### 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 If axioms and their implications have no contradictions, they pass my criterion of truth and existence
###### 5. Theory of Logic / D. Assumptions for Logic / 2. Excluded Middle
 18844 You would cripple mathematics if you denied Excluded Middle
###### 5. Theory of Logic / K. Features of Logics / 1. Axiomatisation
 17963 The facts of geometry, arithmetic or statics order themselves into theories
 17966 Axioms must reveal their dependence (or not), and must be consistent
###### 6. Mathematics / A. Nature of Mathematics / 1. Mathematics
 8717 Hilbert wanted to prove the consistency of all of mathematics (which realists take for granted) [Friend]
 12456 I aim to establish certainty for mathematical methods
 12461 We believe all mathematical problems are solvable
###### 6. Mathematics / A. Nature of Mathematics / 2. Geometry
 13472 Hilbert aimed to eliminate number from geometry [Hart,WD]
###### 6. Mathematics / A. Nature of Mathematics / 5. The Infinite / a. The Infinite
 9633 No one shall drive us out of the paradise the Cantor has created for us
 12460 We extend finite statements with ideal ones, in order to preserve our logic
 12462 Only the finite can bring certainty to the infinite
###### 6. Mathematics / A. Nature of Mathematics / 5. The Infinite / d. Actual infinite
 12455 The idea of an infinite totality is an illusion
###### 6. Mathematics / A. Nature of Mathematics / 5. The Infinite / j. Infinite divisibility
 12457 There is no continuum in reality to realise the infinitely small
###### 6. Mathematics / B. Foundations for Mathematics / 2. Proof in Mathematics
 17967 To decide some questions, we must study the essence of mathematical proof itself
###### 6. Mathematics / B. Foundations for Mathematics / 3. Axioms for Geometry
 9546 Euclid axioms concerns possibilities of construction, but Hilbert's assert the existence of objects [Chihara]
 18742 Hilbert's formalisation revealed implicit congruence axioms in Euclid [Horsten/Pettigrew]
 18217 Hilbert's geometry is interesting because it captures Euclid without using real numbers [Field,H]
 17965 The whole of Euclidean geometry derives from a basic equation and transformations
###### 6. Mathematics / B. Foundations for Mathematics / 4. Axioms for Number / a. Axioms for numbers
 17964 Number theory just needs calculation laws and rules for integers
###### 6. Mathematics / C. Sources of Mathematics / 4. Mathematical Empiricism / c. Against mathematical empiricism
 17697 The existence of an arbitrarily large number refutes the idea that numbers come from experience
###### 6. Mathematics / C. Sources of Mathematics / 6. Logicism / d. Logicism critique
 17698 Logic already contains some arithmetic, so the two must be developed together
###### 6. Mathematics / C. Sources of Mathematics / 7. Formalism
 10113 The grounding of mathematics is 'in the beginning was the sign'
 10115 Hilbert substituted a syntactic for a semantic account of consistency [George/Velleman]
 12459 The subject matter of mathematics is immediate and clear concrete symbols
###### 6. Mathematics / C. Sources of Mathematics / 8. Finitism
 10116 Hilbert aimed to prove the consistency of mathematics finitely, to show infinities won't produce contradictions [George/Velleman]
 18112 Mathematics divides in two: meaningful finitary statements, and empty idealised statements
###### 11. Knowledge Aims / B. Certain Knowledge / 1. Certainty
 9636 My theory aims at the certitude of mathematical methods
###### 26. Natural Theory / D. Laws of Nature / 8. Scientific Essentialism / d. Knowing essences
 17968 By digging deeper into the axioms we approach the essence of sciences, and unity of knowedge