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- 1 There are no axioms or postulates in Euclid's Elements
- 2 GA Review
- 3 Using Heath's translation of axioms
- 4 Euclid and physical space
- 5 Euclidean geometry was not the only geometry "for 2000 years"
- 6 Section: Logical Basis
- 7 Pronunciation?
- 8 Euclidean geometry is an invention of historians?!? pls review
- 9 Earlier textbooks existed and were 'systematic'
- 10 "Lost Girl" writers use Wikipedia for script?
- 11 First 28 propositions
- 12 Misunderstanding or nonsense?
- 13 Broader concequences of euclidean geometry / euclids elements outside geometry
- 14 Euclid's "line" = curve or line?
- 15 Pasch's axiom
There are no axioms or postulates in Euclid's Elements
This article is factually wrong. There are no axioms or postulates in Euclid's Elements.
There are no axioms or postulates in Euclidean geometry. Every "requirement" is well defined from the first, that is, a straight line can be drawn between any two points.
Read my article:
Play with the dynamic applets:
- This review is transcluded from Talk:Euclidean geometry/GA1. The edit link for this section can be used to add comments to the review.
Hello. I am going to have to fail this article's GA nomination, due mainly to referencing issues, but also to a few other things. Here is a list of the major issues that need to be addressed:
- The main problem is that the article is severely under-referenced. There are many sections that completely lack references, and most sections have at least one paragraph that is unreferenced.
- Web references need to be formatted with a publisher and access date at the very least.
- The references section has two books listed in the bullet point part that aren't present in the in-line references. Books that aren't used for in-line references should be removed altogether or moved to a "Further reading" section.
- There are a lot of really short paragraphs and really short sections. These need to be either expanded or combined. As they are, they make the article look very choppy and harder to read.
- The Euclid proof image in the Axiomatic treatment section needs some tweaks made to its licensing.
- Can you explain what the necessary tweaks are? It's not obvious to me what the problem is.--188.8.131.52 (talk) 22:14, 25 February 2009 (UTC)
- If you go to the image page, the license says "Note: This tag should not be used anymore!", and then gives a list of licensing options to choose from to replace the deprecated tag. Dana boomer (talk) 00:13, 26 February 2009 (UTC)
- The first two images in the gallery in the Some important or well known results section need descriptions added to their image pages.
- The formatting of the image gallery in the Applications section needs to be tweaked - the last image falls off the page.
- Changed formatting. This will of course depend on your browser and screen.--184.108.40.206 (talk) 22:14, 25 February 2009 (UTC)
Once these issues are taken care of and a full copyedit is completed, please feel free to renominate this article at GAN. Please let me know if you have any questions or comments. Dana boomer (talk) 18:43, 24 February 2009 (UTC)
- Thanks for the comments! Some of these I'll fix, others I'll leave to other people.--220.127.116.11 (talk) 22:06, 25 February 2009 (UTC)
Thanks, Dana boomer, for the helpful comments on the article. I think most of them are now pretty much taken care of, with the possible exception of references (not sure if they're sufficient now), and the definite exception of the choppiness, short paragraphs and short sections. I assume the latter was mostly referring to the section on applications, and I think that's a symptom of the fact that the application section is thin, and lacks any kind of narrative thread, transitions between topics, etc. That's not something that can be taken care of with a quick fix.--18.104.22.168 (talk) 07:21, 27 February 2009 (UTC)
Using Heath's translation of axioms
I've replaced the statement of the axioms with Heath's translation. Although rewording them may help in making them a little easier to read and a little more consistent with the typical terminology used in high school geometry courses (e.g., "line segment" rather than "finite line"), I think we run into all kinds of problems with homebrewed formulations of them. E.g., there was some discussion a while back about whether Playfair's axiom was equivalent to Euclid's fifth postulate, and this hinged on the exact interpretation of what the postulates said -- but the discussion was being based on the formulation in the WP article, which differed in subtle ways from Euclid's.--22.214.171.124 (talk) 19:19, 27 February 2009 (UTC)
- Yes, there is a problem with the common assumption that Postulate 5 is in some sense "equivalent" to the Parallel Postulate, and regardless of the form of words used. The difficulty is that a negation of the Parallel Postulate does not entail a negation of Postulate 5. To amplify a little: if one were to assert, as an axiom, "there is NO parallel line in Euclidean Geometry" (ie all lines either converge or diverge, and there is no line separating the two classes, such that it neither converges nor diverges), Postulate 5 would continue to be true. And since it remains equally true, whether there is or is not such a thing as a parallel line, the two postulates are therefore logically independent and can have no necessary connection with each other. Alan1000 (talk) 03:39, 21 March 2016 (UTC)
Euclid and physical space
In the section "The 20th century and general relativity," there is an image that has a caption which contains the words: "A disproof of Euclidean geometry as a description of physical space." Is there any passage in Euclid's books where he claimed that his plane geometry is a description of physical space? I was under the impression that Euclid's geometry purported to be a pure, a priori exercise in the mathematics of forms. How can a book that starts by saying "A point is that which has no parts" be considered to be a book about physical space? Lestrade (talk) 18:40, 28 April 2009 (UTC)Lestrade
- I think you're imposing modern mathematical attitudes on the ancients. It's a very modern idea that mathematics doesn't have to describe anything about the real world.
