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Ideas of David Hilbert, by Text

[German, 1862 - 1943, Professor of Mathematics at Königsberg, and the Göttingen.]

1899 Foundations of Geometry
p.9 Euclid axioms concerns possibilities of construction, but Hilbert's assert the existence of objects [Chihara]
p.17 Hilbert's formalisation revealed implicit congruence axioms in Euclid [Horsten/Pettigrew]
p.25 Hilbert's geometry is interesting because it captures Euclid without using real numbers [Field,H]
p.42 Hilbert aimed to eliminate number from geometry [Hart,WD]
1899 Letter to Frege 29.12.1899
p.51 If axioms and their implications have no contradictions, they pass my criterion of truth and existence
1900 On the Concept of Number
p.183 p.129 Hilbert said (to block paradoxes) that mathematical existence is entailed by consistency [Potter]
1900 works
p.148 The grounding of mathematics is 'in the beginning was the sign'
p.153 Hilbert substituted a syntactic for a semantic account of consistency [George/Velleman]
p.156 Hilbert aimed to prove the consistency of mathematics finitely, to show infinities won't produce contradictions [George/Velleman]
6.7 p.154 Hilbert wanted to prove the consistency of all of mathematics (which realists take for granted) [Friend]
1904 On the Foundations of Logic and Arithmetic
p.130 p.130 The existence of an arbitrarily large number refutes the idea that numbers come from experience
p.131 p.131 Logic already contains some arithmetic, so the two must be developed together
1918 Axiomatic Thought
[03] p.1108 The facts of geometry, arithmetic or statics order themselves into theories
[05] p.1108 The whole of Euclidean geometry derives from a basic equation and transformations
[05] p.1108 Number theory just needs calculation laws and rules for integers
[09] p.1109 Axioms must reveal their dependence (or not), and must be consistent
[53] p.1115 To decide some questions, we must study the essence of mathematical proof itself
[56] p.1115 By digging deeper into the axioms we approach the essence of sciences, and unity of knowedge
1925 On the Infinite
p.184 p.66 My theory aims at the certitude of mathematical methods
p.184 p.184 I aim to establish certainty for mathematical methods
p.184 p.184 The idea of an infinite totality is an illusion
p.186 p.186 There is no continuum in reality to realise the infinitely small
p.191 p.65 No one shall drive us out of the paradise the Cantor has created for us
p.192 p.192 The subject matter of mathematics is immediate and clear concrete symbols
p.195 p.195 We extend finite statements with ideal ones, in order to preserve our logic
p.196 p.174 Mathematics divides in two: meaningful finitary statements, and empty idealised statements
p.200 p.200 We believe all mathematical problems are solvable
p.201 p.201 Only the finite can bring certainty to the infinite
1927 The Foundations of Mathematics
p.476 p.285 You would cripple mathematics if you denied Excluded Middle