Ideas from 'Foundations of Geometry' by David Hilbert [1899], by Theme Structure

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6. Mathematics / A. Nature of Mathematics / 2. Geometry
Hilbert aimed to eliminate number from geometry
                        Full Idea: One of Hilbert's aims in 'The Foundations of Geometry' was to eliminate number [as measure of lengths and angles] from geometry.
                        From: report of David Hilbert (Foundations of Geometry [1899]) by William D. Hart - The Evolution of Logic 2
                        A reaction: Presumably this would particularly have to include the elimination of ratios (rather than actual specific lengths).
6. Mathematics / B. Foundations for Mathematics / 3. Axioms for Geometry
Euclid axioms concerns possibilities of construction, but Hilbert's assert the existence of objects
                        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.
                        From: report of David Hilbert (Foundations of Geometry [1899]) by Charles Chihara - A Structural Account of Mathematics 01.1
                        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.
Hilbert's formalisation revealed implicit congruence axioms in Euclid
                        Full Idea: In his formal investigation of Euclidean geometry, Hilbert uncovered congruence axioms that implicitly played a role in Euclid's proofs but were not explicitly recognised.
                        From: report of David Hilbert (Foundations of Geometry [1899]) by Horsten,L/Pettigrew,R - Mathematical Methods in Philosophy 2
                        A reaction: The writers are offering this as a good example of the benefits of a precise and formal approach to foundational questions. It's hard to disagree, but dispiriting if you need a PhD in maths before you can start doing philosophy.
Hilbert's geometry is interesting because it captures Euclid without using real numbers
                        Full Idea: Hilbert's formulation of the Euclidean theory is of special interest because (besides being rigorously axiomatised) it does not employ the real numbers in the axioms.
                        From: report of David Hilbert (Foundations of Geometry [1899]) by Hartry Field - Science without Numbers 3
                        A reaction: Notice that this job was done by Hilbert, and not by the fictionalist Hartry Field.