14157 | Modern geometries only accept various parts of the Euclid propositions [Russell on Euclid] |
10250 | Euclid needs a principle of continuity, saying some lines must intersect [Shapiro on Euclid] |
8673 | Euclid's parallel postulate defines unique non-intersecting parallel lines [Friend on Euclid] |
13007 | Archimedes defined a straight line as the shortest distance between two points [Leibniz on Archimedes] |
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] |
18742 | Hilbert's formalisation revealed implicit congruence axioms in Euclid [Horsten/Pettigrew on Hilbert] |
10052 | Geometry is united by the intuitive axioms of projective geometry [Musgrave on Russell] |
10302 | Euclid says we can 'join' two points, but Hilbert says the straight line 'exists' [Bernays] |
10157 | Tarski improved Hilbert's geometry axioms, and without set-theory [Feferman/Feferman on Tarski] |
8997 | There are four different possible conventional accounts of geometry [Quine] |
18156 | Modern axioms of geometry do not need the real numbers [Bostock] |
18217 | Hilbert's geometry is interesting because it captures Euclid without using real numbers [Field,H] |
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] |
9546 | Euclid axioms concerns possibilities of construction, but Hilbert's assert the existence of objects [Chihara] |
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] |