A Historical and Mathematical Study on Continuum : Focusing on the Composition and Infinite Division
**Professor, Seoul National University, South Korea yochoi@snu.ac.kr
Abstract
This study conducted historical and mathematical analyses of the composition of continuum and infinite division. Regarding the results of the historical analysis, first, it can be seen that mathematicians struggled with the problems related to Zeno’s “paradox of plurality” in the composition and infinite division of the continuum. Second, historically, infinite small appeared frequently. Third, the problem of continuum, which was a geometric problem, was not explained by geometry alone, and mathematical answers to this problem became possible after the straight line became homomorphic with the real set, following the arithmetization of mathematics. Through mathematical analysis, Zeno's “paradox of plurality”, which includes questions about the composition and infinite division of a continuum, could be explained depending on the uncountability of real numbers and the countable additivity of Lebesgue measure. In addition, it was confirmed that it is inappropriate to divide a straight line to a point. The historical and mathematical analyses of this study suggest that students may have cognitive difficulties when dealing with the composition and infinite division of a continuum in relation to area, volume, and integral in school mathematics. The continuum is a concept closely related to the limit and calculus of school mathematics, and the historicalㆍmathematical analysis of this study could serve as a foundation for teaching and learning plans of limits and calculus.
Keywords:
Continuum, Zeno, Paradox of Plurality, Infinitely Divisible, Uncountability of Real Number Set, Lebesgue MeasureReferences
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