The ancients didn't conceive of the postulates as arbitrarily chosen statements for building an arbitrary logical system, they conceived of them as statements that were obviously true, so that they could be used to prove other things that were equally true in an absolute sense, but not as obvious.--126.96.36.199 (talk) 18:51, 29 May 2009 (UTC)
- You are absolutely right; the conception that postulates can be arbitrarily chosen to build an arbitrary logical system is an entirely modern conception.
- Alan1000 (talk) 15:38, 21 March 2016 (UTC)
- Euclid would have been horrified by the suggestion that his geometry was a description of physical space. One must understand him in his philosophical context. His geometry is a blend of Platonism and Atomism, neither of which had any connection with the sensual perception of space. I would only quibble with 'the mathematics of forms'; I would prefer, 'the Forms of mathematics'!
- Alan1000 (talk) 15:38, 21 March 2016 (UTC)
The following statement: "A disproof of Euclidean geometry as a description of physical space. In a 1919 test of the general theory of relativity, stars (marked with short horizontal lines) were photographed during a solar eclipse. The rays of starlight were bent by the Sun's gravity on their way to the earth. This is interpreted as evidence in favor of Einstein's prediction that gravity would cause deviations from Euclidean geometry." is not true.
- I have read - although I cannot quote a reference - that the 1919 experimental results were later found to be within the predictable margin of error for the experiment, and thus proved nothing at all. Can anyone verify this?
- Alan1000 (talk) 15:38, 21 March 2016 (UTC)
Mathematically the Euclidean geometry is true, as well as the other 2 plane geometries are true. Mathematics itself is not about the physical world. One cannot prove or disprove mathematical theorems with experiments. What is more, here we have an axiom, and to prove an axiom is a nonsense. For this specific photography a text for a disproof of the Newton's theory of classical physics is more appropriate. But then we have to talk about physics, not about mathematics
How about a line having an infinite number of points as constituents ? this lead Zenon to postulate that movement is impossible , because to traverse an infinity of points towards your destination would take an infinity of time .Only this single paradox is sufficient to dismiss Euclidean mathematics from the true world.It does make it easier for children up to 10 years old to learn mathematics , but once they can throw a curve ball , i guess the veil is broken. —Preceding unsigned comment added by Pef333 (talk • contribs) 04:19, 7 January 2010 (UTC)
- OK, so you're saying that when your wife alters the hem on a dress, she uses the Theory of General Relativity? No, I thought not. The mathematical fact is that Euclidean Geometry is both true and valid where the gravitational field is zero, or where the effects of gravity can safely be ignored. Alan1000 (talk) 14:37, 21 March 2016 (UTC)
Also in the second paragraph of the article it is stated that there are many geometries. In fact, apart from the Euclidean geometry, there are only 2 other plane geometries, which is not many.
