Talk:Parallel postulate

message: The following sentence is not true.  It's true for hyperbolic geometry, but not elliptic geometry.
:The parallel postulate is the only postulate of Euclidean geometry which fails for non-Euclidean geometry.
I'm removing the image, because it displays the Corresponding Angles Postulate, not the Parallel Postulate.--DroEsperanto 01:01, 24 June 2006 (UTC)
thread_title: Untitled

message: It's a bit of a nitpick, but theorems aren't "proved".  They're "proven".  Saying that a theorem is "proved" rather than "proven" is like saying that a toaster is "broke" rather than "broken".--Flarity 06:14, 23 October 2006 (UTC)
thread_title:  Pedantic Note 

message: I looked on google & aol search and couldn't find 1 reference for that film, so i'm not to sure it 
even exists.No references for the director either Dinonerd 17:39, 5 June 2007 (UTC)
thread_title: Popular Culture

message: I tagged the statement by Archimedes due to concerns about its accuracy.  The full list of his treatises is given in the article Archimedes, and none is called On Parallel Lines.  Some clarification is needed here or this information may be removed. --Ianmacm 18:13, 12 August 2007 (UTC)
thread_title: Archimedes

message: I remember a lot from my high school geometry class, and I really loved that class, but looking at the definition now, it seems strange to me that this should be called the parallel postulate, as it doesn't really define parallel lines, but rather defines non-parallel lines.  Also, I'm very sorry for just rambling, but I remember from my logic class, that given a statement, that the inverse or the converse are not always necessarily true.  Please correct me if I am wrong.  I realize that this may be the base for some of the non-Euclidean geometries, that given this postulate it must be proven otherwise that the inverse (if the sum of the interior angles are not less than two right angles, then the lines will not meet if extended to infinity) is also true, and cannot simply be taken as an assumption.  I'm no expert, and I have a lot to study yet about mathematics, but I was wondering if anyone else has some thoughts on this. Traveling matt (talk) 23:50, 16 March 2010 (UTC)
thread_title:  Silly argument 

message: is one of the main contributors to Wikipedia (over 67,000 edits; he's ranked 198 in the number of edits), and practically all of his edits have to do with Islamic science, technology and philosophy. This editor has persistently misused sources here over several years. This editor's contributions are always well provided with citations, but examination of these sources often reveals either a blatant misrepresentation of those sources or a selective interpretation, going beyond any reasonable interpretation of the authors' intent. Please see: Wikipedia:Requests for comment/Jagged 85. That's an old and archived RfC. The point is still valid though, and his contribs need to be doublechecked. I searched the page history, and found 5 edits by Jagged 85 in October 2008. Tobby72 (talk) 19:49, 15 June 2010 (UTC)
thread_title:  Misuse of sources 

message: In section "Logically equivalent properties", these two statements:
:11. Given two parallel lines, any line that intersects one of them also intersects the other.
:15. Proclus' axiom, which states "if a line intersects one of two parallel lines, both of which are coplanar with the original line, then it must intersect the other also", is also equivalent to the parallel postulate.
should literally mean the same. I suggest one should be removed. --210.240.195.212 (talk) 11:45, 10 December 2010 (UTC)
replies: text: Axiom 11 is not even true in three dimensions, whereas 15 is a fair representation of the correct axiom. this section is muddled in that 11 refers only to two dimensions whereas 15 would be true in n-dimensional euclidean space. This needs to be made clearer. 51kwad (talk) 09:47, 24 January 2011 (UTC)
thread_title: Literally mean the same

message: I propose that the Criticism section be removed. It is now universally acknowledged that the parallel postulate is independent of the other postulates, and thus can be added or negated as a consistent additional axiom. The argument from direct perception is blatantly false in a world where General Relativity holds and where the parallel postulate is untrue. 51kwad (talk) 09:43, 24 January 2011 (UTC)
thread_title:  Criticism 

message: This appears to have been lifted from . 51kwad (talk) 10:10, 24 January 2011 (UTC)
replies: text: In fact the opposite, the WP stuff has been copied into the paper. 51kwad (talk) 13:03, 27 January 2011 (UTC)
thread_title:  Factual accuracy of the "History" section 

message: Traveling Matt is absolutely correct.  Euclid's Postulate 5 is not synonymous with the "postulate of the parallels";  the latter was a separate inference.  Postulate 5 deals explicitly, specifically and literally with internal angles which add up to LESS THAN two right angles.  This clearly rules out parallelism - a point which (if I may echo Russell), with the exception of Matt, has proved too subtle for the philosophers and mathematicians to grasp.
In Euclid's time, mathematicians were fully apprised of the special philosophical difficulties surrounding the concept of parallelism.  Euclid appreciated that geometry as he understood it depended upon a "postulate of the parallels", but he wanted to short-circuit the philosophical difficulties as much as possible.  So he asserted as much as he dared about parallels in Definition 23, and in Postulate 5, said only as much as would enable the reader to INFER a Postulate of Parallels.  This was philosophically dishonest, but what was the guy supposed to do???
Incidentally, in the context of classical Greek mathematics, there is a world of difference between a "postulate" and an "axiom".  Euclid's postulates are not axioms (as he understood the word).  In fact there are, strictly speaking, no "axioms" in Euclid.
Alan1000 (talk) 15:09, 1 February 2011 (UTC)
thread_title:  Another Pedantic Note 

