ESHS/HSS Annual Meeting

A Study on the Correlation Between the Symbolic System and Innovative Achievements of Japanese Mathematics

• In Event: Diagrammatic Computations and Their Symbolic Aspects (I)

English Abstract

During the Edo period, Wasan (和算, Traditional Japanese mathematics) developed a distinctive symbolic and theoretical system in a relatively isolated academic environment. While the mathematical advancements of this period were influenced by classic Chinese mathematical works such as Introduction to Suan Xue Qi Meng (《算學啟蒙》, the Introduction to Mathematics ), Wasan also achieved breakthroughs through localized innovations. As the carrier of mathematical expression, the evolution of the symbolic system is closely intertwined with the emergence of innovative achievements. Japanese scholars drew on the symbolic features of Tianyuan Shu (天元術, the Celestial element procedure) to construct Bōsho-hō(傍書法, the Side-writing method. This is a notation that represents polynomials using the numerical coefficients of arithmetic trees and Chinese character textual coefficients, through this approach, multiple unknowns can be handled simultaneously) and Endan-jutsu (演段術, the Elimination technique, this is a special mathematical method used by Japanese scholars for eliminating unknown variables, some scholars interpret it as an explanation of an algorithmic technique) symbolic systems, thereby accomplishing significant mathematical innovations. For instance, Seki Takakazu (1640?-1708,関孝和, a renowned mathematician from Japan during the 17th and 18th centuries) first introduced a dot to represent unknown quantities in Kaikendai no Hō (解見題之法, Methods of Solving Explicit Problems, What Japanese scholars consider to be the mathematical work written by Seki Takakazu in 1683 has no published edition; only handwritten copies exist), pioneering the abstraction of Wasan’s symbolic system. Subsequently, Takebe Katahiro(1664-1739,建部賢弘, a renowned mathematician from Japan during the 17th and 18th centuries, who was a student of Seki Takakazu) further expanded Seki’s symbolic framework in Hatsubi Sampō Endan Genkai (発微演算法演段諺解, a mathematical work written by mathematician Takebe in 1687, it provides a comprehensive explanation of the mathematics book Hatsubi Sampō by teacher Seki Takakazu). By analyzing the symbolic innovations of representative scholars such as Seki and Takebe, this study reveals the intrinsic connections between these symbolic advancements and major breakthroughs in fields including algebra, calculus, and geometry. It concludes that the original design of Wasan symbols directly enhanced computational efficiency, facilitated the solution of complex problems, and promoted the construction of theoretical systems, thereby providing crucial support for the leapfrog development of Japanese mathematics within the Eastern intellectual tradition.

Author

• Rina Sa