The Vicious Circle in Calculating Circle Area and the Local Organization

원의 넓이에 관련된 순환논법과 국소적 조직화

Proofs in school mathematics are regarded as the procedures to examine a proposition's truth or falsehood. However, they are not based on an axiomatic system in general. This implies the possible existence of vicious circles in proofs in school mathematics. The concept of proof can be more completely acquired when accompanied with concept of circular reasoning and necessity of axiomatic system. Therefore, it is necessary to discuss on the axiomatic system in school mathematics curriculum. The vicious circle can be found in computing an area of a circle by using definite integral in differentiation/integration part of high school textbooks. This paper will first illustrate this in detail and then offer several comments on the axiomatic methods related to the dissolution of that circular reasoning. To further the discussion, Archimedes' derivation for the area of a circle will be considered next. Finally, several arguments on circular reasoning, local organization, and axiomatic system in school curriculum will be presented at the end of the paper.

본 논문에서는 학교수학에서 발견할 수 있는 순환논법의 예로서 고등학교 미분과 적분 교과서에서 정적분을 통해 원의 넓이를 구하는 과정에서 발견되는 순환논법을 수학적으로 분석하고, 학교수학의 수준에서 원의 넓이에 관련된 몇 가지 증명방법들의 의미를 비교 분석한다. 특히 원의 넓이에 대한 아르키메데스의 증명과정을 고찰하여 학교수학에서 국소적 조직화의 의미와 가치를 재조명한다.