Social Networks 16 (1994) 181-189 North-Hoiland

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Book review

FE. Tjon Sie Fat, Representing Kinship, S~rnp~e aided of E~ernen~u~ Structures (Leiden: Faculty of Social Sciences, Universi~ of Leiden, 19901

This book raises the level of kinship analysis, and will be the impetus for a new series of studies in a field that had grown stagnant. The fundamental new feature of this work is that it integrates relative age into algebraic models of kinship. In most traditional societies, old men have the power to marry several younger women. Never mind that it makes more sense sexually to do it the other way around, since the realities of mate competition revolve around reproductive success.

On the other hand, there are several aspects of this work that require comment. Fat uses the familiar symbols f, m, and h as generators for kinship mappings, where f stands for ‘father’; m, for ‘mother’; and h, for ‘husband’. However, Fat’s interpretation of each function is the inverse of the standard English usage, where f(x) would mean ‘x’s father’. Instead, Fat takes f(x) to be the mapping from a male x to his children. White (1963) denotes this very mapping by C, read ‘Child of a man’, while Boyd (1969) uses f-r to mean the same thing. The reviewers finally realized, however, that if one thinks of f as the ‘result of fatherhood’, then it makes sense. Similar remarks apply to m as the ‘result of motherhood’. However, regarding h as the ‘result of husbanding’ is something of a stretch.

Fat’s notation is troublesome for the following reasons. It is desir- able that algebraic notation for kinship mappings be easily translat- able to standard kinship notation. Consider the relatives MMBDC, read as ego’s Mother’s Mother’s Brother’s Daughter’s Child, and FMBSC read as ego’s Father’s Mother’s Brother’s Son’s Child. These

Elsevier Science B.V., ~s~erd~ ssm 0378-8733(93)00235-0

182 Book reuiew

two kintypes are used to compare Fat’s system of notation with the other systems:

Anthro Fat MMBDC = m-*fm

Boyd White = m*f-‘m-l = (C-1W)2CJj7-‘C

FMBSC = f-‘m-‘f * = fmfp2 = C-*WCC.

Note that to translate from Boyd to White, replace f by C-‘, and m by CIW. In any event, it seems much more intuitive to translate MM into m* than m-*. With a little practice, however, one becomes accustomed to Fat’s system.

Fat extends the usual formalized approaches to analyzing kinship structures. The first chapters construct an algebraic structure within the context of group theory for describing and comparing Australian marriage class systems, such as the Kariera and Aranda. He also derives models that have, as yet, no empirical counterparts.

Fat’s method offers a compact and encyclopedic formalization of kinship structures. He implies that his philosophical viewpoint is a radical revision of the subject, a ‘reconstruction’ that is “fundamental to a critical assessment of the dynamics of theory change” (p. xi). Fat presents this viewpoint as a paradigmatic shift from logical positivism, referred to as the ‘Received View’, to a form of structuralism pio- neered by Claude Levi-Strauss, which is based loosely on structural linguistics. As a philosophy of science, it ranges from the conservative, which uses discrete mathematical structures as a useful tool in orga- nizing and systematizing empirical data, to the radical, which postu- lates specific innate unconscious structures in the human mind that determines the nature and variety of language, mythology, and social organization. Fat leans toward the conservative side of this contin- uum. Although he offers formalized models of the development of kinship systems, he seems more concerned with devising a tool for modelling social structure than postulating about its origins and na- ture. For example, Fat does not use one of the main structuralist principles, the distinction between marked and unmarked categories, which Epling et al. (1973) have applied to advantage in kinship studies. For example, in the male-female distinction, ‘male’ is typi- cally unmarked, whereas ‘female’ is a marked category. This means that the word for female is formed by adding an affix to the word for

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male. In addition, the male term can be substituted for the female, but not vice versa, unless you are politically correct.

It not clear that Fat is offering us something “altogether different (from logical positivism)” when he adopts his version of the structural- ist, or ‘non-statement’ approach, which: “. . . defines theories directly as set-theoretic structures by means of their models and intended applications” (p. 28).

The difference between logical positivism and Fat’s approach ap- pears to be the following. Positivism regards scientific theories as classes of statements, some ‘theoretical’, some ‘observational’. But even theoretical statements are given an observational interpretation by means of correspondence rules. Fat’s structuralism, on the other hand, attempts to avoid the philosophical difficulties implicit in the observational/theoretical dichotomy. Theories are now defined as classes of models, some of which are isomorphic to the real-world phenomena we wish to study. Thus, we are no longer concerned with giving observational interpretations to euety aspect of our theory. A theory is considered adequate if SQLW of its models have real-world interpretations.