- To prove or disprove an axiom as a model of something physical is not nonsense. Re "In fact, apart from the Euclidean geometry, there are only 2 other plane geometries, which is not many," you're incorrect. For example, see: Euclidean geometry, hyperbolic geometry, elliptic geometry, Dehn plane, Taxicab geometry, Discrete geometry, affine geometry, projective geometry.--188.8.131.52 (talk) 18:51, 29 May 2009 (UTC)
- This is a mere quibble over terminology. There are indeed many different geometries, but they can all be classified under the general headings of "Open" or "Closed". Euclidean geometry is merely a special case of a geometry in the unique space between open and closed geometries (which means that, statistically speaking, it will be vanishingly rare in the Universe; but nonetheless valid for all that).
A final thought: suppose you leap from the fiftieth storey of a building. By the time you reach terminal velocity, Euclidean geometry will be true. Follow that to the bus stop, baby!
Euclidean geometry was not the only geometry "for 2000 years"
The statement "For over two thousand years, the adjective "Euclidean" was unnecessary because no other sort of geometry had been conceived." is obviously incorrect, since mariners have been using spherical trigonometry, which is a non-Euclidean geometry, for a considerable length of time. The thing is that I'm not entirely sure when navigation by means of spherical trigonometry was introduced as part of the standard tool-set of naval and merchant navy officers. Certainly by the middle of the 18th century, but when precisely? —Preceding unsigned comment added by Recoloniser (talk • contribs) 18:06, 16 December 2009 (UTC)
- That's an interesting point. However: (1) 2000 years after Euclid would be ca. 1700, which is earlier than you're talking about; (2) it's not so clear that people using spherical geometry understood it to be a logically independent system, or merely a system defined on the surface of a sphere embedded in a surrounding Euclidean space. For example, they might have conceived of a spherical angle as a subsidiary concept, rather than as a notion of angle that was complete in and of itself. Did anyone formalize spherical geometry as an independent axiomatic system during this period?--184.108.40.206 (talk) 02:09, 18 December 2009 (UTC)
Section: Logical Basis
This article needs attention from an expert on the subject.
I have tried to makes some sense of this section, which consisted mainly of a paragraph on Tarski that originally seemed disconnected from the subject. It appears that the logic behind Euclidean geometry has shifted focus over the years, but still remains the motivator for deep thought about what is math.
- Hi- I'm the person who wrote the original version "Logical basis" section, which has now been greatly expanded on by other people. You put the expert-subject template both at the top of the article and in that section, but I think it only belongs in that section, so I've removed
- - on whose authority? Mount your argument first, and remove later, if no adequate counter-argument is forthcoming. Nobody appointed you Commissar of Correct Thinking.
the one at the top of the article. Usually the expert-subject template is used in cases where the article appears to contain wrong information, -
- who defines "wrong information?"
- and isn't going to be substantially correct unless an expert works on it; is this your opinion about the current version of the "Logical basis" section? I would consider myself at least somewhat of an "expert" on Euclidean geometry. (I have an undergraduate degree in math and physics, PhD in physics.)--220.127.116.11 (talk) 00:46, 2 July 2010 (UTC)
- The appeal to institutional authority is the badge of fascism. Try addressing the arguments instead.
- My view of this section is that it could use some mature perspective from someone versed in the logical developments and historical events involved in the progression of geometry from Euclid, through Gauss, through Hilbert, Tarski and through the more recent developments in constructive type theory. I have put together some treatment of these matters, but it is an amateur effort. Brews ohare (talk) 01:18, 2 July 2010 (UTC)
- Presenting Tarski's formalism for Euclidean geometry is ill-motivated in this context. Focusing on the Hilbert/Brikhoff program,and the development of neutral geometry is more in the spirit of what this section seems to be for. That is to say how Hilbert and others developed neutral geometry and essentially worked until they couldn't do anything with invoking the parallel pos in some form. The text "Roads to Geometry" by Wallace, and West presents this approach. There is also a wealth of finite geometries whose study was prompted by the question of independence of the parallel pos. DifferentiableF (talk) Jul 10, 2010, 3:13 AM —Preceding undated comment added 07:14, 10 July 2010 (UTC).
- Wiki, say something to convince me that you are not an academic joke. I've gathered that you forbid original thinking, and you forbid quoting from original sources. Thus far your anti-intellectual credentials are well established. Does this explain why Wikipedia is never cited on any reputable scientific site?
