第五公设的背景及其对数学发展的意义

The Background of the Fifth Postulate and Its Significance to the Development of Mathematics

欧几里德仅仅依靠形式思维,通过图形佐证形成体系,在第29个命题中首先运用第五公设。由于第五公设的某些矛盾性以及与其他公理、定义、公设的独立性,历代数学家积极探索,或试图弥补,或试图拓展新领域,大大发展了数学。而第五公设对数学发展的最大意义恰恰来自于对第五公设的否定即非欧几何的诞生。

Depending only on formal thinking, Euclid formed system through figure proof, firstly applied the fifth postulate in the 29th proposition. For the self- contradiction and the independence with other axioms, definitions and postulates, modern and past mathematicians have been actively explored it, making up it or expanding new filed, which greatly developed mathematics. While, the most significance lies in the negation to the fifth postulate, that is, the birth of non Euclidean geometry.