初等几何的实践性基础及其应用——几何学的产生

Practicalities Foundations of Elementary Geometry and Its Applications———The Produce of Geometry

现行几何理论与实践是有差距的。现行几何理论中的点、直线、平行线、射线、平面都具有理想性。它们分别是误差界趋向于零时近似点、近似直线、近似射线、近似平行线、近似平面序列的极限。极限具有不可达到的性质,所以理想点、理想直线、理想射线、理想平行线、理想平面的存在唯一性和理想合同性都需要用公理的方法去确定。在三维现实空间研究中,对于理想几何元素,欧几里德体系下的初等几何是适当的、需要的;但在应用这个理论于现实问题时,需要有一个“否定之否定”式的过程。几何公理体系的无矛盾性、公理的实际意义都是形式逻辑和数理逻辑无法解决的问题。解决这两个问题都必须使用唯物辩证法。

There is a gap between the current geometry theory and its practice applications. The point, straight line, ray, parallel line and plane in current geometry possess the character of ideal all. They are the limit of se- quence of approximation point, straight line, ray, parallel line and plane separately, when the error bounds tend to zero. But the limit possesses the characteristic that could not be arrived at. Therefore, the character of only existence and ideal congruence of ideal point, straight line, ray, parallel line and plane must apply axi- oms to affirm them. In researching three-dimensional actual space, for ideal geometry element, the system of Euclidean geometry is suitable and necessary;but it needs a process of “negation of the negation” to apply it in reality question. Two questions of consistency of the axiom system and practice function of axioms could not be settled by the method of formal logic and the method of mathematical logic. The settlement of the two questions must apply the method of materialist dialectics.