Ideas of David Hilbert, by Theme
[German, 1862 - 1943, Professor of Mathematics at Königsberg, and the Göttingen.]
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3. Truth / G. Axiomatic Truth / 1. Axiomatic Truth
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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5. Theory of Logic / D. Assumptions for Logic / 2. Excluded Middle
18844
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You would cripple mathematics if you denied Excluded Middle
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5. Theory of Logic / K. Features of Logics / 1. Axiomatisation
17963
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The facts of geometry, arithmetic or statics order themselves into theories
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17966
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Axioms must reveal their dependence (or not), and must be consistent
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6. Mathematics / A. Nature of Mathematics / 1. Mathematics
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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12456
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I aim to establish certainty for mathematical methods
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12461
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We believe all mathematical problems are solvable
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6. Mathematics / A. Nature of Mathematics / 2. Geometry
13472
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Hilbert aimed to eliminate number from geometry [Hart,WD]
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6. Mathematics / A. Nature of Mathematics / 5. The Infinite / a. The Infinite
9633
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No one shall drive us out of the paradise the Cantor has created for us
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12462
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Only the finite can bring certainty to the infinite
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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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6. Mathematics / A. Nature of Mathematics / 5. The Infinite / d. Actual infinite
12455
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The idea of an infinite totality is an illusion
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6. Mathematics / A. Nature of Mathematics / 5. The Infinite / j. Infinite divisibility
12457
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There is no continuum in reality to realise the infinitely small
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6. Mathematics / B. Foundations for Mathematics / 2. Proof in Mathematics
17967
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To decide some questions, we must study the essence of mathematical proof itself
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6. Mathematics / B. Foundations for Mathematics / 3. Axioms for Geometry
9546
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Euclid axioms concerns possibilities of construction, but Hilbert's assert the existence of objects [Chihara]
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18742
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Hilbert's formalisation revealed implicit congruence axioms in Euclid [Horsten/Pettigrew]
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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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17965
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The whole of Euclidean geometry derives from a basic equation and transformations
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6. Mathematics / B. Foundations for Mathematics / 4. Axioms for Number / a. Axioms for numbers
17964
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Number theory just needs calculation laws and rules for integers
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6. Mathematics / C. Sources of Mathematics / 4. Mathematical Empiricism / c. Against mathematical empiricism
17697
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The existence of an arbitrarily large number refutes the idea that numbers come from experience
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6. Mathematics / C. Sources of Mathematics / 6. Logicism / d. Logicism critique
17698
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Logic already contains some arithmetic, so the two must be developed together
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6. Mathematics / C. Sources of Mathematics / 7. Formalism
10115
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Hilbert substituted a syntactic for a semantic account of consistency [George/Velleman]
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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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10113
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The grounding of mathematics is 'in the beginning was the sign'
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12459
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The subject matter of mathematics is immediate and clear concrete symbols
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6. Mathematics / C. Sources of Mathematics / 8. Finitism
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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18112
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Mathematics divides in two: meaningful finitary statements, and empty idealised statements
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11. Knowledge Aims / B. Certain Knowledge / 1. Certainty
9636
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My theory aims at the certitude of mathematical methods
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26. Natural Theory / D. Laws of Nature / 8. Scientific Essentialism / d. Knowing essences
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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