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竭诚请教网友 -- 芦笛 - (4616 Byte) 2010-5-06 周四, 上午8:28 (758 reads) |
唐好色
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作者：唐好色 在 驴鸣镇 发贴, 来自 http://www.hjclub.org
The concept of infinitesimals was introduced early by Leibniz and perhaps Newton also, but it was not rigorously defined. They were meaningless mathematically and everyone knew it. It is a kind of crutch that if used correctly produces the right results. Feynman's path integral is similar today.
In other words, everyone was talking nonsense when they talked about the infinitesimals back then. It is unfair to fault Marx for talking nonsense just like everyone else. Eventually, they figured out how to define limits rigorously and the whole idea of infinitesimals was abandoned. If you read the definition of limits, there is no mention of infinitesimals. The only thing one can say about Marx is that he wasn't really on top of math research because, by that time, mathematics had pretty much abandoned the concept of infinitesimals.
However the idea of infinitesimals is intuitive and can help undergrads learn the mechanics of calculus quickly. So they talk about infinitesimally close and what not while being vague the whole time. This is enough for most of the engineering students and some physicists also to compute things. In classical analysis, dx and dy are not well defined (on the other hand, dy/dx is well defined).
To do calculus correctly, one needs to learn the rigorous definition of limits (the epsilon-delta construction). This is usually taught in a baby real analysis class in the 3rd year.
The idea of infinitesimals was resurrected in the 1960's and this time put on firm and rigorous mathematical grounds, but the concept isn't needed and they are not part of general mathematical education. Only people who want to specialize in that field learn it. To the vast majority of mathematicians, limits are all they need and infinitesimals are just curiosities.
作者：唐好色 在 驴鸣镇 发贴, 来自 http://www.hjclub.org |
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