# Euclid's 5th Postulate Proven? What?! **URL:** \ **Category:** Factual Questions\ **Created:** [November 29, 2004, 4:05am UTC](https://boards.straightdope.com/t/euclids-5th-postulate-proven-what/277088 "2004-11-29T04:05:35Z")\ **Posts on this page:** 20\ **Page:** 2
**Author:** ![Giles](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/giles/32/60_2.png) [@Giles](https://boards.straightdope.com/u/Giles)\ **Post date:** [November 29, 2004, 2:46pm UTC](https://boards.straightdope.com/t/euclids-5th-postulate-proven-what/277088/21 "2004-11-29T14:46:01Z")
> [@Mathochist](#): > > You aren’t taking this seriously, are you? > > Look, you can build models of any geometry with an altered version of the fifth from a Euclidean one. You can build a model of Euclidean from foundations. Drop by Dunham Labs (in beautiful New Haven, CT) and I’ll whip it out in about an hour or two on the board. > > What this means is that you can build a model for any geometry with the first four postulates and any variant of the fifth directly form set theory and algebra (using intuitionistic logic, even). A proof of the fifth postulate from the first four would, as you say, disprove itself as well. This would mean that no models are possible. This would mean that there’s something wrong with the _foundations_, and _all_ of mathematics would be worthless. No, I think arithmetic and set theory would be safe. It just means there’s something wrong with the first four Euclidean postulates. But no, I’m not taking this seriously, because if the claim is true it’s more than just proving some old theorem that no one was clever enough to prove before. It means that the foundations of geometry, and hence most of the foundations of mathematical physics, are at risk. It is possible for people to find problems in the foundations that have not been seen before, but I don’t think that it’s happened here. ---
**Author:** ![Peter\_Morris](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/peter_morris/32/359_2.png) [@Peter\_Morris](https://boards.straightdope.com/u/Peter_Morris)\ **Post date:** [November 29, 2004, 3:48pm UTC](https://boards.straightdope.com/t/euclids-5th-postulate-proven-what/277088/22 "2004-11-29T15:48:15Z")
Hang on, I don’t get this. (I’m not a mathematician.) You’rew saying that the postulate is assumed to be true, everyone belies it to be true, much is based on the assumption that it’s true - but if anyone ever actually \*proves \*it all other maths will be undermined? How can this be? ---
**Author:** ![ultrafilter](https://avatars.discourse-cdn.com/v4/letter/u/3d9bf3/32.png) [@ultrafilter](https://boards.straightdope.com/u/ultrafilter)\ **Post date:** [November 29, 2004, 4:20pm UTC](https://boards.straightdope.com/t/euclids-5th-postulate-proven-what/277088/23 "2004-11-29T16:20:23Z")
> [@Peter Morris](#): > > Hang on, I don’t get this. (I’m not a mathematician.) You’rew saying that the postulate is assumed to be true, everyone belies it to be true, much is based on the assumption that it’s true - but if anyone ever actually \*proves \*it all other maths will be undermined? How can this be? It’s only true for Euclidean spaces. As **Mathochist** pointed out, there are spaces where the first four axioms hold and the fifth does not. ---
**Author:** ![KellyM](https://avatars.discourse-cdn.com/v4/letter/k/57b2e6/32.png) [@KellyM](https://boards.straightdope.com/u/KellyM)\ **Post date:** [November 29, 2004, 4:31pm UTC](https://boards.straightdope.com/t/euclids-5th-postulate-proven-what/277088/24 "2004-11-29T16:31:25Z")
> [@Peter Morris](#): > > Hang on, I don’t get this. (I’m not a mathematician.) You’rew saying that the postulate is assumed to be true, everyone belies it to be true, much is based on the assumption that it’s true - but if anyone ever actually \*proves \*it all other maths will be undermined? How can this be? Because we’ve proven, using mathematics as we best understand it, that it’s impossible to prove the parallel postulate from the axioms of neutral geometry. Therefore, if someone were to subsequently prove the parallel postulate from the axioms of neutral geometry, the metatheory of geometry would contain as proven theorems a statement (“the parallel postulate cannot be proven”) and also its contradiction (“the parallel postulate can be proven”). That would make the metatheory of geometry inconsistent – and that would shake the foundations of all mathematical thought. ---
