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Single Idea 9546

[filed under theme 6. Mathematics / B. Foundations for Mathematics / 3. Axioms for Geometry ]

Full Idea

Hilbert's geometrical axioms were existential in character, asserting the existence of certain geometrical objects (points and lines). Euclid's postulates do not assert the existence of anything; they assert the possibility of certain constructions.

Gist of Idea

Euclid axioms concerns possibilities of construction, but Hilbert's assert the existence of objects

Source

report of David Hilbert (Foundations of Geometry [1899]) by Charles Chihara - A Structural Account of Mathematics 01.1

Book Ref

Chihara,Charles: 'A Structural Account of Mathematics' [OUP 2004], p.9

A Reaction

Chihara says geometry was originally understood modally, but came to be understood existentially. It seems extraordinary to me that philosophers of mathematics can have become more platonist over the centuries.

The 21 ideas with the same theme [formal starting points for deriving geometry]:

22278 Euclid relied on obvious properties in diagrams, as well as on his axioms [Potter on Euclid]
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]
10302 Euclid says we can 'join' two points, but Hilbert says the straight line 'exists' [Euclid, by Bernays]
14157 Modern geometries only accept various parts of the Euclid propositions [Russell 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]