和算關流分式符號表徵的發展、過渡與概念意義

The Development, Transition and Conceptual Meanings of the Dymbolic Representations about Fractions in the Seki School of Wasan

本研究考察江戶時期重要關流和算家的著作，分析從傍書法發展、過渡至點竄符號系統的過程中，和算家用以表示分式的表徵與符號，並探討相關特色與概念上的意義。研究中發現，分式表徵的發展過程中，共出現了五種表示分式的方式：1.以文字記錄乘除法操作、2.分開記錄被除式與除式、3.將除法運算記於「｜」右邊、4.將除法運算記於「｜」左邊、5.籌式符號「｜」的左右側皆為式或數，左側作為分母、右側作為分子，此分式形成一個數學物件，並可對其執行運算與操作。和算的分式表徵與符號的發展，主要從具程序與操作特色的文字敘述與除法，過渡至數學物件「分式」，即操作性的程序面向先於結構性的物件面向。特別地，分式表徵與符號的發展歷程並非線性，出現了多元的表徵方式，且和算家在分式表徵的使用上，也反應出數學概念具有「程序－物件」二元性並存的特色。而本文釐清了和算家在分式符號使用上的流變，對於判定文本作者、抄寫者或者成書時期，可以當成一項重要的依據。

In this paper, I investigate the contents of the important texts by authors in the Seki school in the Japan's Edo period and analyze the symbolic representations of fractions that Wasan mathematicians used in the process of the development and transition from Boshoho(傍書法) totenzan (點竄), and their related conceptual meanings. Through the analysis, I find that there are five different ways used to represent fractions in the process of the development of symbolic representations: 1.recording the division operation by words; 2.recording the divisor and the dividend separately; 3. recording the division operation on the right hand side of '|'; 4. recording the division operation on the left hand side of '|'; 5. the left hand side and right hand side of '|' being both numbers or rod expressions, where the left hand side indicates the denominator and the right the numerator. By this way, the fraction becomes a mathematical object which can be operated and manipulated. The symbolic representations of the fraction are developed and transited, from the rhetorical mathematical expressions and the kind of division with procedural and operational features, to the structural mathematical object ＂fraction＂, and thus the operational process precedes the structural object. In particular, the development of the symbols of the fraction is not linear, and multiple representations emerged during the process. The symbols of the fraction that Wasan mathematicians used reveal the duality of mathematical conceptions. Finally, the clarification of the development and the shift about the symbols of the fraction used by the Wasan mathematicians turns into an important evidence for modern scholars to determine the author, scriber and the date of a text.