Perhaps the pronunciation of 'Euclidean' should be added to the article - I had to look this up elsewhere (<http://dictionary.reference.com/browse/Euclidean>). --18.104.22.168 (talk) 19:13, 29 September 2010 (UTC)
Euclidean geometry is an invention of historians?!? pls review
Euclidean geometry is an invention by young acient historinas in the 70s. "They teach Davenportian geometry in high schools now, though of course they call it Euclidean", accordingt to Professor Gene Haddlebury (and redisributed by the University teacher mr. Blank (M.A) at the Univerity in Mainz.) Source: http://www.theonion.com/articles/historians-admit-to-inventing-ancient-greeks,18209/) —Preceding unsigned comment added by 22.214.171.124 (talk) 08:19, 31 January 2011 (UTC)
- Please stop adding this joke to the article, as theonion.com produces satire. DVdm (talk) 08:22, 31 January 2011 (UTC)
Earlier textbooks existed and were 'systematic'
According to Heath in  vol. 1 p. 117 Aristole mentions as examples in his text numerous geometric propositions and definitions which Heath belives come from a textbook used by the Platonic Academy written by Theudius of Magnesia who studied closely at the Academy along Eudoxus.
-  The Thirteen Books of Euclid's Elements, translation and commentaries by Heath, Thomas L. (1956) in three volumes. Dover Publications. ISBN 0-486-60088-2 (vol. 1), ISBN 0-486-60089-0 (vol. 2), ISBN 0-486-60090-4 (vol. 3)
- But the article doesn't claim that no earlier, systematic textbook existed...?--126.96.36.199 (talk) 01:48, 5 January 2012 (UTC)
"Lost Girl" writers use Wikipedia for script?
HA! I noticed something while watching Lost Girl... at 30min, 45seconds in on Season 2, Episode 16, Kenzie WORD FOR WORD repeats part of this article! Not while looking at Wikipedia or using any computer either, she repeats this part:"Euclid's axioms seemed so intuitively obvious that any theorem proved from them was deemed true in an absolute, often metaphysical, sense" part. Just popping in to pass on that discovery. TinyEdit (talk) 01:15, 9 February 2012 (UTC)
- Curious, but not really something we can include. It doesn't tell us anything about the theorem (or at least no more than however many textbooks on it that we also don't mention). It's just someone using WP, as people are quite legally allowed to do, and which happens more and more. I from time to time use images from WP in my blog. Other people mirror WP wholesale, or copy and paste it into books they then sell as their own work.--JohnBlackburnewordsdeeds 01:35, 9 February 2012 (UTC)
First 28 propositions
The article claims that "the first 28 propositions [Euclid] presents are those that can be proved without [the 5th postulate]." I take this to mean that they can be proved from the first four postulates. However as pointed out at this page, "The first 15 propositions in Book I hold in elliptic geometry, but not [Proposition 16]." (Consider a triangle whose interior angles are each greater than a right angle, in which case its exterior angles are each less than a right angle.) Unless Postulates 1-4 somehow rule out elliptical geometry I don't see how Proposition 16 could follow from them. If 1-4 do achieve this I'd be fascinated to see a reputable source with a proof to that effect.
- Absolutely. Such a claim appears to me to be fantastical. To begin with, the first 23(?) propositions are offered as definitions (or stipulations), and are therefore in no need of proof (at least within the context of Euclidean geometry).
Incidentally there are restatements of Postulates 1 and 2 that rule out elliptical geometry. The requirement of uniqueness for Postulate 1 would achieve this, as would the requirement for Postulate 2 that no extension ever self-intersect, but that's not how they're worded. --Vaughan Pratt (talk) 04:38, 17 August 2013 (UTC)
- You do not need to restate the postulates. Postulate 2 as originally stated rules out elliptic geometry since it can be used to prove that there exist parallel lines -
- This would astonish me greatly. Please develop your arguments?
- (you only need postulate 5 to provide uniqueness in the Playfair formulation) -
- i'm afraid you will need to explain what this means. You seem to be coming dangerously close to asserting that on the basis of some axioms, it is possible to assert additional facts about other axioms. This is a logical minefield. Since the definition of "axiom" precludes logical relationships with other axioms, you will need to develop your arguments.
