Kazuhiko Kotani

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.

Plato’s concept of “One” is a fundamental philosophical idea characterized by indivisibility, invariability, and equality. This paper explores the relationship between Plato’s One and modern scientific disciplines, including digital information theory, biology, and neuroscience. Firstly, the paper examines how numerical representation and digital data processing reflect the properties of Plato’s One. Secondly, the paper extends this discussion to biological systems, arguing that life exhibits the characteristics of Plato’s One through natural selection. Thirdly, the paper examines how natural selection gives highly conserved DNA bases properties close to Plato’s One. Finally, the paper analyzes how neural activity, specifically action potentials, aligns with the properties of Plato’s One and serves as the foundation for neural activity. By integrating insights from philosophy, evolutionary biology, and cognitive neuroscience, this study proposes that Plato’s One is not merely a metaphysical abstraction but an intrinsic principle underlying information processing, life, and cognition. The implications of this perspective suggest a deep structural connection between classical philosophical thought and contemporary scientific understanding. In conclusion, this paper provides the concept of Plato’s One as a common ground for philosophy, mathematics, information science, biology, and neuroscience.

Plato placed great importance on the natural number one. Plato stated that the three properties of Plato’s one are indivisible, invariable, and equal and that an ideal one has no physical properties. If we regard differences among lives as differences among genes and gene products and life as the container of genes, then life has properties similar to Plato’s one. Furthermore, life has properties of multiplying when it survives and disappearing when it dies. These properties of life are prerequisites for natural selection. Natural selection makes DNA become digital information, enabling accurate copies of DNA. Furthermore, DNA bases, essential for survival, become close to Plato’s one by natural selection. In addition, the information content of life must be sufficiently large for death to be irreversible, and the irreversibility of death is a prerequisite for natural selection.

What is number? This question is difficult to answer. Because the number is one of the most basic concepts, it is difficult to define the natural number with other concepts. Still, this problem is worth trying to answer. Now, everything is digitized and processed on computer. The importance of the number is increasing day by day. Now is time to consider what number is. Throughout the history of humankind, the ancient Greeks considered this question most profoundly. In particular, Plato defined the natural number one. The natural number one is equal, invariable and indivisible. These properties are intuitively acceptable. However, we have never seen or touched the natural number one itself. How can we know it? Socrates said that we know it before birth. This claim is called anamnesis. In this paper, we use a method, in which Socrates’ anamnesis is studied by the contemporary science. From a modern viewpoint, we could take Socrates’ anamnesis to mean that the natural number one is written in our genes. This article considers whether there is a biological entity corresponding to the natural number one. As a result, we find that a life itself is the prototype of the natural number one, and then properties of life make a critical base of DNA similar to the natural number one through natural selection. A life is an integrated and indivisible system, which resists the law of entropy. Furthermore, the basic properties of life enable natural selection, which conserves genetic information despite the law of entropy. The source of the power, which enables life to resist the law of entropy, is the genetic information. In conclusion, a life is a prototype of the natural number one. Furthermore, a life recognizes nature using natural numbers and resists the law of entropy using natural numbers.

The derivative is a basic concept of differential calculus. However, if we calculate the derivative as change in distance over change in time, the result at any instant is 0/0, which seems meaningless. Hence, Newton and Leibniz used the limit to determine the derivative. Their method is valid in practice, but it is not easy to intuitively accept. Thus, this article describes the novel method of differential calculus based on the double contradiction, which is easier to accept intuitively. Next, the geometrical meaning of the double contradiction is considered as follows. A tangent at a point on a convex curve is iterated. Then, the slope of the tangent at the point is sandwiched by two kinds of lines. The first kind of line crosses the curve at the original point and a point to the right of it. The second kind of line crosses the curve at the original point and a point to the left of it. Then, the double contradiction can be applied, and the slope of the tangent is determined as a single value. Finally, the meaning of this method for the foundation of mathematics is considered. We reflect on Dehaene’s notion that the foundation of mathematics is based on the intuitions, which evolve independently. Hence, there may be gaps between intuitions. In fact, the Ancient Greeks identified inconsistency between arithmetic and geometry. However, Eudoxus developed the theory of proportion, which is equivalent to the Dedekind Cut. This allows the iteration of an irrational number by rational numbers as precisely as desired. Simultaneously, we can define the irrational number by the double contradiction, although its existence is not guaranteed. Further, an area of a curved figure is iterated and defined by rectilinear figures using the double contradiction.