## A Non-Commutative Marriage System in the South PacificThe Ambrym system has at its heart a fundamentally non-trivial mathematical object: a non-commutative group...Tony Phillips
## IntroductionThe social structure of isolated indigenous populations has given anthropologists a useful quantitative tool for studying how people have evolved and dispersed. These structures often incorporate general abstract principles, but the precise way these principles are worked out can vary enormously from society to society. Marriage patterns are especially complicated and especially stable. An analogy from the natural world is the organization of the flowering parts (the reproductive organs) of plants. The variety is enormous, but because the structure is integral to the way one generation produces the next, it tends to be very stable in time; its analysis is an essential part of the study of species and their relatedness. The island populations of the South Pacific, and in particular the the inhabitants of Ambrym (part of the Vanuatu archipelago, roughly 1500 miles northeast of Australia), have received sustained attention in this respect since the late 19th century. Ambrym is prominent in Lévi-Strauss' One aspect of the marriage system on Ambrym that anthropologists have noted, besides its intricacy, is the lucidity with which the people themselves explain its workings. Bernard Deacon (1927) observed: "It is perfectly clear that the natives (the intelligent ones) do conceive of the system as a connected mechanism which they can explain by diagrams. The way they could reason about relationships from their diagrams was absolutely on a par with a good scientific exposition in a lecture-room." So it is especially satisfying to note that the Ambrym system has at its heart a fundamentally non-trivial mathematical object: a non-commutative group. ## The male perspectiveThe following diagram is adapted from Rio's Figure 3.1. Blue triangles, males; red circles, females. This cylindrical diagram shows the division of Ambrym society into three patrilineal lines, and the possible marriage relationships between them, in the style of a genealogical tree. The three male lines are marked here as M, M' and M''. In each of these lines there is an alternation between generations: those who marry "right" (+) and those who marry "left" (-) in this diagram. Thus an M'_{+} chooses a wife among the sisters of M''_{-} individuals, whereas an M'_{-} marries a sister of of an M_{+}.
Note that there is a more genealogical distinction between the "+" and the "-" generations. A member of a "+" generation marries into a slightly Implicit in this diagram is a non-commutative group with 6 elements, generated by the kinships The equations S_{3} of symmetries of an equilateral triangle. Each symmetry corresponds to a permutation of the vertices, labeled here as 1, 2, 3. Matching the "son" kinship S with the permutation (12) [flip about the axis through vertex 3] and the "son-in-law" kinship T with the permutation (123) [counterclockwise rotation by 120^{o}] extends to a complete identification of the two systems. The diagram on the left is in fact quite similar to that drawn in the sand by Deacon's native informant in 1927 (Rio, p. 49).
The group ## The female perspectiveRio's diagram can be redrawn to show the female lineages.
Ambrym society has two female lineages, here labeled W _{+} and W_{-}, and in each lineage there are functionally three generation-classes, here distinguished by 1, 2, 3. The "+" lineage only marries males from the "+" generations; the "-" lineage only marries males from the "-" generations. The kinships D : mother --> daughter and E : mother --> daughter-in-law satisfy D^{3} = I, E^{2} = I and D^{2} = EDE, generating another copy of S_{3}. Observe a complication in the diagram: the generations W1_{+}, W2_{+}, W3_{+} pair with M'_{+}, M_{+}, M''_{+} respectively, whereas W1_{-}, W2_{-}, W3_{-} pair with M_{-}, M'_{-}, M''_{-} respectively (different order).
The "male perspective" is more biological because there are six male genotype clusters (Y-chromosome, mitochondria) which follow the ## Ambrym and the French intelligentsia ...The publication of Deacon's diagram in ## ... including André WeilIn a completely different direction, Lévi-Strauss enlisted none other than André Weil in the mathematical elucidation of kinship structures; Weil contributed an eight-page appendix "Sur l'étude algébrique de certains types de lois de mariage (Système Murngin)" to the first part of Weil understood the essence of a marriage-kinship system to be incorporated in a set of permitted marriage-types, satisfying the following axioms: - (A) Every member of society, male or female, fits in exactly one permitted marriage-type.
- (B) For each member of society, the designated marriage-type is a function only of that individual's sex and the marriage-type of that individual's parents.
So the marriage-kinship system can be condensed into two functions, Applying Weil's analysis to Ambrym, we find six marriage-types: - A = (M
_{+}, W2_{+}) - B = (M
_{-}, W1_{-}) - C = (M'
_{+}, W1_{+}) - D = (M'
_{-}, W2_{-}) - E = (M''
_{+}, W3_{+}) - F = (M''
_{-}, W3_{-}).
These and the permutations
Since Unfortunately for us, the system Weil analyzed (the Murngin inhabit the northern tip of Australia) is mathematically less interesting than that of Ambrym. In the Murngin marriage code, the only sanctioned marriages are within a special class of "cross cousins:" between a man and one of his mother's brothers' daughters (equivalently, a woman and one of her father's sisters' sons). Weil encodes this as an additional axiom: - (C) A man must be able to marry the daughter of the brother of his mother.
and then proves that Axiom C forces the group generated by The rest of "Sur l'étude algébrique ..." only examines commutative systems. Tony Phillips Those who can access JSTOR can find some of the papers mentioned above there. For those with access, the American Mathematical Society's MathSciNet can be used to get additional bibliographic information and reviews of some these materials. Some of the items above can be accessed via the ACM Portal , which also provides bibliographic services. |
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