“Intuition and Reflection” in Tanabe’s Early Philosophy of Mathematics.

Abstract

This paper considers the early philosophy of mathematics of Tanabe Hajime (1885-1962). Tanabe began his philosophical studies with an interest in the foundation of mathematics. He obtained a Ph.D. in 1918 and published his dissertation titled An Inquiry into the Philosophy of Mathematics (1925). Although this fact is well known in general, there is very little research on his early philosophy of mathematics and thus the contents and importance of this work have not been fully appreciated. This paper examines Tanabe’s early works on mathematics from the perspective of Nishida Kitarō, who was Tanabe’s Ph.D. supervisor. Tanabe was specially interested in Nishida’s concepts of “intuition” and “reflection”, and established the foundation of mathematics through his theory of genetic cognition, that was organized logically according to mathematical rationality. Section one situates and introduces Tanabe’s early philosophy. Section two provides an overview of the problem of genetic cognition and the generation of numbers. Section three examines how Tanabe establishes the foundation for natural numbers and then considers the significance of Tanabe’s philosophy from the perspective of intuition and reflection. Section four focuses on Tanabe’s theory of real numbers in his text “Continuous, Differential and Infinite” (1916). Here, I argue that the transcendental meaning of irrational numbers could be understood as “intuition”.