Euclidean and Non-Euclidean Geometries¶
Greenberg, M. J. (2008). Euclidean and Non-Euclidean Geometries: Development and History.
Cited by¶
2 citations across 2 artifacts.
Each citation links to the sentence it supports in the citing article.
Primes¶
- Axiom
- The independence question is exactly what those attempts were probing, and the resolution was structural: nineteenth-century work exhibited consistent models (hyperbolic geometry, where many parallels exist; elliptic geometry, where none do) that satisfy the other four postulates while differing on the fifth.
This sourceStandard history and proof treatment of the parallel postulate's independence (ch. 7), the consistent hyperbolic and elliptic models, and their differing triangle angle-sums.
- The independence question is exactly what those attempts were probing, and the resolution was structural: nineteenth-century work exhibited consistent models (hyperbolic geometry, where many parallels exist; elliptic geometry, where none do) that satisfy the other four postulates while differing on the fifth.
Mechanisms¶
- Axiom Boundary Statement
- The canonical demonstration is the parallel postulate: for two millennia mathematicians tried to prove it, until denying it turned out to yield perfectly consistent non-Euclidean geometries — proof that it was a choice all along, examinable only from outside the system that adopts it.
This sourceRecounts the two-millennia proof effort and explains the parallel postulate's independence and its metamathematical assessment outside the object system.
- The canonical demonstration is the parallel postulate: for two millennia mathematicians tried to prove it, until denying it turned out to yield perfectly consistent non-Euclidean geometries — proof that it was a choice all along, examinable only from outside the system that adopts it.
Verification¶
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Links previously used in the corpus¶
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