message: On consideration, I can't resist adding another couple of arguments to show that Postulate 5 is not equivalent to a 'Postulate of the Parallels'.
(1) If it is thus equivalent, then Definition 23 is redundant. Euclid would have realised that (he was quite a bright chap); that he chose to leave Definition 23 in place is, prima facie, evidence that he himself did NOT regard P5 and PP as logically equivalent.
(2) Every conceivable example, or manifestation, of Postulate 5 may be understood in terms of finite distances (since the lines must intersect somewhere). However, the case of parallel lines cannot be understood solely in terms of finite distances (or even 'indefinite' distances, by which Euclid means no more than 'any and all distances you can imagine'). So PP imports something new which was not present in P5. This shifting of the logical goalposts is something else which seems to have slipped under most commentators' radar.
I am not denying, obviously, that P5 irresistibly suggests PP. But that's just a matter of overwhelming psychological plausibility which, as we both know, dear reader, is the truth-seeker's worst enemy. Unless, of course, it's an axiom.
I refrain from making any alteration to the first sentence of the main article, even though I consider it flawed, because I am not a mathematician by training. I strongly believe that only those with the appropriate level of academic expertise should make such alterations.
Alan1000 (talk) 16:00, 3 February 2011 (UTC)
thread_title:  Footnote to Another Pedantic Note 

message: Perhaps I was a little too vague in the previous section, so let me be more precise. It is stated – with two citations – that the Pythagorean theorem is equivalent to the parallel postulate. Having never heard this before I decided to check the references. The Weisstein reference seems to be based on an unpublished, unrefereed paper by Brodie which appears on the cut-the-knot site (not a place that I have found to be all that reliable). The second citation is to Pruss who wrote a philosophy (not mathematical) book in which this statement is made as an example of what he means by logical equivalence. There is no proof there ... I am sure he just made this up so that he didn't have to say "If X can be proved assuming Y, and Y can be proved assuming X, then X and Y are logically equivalent." These citations make me think that this claim is bogus. I could be wrong. If someone has a real reference I'd like to see it. If someone would like to go through Brodie's paper and verify it, that would also be nice - but I am not willing to spend the time to do it myself. Bill Cherowitzo  (talk) 03:37, 1 October 2011 (UTC)
replies: text: Bill its mentioned here as a consequence of the parallel postulate  I'll look for somthing better J8079s (talk) 21:12, 23 August 2012 (UTC)
thread_title:  Pythagorean Theorem???? 

message: The section regarding the converse of the given parallel postulate does not make mention of the converse.
The parallel postulate is given here, loosely, as "if not right angles, then not parallel"  The converse is "if not parallel, then not right angles."   The statement offered in the section titles converse is not that one, and it's killing me.  Somebody please fix this.  — Preceding unsigned comment added by 70.171.24.73 (talk) 03:46, 10 March 2012 (UTC)
thread_title:  Misuse of "converse" 

message: What is the purpose of the section "Converse of Euclid's parallel postulate"? It looks like nonsense to me.
"The Elements contains the proof of an equivalent statement (Book I, Proposition 27): If a straight line falling on two straight lines make the alternate angles equal to one another, the straight lines will be parallel to one another. As De Morgan[19] pointed out, this is logically equivalent to (Book I, Proposition 16)." This is completely unrelated to the parallel postulate. Both of these are true in neutral geometry. 
"Euclid did not postulate the converse of his fifth postulate, which is one way to distinguish Euclidean geometry from elliptic geometry" This is misleading. The 5th postulate is the way to distinguish between Euclidean, Elliptic, and Hyperbolic geometry. It also mentions changing other axioms, which then gets into not Euclidean geometries which are neither Euclidean nor non-Euclidean.
Jbeyerl (talk) 01:12, 16 July 2014 (UTC)
replies: text: This pretty much depends on what you think the fifth postulate is saying. Euclid's formulation is a bit awkward to work with. In the Playfair version, the fifth postulate is saying that there is at most one parallel to a given line through a non-incident point. The converse statement is that there is at least one. As you say, this converse is true in neutral geometry ... parallels exist in neutral geometry. Logically unrelated, yes, but most students think of Playfair's axiom as an if and only if statement (and many authors present it that way). Making this statement explicit is a good way to combat that. As to distinguishing Euclidean geometry from elliptic ... it is the converse which does this. Euclidean geometry, being a neutral geometry, has parallels while elliptic geometry does not and so can not be a neutral geometry. Saying that there is at most one parallel in this Playfair form does not distinguish these two geometries (but it does distinguish Euclidean from hyperbolic, both of which are neutral geometries and so have parallels.) As to changes in other axioms ... elliptic geometry is non-Euclidean (by definition), but it is not a neutral geometry, so changes to axioms other than the fifth must occur if you want to get elliptic geometry. Bill Cherowitzo  (talk) 03:47, 16 July 2014 (UTC)
thread_title:  Converse of Euclid's parallel postulate 

