Abstract
The mathematical competence of non-literate cultures expressed in the design of complex sociological structures has been recognized since a path-breaking Appendix by the mathematician André Weil to Lévi-Strauss’s treatise on kinship structures. The import of Weil’s contribution was to highlight the role of symmetries underlying kinship structures and the algebraic concept of a group which can be seen as a general theory of symmetry. The kinship structure of the Cashinahua people who inhabit the south-western Brazilian Amazon is a unique example of symmetry in social organization. This point is illustrated here by means of a correspondence between the group of actions of Cashinahua kinship terms on Cashinahua name-sake classes, and of symmetries connecting graphic patterns, showing an underlying non-trivial structure known as a dihedral group.
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Notes
- 1.
This is tradition of functionalist theories of kinship, represented by Malinowski, Leach and Fortes. In Leach´s terms, “kinship is a language for expressing property rights” (Leach, 1961).
- 2.
This program began with André Weil´s Appendix to Lévi-Strauss’ Elementary Structures of Kinship (1967), followed by an extensive literature (Courrège, 1971; Gregory, 1986, 2015; Lorrain, 1975; Samuel, 1967; Weil, 1967; White, 1963; Tjon Sie Fat, 1990), to which a further flood of papers was added since Louis Dumont applied Lévi-Strauss´ marriage theory as exchange to so-called cognatic societies of South India, where no descent categories are named (Dumont, 1953; Trautmann, 1981; Overing (Kaplan) 1975; Viveiros de Castro, 1998).
- 3.
- 4.
Ballonoff tried to connect the two domains in a series of papers, unfortunately written in mathematical language inaccessible to cultural anthropologists (Ballonoff, 2017).
- 5.
- 6.
This is not a trivial condition. It is not valid in the Western kinship terminology.
- 7.
Sian was at the State University of Campinas under a scholarship for cinema studies, with no commitment to any research on kinship.
- 8.
Among the Kayapo (Mebengekrore) people of Central Brazil, there are “triadic” terms translated, in the context of a woman addressing her husband, as “your daughter”, meaning “your daughter who is also my daughter” – carrying a different connotation from “my daughter” (Lea, 2004).
- 9.
The symbol * stands for the group operation of composing two kinship terms. Gender signs were omitted with loss, since only male relations are included in Fig. 3.
- 10.
- 11.
This is in a nutshell the difficulty faced by Dumont´s interpretation of “Dravidian” cognatic kinship terminology in terms of the opposition between “consanguines” and “affines”: a father and a mother are consanguineous (“cognatic” assumption), as a “brother” and “sister” are. Now, a “father´s sister” is a composition of a “consanguineous with a consanguineous”, but it results in an “affine” (non-consanguineous) relationship, and a “mother´s brother” relationship is a composition of “consanguineous relationships with a consanguineous relationship”, but it produces an “affine relationship” as well. This could suggest an algebraic rule as “consanguineous plus consanguineous = “affine”. But this is inconsistent with “a brother´s brother” is a brother, and “a sister´s sister” is a sister (Almeida, 1990). Thus, an algebra of consanguinity and affinity is not trivial.
- 12.
In Fig. 10, I will present a different set of generators: a reflection and a rotation.
- 13.
As it happens, this is the group of symmetries of a square. There are eight symmetries – transformations that leave the square invariant: two reflections (up-down, left–right), two diagonal reflections (along the south-west/north-east axis, and along the north-west/south-east axis); and four 90o rotations, totalling eight symmetries. These symmetries are represented as the eight transformations of a thorn design in the following graphs.
- 14.
An isomorphism of groups is a one-to-one correspondence T between two sets A and B with operations * and ^ respectively, such that T(a*b) = T(a) ^ T(b). An example of isomorphism is the correspondence between multiplication of real numbers and sum of their logarithms: in this case, log(ab) = log(a) + log(b). A numerical example with base-2 logarithms:
log 2 (4 × 16) = log 2 (64) = 6, and log 2 (4) + log 2 (16) = 2 + 4 = 6.
An homomorphism is a many-to-one correspondence that preserves the structure. An example is the correspondence S between the set of integers (0, 1, 2, …) with addition + and the set {0,1} with “computer” addition +’. Here the correspondence maps even numbers to 0 and odd numbers to 1. Under this transformation,
S(n + m) = S(n) +’ S(m). Numerical example: S (2 + 3) = S (5) = 1, S (2) +’ S (3) = 0 + 1 = 1.
An homorphism is a transformation that collapses distinctions while preserving structure. In this text, it collapses gender distinctions while preserving the structure of kinship operations. This is not a universal feature of kinship terminologies.
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Acknowledgements
This work is indebted to Viveiros de Castro’s exhaustive review of Dravidian Systems literature (1998) and to Thomas Trautmann’s monumental “Dravidian Kinship” (1981), as well as to Tjon Sie Fat’s complete synthesis and expansion of the mathematical theory of marriage systems (1990). I am also indebted to Dwight Read’s critiques and to Paul Ballonoff’s reception to my early paper on this theme in the Journal of Mathematical Anthropology and Cultural Theory. Tom Trautmann’s work and personal encouragement was fundamental, as well as the intellectual inspiration and friendly support of Marshall Sahlins and of my wife Manuela Carneiro da Cunha.
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Almeida, M.W.B. (2022). Indigenous Mathematics in the Amazon: Kinship as Algebra and Geometry Among the Cashinahua. In: Vandendriessche, E., Pinxten, R. (eds) Indigenous Knowledge and Ethnomathematics. Springer, Cham. https://doi.org/10.1007/978-3-030-97482-4_8
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