- and elliptic geometry has no parallel lines. Postulate 3 also needs to be modified before it can be used in the elliptic geometry setting. Part of the problem here is that Euclid's postulates are not sufficient to give Euclidean geometry (this has been known for over a century) and one must work with a complete set of axioms rather than attempting to repair Euclid's. Neutral geometry (in which the first 28 propositions of Euclid are valid) is obtained by removing the parallel postulate from a complete set of axioms for Euclidean geometry, such as Hilbert's axioms. Bill Cherowitzo (talk) 18:20, 17 August 2013 (UTC)
- You seem to contradict yourself. On the one hand you say the postulates don't need to be restated. On the other you say that you need to start from neutral geometry. How is that not a restatement?
- Since elliptical geometry is a model of the five postulates as originally stated, it is not possible to prove the existence part of Playfair's formulation from them. The original statement of Postulate 2 merely promises the ability to produce a finite straight line continuously in a straight line, which clearly holds on the sphere unless Postulate 2 is strengthened in some way. Furthermore nothing in Euclid's proof of Proposition 16 uses anything to do with such a strengthening, so it's clear Euclid simply made a mistake in his proof, thereby invalidating all subsequent propositions that depend on Proposition 16. --Vaughan Pratt (talk) 16:29, 24 August 2013 (UTC)
- As I've said, elliptical geometry is NOT a model for Euclid's five postulates. This has been proved over and over again, dating back to Omar Khayyám, Girolamo Saccheri and J. H. Lambert. The main problem is Postulate 2, which as stated is ambiguous in meaning. The two interpretations are 1) straight lines are infinite in extent (length) or 2) straight lines are boundary-less. Of these, only interpretation 2 works on a sphere, -
- Please expound; meaning not entirely clear.
- while Euclid implicitly uses interpretation 1 in his proof of Prop. I, 16. (See, Faber, Foundations of Euclidean and Non-Euclidean Geometry, 1983, Marcel Dekker, p.113) It was Riemann in 1854 who clarified these distinctions and showed that they are not equivalent. Another problem that arises is that you can not make Postulate 3 work on a sphere. Bill Cherowitzo (talk) 18:00, 24 August 2013 (UTC)
- In logic there is no model-theoretic difference between "more ambiguous" and "weaker" since both mean "admitting more interpretations". You appear to be agreeing with me that Postulate 2 (as worded in the article) must be strengthened before it can be used to prove Proposition 16. The article itself says as much when it points out in Section 7 that postulates 1-4 are "consistent with either infinite or finite space (as in elliptic geometry)". The article is therefore incorrect when it claims without qualification that this and later propositions can be proved from the postulates listed in the article. --Vaughan Pratt (talk) 19:02, 25 August 2013 (UTC)
- I thought that I was fairly clear, but it does look like I have to spell it out. Postulate 2 had only one meaning for Euclid and for two thousand years this was the way it was interpreted. Interpretation 1 is a fairly modern point of view due to Riemann. You are not allowed to pick and choose which interpretation you wish to use to make your arguments. If you go with interpretation 1 you do not get Euclidean geometry nor Elliptic geometry (without making other changes). With interpretation 2 you get Euclidean geometry and not Elliptic geometry. You are being selective in pointing out what section 7 says by leaving out the condition that says in effect, "When interpreted as a basis for physical space, ...". This statement shows that what follows is not mathematics, which does not involve itself with such interpretations of axioms. This was probably written by a physicist and is only consistent with interpretation 1. The parenthetical remark is clearly false. While this discussion has been fun, it is now beyond the realm of discussing how to improve this article, and so is no longer appropriate for this talk page. If you have a reliable secondary source for your point of view we can discuss its merits, otherwise I bid you farewell. Bill Cherowitzo (talk) 01:47, 26 August 2013 (UTC)
- You're perfectly clear, but that wasn't my complaint, which is that the article is not at all clear on this point of interpreting Postulate 2.
