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I am wondering about the axiomatic method here and Euclid's 5th often comes up. I don't understand why they said that parallel lines could meet after all in Non-Euclidean Geometry. Surely Euclid did not have this exact definition of a line in mind when he made the axiom, no? I simply don't see how the line we have in Non-Euclidean geometry is what Euclid meant, and therefore why this lead people to reconstruct mathematics (along with many other things in history).

It seems like a just a small terminological conflict that can be resolved by realizing that what we have here in Non-Euclidean geometries are geodesics, not lines.

Some mathematical clarification would be helpful on why exactly what we have in say, hyperbolic space, is considered a line. Why (mathematically) did this shake the groundwork of everything so much? Did we actually for a brief time in history think that it was true of Non-Euclidean spaces, and did anything go seriously wrong in the real world over this?

Pineapple Fish
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    Right and wrong are not the issue. The issue was whether or not the Fifth Postulate could be deduced from the other Four. As it turns out, it can't be. There are consistent Geometric theories that assume the Fifth Postulate, and consistent theories that do not. – lulu Jun 08 '20 at 00:13
  • @lulu oooooooooooh, of course. Idk how I missed that after so much thinking about it – Pineapple Fish Jun 08 '20 at 00:15
  • The conflict is not an issue either. We can avoid all conflict by giving everything new names every time we do anything new. That's not the point though. The point is knowing what can be concluded and what systems can we generalize and are they complete. All geometries are consistent with the first four postulates. The study is what conclusions we can make in these systems and which are internal to the system. It would be BAD if we avoided calling geodesics lines because they are equivalent to lines in an alternate system that is exactly the same but with only one tweak. – fleablood Jun 08 '20 at 00:23

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