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为了数学的明天,,穿越时空,重返南大(III)

(2019-08-16 19:41:21)

为了数学的明天,,穿越时空,重返南大(III

    进入二十一世纪,非阿基米德数学(比如:含有无穷小的连续统)逐渐兴起,我们用该如何面对?

    这是一个基本问题,必须彻底搞清楚,事实求是.

  请见本文附件。

 

袁萌  陈启清   818

附件:

Continuity and Infinitesimals

First published Wed Jul 27, 2005; substantive revision Fri Sep 6, 2013

The usual meaning of the word continuous is “unbroken” or “uninterrupted”: thus a continuous entity—a continuum—has no “gaps.” We commonly suppose that space and time are continuous, and certain philosophers have maintained that all natural processes occur continuously: witness, for example, Leibniz's famous apothegm natura non facit saltus—“nature makes no jump.” In mathematics the word is used in the same general sense, but has had to be furnished with increasingly precise definitions. So, for instance, in the later 18th century continuity of a function was taken to mean that infinitesimal changes in the value of the argument induced infinitesimal changes in the value of the function. With the abandonment of infinitesimals in the 19th century this definition came to be replaced by one employing the more precise concept of limit.

Traditionally, an infinitesimal quantity is one which, while not necessarily coinciding with zero, is in some sense smaller than any finite quantity. For engineers, an infinitesimal is a quantity so small that its square and all higher powers can be neglected. In the theory of limits the term “infinitesimal” is sometimes applied to any sequence whose limit is zero. An infinitesimal magnitude may be regarded as what remains after a continuum has been subjected to an exhaustive analysis, in other words, as a continuum “viewed in the small.” It is in this sense that continuous curves have sometimes been held to be “composed” of infinitesimal straight lines.

Infinitesimals have a long and colourful history. They make an early appearance in the mathematics of the Greek atomist philosopher Democritus (c. 450 B.C.E.), only to be banished by the mathematician Eudoxus (c. 350 B.C.E.) in what was to become official Euclidean” mathematics. Taking the somewhat obscure form of “indivisibles,” they reappear in the mathematics of the late middle ages and later played an important role in the development of the calculus. Their doubtful logical status led in the nineteenth century to their abandonment and replacement by the limit concept. In recent years, however, the concept of infinitesimal has been refounded on a rigorous basis.

1. Introduction: The Continuous, the Discrete, and the Infinitesimal

2. The Continuum and the Infinitesimal in the Ancient Period

3. The Continuum and the Infinitesimal in the Medieval, Renaissance, and Early Modern Periods

4. The Continuum and the Infinitesimal in the 17th and 18th Centuries

5. The Continuum and the Infinitesimal in the 19th Century

6. Critical Reactions to Arithmetization

7. Nonstandard Analysis

8. The Constructive Real Line and the Intuitionistic Continuum

9. Smooth Infinitesimal Analysis

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