structure for 'Mathematics'    |     alphabetical list of themes    |     expand these ideas

6. Mathematics / B. Foundations for Mathematics / 3. Axioms for Geometry

[formal starting points for deriving geometry]

20 ideas
Modern geometries only accept various parts of the Euclid propositions [Russell on Euclid]
Euclid says we can 'join' two points, but Hilbert says the straight line 'exists' [Euclid, by Bernays]
Euclid's parallel postulate defines unique non-intersecting parallel lines [Euclid, by Friend]
Euclid needs a principle of continuity, saying some lines must intersect [Shapiro on Euclid]
Archimedes defined a straight line as the shortest distance between two points [Archimedes, by Leibniz]
We shouldn't just accept Euclid's axioms, but try to demonstrate them [Leibniz]
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]
The whole of Euclidean geometry derives from a basic equation and transformations [Hilbert]
Euclid axioms concerns possibilities of construction, but Hilbert's assert the existence of objects [Hilbert, by Chihara]
Hilbert's formalisation revealed implicit congruence axioms in Euclid [Hilbert, by Horsten/Pettigrew]
Hilbert's geometry is interesting because it captures Euclid without using real numbers [Hilbert, by Field,H]
Geometry is united by the intuitive axioms of projective geometry [Russell, by Musgrave]
Tarski improved Hilbert's geometry axioms, and without set-theory [Tarski, by Feferman/Feferman]
There are four different possible conventional accounts of geometry [Quine]
Modern axioms of geometry do not need the real numbers [Bostock]
'Metric' axioms uses functions, points and numbers; 'synthetic' axioms give facts about space [Field,H]
Euclid has a unique parallel, spherical geometry has none, and saddle geometry has several [Hart,WD]
Analytic geometry gave space a mathematical structure, which could then have axioms [Chihara]
The culmination of Euclidean geometry was axioms that made all models isomorphic [McGee]
In non-Euclidean geometry, all Euclidean theorems are valid that avoid the fifth postulate [Walicki]