和算知識中的術、法、表之意義與特色

The meanings and characteristics of mathematic knowledge in Wasan-Jutsu, hō and Hyō

知識從事者認為重要而值得探討的問題，及他們所尋求適當答案的種類，往往因社群而異。由這些社群所生產出來的知識種類也展現出差異。本研究貼近和算文本與脈絡，闡述和算家所發展的數學知識類型－術、法、表－探討它們的意義與功能，並比較之間的差異。「術」為程序性且機械性的演算法，可視為將問題的數據逐步作運算、操作、轉化，最終獲得答案的作業流程，有時也表徵了某些公式或定理。它既是和算家尋求的答案類型，也是求得數值解的重要媒介，並揭示了數量之間的抽象關係。「法」是程序性的數學運算或數學方法，藉以計算求得數值，或常被鑲嵌於術文中，作為執行演算法的子程式或作業流程中的子程序。「表」則記錄、濃縮了數學知識，呈現出數學概念與數學物件之間的靜態關係與結構，並作為重要認知工具，被用於求解問題以及探索、核證知識的過程中。和算家立法則、創諸表作為基礎數學工具，提出數學「問題」後，存在研究與實作兩種知識活動面向：利用相關知識輔以「法」的使用與「表」的檢索，造出概括性的演算法「術」。再者，據已知數（量），依「術」的流程，搭配「法」或「表」，利用計算工具，在紙或計算表面上機械化地逐步操作、計算，求得相應的數值解。 The problems that practitioners consider crucial and worth researching as well as the appropriate solutions they seek are vary by community. As a result, the types of knowledge produced by communities also differ. This paper aims to explain three types of knowledge, namely jutsu (術), hō (法), and Hyō (表), in the context of Wasan by identifying and discussing their meanings and functions and comparing them. Jutsu is considered a procedural and mechanical algorithm and can also be seen as a procedure by which operations and computations are applied to the values in mathematics problems step by step to result in an answer. Jutsu is an instrument for obtaining numerical solutions and the types of solution preferred by Wasan mathematicians. Sometimes, Jutsu connotes a formula or theorem and reveals an abstract relationship between values or magnitudes. Hō is a procedural mathematical operation or method through which Wasan mathematicians can calculate values, and it is often a subprogram or a subprocedure of the primary algorithm in the procedure of Jutsu. Hyō records and condenses mathematical knowledge and reveals the structures of and relationships between mathematical concepts and mathematical objects, and it is treated as an instrument for problem solving and is used in the processes of exploring and justifying knowledge. When Wasan mathematicians propose mathematics problems, two aspects of knowledge activities related to mathematics research and practice are considered: They construct Jutsu using the hō or the hyō or mathematical knowledge relevant to problems, and they operate and calculate step by step on paper or another surface used for calculating to obtain the numerical solutions according to the steps of Jutsu through the use of calculators.