls what we have termed Pasch's axiom Pasch's theorem (since he can prove it from another axiom which does not appear in Hilbert). I suspect that there is no reference which actually calls this Pasch's theorem (except in the generic sense since he did prove the theorem). If I am correct, then this article would fail notability, but I won't prod it yet since there are still a few leads to track down. Bill Cherowitzo (talk) 05:00, 14 August 2013 (UTC)
According to our page Hilbert's axioms, the Pasch theorem is the axiom that Hilbert had in the early editions of the Grundlagen that was later proved to be a theorem (by E.H. Moore and improved on by R.L. Moore). Unfortunately, there is no reference given for this terminology. However, this result is notable whereas the one in this article is not (IMHO). This would certainly not be the first time that MathWorld got it wrong. I am strongly tempted to pull the current statement and replace it by this axiom turned theorem, but I really would like to see a reference that uses this terminology before I do it. Bill Cherowitzo (talk) 19:50, 14 August 2013 (UTC)
The deeper I dig into this the worse it gets. It appears that what we call Pasch's axiom has been, rather willy-nilly, called Pasch's theorem, Pasch's lemma, and Pasch's postulate in the scholarly literature. I have not yet found any reliable source that refers to the result on this page as Pasch's theorem, but there is one possibility that I still need to track down. Bill Cherowitzo (talk) 03:30, 16 August 2013 (UTC)
Today's installment. A new wrinkle in this story. In the early paper by E.H. Moore and the refined argument by R.L. Moore it is shown that the result ([ABC] and [ACD] imply [BCD]) implies Hilbert's old axiom II.4. As the result is provable from earlier axioms, they conclude that old axiom II.4 is really a theorem and not an axiom. The result in question is found in Greenberg, stated in Coxeter, appears with proof in Veblen, and of course is in the Moores' work. None of these refer to the result as Pasch's theorem. An argument can be made that either this result (12.277 in Coxeter) or Hilbert's old axiom II.4 should be called Pasch's theorem, but this terminology does not appear in any of the older sources. Bill Cherowitzo (talk) 18:47, 16 August 2013 (UTC)

Clarification needed[edit]

The lead currently begins

In geometry, Pasch's theorem, stated in 1882 by the German mathematician Moritz Pasch, is a result of Euclidean geometry which cannot be derived from Euclid's postulates.

On the face of it this is self-contradictory. There needs to be a clarification of what is meant here by "Euclidean geometry", which to me means "geometry derivable from Euclid's postulates". Loraof (talk) 15:57, 7 December 2015 (UTC)

You are right, this needs fixing. We could add the clause, "thus showing that the original set of postulates needed to be expanded" or something like that. I had not fixed this myself because I got side-tracked into the issue of who actually calls this result Pasch's theorem. Since my comments above I have tracked down a fairly authoritative source who used this term in several papers. When I asked about the origins of the term he ultimately admitted that he could find no previous citations for it. In his work, Pasch's theorem is not the result given in this article. I am pretty sure that this is another example of something that sounded right but has no actual reliable source to back it up − but I haven't been able to prove that to my own satisfaction yet. Bill Cherowitzo (talk) 17:03, 7 December 2015 (UTC)
It seems to me that this is not a "result" at all, but rather a postulate. Loraof (talk) 20:15, 7 December 2015 (UTC)
Personally, all these betweeness results, out of context, look like postulates to me. What one calls a postulate versus a result depends on the author's development of the topic, so you can't really tell one from the other without a "scorecard". If we ever do track this down, the context will have to be part of the article. Bill Cherowitzo (talk) 20:07, 9 December 2015 (UTC)