What mathematics can and cannot settle about kinship

Kinship looks as if it had been designed for a mathematician. There is a small alphabet of relations (F father, M mother, B brother, Z sister, S son, D daughter, H husband, W wife), strings that read outward from Ego, so that MB is the mother's brother, and a rule for composing them: a father's father is FF. Every language then sorts the endless supply of strings into a short list of words. Yet for several decades some of the most influential figures in social anthropology argued that formal models of kinship were misleading or beside the point, while the formalists went on producing results. Both sides had a case. The clearest way to see why is to take one question that mathematics answers well and one it cannot answer at all.

A question mathematics answers well

Take three men in the generation above Ego: F, FB and MB. In how many ways can a language sort them into terms? Each way is a partition of a three-element set, and a three-element set has exactly five partitions (the Bell number B3 = 5):

Robert Lowie named four of these arrangements in 1928, and English belongs to the lineal one: father stands alone, and FB and MB are both uncle. The fifth, one word for father and mother's brother and another for father's brother, is just as available on paper, and it is the one that does not turn up as a regular type. Françoise Héritier treated its absence as support for a sociological law of kinship. Per Hage (1997) offered a different explanation, following the linguist Joseph Greenberg: it is a case of a general tendency to avoid disjunctive categories.

You can check what "disjunctive" means by hand. Describe each man with two features:

A term is conjunctive when a single list of feature values picks out exactly its members. "Collateral" picks out FB and MB; "parallel" picks out F and FB; "lineal" picks out F alone; "cross" picks out MB alone. Every term in the first four arrangements can be named this way. F and MB together cannot: they share no value on either feature, and the one thing they do have in common, being a man of the parents' generation, is true of FB as well. Only the fifth arrangement needs an "or" in its definition.

The counting grows quickly. Sibling terms can attend to three two-way features (the sibling's sex, older or younger than Ego, and same sex as Ego or not), which gives 2 × 2 × 2 = 8 sibling types. The number of ways to partition eight things is the Bell number B8 = 4,140; of those, 127 use exactly two terms and 966 use three. Sara Nerlove and A. Kimball Romney (1967) worked in this space when they built a typology of sibling terminologies, and they tied one distinction, same-sex against opposite-sex siblings, to social lives that put weight on relations between brothers and sisters. Only a small fraction of the 4,140 possibilities turn up with any frequency.

This is what the mathematics is good at. It measures the space of possibilities, shows which regions of it can be described simply, and turns "why do we never see that one?" into a precise claim that a single new terminology could refute.

A question it cannot answer

Now a different kind of question. When David Schneider worked on Yap in 1947 and 1948, his hosts told him that intercourse played no part in conception. One influential definition of kinship, Harold Scheffler and Floyd Lounsbury's of 1971, required local theory to treat intercourse as necessary for engendering a child. Taken literally, the Yapese then had no kinship until some twenty years later, when, under the influence of American schooling, they adopted a different theory of conception. Schneider (1984) used the case to argue that the genealogical definition had carried a Euro-American idea of blood relationship into what was meant to be a neutral tool.

No theorem settles this. A formal model takes its primitive relations as given. Supply genealogical positions and it describes the Yapese terms genealogically; supply household obligations and it describes them that way, with the same algebra. Whether the Yapese themselves reckon by genealogy is a question about the Yapese, answered by fieldwork and argument. Even the formalists built this into their position: Scheffler and Lounsbury held that a genealogical grid is justified only if it is part of the culture being described, which makes its presence an empirical claim rather than a mathematical one. A model can tell you what follows from a definition. It cannot choose the definition. The note on formal analysis of kin terms follows that argument further.

The long argument, in brief

The research areas page gives the timeline. The shape of the dispute is what matters here.

