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
12462

Only the finite can bring certainty to 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

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]

22293

Hilbert said (to block paradoxes) that mathematical existence is entailed by consistency [Potter]

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
