The Four Names That Were Not Enough

In 1899 Baldwin Spencer and Frank Gillen printed a table of who may marry whom in Central Australia, and in the next paragraph said it was incomplete. Their own published tables say exactly what it was missing. Four categories commute, so a man's first cross-cousin sits in the category he must marry. Eight categories do not commute: they carry the algebra of the eight symmetries of a square, and the first cross-cousin lands one subsection short. The general form of that fact was proved by Andre Weil, in an appendix, at Claude Levi-Strauss's request, in 1949. What the counted marriages then show is stranger than either: the coarse rule is kept almost always, and the fine one almost never.

Every table here is transcribed from a named published source. Nothing about a system is asserted that your browser has not just recomputed from it.

A system of social categories is a strange kind of machine. Every person carries one word. The word is not chosen: it is computed, falling out of the words their mother and father carried. And from two of those words, two strangers who have never met and can trace no shared genealogy can read off the category of their relationship to each other.

Across much of Aboriginal Australia these words are called skin names, and they are in daily use now. Some peoples use four, some eight, some none. This page is about a question that the difference between four and eight raises, and about the fact that the two men who first tabulated an eight-name system in print noticed the answer and could not state it.

The four-name table, 1899

Spencer and Gillen were working among the Arrernte, whom they called the Arunta. In the southern part of the country they surveyed, people used four names. The rule, in their words:

That is a complete specification. It fixes who marries whom, and it fixes what the children are called. Below, the machine is running on that table and nothing else. Pick anyone and walk.

Two things are worth noticing before anything else. Walk FF, a person's father's father, and you come home: in all three of these systems a man's son's son carries the man's own name. Now walk the mother's line. In the four-name systems you are home after two steps. In the eight-name system it takes four.

The paragraph that gives the game away

Having printed the four-name diagram, Spencer and Gillen immediately undercut it. This is the sentence the rest of this page is about:

Read that slowly. A Panunga man must marry a Purula woman. But the Purula women are not interchangeable to him. Some he may marry and some he may not, and the four-name system, whose entire job is to tell him whom he may marry, does not distinguish them. The name is doing arithmetic, and the arithmetic is coming out coarse.

In the north, they report, people used eight names, and the eight-name system had separate words for the two groups. They tabulate it, and add a remark we will come back to at the end:

What the algebra says the missing distinction is

Take the table as a piece of mathematics and nothing else. There are two maps from categories to categories. Call w the marriage map, so w(x) is the category a person of category x must marry. Call m the descent map, so m(x) is the category of the children of a woman of category x. Everything else follows. The children of a man of category x are m(w(x)), because he marries a woman of w(x) and their children take what her line gives them.

Compose those two maps in every order and you get a group. Here is the group of each of the three published tables, generated in your browser from the tables above:

The four-name systems give the Klein four-group, which is commutative. The eight-name system gives the group of the eight symmetries of a square: four rotations and four reflections, and it is not commutative. The matrilineal cycle is the rotation. Marriage is a reflection. That is not an analogy. It is the same group, and it was identified the same way by Gisele De Meur and Alain Gottcheiner in 2000, who write that for the Arrernte system "m and f are represented in this group by a 90 degree rotation and a mirror, respectively."

Now the point, which is one line long.

So a commutative system is a first-cross-cousin system, and a non-commutative one is not. The four-name table commutes, so a Panunga man's first cross-cousin is Purula, the section he must marry. The eight-name table does not commute, so his first cross-cousin is not in the subsection he must marry. She is one subsection off, and the subsection she is in is the other half of the old Purula.

That is the distinction Spencer and Gillen said their diagram could not show.

Weil's appendix, 1949

The theorem above is not new here. It is the central move of a seven-page appendix that Claude Levi-Strauss commissioned from Andre Weil, one of the foremost mathematicians of the century, for Les structures elementaires de la parente. Weil opens:

In these few pages, written at Levi-Strauss's request, I propose to show how a certain type of marriage laws can be interpreted algebraically, and how algebra and the theory of groups of substitutions can facilitate its study and classification.

