Talk:Parallel postulate
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Playfair's axiom has been redirected to here. The Anome 20:21 11 Jun 2003 (UTC)
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)
postulate I postulate II postulate III postulate IV postulate V
Pedantic Note
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)
Popular Culture
A good way to ruin a Wikipedia article is to show how the topic relates to what is euphemistically known as "popular culture." It is possible that contemporary "popular culture" is approaching an all-time nadir and is the most vile and decadent expression since the fall of the Roman Empire.Lestrade 15:39, 5 March 2007 (UTC)Lestrade
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)Dinonerd
It's failed the google test so I am going to go ahead and remove it. As far as I can see it's just a bit of promotion for an otherwise unnotable minor project. Whoever wrote it up could come back with some references if they wish to put it back up, but it'll take a lot of convincing to show that this film/director which so far as I can tell, is basically unknown is 'popular' or notable. --I 05:34, 8 June 2007 (UTC)
Archimedes
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)
More questions on the 5th postulate
If, as mentionned in the text, spherical, projective or elliptic geometry is allowed by Euclid 5th postulate then the theorems contained in Euclid's elements are not only valid for usual euclidean geometry but also for projective geometry. So for instance, no pythagoras theorem or anything of this kind. I never read the elements but I will be quite surprised this to be true. Can you please be precise on that point. I also checked on internet most of the things which are written are imprecise or simply wrong, as you mentionned below. MM November 2008
Afterthought: it is plausible that the statement is equivalent to the parallel axiom. For 1. in spherical geometry by two antipodal points there are infinitely many lines. 2. In the projective plane the postulate is ambiguous because the complement of a line is connected.
I strongly agree with you: see my comments below. So, mentionning spherical geometry here is a misconception. Alain Gen
Equivalence
There's a contradiction between this article and the one on Euclidean geometry. The latter says that Playfair's axiom is equivalent to Euclid's parallel postulate, and this article says that's not true. I did a lot of googling and reading to clarify this, but I'm not sure whether I've missed something and I'd like to discuss this here first before making changes.
The question is whether it follows from the other four postulates that there is at least one parallel. This articles denies this, whereas the German article on the parallel postulate explicitly affirms it.
As the article states, Euclid's Proposition 16, the Exterior Angle Theorem, plays a central role, since it is used in proving that there is at least one parallel. Cut the Knot has a clear analysis of Proposition 16 and the implicit assumption that Euclid made in proving it. (See also [1], p. 163.)
Clarification of the status of Proposition 16 is complicated by the ambiguity in the use of the term absolute geometry. The general idea is that absolute geometry is the geometry that results from Euclid's first four postulates without assuming the fifth (in fact this is how Mathworld defines absolute geometry); however, due to the recognition that there are some problems in Euclid's approach, various other axiom systems have been developed for absolute geometry -- see [2], which is the most extensive source on this I found.
Proposition 16 does not hold in elliptical geometry. Almost everyone seems to be in agreement that elliptical geometry is not an absolute geometry, but they rarely say which postulates they're assuming for absolute geometry. Indeed, many of the other axiom systems contain incidence axioms or plane separation axioms that can be used to prove Proposition 16 (see [3] ("Elliptic Geometry") and [4] (step 11)) and that are not satisfied in elliptical geometry. This does not, however, decide the question whether Proposition 16 follows from the first four of Euclid's original postulates.
Cut the Knot is the only source I found that explicitly says that elliptic geometry is an absolute geometry -- see [5]. (Interestingly, though they also say that Playfair's Axiom is equivalent to Euclid's parallel postulate: [6].) Also, this page takes the view that Proposition 16 is not a theorem of absolute geometry.
The Wikipedia articles on non-Euclidean geometry, elliptic geometry and hyperbolic geometry only talk about these geometries violating the parallel postulate; they make no statement about the other postulates, but it seems implicit that only the parallel postulate is violated.
The article on absolute geometry, which assumes only the first four postulates, mentions only hyperbolic geometry, not elliptic geometry, as an example, and states that Euclids first 28 propositions are valid in absolute geometry.
Another central question is whether Euclid's second postulate excludes elliptic geometry. (The other three of the first four apparently don't.) This depends on how you interpret "can be extended indefinitely in a straight line". One might need to understand the Greek original in order to tell whether "indefinitely" is meant here in a way that excludes great circles on a sphere that are finite but without ends. (This page takes the view that Proposition 16 is based on the second postulate.)
