Note on Non-Euclidean geometries

Non-Euclidean geometries are developments of Euclidean geometry and not critiques of it. Oddly enough, if we could prove the Fifth Postulate, Euclid would have been wrong about it being a postulate and this would have counted as a critique. Again, if all attempts to deny the Fifth Postulate ended up assuming it, then it would been a self-evident axiom, and Euclid would have been wrong again. But non-Euclidean geometries developed as failures to achieve either of these ends, and so they are, in fact, deeper understandings of the Fifth postulate as a postulate.

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