message: The article claims that this is equivalent to the parallel postulate.
"5.There exists a pair of similar, but not congruent, triangles."
This axiom is easily proven by extending two sides of a given triangle away from the point at which they meet, and then using SAS similarity.108.85.152.134 (talk) 19:45, 9 February 2016 (UTC)
replies: text: Afraid not. This can be tricky business. If you are assuming SAS similarity as an axiom, then it is equivalent to the parallel postulate. If you are proving SAS similarity, you need the parallel postulate to do so. In hyperbolic geometry, where the parallel postulate is not true, if two triangles have the same angle measures, then they are congruent, i.e., the corresponding sides have the same lengths. In that geometry the only way that two triangles can be similar is if they are congruent. Bill Cherowitzo  (talk) 21:47, 9 February 2016 (UTC)
thread_title:  Already proven axiom 

message: Substituted at 02:26, 5 May 2016 (UTC)
thread_title: Assessment comment

message: I don't know how this page got on my watchlist; I am interested in the parallel postulate but am not an expert.  I just reverted revision 766900271 by 2A01:E35:2E78:9EA0:1493:7C48:B7F1:D585 (talk) because matworld.wofram.com claims it is equivalent and because an IP editor should at first raise such an issue in the talk page.  I will let those more qualified than I consider this editor's suggestion.----Guy vandegrift (talk) 22:13, 22 February 2017 (UTC)
replies: text: I too am interested in this ... see my remarks above (Pythagorean theorem ???). The wolfram claim is based on Bodie's article at cut-the-knot, so it isn't an independent source. I would hope to hear more from the IP editor about who the "genuine geometer" is and what was the uncovered error. Even if Bodie's proof is wrong, that does not mean the result is false, but it would strengthen my feelings on the matter.--Bill Cherowitzo  (talk) 00:52, 23 February 2017 (UTC)
replies: text: Snooping around the internet I found a student's term paper that points to page 128 of this  book that claims an equivalence of the 5th postulate to Hilbert's parallel postulate: Greenburg, Marvin J. Euclidean and Non-Euclidean Geometries: Development and History. W.H. Freeman and Company. --Guy vandegrift (talk) 15:36, 23 February 2017 (UTC)

text: Greenburg only shows that the parallel postulate implies the pythagorean theorem (a fact known to Euclid!) and does not give an equivalence. The student's work, showing that the pythagorean theorem does not hold in hyperbolic geometry (and I haven't looked at it carefully to even guess as to whether or not it is valid) would not give the equivalence she claims even if the argument is valid (you would need to show that the pythagorean theorem did not hold in any geometry in which the parallel postulate fails.)--Bill Cherowitzo  (talk) 19:54, 23 February 2017 (UTC)
replies: text:  Given absolute geometry  any statement true in euclidean geometry but false in hyperbolic geometry can serve as replacement of the parallel postulate see http://math.stackexchange.com/q/906918/88985answer by Julien Narboux http://math.stackexchange.com/a/1277051/88985 refers to: Wanda Szmielew. Elementary hyperbolic geometry. In P. Suppes L. Henkin and A. Tarski, editors, The axiomatic Method, with special reference to Geometry and Physics, pages 30–52, Amsterdam, 1959. North-Holland. WillemienH (talk) 23:15, 23 February 2017 (UTC)

text: Hmmm. My bad, and I have struck out the offending comment above. In any real discussion of the equivalence of theorems, the axiomatic context has to be taken into account. These two statements are not logically equivalent since there are examples of geometries where one holds and the other doesn't, but if you restrict to the realm of Euclidean and hyperbolic geometry, then the above argument is valid and you get the equivalence (in context). I don't usually restrict my geometric viewpoint in this way and that caused me to overlook this fact. The Euclidean setting is the natural one for this question, but I now find the following question to be more interesting. What is the largest class of geometries in which the two statements are equivalent? Is this just a Euclidean geometry property or can the axiom set be loosened to permit more general settings for which the equivalence is true? These are not questions for Wikipedia, but I did want to point out why I might have fallen into this error. --Bill Cherowitzo  (talk) 17:21, 25 February 2017 (UTC)
thread_title:  I reverted an IP edit that "felt" wrong to me.  An expert should verify. 

message: This could be better formulated.  A postulate is, after all, an obvious, or even a counter-intuitive (such as that upon which non-Euclidean geometry is based), statement for which there is no proof.  There is a much better discussion of this in Heath, who lays out the history of both ancient and modern mathematicians who attempted to prove this postulate, and an acknowledgment of Euclid's genius in recognizing that it could not be proved.  I think a short discussion of this history would improve the article. Contraverse (talk) 14:57, 19 April 2022 (UTC)
thread_title:  "The postulate was long considered to be obvious or inevitable, but proofs were elusive."