- Euclid uses the terms the terms "πεπερασμένος" and "άπειρος," which are usually translated as "finite" and "infinite." I'm not certain, but I believe the root of both words is the same as the root of "perimeter," i.e., they literally mean something like "bounded" and "unbounded." The second postulate reads: Καὶ πεπερασμένην εὐθεῖαν κατὰ τὸ συνεχὲς ἐπ εὐθείας ἐκβαλεῖν, which Fitzpatrick translates as "And to produce a finite straight-line continuously in a straight-line," Heath as "To produce a finite straight line continuously in a straight line." Note that it really doesn't say "indefinitely," it says "continously," συνεχὲς. Only in the fifth postulate does he use the phrase "if produced indefinitely." I don't think that's a coincidence; one main reason the ancients didn't like the parallel postulate is that they thought it was suspect to have to talk about a potentially infinite process. (See the footnote at "One reason that the ancients..." in Euclidean geometry for the source for this statement.) Re whether postulate 2 "excludes great circles on a sphere that are finite but without ends," it's a question that has bugged me as well. I think you have to stop for a moment and and consider that this case is one that occurs naturally to us, today, because noneuclidean geometry has already been developed, but it's not one that would have occurred to Euclid. Because the Elements aren't written in a formally defined mathematical language like, e.g., Tarski's axioms, you just can't always determine unambiguously what they would mean in a context Euclid didn't anticipate. A similar point comes up in postulate 3. Heath (pp. 199-200) interprets it as meaning that space is infinite, since you can draw a circle with *any* radius, with no limit on how big it can be. Realistically, Euclid's original expression of his axioms is just ambiguous, and also inconvenient if you want to do noneuclidean geometry. From a modern point of view, you'd clearly want more of a strict separation between the postulates that definitely describe absolute geometry and the ones that are definitely not absolute.--76.167.77.165 (talk) 16:21, 27 February 2009 (UTC)
To summarize, there's a strong consensus on the Net that Playfair's axiom is equivalent to Euclid's parallel postulate (see e.g. [7], [8], [9], [10] / [11]) -- but it's not clear what this equivalence is relative to. It is certain that the equivalence holds given some of the modern axiom sets used in studying absolute geometry, but is unclear whether it holds given the first four of Euclid's original postulates.
- It might be useful as a starting point to look for whether there's a consensus on the net, but that's only going to get you so far. One problem is that sources on the net are typically not as reliable as print sources. Statements like these are also likely to be context-dependent, so seeming contradictions are not necessarily contradictions. One author may be focusing on how the axioms were historically interpreted in a particular period. Another may be interested in explaining noneuclidean geometry to the general reader without getting bogged down in technicalities. Yet another author may actually have in mind a particular modern set of axioms that's similar to, but not identical to, Euclid's.--76.167.77.165 (talk) 16:42, 27 February 2009 (UTC)
Comments would be much appreciated.
Joriki (talk) 15:13, 9 January 2008 (UTC)
- Given Euclid's first four postulates as a base, Playfair's axiom is equivalent to the conjunction of Euclid's parallel postulate and its converse. I think those that claim that Playfair's axiom is equivalent to Euclid's parallel postulate are either:
- using as a base Euclid's first four postulates plus his unstated assumption (see the converse section)
- confusing Euclid's parallel postulate with some other formulation
- misreading the "if" in Euclid's parallel postulate as "if and only if".