**Author:** ![Orbifold](https://avatars.discourse-cdn.com/v4/letter/o/779978/32.png) [@Orbifold](https://boards.straightdope.com/u/Orbifold)\ **Post date:** [November 29, 2004, 4:43pm UTC](https://boards.straightdope.com/t/euclids-5th-postulate-proven-what/277088/25 "2004-11-29T16:43:23Z")
Hyperbolic geometry (specifically, the geometry of hyperbolic 3-manifolds) is my field of research. I can provide at least three complete descriptions of hyperbolic geometry, and demonstrate quite conclusively that hyperbolic geometry satisfies the first four of Euclid’s axioms but not the fifth. And there are plenty of other mathematicians who can do the same. I think I can safely say that this fellow Matta has no more “discredited” hyperbolic geometry than he has “discredited” the colour green. ---
**Author:** ![Reno\_Nevada](https://avatars.discourse-cdn.com/v4/letter/r/6f9a4e/32.png) [@Reno\_Nevada](https://boards.straightdope.com/u/Reno_Nevada)\ **Post date:** [November 29, 2004, 4:54pm UTC](https://boards.straightdope.com/t/euclids-5th-postulate-proven-what/277088/26 "2004-11-29T16:54:09Z")
I am never forget the day I first meet the great Lobachevsky. In one word he told me secret of success in mathematics: Plagiarize! And, just so my post is not completely worthless > [@Peter Morris](#): > > Hang on, I don’t get this. (I’m not a mathematician.) You’rew saying that the postulate is assumed to be true, everyone belies it to be true, much is based on the assumption that it’s true - but if anyone ever actually \*proves \*it all other maths will be undermined? How can this be? All \*postulates \* are \*assumed \* to be true. That is what a postulate means. In the case of Euclid’s Fifth Postulate, a lot of energy has gone into proving it from the first Four–because the first four are so simple and neat, and the Fifth is so unwieldly. However, it has been proved that the Fifth is a true postulate, and so must be assumed. If that proof is false, a lot of other mathematics must also be false. ---
**Author:** ![Mathochist](https://avatars.discourse-cdn.com/v4/letter/m/c89c15/32.png) [@Mathochist](https://boards.straightdope.com/u/Mathochist)\ **Post date:** [November 29, 2004, 5:27pm UTC](https://boards.straightdope.com/t/euclids-5th-postulate-proven-what/277088/27 "2004-11-29T17:27:11Z")
> [@KellyM](#): > > that would shake the foundations of all mathematical thought. To put it more strongly (and to directly refute **Giles** ’ assertion that set theory would be safe), we can construct consistent models of Euclidean geometry (and thus non-Euclidean) directly from the axioms of set theory. Start with the assumption that there exists a consistent model of Zermelo-Fraenkel set theory. This has a natural numbers object ( **N** ). From here there’s a very standard construction of integers and then rational numbers. Then define real numbers as Cauchy-equivalent Cauchy sequences of rational numbers (Dedekind cuts have trouble with intuitionistic logic sometimes). Now, **R** [sup]n[/sup] provides a consistent model of Euclidean geometry _as long as we started with a consistent model of set theory_. That is, if Euclidean geometry is inconsistent it cannot have a consistent model, and the only way for this construction to fail is to not have a consistent model of set theory. Yes, technically the ZF axioms may be consistent, but there would be no models and thus it’d be a rather useless concept. ---
**Author:** ![Giles](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/giles/32/60_2.png) [@Giles](https://boards.straightdope.com/u/Giles)\ **Post date:** [November 29, 2004, 6:43pm UTC](https://boards.straightdope.com/t/euclids-5th-postulate-proven-what/277088/28 "2004-11-29T18:43:56Z")
> [@Mathochist](#): > > To put it more strongly (and to directly refute **Giles** ’ assertion that set theory would be safe), we can construct consistent models of Euclidean geometry (and thus non-Euclidean) directly from the axioms of set theory. Yes, **R** [sup]n[/sup] with you usual interpretation of coordinate geometry provides a very satisfactory model or Euclidean geometry. So if Euclidean geometry fails, the **R** (the real numbers) fail, and so does set theory. ---
**Author:** ![Chronos](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/chronos/32/134_2.png) [@Chronos](https://boards.straightdope.com/u/Chronos)\ **Post date:** [November 29, 2004, 9:43pm UTC](https://boards.straightdope.com/t/euclids-5th-postulate-proven-what/277088/29 "2004-11-29T21:43:19Z")
It is important to note, however, that while the Fifth is independent, some of the other postulates are not. One can, for instance, extend a straight line segment indefinitely using only the other three absolute postulates. But this guy is a crank at best, or a con artist at worst. What he has proven is that bears use outhouses, water isn’t wet, Santa Clause is an evil martian bent on world domination, and that I’m the Pope. Seriously. All of those statements (and many, many more) are consequences of a proof of the Fifth Postulate. ---
