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Single Idea 18156

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

Full Idea

A modern axiomatisation of geometry, such as Hilbert's (1899), does not need to claim the existence of real numbers anywhere in its axioms.

Gist of Idea

Modern axioms of geometry do not need the real numbers

Source

David Bostock (Philosophy of Mathematics [2009], 9.B.5.ii)

Book Ref

Bostock,David: 'Philosophy of Mathematics: An Introduction' [Wiley-Blackwell 2009], p.295

A Reaction

This is despite the fact that geometry is reduced to algebra, and the real numbers are the equivalent of continuous lines. Bostock votes for a Greek theory of proportion in this role.

Related Idea

Idea 18150 Actual measurement could never require the precision of the real numbers [Bostock]

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]