Theory of Groups of Permutations, Matrices and Kinship

論親屬結構的數學研究

從十九世紀末以來，社會人類學者開始注意應用數學於親屬結構研究的問題，但由於人類學者的不嫺熟於數學，他們的嘗試一直沒有成功。一九四九年，法國著名的數學者Arxdre Weil 發表一篇婚姻方法代數學研究之論文後，親屬結構數學研究的可能性始被實現。Weil 著眼於婚姻類型（mairiage type）的特殊性質，指出可應用數學的排列群論（theory of groups of permutations） 來從事分組制的研究。之後R.R.Bush，將Weil的方法再加發展，認為使用數學中的排列代數（algebra of permutation）或群論中的一些特別的定理，或矩陣代數學（matrix algebra）等，更適宜於研究婚姻與分組制。此外，他更導入了運算數（operator）的概念，來證實排列矩陣更有益於分析之用。其後又有三個數學者Kemeny, Snell及Thompson，共同發展前人的方法，他們收集了一些調查過的實例，將之系統化而組成一套公設（axiom），確立分組制的理論體系。到了一九六三年，H.C. White推出以分組制為研究對象的鉅著。書中對Kemeny- Snell-Thompson公設有所修正並能滿足這些公設的各種親屬結構，給與有系統的描述。從Weil到White，親屬數學在方法上發展的神速，是使人瞠目的。但他們在資料的處理上有些瑕瑾，致使數學方法的效果大大的減小。Weil —開始就把舅姪婚制的四分組社會誤做母系交表婚制的社會來討論，使他的數學方法建構在錯誤的資料上。錯誤的主因是他不能把分組與嗣系群分清，反而把他們相混合，或者缺乏對分組制婚姻規則與嗣系群數目間有一定關係的認識之故。後來數學者所發展的方法，譬如運算數及排列矩陣的應用，是容易指出Weil的錯誤的，但數學家們反而把Weil的錯誤繼承下來，使他們的論著中把父女婚、母子婚、祖孫婚等荒唐的婚姻制當作母系交表婚制討論。

The applicability of mathematics to the study of Anthropology, especially to kinship structure, has long been discussed by social anthropologists. Studies to this aim have been undertaken especially in connection with the analysis of kinship terminology or section systems. These began toward the end of the last century so the term 'kinship algebra' has a long history. Unfortunately, because most anthropologists are unfamiliar with mathematics, only pseudo-mathematics has been applied. Symbolic notations have been used, but these involve only description and no manipulation. Many papers have brought forth brilliant analyses and ingenious models, but a real scientific base for the mathematical approach has never been established. Not unexpectedly, the 'mathe- matical' results were extremely vague and scanty, and this has caused anthropologists to neglect the mathematical approach. The lack of a foundation in pure mathematics also made the leading scholars condemn kinship algebra as pseudoscience (e.g., Malinowski 1930). Thus for a long time kinship algebra has been treated as a step-child and subordinated to the traditional discipline, and no progress has been made in it. This stagnation continued until very recently.