大卫·希尔伯特

作品

• 1899几何基础

  从五组公理出发，对未加定义的“点”“直线”和“平面”进行欧几里得几何的公理化构造；公理的一致性和独立性通过模型来证明。弗雷格反对说，公理必须对给定的对象为真。

• 1900数学问题

  在巴黎国际数学家大会上的演讲；其印刷版提出了23个未解决的问题（演讲本身只讲了十个），其中包括连续统假设和算术公理的一致性，并坚信在数学中没有“我们将不知道”（ignorabimus）。

• 1928数理逻辑基础

  与威廉·阿克曼根据希尔伯特的讲课合写的教科书：它阐述了命题逻辑和一阶逻辑，并提出两个悬而未决的问题——一阶逻辑是否完全，以及是否存在判定程序——分别由哥德尔（1929）以及丘奇和图灵（1936）作出解答。

立场

Hilbert is on the map for the foundations of mathematics and logic. His “Foundations of Geometry” (1899) built Euclidean geometry on groups of axioms whose basic terms – point, line, plane – receive their meaning only from the axioms; he proved the independence of axioms by models and held that the consistency of axioms suffices for the existence of the objects they define. Frege objected in letters and two articles that axioms must be true statements about given objects; the exchange is a classic of the philosophy of mathematics. In response to the paradoxes of set theory and to Brouwer’s intuitionism, Hilbert proposed in the 1920s to formalise mathematics completely and to prove the consistency of these formal systems by “finitary” means, reasoning about finite strings of signs that he regarded as secure as elementary arithmetic. Infinitary mathematics would thus be justified as an instrument, and no one should drive mathematicians “from the paradise that Cantor created”. With Ackermann he also posed the decision problem: is there a mechanical procedure that decides whether a formula of logic is valid? Gödel’s incompleteness theorems (1931) showed that the consistency of arithmetic cannot be proved within arithmetic itself, so the programme could not succeed in its original form; Church and Turing (1936) answered the decision problem in the negative. Modified versions of the programme continue in proof theory. His Königsberg address of 1930 ended with the words “We must know – we will know”, against the “ignorabimus” of Emil du Bois-Reymond.

生平与背景

David Hilbert was born on 23 January 1862 in Königsberg in East Prussia (according to some sources in nearby Wehlau) and studied mathematics there, where Hermann Minkowski and Adolf Hurwitz became his friends; he took his doctorate in 1885 under Ferdinand Lindemann. He became professor in Königsberg in 1893 and moved to Göttingen in 1895, where he stayed until his retirement in 1930 and helped make the university a centre of mathematics; Edmund Husserl was his colleague there from 1901 to 1916. At the International Congress of Mathematicians in Paris in 1900 he spoke on mathematical problems; the printed version of the lecture set out 23 problems that shaped much research in the century that followed. He supported Emmy Noether’s habilitation against resistance in the faculty. In the 1920s he worked with Paul Bernays and Wilhelm Ackermann on the foundations of mathematics. After 1933 many of his colleagues and students were driven out by the Nazi regime. He died in Göttingen on 14 February 1943.

其他主要作品

• Grundlagen der Geometrie (1899)
• Mathematische Probleme (Paris lecture, 1900)
• Neubegründung der Mathematik (1922)
• Über das Unendliche (1926)
• Grundzüge der theoretischen Logik (with W. Ackermann, 1928)
• Grundlagen der Mathematik (with P. Bernays, 2 vols., 1934–39)