**Author:** ![Mathochist](https://avatars.discourse-cdn.com/v4/letter/m/c89c15/32.png) [@Mathochist](https://boards.straightdope.com/u/Mathochist)\ **Post date:** [November 29, 2004, 10:14pm UTC](https://boards.straightdope.com/t/euclids-5th-postulate-proven-what/277088/30 "2004-11-29T22:14:37Z")
> [@Chronos](#): > > All of those statements (and many, many more) are consequences of a proof of the Fifth Postulate. But how many? countable? uncountable? weakly or strongly inaccessible? 😃 ---
**Author:** ![Giles](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/giles/32/60_2.png) [@Giles](https://boards.straightdope.com/u/Giles)\ **Post date:** [November 29, 2004, 10:23pm UTC](https://boards.straightdope.com/t/euclids-5th-postulate-proven-what/277088/31 "2004-11-29T22:23:21Z")
> [@Mathochist](#): > > But how many? countable? uncountable? weakly or strongly inaccessible? > > 😃 Assuming the statements are made with a finite number of symbols, and have finite length, surely the number of statements is countable :D1 :D2 :D3 … ---
**Author:** ![ultrafilter](https://avatars.discourse-cdn.com/v4/letter/u/3d9bf3/32.png) [@ultrafilter](https://boards.straightdope.com/u/ultrafilter)\ **Post date:** [November 29, 2004, 10:24pm UTC](https://boards.straightdope.com/t/euclids-5th-postulate-proven-what/277088/32 "2004-11-29T22:24:36Z")
> [@Giles](#): > > Assuming the statements are made with a finite number of symbols, and have finite length, surely the number of statements is countable :D1 :D2 :D3 … Hey, we’ve already blown consistency out of the water. Not only is the number countable, but it’s strongly inaccessible too. ---
**Author:** ![Mathochist](https://avatars.discourse-cdn.com/v4/letter/m/c89c15/32.png) [@Mathochist](https://boards.straightdope.com/u/Mathochist)\ **Post date:** [November 29, 2004, 10:32pm UTC](https://boards.straightdope.com/t/euclids-5th-postulate-proven-what/277088/33 "2004-11-29T22:32:59Z")
> [@ultrafilter](#): > > Hey, we’ve already blown consistency out of the water. Not only is the number countable, but it’s strongly inaccessible too. WOOHOO! Just the _thought_ of a separable Hilbert space with a strongly inaccessible base… I mean, capture the flag would be tough, but… ---
**Author:** ![John\_Kentzel-Griffin](https://avatars.discourse-cdn.com/v4/letter/j/ccd318/32.png) [@John\_Kentzel-Griffin](https://boards.straightdope.com/u/John_Kentzel-Griffin)\ **Post date:** [November 30, 2004, 6:44am UTC](https://boards.straightdope.com/t/euclids-5th-postulate-proven-what/277088/34 "2004-11-30T06:44:16Z")
> [@Mathochist](#): > > That is, if Euclidean geometry is inconsistent it cannot have a consistent model, and the only way for this construction to fail is to not have a consistent model of set theory. Yes, technically the ZF axioms may be consistent, but there would be no models and thus it’d be a rather useless concept. I understand that an inconsistent system cannot have a model. I don’t understand what it means to have a consistent system with no models. Is that like a statement being true vacously? ---
**Author:** ![Mathochist](https://avatars.discourse-cdn.com/v4/letter/m/c89c15/32.png) [@Mathochist](https://boards.straightdope.com/u/Mathochist)\ **Post date:** [November 30, 2004, 2:00pm UTC](https://boards.straightdope.com/t/euclids-5th-postulate-proven-what/277088/35 "2004-11-30T14:00:20Z")
> [@DrMatrix](#): > > I understand that an inconsistent system cannot have a model. I don’t understand what it means to have a consistent system with no models. Is that like a statement being true vacously? Not quite. More like the statements don’t contradict each other, but there still aren’t any examples. Whether this is even _possible_ is actually a philosophical matter. Some people take the position that any structure (search for my posts using the word “structuralism”) which is consistent has an instantiation (“model”) simply by virtue of being consistent. ---
**Author:** ![Spectre\_of\_Pithecanthropus](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/spectre_of_pithecanthropus/32/12343_2.png) [@Spectre\_of\_Pithecanthropus](https://boards.straightdope.com/u/Spectre_of_Pithecanthropus)\ **Post date:** [November 30, 2004, 2:36pm UTC](https://boards.straightdope.com/t/euclids-5th-postulate-proven-what/277088/36 "2004-11-30T14:36:10Z")
[QUOTE=FrylockThis would be… huge… right? Like, as in not only math but what we know of Physics would have to be completely rethought, right? [/QUOTE] Huge? Meh. Just imagine if the Fifth Postulate had been _disproved!_!! :eek: ---
