무한개념의 발달

Abstract

Ⅰ. Introduction It is necessary to study the development of "the Infinity" in order to comprehend the today mathematics theory. The major purposes of this thesis are 1. to study the history of the mathematics from the view-point of the "paradigm" theory 2. to study the development of the "Infinity" in relation to the change of the mathematical paradigms in the mathematical history Ⅱ. The change of the paradigms in the mathematical history The structure of the development in the mathematical history can be appropriately analyzed by means of the T.S Kuhn's "paradigm" theory. The mathmatical history can be categorized macroscopically as the old, the medieval, the modern (16C-19C), todays(19C- ). Each period has its specific characters as following table. ◁표삽입▷ (원문을 참조하세요) Ⅲ. The development of the infinity In this chapter, the development of the infinity is presented as the followings-unlimitedness, infinitesimal, limit, infinite set. 1. Unlimitedness The concept of "the infinity" meant "large" in the old times and was perceived "finity that has the limit", "partial enlargement of the finity", "summary of the finity". 2. Infinitesimal In the eucleides geometry of the Greek ages, the whole should be larger than the part. A part can be divided into many parts endlessly, and is consisted of many indivisible atoms. They might recognize the concept of the infinitesimal, which was more finite than infinite. In the mordern times the concept of the infinitesimal was used in the method of the computation areas. Cavalieri defined that line and plane was the indivisible quantity of the plane and volume in computing areas. Cavalieri's infinity was bounded in the finite, never to be the infinity which transcended the boundary of the finite. 3. Limit Cauchy defined the sum of the infinite series. He also defined the limit of series as "ε-δ" which meant that he considered the infinity as transcendental. However Cauchy's infinity was bounded to "the pro-time infinity" 4. Infinite set Cantor developed the infinite set as an axiom and recognized the natural number set as bounded infinite, therefore he could arithmetize the infinity. Hillbert developed the Cantor's concept of the infinit set into the infinite-dimensional space. Today all mathematical concept is constructed under the base of the infinity concept. Ⅳ. Analysis and infinite Among the correlation between the development of the Analysis and the infinite concept, the development of series, function, Fourier series, Rieman integral, Cantor's set theory are studied. Ⅴ. Conclusion Now the technical computation is treated as more primarily than the infinity concept in the highschool curriculum, which makes most students neglect the infinity concept. If a teacher comprehend the infinity(its concept, its development process, etc..) he could pump-prime students' concern of the infinity concept. Therefore, his students could understand the infinity more easily than the others' students.;오늘날의 수학은 무한개념을 빼고서는 성립할 수 없다. 본 논문의 목적은 수학사에 있어서 Paradigm의 변천과 이에 따른 무한개념의 발전과정이 어떠하였는가를 고찰하는데 있다. 먼저 본 논문의 기초적 개념이 되는 T.S Kuhn이 제시한 Paradigm의 의의를 알아보고, 고대, 중세, 근대, 현대의 시대별로 수학적 Paradigm의 변천과 그 시대의 무한에 대하여 기술하였다. 그리고 무한개념의 발달과정별로 무한대, 무한소, 극한, 무한집합등을 분석 고찰함으로써 무한개념 및 그 개념의 발달경위등을 상세히 전개하여 논술하였다. 또한 16세기 이후부터 연구되어온 해석학의 발달과 무한개념과의 관계를 알아보았다. 이상으로 본 연구는 무한개념의 발달과정이 전시대의 누적된 연구의 산물인 아니라 새로운 사고방식 또는 사상에 의하여 창조된 것이라는 점을 부각시켰고, 무한개념은 현대수학의 사상적 기반이 되고 있음을 고찰하였다.