为了数学的明天，，穿越时空，重返南大（III）
2019-08-16 19:41阅读：
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
8月18
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
na
tura 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
Bibliography
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