Fat says that his “theories are defined directly as set-theoretic structures by means of their models, together with a collection of distinct intended applications, no logical concepts of formal linguistic structure are involved - hence the characterization as ‘non-statement’ or “structuralist”‘. Yet set-theory requires a formal language, such as the first order predicate calculus. ‘Logical concepts’ therefore seem inescapable. At best, Fat’s approach reduces logical statements from being directly involved in scientific theory (the logical positivist ap- proach of theories as logical sentences) to being only indirectly in- volved (theories as collections of set theoretic entities, themselves definable in terms of logical statements).

The mathematical models Fat presents are ingenious and sophisti- cated. He has effectively shown that kinship studies are far from complete and closed. His book may prove to be a source of future inspiration. Unfortunately, he presents his models with minimal refer- ence to the theory of permutation groups, which would have simpli- fied his notation and proofs. The result is a thicket of notation that is needlessly hard to follow. On the other hand, developing the relevant parts of group theory might have put off other readers. Boyd’s (1991) book is perhaps an example of the opposite extreme of putting in too

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much math. On the other hand, Brink and Pretorius (1992) develop a mathematical framework that would have been quite appropriate for Fat’s book.

At other times, Fat seems reluctant to use elementary set theory or to make clear the intended interpretations for his notation. For example, on pp. 91ff Fat defines ‘objects’ in a very unclear way. We think he means a Cartesian product 2 x I, where 2 is the integers representing generations and where I = { 1,. . . , n) represents patrilines (or matrilines).

The section ‘Sets and relations’ (pp. 71ff) contains several errors. For example, a mapping from S into T is defined as a relation f from S to T “such that (s, tr) and (s, t2) are in f if and only if t, = t,, and for each s in S there exists and element (s, t) in f”. The “if and” should be omitted, else no mapping could exist into a set with more than one element.

However, we question the wisdom of presenting Lemma 1 without a proof (p. 92). This lemma asserts that for all positive integers n, the set of all integers between one and n - 1 that are coprime to n form a group under multiplication mod fz. While this is correct, it seems difficult enough to warrant either a proof or a reference to one.

Fat uses two permutations to generate his model of exchange. For a given ‘row’ of 2 x I, the cyclic permutation c maps an element in the ith column onto an element in the i + 1-th row for i < n, and c(n) = 1, making an infinite cylinder. An arbitrary row of this infinite cylindrical matrix forms the starting generation. In this generation, the husband mapping is simply defined as the c permutation, i.e. the ith patriline obtains its wives from the i f 1-th patriline. Using this starting row as the 0th generation, the husband mapping in the ith generation is specified by the c permutation taken to the k to the ith power, where k is a positive integer less than and coprime to n.

Lemma 1 (two paragraphs above) means that choosing k coprime to n has obvious advantages, since it enables Fat to analyze his model in a group theoretic context. However, there is another reason for k to be coprime to n, which Fat does not mention. One can construct a few rows of this matrix, allowing k to share divisors with n. The model invariably stabilizes into a periodic behavior where either (i) the patrilines are partitioned into subgroups, connected by the hus- band mapping, and stay in these subgroups for all subsequent genera-

Book review 185

tions, or (ii) the patrilines all map onto themselves, and stay that way for all subsequent generations.

A model that exhibits such behavior is not a kinship system. Case (8 produces an endogenous caste system; in a sense, this is n disjoint societies. Case (ii) implies inbreeding. Choosing k such that it is coprime to n prevents such behavior. Since the coprime elements form a group, we are also assured that the model will never drift into these undesired behaviors.

In Chapter 3, ‘Age metrics and twisted cylinders: predictions from a structural model’, Fat constructs a model that incorporates age differ- ences between a husband and his spouse. In most societies, men marry younger women. This implies that a man’s MBD is, on average, just the right amount younger for marriage, whereas the patrilateral cross cousin, P’.ZD, is usually too old for our male ego. The usual model of matrilateral cross-cousin marriage is the ‘circulating cunnu- bium’, where at least three lineages marry around in a circle. Fat, however, shows by means of a simple transitivity argument that this cannot produce younger wives for all the men. This is a real contribu- tion to anthropolo~ - a sort of ‘impossibili~ theorem’.

To construct a deterministic model of men marrying younger women, Fat again introduces a series of nodes that are mapped on to an infinite cylinder, where the columns are y1 patrilines, and the rows represent the progression of time. Now, however, the husband map- ping does not map nodes onto other nodes in the same row. The nodes are mapped diagonally downward, to represent the fact to be explained: husband-are generally older than their wives.

By virtue of this model, a man from the ith patriline will always obtain his wife from the i + 1-th patriline. Since i + 1-th is reduced module n, choosing an initial node, which Fat denotes Nf’p q), and iteratively applying the husband mapping will generate a helix travel- ling down the cylinder. Fat entertains the possibility of two or more such helices travelling down the cylinder, forming patterns at least as complicated as the DNA double helix.

