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Why did it take mathematicians so long to realise that Euclid's fifth postulate is independent of the other 4?

Why didn't people like Lagrange notice that a sphere is a model for a non-Euclidean geometry (first 4 axioms satisfied, the fifth not satisfied)? They had ships and cartography long before Gauss and Bolyai were born.

Am I misunderstanding something?

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    $\begingroup$ @Casteels but they didn't observe that it was a model for Euclid's first four postulates in which the fifth failed. $\endgroup$ – hunter Mar 17 '15 at 12:36
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    $\begingroup$ They didn't because they didn't. Why would they have? If you think it's obvious, I would simply respond that the fact that it went undiscovered for so long is in and of itself evidence that it isn't. $\endgroup$ – Jack M Mar 17 '15 at 18:28
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    $\begingroup$ @JackM: That is about the least helpful comment I have seen on this site! $\endgroup$ – TonyK Mar 17 '15 at 18:48
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    $\begingroup$ @TonyK I think it cuts right to the heart of the fundamental problem with "why" questions. Any "why X" question implies that there is something surprising about X which needs to be explained. Whether or not this particular X is surprising is at best subjective, which I think makes the question somewhat unanswerable. $\endgroup$ – Jack M Mar 17 '15 at 21:54
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    $\begingroup$ I would further expand. IMNSHO the reason for not having discovered spherical geometry is philosophical: nobody thought that our world could be other that Euclidean, so even if they had a practical example they could not reframe it to notice that there was a sensible way to remap the concept of a point and a line. $\endgroup$ – mau Mar 18 '15 at 6:13
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In spherical geometry, a line is not a line as the ancients understood it, but a great circle. Ok, we say in hindsight, but a line is an undefined term, and great circles satisfy all the axioms that lines should. But this sort of ontological issue would have been extremely confusing before we had the right language to think about math in.

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  • $\begingroup$ So what did the ancients think a line was? What sort of strange geometry were they looking for? $\endgroup$ – Noppawee Apichonpongpan Mar 17 '15 at 13:38
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    $\begingroup$ @NoppaweeApichonpongpan The ancients thought a line was a line. $\endgroup$ – Jack M Mar 17 '15 at 18:29
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    $\begingroup$ @JackM: Will you cut that out?! $\endgroup$ – TonyK Mar 17 '15 at 18:49
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    $\begingroup$ @TonyK It's an accurate answer. Ancient geometers thought of a line as a line. Expecting ancient mathematicians to think in terms of undefined terms obeying formal laws is anachronistic. $\endgroup$ – Jack M Mar 17 '15 at 21:56
  • $\begingroup$ But if your not using the 5th postulate then what makes a "line" a line as we know it. Basically if all your allowed to use are the 1st 4 postulates then all you know is that points are some set and lines are sets of points such that for any two points there is one line going through them. Correct me if I'm wrong but is there even any notion of distance from purely the 1st 4 axioms? $\endgroup$ – Hao Sun Jul 6 '18 at 0:16
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Euclid's first two postulates arguably also fail on the sphere, even if we allow that great circles are lines.

Euclid's first postulate essentially says that there is a line between any two points, and one could argue that a unique line is meant. This is false on the sphere where antipodal points are connected by many lines.

Euclid's second postulate essentially says that a line segment can be extended indefinitely, which could be taken to mean that space must "go on forever", which the surface of the sphere does not (without repeating itself).

Furthermore, the identification of great circles with lines is itself problematic since it assumes a non-trivial definition of a straightness other than that of Euclid. Euclid's definition that "a straight line is a line which lies evenly with the points on itself" is hardly sufficient to single out great circles, and even Archimedes's definition "that among lines which have the same limits, the straight line is the smallest" is insufficient by itself since great circle arcs greater than half the circumference are not the shortest distance between its endpoints.

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  • $\begingroup$ Nice answer. Also, the fifth postulate itself, such as stated here, for instance -- doesn't make sense on a sphere, (and you cannot even say it's false). Moreover, uniqueness of the line connecting two points fails not only for antipodal points. $\endgroup$ – Peter Franek Mar 31 at 16:48
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In a sense, Euclid himself realized this. He set the 5th postulate apart from the other four, was not completely satisfied with it, and invoked it only after his first 28 propositions. It was debated in his time about whether the fifth postulate was necessary. In and after his time the fifth postulate was not considered as intuitive or as central as the other four postulates.

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    $\begingroup$ What I am confused about is why did people like Lagrange and Gauss (geniuses) attempted to prove the fifth postulate from the other four when they could see that the fifth postulate is independent of the other 4 from spherical geometry. I think I am misunderstanding something. $\endgroup$ – Noppawee Apichonpongpan Mar 17 '15 at 19:21
  • $\begingroup$ I never read that Gauss attempted to derive the fifth postulate from the other four - he searched whether the sum of the angles of a very large triangle is actually 180 degrees, which amounts to say that the fifth postulate is satisfied. $\endgroup$ – mau Mar 18 '15 at 6:09
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Hubris. They did not think of postulates as the rules of a game of logic. They thought of them as representations of physical truths. Accepting a different postulate was tantamount to saying they got "God" wrong.

addendum

I don't mean to imply the Christian God. I am trying to say that they thought that we were born with the fundamental truths wired into our beings and all we had to do was "think" about them to discover them. More to the point, since we assumed that these a-priori truths had to actually be truths, then the idea of competing models of reality was brushed aside as nonsense.

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