**Author:** ![Giles](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/giles/32/60_2.png) [@Giles](https://boards.straightdope.com/u/Giles)\ **Post date:** [November 30, 2004, 2:45pm UTC](https://boards.straightdope.com/t/euclids-5th-postulate-proven-what/277088/37 "2004-11-30T14:45:44Z")
> [@Spectre of Pithecanthropus](#): > > [Just imagine if the Fifth Postulate had been _disproved!_!! :eek: Not really a problem, as far as physics goes, since Einstein’s universe is a non-Euclidean one. ---
**Author:** ![Mathochist](https://avatars.discourse-cdn.com/v4/letter/m/c89c15/32.png) [@Mathochist](https://boards.straightdope.com/u/Mathochist)\ **Post date:** [November 30, 2004, 2:49pm UTC](https://boards.straightdope.com/t/euclids-5th-postulate-proven-what/277088/38 "2004-11-30T14:49:07Z")
> [@Spectre of Pithecanthropus](#): > > [QUOTE=FrylockThis would be… huge… right? Like, as in not only math but what we know of Physics would have to be completely rethought, right? Huge? Meh. Just imagine if the Fifth Postulate had been _disproved!_!! :eek: [/QUOTE] As we’ve covered, a proof also amounts to a disproof so really it wouldn’t be any worse. As for effects on physics and stuff, of course physics wouldn’t be altered since it’s what really happens. Our description of it would be shot, though, since a disproof of the fifth postulate leads to an contradiction in set theory and thus brings down all of mathematics. ---
**Author:** ![ZenBeam](https://avatars.discourse-cdn.com/v4/letter/z/3ab097/32.png) [@ZenBeam](https://boards.straightdope.com/u/ZenBeam)\ **Post date:** [November 30, 2004, 5:36pm UTC](https://boards.straightdope.com/t/euclids-5th-postulate-proven-what/277088/39 "2004-11-30T17:36:21Z")
From [**Snarky\_Kong** ’s Wikipedia Link](http://en.wikipedia.org/wiki/Parallel_postulate) > [@](#): > > Some of the other statements that are equivalent to the parallel postulate appear at first to be unrelated to parallelism. Some even seem so self-evident that they were unconsciously assumed by people who claimed to have proved the parallel postulate from Euclid’s other postulates. Here are some of these results: > […] > 4. Every triangle can be circumscribed. 😕 I can almost see how this could fail in spherical geometry, since things get a bit squirrelly when the triangle takes up over half the space, and maybe the definition of circumscribe is restrictive, but I really can’t see how a triangle in hyperbolic space can’t be circumscribed. Anyone got an explanation? ---
**Author:** ![Orbifold](https://avatars.discourse-cdn.com/v4/letter/o/779978/32.png) [@Orbifold](https://boards.straightdope.com/u/Orbifold)\ **Post date:** [November 30, 2004, 5:45pm UTC](https://boards.straightdope.com/t/euclids-5th-postulate-proven-what/277088/40 "2004-11-30T17:45:17Z")
> [@ZenBeam](#): > > From [**Snarky\_Kong** ’s Wikipedia Link](http://en.wikipedia.org/wiki/Parallel_postulate)😕 I can almost see how this could fail in spherical geometry, since things get a bit squirrelly when the triangle takes up over half the space, and maybe the definition of circumscribe is restrictive, but I really can’t see how a triangle in hyperbolic space can’t be circumscribed. > > Anyone got an explanation? In Euclidean geometry, you find the circumcentre (the centre of the circumscribed circle) by constructing the perpendicular bisectors to each of the three sides of the triangle and seeing where they intersect. In hyperbolic geometry, you can still construct the perpendicular bisectors to the three sides of the triangle…but those three lines are no longer guaranteed to intersect at all. Instead there are three cases: (a) The three lines do intersect at a common point. In this case, the point of intersection is the circumcentre of the original triangle. (b) The three lines are parallel, i.e. they “meet” at a point on the “circle at infinity”. In this case, the original triangle has no circumcircle. Instead, the three vertices of the triangle lie on something called a horocycle, which is a curve of constant curvature (like a circle) but of infinite length (unlike a circle). (c) The three lines are “ultra-parallel”, i.e. they “meet” at a point _outside_ the “circle at infinity”. In this case, the three lines have a common perpendicular: there is a unique fourth line L which the first three lines meet at right angles. The three vertices of the original triangle lie on a curve C which is at a constant distance from L along its entire length. C is not a line, circle, or horocycle, but it is a curve of constant curvature. Clear as mud? [Previous page](https://boards.straightdope.com/t/euclids-5th-postulate-proven-what/277088.md?page=1) [Next page](https://boards.straightdope.com/t/euclids-5th-postulate-proven-what/277088.md?page=3)