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国内数学守旧派为何惧怕超实微积分?

(2019-08-24 14:52:58)

国内数学守旧派为何惧怕超实微积分?

   坦率地说,进入二十一世纪,数学形态发生巨变,形式化方法广泛流行。

国内数学守旧派,故步自封,沉迷于十九世纪数学,“亲”不够,惧怕

“对接”现时代超实微积分。 

   对超实微积分有兴趣读者可见本文附件文章。

袁萌  陈启清   824

附件:

Hyperreal calculus

Abstract This project deals with doing calculus not by using epsilons and deltas, but by using a number system called the hyperreal numbers. The hyperreal numbers is an extension of the normal real numbers with both innitely small and innitely large numbers added. We will rst show how this systemcanbecreated,andthenshowsomebasicpropertiesofthehyperreal numbers. Then we will show how one can treat the topics of convergence, continuity, limits and dierentiation in this system and we will show that the two approaches give rise to the same denitions and results.

Contents 1 1 Construction of the hyperreal numbers 3 1.1 Intuitive construction .. . . . .. . 3

1.2 Ultralters . 3

1.3 Formal construction . . . . . . .. . . 4

1.4 Innitely small and large numbers . . . . . . . . . . . . . . . . . 5

1.5 Enlarging sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.6 Extending functions . . . . . . . . . . . . . . . . . . 6

 

2 The transfer principle 6

2.1 Stating the transfer principle . . . . . . . . 6

2.2 Using the transfer principle . . . . . . . . . . . . . . . 7

3 Properties of the hyperreals 8

3.1 Terminology and notation . . . . . . . . . . . . . . 8

3.2 Arithmetic of hyperreals .  . . . . . . . . 9

3.3 Halos . . . . . . . . . . . . . . . . . . . . . . . 9

3.4 Shadows . . . . . . . . . . . . . . . . . . . 10

4 Convergence 11 4.1 Convergence in hyperreal calculus. . . . . . .. 11 4.2 Monotone convergence . . . .  . . . . . 12

5 Continuity 13


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