- That said, Euclid also uses unfounded assumptions in propositions 1 and 4; I'm not sure OTTOMH how much of it depends on these. -- Smjg (talk) 22:01, 24 October 2008 (UTC)
- When you say Euclid uses unfounded assumptions, one thing you have to watch out for is that he never intended the list of five postulates to be exhaustive. They were more like a "greatest hits" list or a FAQ. There are places later on in the Elements where he essentially says, "Okay, here's this other fact that I need to bring in for this proof. It's not on my original list, but it's obvious that it's true. I just didn't list it before because it's so obvious." Re the distinction between the parallel postulate and its converse, this all depends on the question of whether two distinct lines can have two points in common (Parallel_postulate#Converse_of_Euclid.27s_parallel_postulate), which I think is logically equivalent to the question of whether two distinct lines can have a common segment (since the segment joining the two points is implicitly unique according to postulate 1). Heath gives a long discussion of this topic, pp. 196-199. His interpretation is that all the constructions referred to in the postulates are implicitly unique, and therefore postulate 2 (extending a line) does imply that distinct lines can't have a common segment. However, "This [...] assumption is not appealed to by Euclid until XI.I.")--76.167.77.165 (talk) 17:03, 27 February 2009 (UTC)
Two remarks: 1. Playfair's axiom is a proposition already considered by Proclus 2. the discussion about elliptic geometry is based on a misconception -it confuses spherical geometry with elliptic geometry. 193.50.42.4 (talk) 15:50, 24 October 2008 (UTC)Alain Gen
- Confuses spherical geometry with elliptic geometry in what way? -- Smjg (talk) 22:01, 24 October 2008 (UTC)
Sorry, I didn't answer immediately. Elliptic geometry is what you get from spherical geometry when the antipodal points get identified. In this way, the "other point" in the text is in fact the same point --there can't be two distict straight line through two distinct points (and in spherical geometry those two points are antipodal). By the way, when you identify a point with its antipodal, the quotient manifold that you get is not orientable and so the very formulation of the parallel axiom makes no sense.
"using as a base Euclid's first four postulates plus his unstated assumption":
it is the usual meaning of "equivalent" (implicitely, equivalent when the other axioms are assumed).
193.50.42.3 (talk) 13:27, 3 November 2008 (UTC)Alain Gen
Summarizing some of the points I made above, I think the upshot of all this is that you can't really get absolute geometry from postulates 1-4 as stated by Euclid, simply omitting 5. Euclid's intended interpretation of postulates 2 and 3 is quite strong, implying that distinct lines can't have more than one point in common and that space is limitless in extent, in the sense that figures can be scaled up arbitrarily. This isn't consistent with elliptic geometry. Therefore I don't think it's really meaningful to try to prove definitively whether Playfair is exactly equivalent to the parallel postulate. Their equivalence or lack of equivalence is something that can only really be meaningfully determined in an axiomatic system hasn't got the assumption of flatness of space sprinkled all over it.--76.167.77.165 (talk) 17:03, 27 February 2009 (UTC)
One of the examples under "Logically equivalent properties" (Given two parallel lines, any line that intersects one of them also intersects the other) is incorrect. If the intersecting line is not coplanar with the two parallel lines, then it need not, and in fact cannot, intersect both of them. If the intersecting line is coplanar with the parallel lines, then this example is a rephrasing of Proclus' axiom, which is discussed separately below. Rickmbari (talk) 22:11, 1 March 2010 (UTC)
Has anyone questioned the stated equivalence with the Pythagorean Theorem? The references given are highly suspect. One is a philisophical work on implication and seems to use this as an example of what equivalence means (in form only, not content). The other, in turn, refers to a Cut-the-Knot article and I have found several inaccuracies at that site in general. I don't believe that this equivalence is valid, but at the moment I am not willing to spend the time to locate the errors in the supposed proof. Wcherowi (talk) 20:29, 23 August 2011 (UTC)
Silly argument
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)
Misuse of sources
Jagged 85 (talk · contribs) 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)
Literally mean the same
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)
- 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)
Criticism
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)
Factual accuracy of the "History" section
This appears to have been lifted from [12]. 51kwad (talk) 10:10, 24 January 2011 (UTC)
- In fact the opposite, the WP stuff has been copied into the paper. 51kwad (talk) 13:03, 27 January 2011 (UTC)
Another Pedantic Note
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)
Footnote to Another Pedantic Note
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)
Work Needed
This article is in need of some serious work. The section on the converse is factually inaccurate and should be removed. The history section needs much weeding and removal of material that is not supported by the references given and where the references are of dubious value. The section on equivalences should be more than just a listing and at least one of those items should be double checked for accuracy. Also, a clear definition of "parallel lines" needs to be given and should not be confused with various properties of such lines which depend upon the geometry in which they live. The criticism section should be either moved into the history section or expanded to include other philosophical issues. Wcherowi (talk) 21:00, 23 August 2011 (UTC)
Pythagorean Theorem????
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)
Misuse of "converse"
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)