- It is not "my point of view" that Postulate 2 as stated admits multiple interpretations not all of which suffice for Proposition 16, it's a matter of established historical record, as you've pointed out yourself. Furthermore the article says nothing at all about this, an obvious shortcoming of the article which could therefore benefit from the appropriate improvement.
- I would however be very interested to see a reliable source for "Postulate 2 had only one meaning for Euclid and for two thousand years this was the way it was interpreted." I'm not sure what you're saying here. Are you claiming that Postulate 2 admits only two interpretations (I can think of others, one of which I've already mentioned above), and that Euclid picked the one that makes his proof sound? Euclid's proof of Proposition 16 is clearly unsound as it stands, since nowhere does it use any details of Postulate 2 that might make it sound, no matter how interpreted. (Whereas one does not need to "pick and choose" an interpretation in order to make this proof-theoretic argument, one does however need to pick one in order to prove Proposition 16.). It follows that Euclid simply overlooked this problem altogether. The prima facie evidence for this is that if he'd noticed the problem his proof would look different. --Vaughan Pratt (talk) 00:02, 27 August 2013 (UTC)
Misunderstanding or nonsense?
"Einstein's theory of general relativity shows that the true geometry of spacetime is not Euclidean geometry"
(A) How (exactly) can a theory (i.e. something theoretically) prove or disprove something which is part of physical reality? (B) If there exists a true geometry, could there be some other?
What did the author want to tell me? Maybe that some (i.e. one of may, not the) noneuclidian geometry (which is not(!) a Einsteinian but, if so, a Minkowski-Geometry) is much more suitable for the scientific description of space and time while presenting a uniform approach. Maybe. — Preceding unsigned comment added by 188.8.131.52 (talk) 17:45, 1 January 2015 (UTC)
Broader concequences of euclidean geometry / euclids elements outside geometry
The concequences of Euclidean geometry / Euclid's Elements were mutch broader than just geometry. For example it had also influence on philosophy via the so called geometrical method used by for instance Hobbes and Spinoza.
Please weigh in here: Talk:Line (geometry)#Line = fr:droite (not fr:ligne). Thanks. Fgnievinski (talk) 04:34, 30 June 2015 (UTC)
- Your request is not clear. There are many two dimensional geometries, the Euclidean plane being one of them, but what do you mean by "plane geometry"? Bill Cherowitzo (talk) 23:34, 25 January 2016 (UTC)
- Most mathematicians understand what "plane geometry" refers to. If no one is willing to write the article, the redirect should not exist. 184.108.40.206 (talk) 23:45, 25 January 2016 (UTC)
- Essentially it is just geometry over R2. Plane (mathematics) 220.127.116.11 (talk) 00:43, 26 January 2016 (UTC)
- That is Euclidean geometry studied by means of coordinates over the reals. Perhaps you are thinking that Euclidean geometry is the geometry given by Euclid's five postulates. This is not the case (but you have to dig fairly deeply into this article to discover that), those axioms are not sufficient to prove all the theorems in Euclid's Elements. A complete set of axioms needed to do this, such as Hilbert's axioms has over twenty items. There are other sets of axioms that also give Euclidean geometry, each with particular strengths and weaknesses (see Foundations of geometry). Birkhoff's system is the simplest and gets the reals involved immediately. However, this system is a bit deceptive since you are assuming all the properties of the reals and this makes things easy to prove because that is a very powerful assumption. Pasch's axiom, for instance, is easy to prove given the properties of the real numbers, but if you don't have that then it must be taken as an axiom, as Hilbert does. The redirect seems appropriate to me, since most people who use the term "plane geometry" are thinking about the Euclidean plane, R2. What you may be thinking about is studying this geometric object analytically (via coordinates) versus studying it synthetically (via axioms). Note that there is also a disambiguation page Plane geometry (disambiguation) that will lead you to pages with other meanings of "plane geometry". Bill Cherowitzo (talk) 04:11, 26 January 2016 (UTC)
- I am thinking to link plane geometry to plane geometry (disambiguation) (or rename the latter) and make of plane geometry more a kind of referal/general information article on 2 dimensional geometry (mostly linking) to other articles. WillemienH (talk) 09:48, 26 January 2016 (UTC)