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Ideas of David Hilbert, by Text
[German, 1862 - 1943, Professor of Mathematics at Königsberg, and the Göttingen.]
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1899
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Foundations of Geometry
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p.9
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9546
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Euclid axioms concerns possibilities of construction, but Hilbert's assert the existence of objects [Chihara]
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p.17
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18742
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Hilbert's formalisation revealed implicit congruence axioms in Euclid [Horsten/Pettigrew]
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p.25
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18217
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Hilbert's geometry is interesting because it captures Euclid without using real numbers [Field,H]
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p.42
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13472
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Hilbert aimed to eliminate number from geometry [Hart,WD]
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1899
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Letter to Frege 29.12.1899
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p.51
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15716
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If axioms and their implications have no contradictions, they pass my criterion of truth and existence
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1900
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On the Concept of Number
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p.183
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p.129
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22293
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Hilbert said (to block paradoxes) that mathematical existence is entailed by consistency [Potter]
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p.148
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10113
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The grounding of mathematics is 'in the beginning was the sign'
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p.153
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10115
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Hilbert substituted a syntactic for a semantic account of consistency [George/Velleman]
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p.156
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10116
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Hilbert aimed to prove the consistency of mathematics finitely, to show infinities won't produce contradictions [George/Velleman]
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6.7
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p.154
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8717
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Hilbert wanted to prove the consistency of all of mathematics (which realists take for granted) [Friend]
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1904
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On the Foundations of Logic and Arithmetic
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p.130
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p.130
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17697
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The existence of an arbitrarily large number refutes the idea that numbers come from experience
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p.131
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p.131
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17698
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Logic already contains some arithmetic, so the two must be developed together
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[03]
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p.1108
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17963
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The facts of geometry, arithmetic or statics order themselves into theories
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[05]
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p.1108
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17965
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The whole of Euclidean geometry derives from a basic equation and transformations
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[05]
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p.1108
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17964
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Number theory just needs calculation laws and rules for integers
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[09]
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p.1109
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17966
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Axioms must reveal their dependence (or not), and must be consistent
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[53]
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p.1115
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17967
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To decide some questions, we must study the essence of mathematical proof itself
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[56]
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p.1115
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17968
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By digging deeper into the axioms we approach the essence of sciences, and unity of knowedge
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p.184
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p.66
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9636
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My theory aims at the certitude of mathematical methods
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p.184
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p.184
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12456
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I aim to establish certainty for mathematical methods
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p.184
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p.184
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12455
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The idea of an infinite totality is an illusion
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p.186
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p.186
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12457
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There is no continuum in reality to realise the infinitely small
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p.191
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p.65
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9633
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No one shall drive us out of the paradise the Cantor has created for us
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p.192
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p.192
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12459
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The subject matter of mathematics is immediate and clear concrete symbols
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p.195
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p.195
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12460
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We extend finite statements with ideal ones, in order to preserve our logic
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p.196
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p.174
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18112
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Mathematics divides in two: meaningful finitary statements, and empty idealised statements
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p.200
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p.200
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12461
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We believe all mathematical problems are solvable
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p.201
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p.201
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12462
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Only the finite can bring certainty to the infinite
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1927
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The Foundations of Mathematics
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p.476
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p.285
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18844
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You would cripple mathematics if you denied Excluded Middle
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