几何基础

简介

Hilbert published “Grundlagen der Geometrie” in 1899 in the Festschrift for the unveiling of the Gauss–Weber monument in Göttingen; the book went through many revised editions. It builds Euclidean geometry on five groups of axioms – of incidence, order, congruence, parallels and continuity – for three systems of undefined things called points, lines and planes, whose properties are fixed by the axioms alone. Hilbert proved the consistency of the axioms relative to arithmetic by a model in numbers and their independence by models in which one axiom fails, and showed which theorems depend on which axioms. As a motto he chose Kant’s sentence that all human knowledge begins with intuitions, proceeds to concepts and ends with ideas. Frege objected in letters (1899–1900) and two articles that axioms must be true statements about given objects; Hilbert replied that consistency suffices. The book made the abstract axiomatic method a model for mathematics.

大卫·希尔伯特 的更多作品

• 1900数学问题
• 1928数理逻辑基础