TITLE:
Differentiation of Elementary Functions by Double Reductio ad Absurdum
AUTHORS:
Kazuhiko Kotani
KEYWORDS:
Double Reductio ad Absurdum, Differentiation, Calculus, Plato’s One, Zeno’s Arrow Paradox
JOURNAL NAME:
Open Journal of Philosophy,
Vol.16 No.1,
February
27,
2026
ABSTRACT: This paper proposes a logical framework for differentiating elementary functions using the double reductio ad absurdum, revisiting the methods established by Eudoxus and Archimedes. In the modern era, digitization using computers is advancing across all fields, including documents, photographs, videos, music, and more. Furthermore, genetic information is digital. The element of digital information is Plato’s One. Consequently, the restoration of Plato’s One is necessary, and this paper demonstrates how the ancient Greeks, starting from Plato’s One, employed the double reductio ad absurdum to expand the mathematical universe from discrete to continuous. Specifically, this paper introduces differentiation by rigorously bounding the derivative between the upper and lower secant slopes. Using this approach, this paper demonstrates how this method determines unique derivatives for power, exponential, and trigonometric functions through the double reductio ad absurdum. While computationally more complex than standard techniques, this approach offers significant philosophical value and serves as an educational tool to provide a logical and intuitive grounding in calculus. Furthermore, this paper resolves Zeno’s arrow paradox, the origin of differentiation, using a double reductio ad absurdum, and conducts a neuroscientific examination of human visual perception, which underpins Zeno’s arrow paradox.