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
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
You would cripple mathematics if you denied Excluded Middle
5. Theory of Logic / K. Features of Logics / 1. Axiomatisation
The facts of geometry, arithmetic or statics order themselves into theories
Axioms must reveal their dependence (or not), and must be consistent
6. Mathematics / A. Nature of Mathematics / 1. Mathematics
I aim to establish certainty for mathematical methods
We believe all mathematical problems are solvable
Hilbert wanted to prove the consistency of all of mathematics (which realists take for granted)
6. Mathematics / A. Nature of Mathematics / 4. The Infinite / a. The Infinite
No one shall drive us out of the paradise the Cantor has created for us
We extend finite statements with ideal ones, in order to preserve our logic
Only the finite can bring certainty to the infinite
6. Mathematics / A. Nature of Mathematics / 4. The Infinite / d. Actual infinite
The idea of an infinite totality is an illusion
6. Mathematics / A. Nature of Mathematics / 4. The Infinite / k. Infinite divisibility
There is no continuum in reality to realise the infinitely small
6. Mathematics / A. Nature of Mathematics / 5. Geometry
Hilbert aimed to eliminate number from geometry
6. Mathematics / A. Nature of Mathematics / 6. Proof in Mathematics
To decide some questions, we must study the essence of mathematical proof itself
6. Mathematics / B. Foundations for Mathematics / 2. Axioms for Geometry
Hilbert's formalisation revealed implicit congruence axioms in Euclid
The whole of Euclidean geometry derives from a basic equation and transformations
6. Mathematics / B. Foundations for Mathematics / 3. Axioms for Number / a. Axioms for numbers
Number theory just needs calculation laws and rules for integers
6. Mathematics / C. Sources of Mathematics / 4. Mathematical Empiricism / c. Against mathematical empiricism
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
Logic already contains some arithmetic, so the two must be developed together
Kant taught that mathematics is independent of logic, and cannot be grounded in it
6. Mathematics / C. Sources of Mathematics / 7. Formalism
The grounding of mathematics is 'in the beginning was the sign'
The subject matter of mathematics is immediate and clear concrete symbols
Hilbert substituted a syntactic for a semantic account of consistency
6. Mathematics / C. Sources of Mathematics / 8. Finitism
Hilbert aimed to prove the consistency of mathematics finitely, to show infinities won't produce contradictions
Mathematics divides in two: meaningful finitary statements, and empty idealised statements
11. Knowledge Aims / B. Certain Knowledge / 1. Certainty
My theory aims at the certitude of mathematical methods
26. Natural Theory / D. Laws of Nature / 8. Scientific Essentialism / d. Knowing essences
By digging deeper into the axioms we approach the essence of sciences, and unity of knowedge