Andre Weil, "On the Algebraic Study of Certain Types of Marriage Laws (Murngin System)", appendix to Claude Levi-Strauss, The Elementary Structures of Kinship, trans. Bell, von Sturmer and Needham, Beacon Press, 1969, p. 221

He works with marriage types rather than categories, and with the maps taking a marriage to the marriage its son and its daughter will contract. Then he introduces a third condition and computes what it costs:

(C) Any man must be able to marry his mother's brother's daughter.

[...] This condition is known, in the theory of groups, as the permutability of the substitutions f and g. The pairs of permutable substitutions can be studied and classified according to known principles. In the language of the theory of groups (which it is unfortunately almost impossible to express in non-technical terms without going into very long explanations), the group of permutations generated by f and g is an Abelian group, which, having two generators, is necessarily cyclic or else is the direct product of two cyclic groups.

Weil, same appendix, pp. 222-223

That is our theorem in his notation. His maps are not ours, so the equivalence is worth checking rather than asserting, and the browser has just checked it on all three tables:

There is a sting in this. Weil's condition (C) is an assumption, adopted to make the algebra tractable, and it is exactly the assumption the Arrernte eight-subsection system does not satisfy. The system Spencer and Gillen had tabulated fifty years earlier is ruled out by Weil's third axiom before the analysis begins. Weil was candid that his hypotheses were chosen for convenience: elsewhere in the same appendix he calls one of them "perhaps unduly restrictive, but which will serve." The rest of the appendix studies only commutative systems.

The last four buttons are a check from outside. The anthropological literature independently names four second-cousin types as the marriageable ones in an Arrernte-type system: MMBDD, MFZDD, FFZSD and FMBSD. None of them went into building anything here, and all four land in the prescribed spouse's subsection when walked across the 1899 table.

The old system is the new one divided by grandmother

There is a clean way to say what the four extra names do. Collapse the eight subsections back onto the four by pairing each subsection with the one two steps along the mother's line, which is Spencer and Gillen's own pairing of the old name with the new one. That collapse turns the eight-name marriage rule into the four-name marriage rule, and the eight-name descent rule into the four-name descent rule. The old system is the new one divided by the map mother's mother. And mother's mother is precisely the centre of the group: the one non-trivial thing in an eight-subsection system that commutes with everything else.

An aside: the algebra settles an argument the book is having with itself

Working through that collapse turns up something small and satisfying. Two pages after tabulating the eight names, Spencer and Gillen say which new name is the other half of the old Panunga, and they say it twice, differently.

The first makes Uknaria the other half of Panunga. The second makes it Ungalla. They cannot both be right, and nothing on either page settles it. The arithmetic does, because only one of the two readings lets the eight-name tables collapse onto the four-name ones. The marriage rule survives either way, which is presumably why the slip was easy to make. The descent rule does not.

So the page 71 reading stands and the page 74 sentence is a slip: Ungalla is the other half of Purula, not of Panunga. It is a small thing, and it is the sort of thing an algebra is good for. The tables kept a consistency their own prose did not, for a hundred and twenty-seven years, until something mechanical went looking.

How many such systems could there be?

Set the ethnography aside and ask what is possible. Suppose only four things, all of them read off the tables above rather than imposed:

• marriage is symmetric, and nobody marries inside their own category;
• a woman's children have one determinate category, and different mothers' categories give different children's categories;
• a man's son's son is back in the man's own category;
• the system is connected: any category can be reached from any other by some chain of marriages and births.

Enumerate every pair of maps on n categories obeying those four rules, and throw away relabellings. The answer is very small.

Two shapes for every even number of categories, and every group that appears is dihedral. With two categories you get the two kinds of moiety and nothing else. With four you get the Kariera shape. With eight you get the Arrernte shape. The last column is the interesting one, because it is arithmetic and it is not obvious:

• four categories: marry a first cross-cousin;
• six categories: marry a third cross-cousin;
• eight categories: marry a second cross-cousin.

