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Single Idea 13474
[filed under theme 6. Mathematics / B. Foundations for Mathematics / 3. Axioms for Geometry
]
Full Idea
There is a familiar comparison between Euclid (unique parallel) and 'spherical' geometry (no parallel) and 'saddle' geometry (several parallels).
Gist of Idea
Euclid has a unique parallel, spherical geometry has none, and saddle geometry has several
Source
William D. Hart (The Evolution of Logic [2010], 2)
Book Ref
Hart,W.D.: 'The Evolution of Logic' [CUP 2010], p.44
The
21 ideas
with the same theme
[formal starting points for deriving geometry]:
8673
|
Euclid's parallel postulate defines unique non-intersecting parallel lines
[Euclid, by Friend]
|
10250
|
Euclid needs a principle of continuity, saying some lines must intersect
[Shapiro on Euclid]
|
14157
|
Modern geometries only accept various parts of the Euclid propositions
[Russell on Euclid]
|
10302
|
Euclid says we can 'join' two points, but Hilbert says the straight line 'exists'
[Euclid, by Bernays]
|
22278
|
Euclid relied on obvious properties in diagrams, as well as on his axioms
[Potter on Euclid]
|
13007
|
Archimedes defined a straight line as the shortest distance between two points
[Archimedes, by Leibniz]
|
12937
|
We shouldn't just accept Euclid's axioms, but try to demonstrate them
[Leibniz]
|
3343
|
Euclid's could be the only viable geometry, if rejection of the parallel line postulate doesn't lead to a contradiction
[Benardete,JA on Kant]
|
17965
|
The whole of Euclidean geometry derives from a basic equation and transformations
[Hilbert]
|
9546
|
Euclid axioms concerns possibilities of construction, but Hilbert's assert the existence of objects
[Hilbert, by Chihara]
|
18742
|
Hilbert's formalisation revealed implicit congruence axioms in Euclid
[Hilbert, by Horsten/Pettigrew]
|
18217
|
Hilbert's geometry is interesting because it captures Euclid without using real numbers
[Hilbert, by Field,H]
|
10052
|
Geometry is united by the intuitive axioms of projective geometry
[Russell, by Musgrave]
|
10157
|
Tarski improved Hilbert's geometry axioms, and without set-theory
[Tarski, by Feferman/Feferman]
|
8997
|
There are four different possible conventional accounts of geometry
[Quine]
|
18156
|
Modern axioms of geometry do not need the real numbers
[Bostock]
|
18221
|
'Metric' axioms uses functions, points and numbers; 'synthetic' axioms give facts about space
[Field,H]
|
13474
|
Euclid has a unique parallel, spherical geometry has none, and saddle geometry has several
[Hart,WD]
|
9553
|
Analytic geometry gave space a mathematical structure, which could then have axioms
[Chihara]
|
18760
|
The culmination of Euclidean geometry was axioms that made all models isomorphic
[McGee]
|
17762
|
In non-Euclidean geometry, all Euclidean theorems are valid that avoid the fifth postulate
[Walicki]
|