现代几何学的二十条公理

    学习微积分需要利用几何学的知识，这些理论知识可以从20条几何公理推导出来。

    也许，当今大型计算机系统能够“理解”这些公理，变得与人类一样聪明。

    在人类历史上，这二十条几何公理是希尔伯特最先总结出来的，这是他对数学的最大贡献。

袁萌  陈启清  3月8日

附件：希尔伯特《几何学基础》第一章第一节内容：

§1. THE ELEMENTS OF GEOMETRY AND THE FIVE GROUPS OF AXIOMS.

Let us consider three distinct systems of things. The things composing the rst system, we will call points and designate them by the letters A, B, C,...; those of the second, we will call straight lines and designate them by the letters a, b, c,...; and those of the third system, we will call planes and designate them by the Greek letters α, β, γ,...The points are called the elements of linear geometry; the points and straight lines, the elements of plane geometry; and the points, lines, and planes, the elements of the geometry of space or the elements of space. We think of these points, straight lines, and planes as having certain mutual relations, which we indicate by means of such words as “are situated,” “between,” “parallel,” “congruent,” “continuous,” etc. The complete and exact description of these relations follows as a consequence of the axioms of geometry. These axioms may be arranged in ve groups. Each of these groups expresses, by itself, certain related fundamental facts of our intuition. We will name these groups as follows:

I, 1–7. Axioms of connection.

II, 1–5. Axioms of order. III. Axiom of parallels (Euclid’s axiom).

IV, 1–6. Axioms of congruence.

V. Axiom of continuity (Archimedes’s axiom).