Edmund Leach, in a 1958 essay on Trobriand clans, argued that kin terms are labels for the social groupings a person is taught to recognize from birth, and he called the Trobriand terminology illogical. Lounsbury's reply of 1965 showed that a few consistent rules, applied to genealogical positions, reproduce the Trobriand usage. Leach's Malinowski Memorial Lecture of 1959, "Rethinking anthropology", gave its title to his 1961 collection. In it he took the Trobriand claim that father and son stand to each other much as male cross-cousins or brothers-in-law do, and nothing like mother and child, as a pattern the analyst has to accept as stated, whatever ideology lies behind it. Whatever his quarrel with genealogical readings, it was not a quarrel with abstraction: the same book reprints his own structural model of matrilateral cross-cousin marriage.

Rodney Needham, introducing the 1971 volume Rethinking Kinship and Marriage, proposed treating kinship as a matter of how rights are allocated and handed down the generations. Later he brought the biologists' idea of polythetic classes, held together by overlapping resemblances rather than one shared feature, into anthropology (Needham 1975), an idea that sits awkwardly with any typology built on a single defining criterion. Schneider went furthest, doubting that "kinship" names a comparable domain across societies at all.

The formal reply took two forms. One was to find order where critics saw none, as Lounsbury did for the Trobriand terms and as Per Hage and the mathematician Frank Harary did across three books of graph-theoretic anthropology between 1983 and 1996. The other was to rebuild the foundations so that the mathematics did not presuppose genealogy, as in Dwight Read's kin-term algebra. The earliest piece of that algebraic line, André Weil's treatment of marriage classes, is covered in our explainer on Weil.

The 2008 essay

David Jenkins's "Anthropology, mathematics, kinship" (2008) is a tribute to Hage, who died in 2004, and to his long partnership with Harary. It surveys their three books and Hage's last project, an account of how kin terminologies change over time built on Greenberg's linguistics, and argues that graph theory let them compare societies without flattening the particular relations each one recognizes. It closes by defending the comparative record against what Jenkins saw as a discipline losing touch with its own past.

Since then

The comparative line did not stop. A 2011 volume edited by Doug Jones and Bojka Milicic, Kinship, Language, and Prehistory, carries the subtitle Per Hage and the Renaissance in Kinship Studies. In 2022 Jones returned to Lounsbury's Trobriand analysis in the journal Kinship, recasting it with Optimality Theory so that the reduction rules are derived from ranked constraints, a tradeoff between saying enough and avoiding marked forms, rather than simply stipulated. The line that followed Schneider went another way: Janet Carsten's Cultures of Relatedness (2000) is one example of work that studies relatedness through people's own statements and practices, many of which fall outside older definitions of kinship.

To watch both approaches work on the same few words, read Leach (1958) on the Trobriand terms, then Lounsbury (1965), then Jones (2022): three generations of argument over one short vocabulary.

Sources and further reading

• Jenkins, David. 2008. "Anthropology, mathematics, kinship: a tribute to the anthropologist Per Hage and his work with the mathematician Frank Harary." Mathematical Anthropology and Cultural Theory 2(3). https://escholarship.org/uc/item/1hs7w49v
• Hage, Per. 1997. "Unthinkable categories and the fundamental laws of kinship." American Ethnologist 24(3): 652-667. https://doi.org/10.1525/ae.1997.24.3.652
• Nerlove, Sara, and A. Kimball Romney. 1967. "Sibling terminology and cross-sex behavior." American Anthropologist 69(2): 179-187. https://doi.org/10.1525/aa.1967.69.2.02a00050
• Lounsbury, Floyd G. 1965. "Another view of the Trobriand kinship categories." American Anthropologist 67(5, part 2): 142-185. https://doi.org/10.1525/aa.1965.67.5.02a00770
• Leach, E. R. 1961. Rethinking Anthropology. London: Athlone Press (London School of Economics Monographs on Social Anthropology 22).
• Needham, Rodney, ed. 1971. Rethinking Kinship and Marriage. London: Tavistock.
• Schneider, David M. 1984. A Critique of the Study of Kinship. Ann Arbor: University of Michigan Press.
• Jones, Douglas. 2022. "Yet another view of Trobriand kinship categories, from optimality to conceptual structure." Kinship 2(1). https://doi.org/10.5070/K72151391