10 展望未来展望任何一门科学的未来都是十分困难的,但是,我们仍然坚信当代最伟大的数理逻辑专家哥德尔(Gdel)的话:“我们有充分的理由相信,以这种或那种形式表示的非标准分析,将成为未来的分析学.”首先,从上述历史和现状的叙述,我们可以看到,微积分初期就处在一种朴素的非标准分析状态.从非标准分析的角度来看,牛顿和莱布尼兹的概念和推理基本上是正确的.虽然莱布尼兹关于微分 dx 及 dy 在不同的地方有不同的解释,但是,在紧要关头,他总是认为 dx 是一个正无穷小,而dy 是无穷小(可能是正无穷小,负无穷小或零).当莱布尼兹把曲线的切线定义为连结曲线上无限接近的两点(即这两点隔开一段距离,而这段距离小于任何指定的长度时)的直线时,他把这两点的横坐标之差称为 dx,把这两点的纵坐标之差称为 10 Looking Ahead The future of any science is difficult, but we remain convinced of what Gdel, the greatest mathematical expert in our day, said: “We have every reason to believe that one or the other The formal analysis of non-standard, will be the future of analytics. ”" First of all, from the history and current status of the narrative, we can see that calculus initially in a simple non-standard state of the analysis from non-standard analysis Newton’s and Leibniz’s concepts and reasoning are basically correct, and although Leibniz has different interpretations of the differences dx and dy in different places, at the crucial moment he always thinks dx is a positive infinitesimal, and dy is infinitesimal (may be infinitesimal, infinitesimal infinitesimal or zero.) When Leibniz defines the tangent of the curve as two points that are infinitely close to the connecting curve (ie, the two points are separated by a distance, And this distance is less than any specified length) when the straight line, he said the difference between the two abscissa called dx, the difference between the two ordinates as