Murngin親屬結構的數學研究

MATHEMATICAL STUDY OF THE MURNGIN SYSTEM

近年來，在親屬結構研究的領域裏，最為突出的一件事，是數學家們在“親屬代數學（kinship algebra）”上的成就，把多年來的懸案，逐步付諸解決。遠在十九世紀末，社會人類學者就已經注意到澳洲土著族甚於制定婚姻（prescriptive marriage） 而形成的分組制度（section system），是適合於用數學的方法來加以分析的。有許多學人曾不斷的從事這方面的嘗試（參閱B. Malinowski 1930），惜限於數學知識，而未曾成功。一九四九年，Andre Weil發表了一篇運用代數學研究Murngin制婚姻法則的論文，開應用新數學來分析婚姻法則的先河。Weil提出三個條件，作為組構分組制度的基本原則：（A）對每一個成員來說，不論男性或女性，都有一種而且是惟一的一種婚姻類型（marriage type），是他（或她）們有權可以履行的。（B）就每一個成員來說，他（或她）可以履行婚姻類型的條件，決定於他（或她）的性別，以及他（或她）由之而所出的婚姻類型關係的兩造。（C）所有的男性，都應可和他們的母親兄弟的女兒結婚。在方法上，Weil著眼於婚姻類型的特殊性質，即子女的婚姻類型的排列為父母婚姻類型的不同順序的排列，指出可應用數學的“排列群論（theory of groups of permutations） ，來從事分組制度的研究。至於Murngin現行的八分組制，由於其婚姻規定和上揭的原則相抵觸，Weil導入了新的數學方法，採用模數為2的n項加法制 （the addition of n-tuples modulo two system）來證明 Claude Levi-Strauss 所提有關Murngin婚姻制度的假設。此後，Robert R. Bush發展Weil的方法，介紹了數學中的“運算數（operatei）' 的概念；證實了 “排列矩陣（permutation matrices）”更有益於分析之用。由於“恒等運算數（identity operator）'的產生和運算數的種種結合的方式，所與社會的嗣系線的世代循環法則及其婚姻法則，可自動的顯示出來（參閱White 1963: 159-72）。

In 1949 Andre Weil proposed a unique concept of 'marriage types' for the analysis of section systems. He wished to prove that the theory of groups of permutations is applicable to the study of prescribed marriage systems. Weil's idea was developed by a group of mathematicians, though sporadically, and thus a kind of new mathematical approach to the kinship structure has been established. Following Weil, Robert Bush in his undated mimeo manuscript introduced the 'permutation matrices' as a more effective tool for analysis. Kemeny, Snell and Thompson (1956) first systematized the properties of the prescriptive marriage systems as an integrated set of axioms. All distinct kinship structures which satisfy the above-mentioned axioms are systematically derived and described by Harrison White (1963), who adopting more practical 'generators' to set the structural analysis on a more concrete bases. In the past two decades the kinship mathematics developed .by the mathematicians has made a remarkable progress, but ther still remain controversial problems, and the effective range of the mathematical approach is obviously limitted. White himself recognized the failure of his structural analysis of those societies such as Murngin or Purums, which practice unilateral cross-cousin marriage. There may exist many reasons for this failure, as discussed by White and Reid (1967)， but it is time to question whether the method they applied is sound or not. If we examine the writings of Weil and his successors, the fatal fact emerges that their method does not allow distinction of the oblique marriage from the cross-cousin marriage. This defect is caused by the fact that the mathematicians neglect important anthropological phenomena, as for example, the regulation of marriage rules among the descent groups (Liu 1968). Here I propose the following hypotheses for a new mathematical approach to the prescriptive marriage systems. (1) The matrilateral cross-cousin marriage system is described by an n-generation cycle or circulation, where the marriage cycle (or circulation connubium) is derived from n hordes or clans, the minimum number for n being 3. (2) Under the same condition the patrilateral cross-cousin marriage system is described by a 2-generation cycle, or the so-called cycle of 'alternating generations.' (3 ) The bilateral cross-cousin marriage system is described by the above men-tioned two principles, the 'n-generation' and' 2-generation' cycles, but in this case n may be 2 or more.