The prescribed spouse is the k-th cross-cousin for the least k with m^2k the identity, so a mother's line of length c needs c to divide 2k. Four categories give c = 2 and so k = 1; eight give c = 4 and so k = 2; six would give c = 3, an odd number, and so k = 3. Six categories would push the prescribed marriage further out than eight do. That is what the arithmetic gives, and no more: we are not claiming it explains why four and eight are the sizes attested in Australia, and we found no source that argues it.

A six-category system does exist, a long way away. On Ambrym in Vanuatu, Bernard Deacon described six marriage classes in 1927, and the structure is non-commutative: the symmetries of a triangle. Our census says the connected regular six-category shape is unique and is exactly that group. We have not transcribed the Ambrym tables ourselves, so this page does not run them; the agreement is between our enumeration and a published description of them.

What the name cannot say

A word carrying eight values cannot encode a genealogy. It is a lossy compression, and the question worth asking is what it throws away. Enumerate every genealogical path of a few steps and count how many come home to the same word.

Every position in that last list is formally marriageable, by the name alone. They are not all marriageable in fact. Which is what both founding sources say, in almost the same words, thirteen years and two thousand kilometres apart. Spencer and Gillen have already been quoted. Here is A. R. Brown, later Radcliffe-Brown, on the Kariera in 1913:

Neither man is describing a system that tells you whom to marry. Both are describing a system that tells you whom you may not marry, and leaves the rest to knowledge the name does not carry. Modern specialists say it more bluntly. Laurent Dousset: "Sections are not marriage classes and thus cannot be taken as the sole element determining marriage partners." And Radcliffe-Brown himself, thirty-eight years after the Kariera paper: the classes "result from the giving of names to kinship divisions which can quite well exist, and do exist, without names as part of the kinship system organized by means of the kinship terminology."

Then how well is the rule actually kept?

This is where the tidy story earns its keep or does not, and the counted marriages give an answer sharper than the question. Two different things get counted, and they come out completely differently.

The pattern is consistent and it is the opposite of what a reader might expect. The skin rule, the coarse category, is kept in something like nine or ten cases out of ten. The fine prescription, the actual mother's mother's brother's daughter's daughter that the anthropological literature so often presents as the rule, is kept in one or two cases out of ten. Denham's Alyawarra corpus makes the contrast inside a single dataset: one wrong marriage in 114 by skin, and twenty per cent compliance with the stated genealogical preference.

Which is precisely what the algebra predicts a lossy compression should do. The name pins down a wide category and people stay inside it. The name does not pin down a person, and people do not marry the person it fails to pin down.

What Weil actually concluded

One last turn, because the founding text of this whole mathematics ends in a failure that is usually left out of the retelling. Weil's target was not the Arrernte but the system W. Lloyd Warner had reported from north-east Arnhem Land, which Warner called Murngin and which is Yolngu country. Weil worked his eight-class model through and found that it does not hold together. The society it describes is reducible: it falls into two populations with no kinship tie between them.

Levi-Strauss's own commentary on the result is remarkable, and it is printed immediately after the appendix:

In connexion with Murngin society properly speaking, it is difficult to imagine that a society should function in the circumstances suggested by the theory and still maintain its individuality. [...] It remains to be seen how the Murngin manage to avoid the danger of their theoretical system, which can only be done by applying it incorrectly.

Claude Levi-Strauss, "Commentary", The Elementary Structures of Kinship, 1969, pp. 227-228

The model works only if the people do not follow it. He is also clear, elsewhere in the same book, that the units the algebra runs on are the analyst's and not anybody else's: of Weil's sixteen marriage types he writes that "one can be sure that in elaborating this system, the aboriginal mind never had recourse to these sixteen categories."

So the honest summary of the mathematics is narrower than its reputation. It is exact about what the published tables imply. It is exact about what a four-name system cannot distinguish and an eight-name one can. It said nothing true about how anyone lives, and the man who commissioned it said so in print.