Grundlagen der Geometrie¶
Hilbert, D. (1899). Grundlagen der Geometrie.
Cited by¶
3 citations across 3 artifacts.
Each citation links to the sentence it supports in the citing article.
Primes¶
- Abstraction
This source(Tier C — bibliography only; existence verified, Teubner/Leipzig confirmed.) Axiomatic abstraction of geometry into undefined primitives and axioms independent of intuitive content. Link-only.
- Completeness
- … proof from the axioms (first-order classical logic is complete with respect to its model-theoretic semantics — Gödel's completeness theorem of 1929 — but Peano arithmetic with respect to its standard model is not complete, because Gödel's incompleteness theorem of 1931 exhibits true-but-unprovable statements);
This sourceTeubner, Leipzig. Develops axiomatic abstraction: geometry is determined entirely by a set of axioms (incidence, betweenness, congruence, continuity) independent of intuitive content, the prerequisite move that made later metalogical completeness questions formulable.
- … proof from the axioms (first-order classical logic is complete with respect to its model-theoretic semantics — Gödel's completeness theorem of 1929 — but Peano arithmetic with respect to its standard model is not complete, because Gödel's incompleteness theorem of 1931 exhibits true-but-unprovable statements);
- Formalization
- The intellectual high-water mark of this move is the early-twentieth-century formalist program in mathematics, where Hilbert (1899/1902) re-grounded Euclidean geometry on a set of explicit, gap-free axioms precisely to expose every assumption that classical practice had carried silently.
This sourceRe-grounds Euclidean geometry on ~20 explicit, gap-free axioms in five groups (incidence, order, congruence, parallels, continuity), exposing assumptions classical practice carried silently and making it possible to ask which theorems depend on which axioms.
- The intellectual high-water mark of this move is the early-twentieth-century formalist program in mathematics, where Hilbert (1899/1902) re-grounded Euclidean geometry on a set of explicit, gap-free axioms precisely to expose every assumption that classical practice had carried silently.
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