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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]