Now, since the husband mapping proceeds diagonally down and around, it follows that only a subset of the nodes in a given column will be touched. Suppose we have iz patrilines and only one helix. Then, starting with an initial node in, say, the first column, every subsequent nth node in this column will be touched by the husband mapping. It thus becomes obvious that the patrilines cannot be

186 Book review

defined simply as the columns of the cylindrical matrix. Instead, they must be defined as subsets of the columns.

When we have more than one helix, things get a little more complicated. Fat builds his model so that the vertical and horizontal distance between the node touched by a given helix and the nearest node touched by any other distinct helix must be the ratio of patrilines to helices. Specifically, suppose n, and n2 are nodes on the same row, such that they are touched by different helices. Then the shortest horizontal distance between these nodes (travelling around the cylin- der) is the required ratio. Similarly, the distance between a node touched by helix in a given column and the closest node in the same column touched by another helix is the above ratio. Fat uses the letter b to denote this ratio. In a given helical exchange model, beginning with an arbitrary node in the ith column that is touched by an exchange helix, every subsequent bth node in this column will be touched by some (possibly distinct) exchange helix.

Without this equal spacing between nodes in a column, it would be extremely difficult to derive a formal definition of the successor function. Fat can now define .s as the function that maps N(p, q) (our arbitrary starting node) onto N(p + b, 4). Successive powers of s will thus map N(p, q) onto N(p + bx, q), where x is defined as the set of nodes in the ith column that are mapped onto by some combination of the husband and successor mappings as applied to the starting node.

Note that specifying b will fix the number of exchange helices, assuming, of course, that b is a divisor of yz. Similarly, the number of helices, denoted t, will only determine b if t is a divisor of ~1. Now we have a model in which a man in the ith patriline obtains his spouse from the i + 1-th patriline, and his children ‘hop’ onto the nearest exchange helix beneath the father’s node.

Fat specifies that the number of matrilines be equal to the number of patrilines minus the number of helices. Thus, there will always be fewer matrilines than patrilines. Perhaps this property is an artifact of the positive age difference between a husband and his wife.

It is also interesting to note that specifying the number of matrilines in this manner insures that a son will be in a different matriline from his father. In fact, if m is the number of matrilines, and if we were to obtain a sample of m adjacent ‘touched’ nodes in a given column, each of these m nodes would be in a different matriline. Labelling the

Book review 187

matrilines as, say, 1, 2, 3,. . . , m, travelling down a column would produce a cycle travelling step by step through all the matrilines. This would not hold if the number of matrilines equalled the number of patrilines. We conjecture that such a cycle will hold if and only if the number of matrilines equals the specified difference. Perhaps this feature of the model is a precaution against inbreeding.

Chapter 4, ‘Symmetries of restricted exchange: the twofold path to complexity’, offers an even more sophisticated model. The models in this chapter are intended to apply to kinship systems where certain negative rules apply as to whom a particular person can marry. Again, Fat refrains from supplying any reasons underlying the construction, save, perhaps, the intuitively obvious considerations of mathematical elegance. This chapter returns to the simple cylindrical matrix of Chapter 2.

In this model, the number of patrilines is constrained to be an even number, say, 2m for some integer m. A permutation r is a series of m disjoint cycles:

r = (1 2) (3 4) . . . (2m - lm),

where (x y> is the permutation that interchanges x and y. The evolution of the exchange cycles over time is defined in terms of an automorphism that does not leave r invariant. Beyond this constraint, the choice of the automorphism is arbitrary. Two additional permuta- tions are defined:

4 = (2m 1) (2 3) . . . (2m - 2 2m - 1)

and a = rq. The basic permutations r and a generate the exchange structure. The automorphism is allowed to leave some points invari- ant.

Varying r will allow a sufficient amount of variety from generation to generation (i.e. as we travel down the rows of the infinite matrix), avoiding the problems of schisms and inbreeding. But is there a motivating reason to choose r as the permutation that must vary under the automorphism? Why not a?

In the model of Chapter 3, a choice of k that is not coprime to iz could generate models that converge upon a ‘caste-system’ of disjoint cycles. The model in Chapter 4 allows a particular model to enter and

188 Book review

leave such cycles. The earlier models could not enter such states without being trapped in them. This new model is thus much stronger.

It would be interesting to know which aspects of Fat’s model stem solely from theoretical, versus empirical, concerns. For example, is it a necessary consequence of restricted exchange systems that the number of patrilines be an even number? If so, what would happen if one of the patrilines were to die out? Would the kinship system change irrevocably?

Fat’s writings on chaos also provoke some interesting speculation. In Chapter 5, he offers some illustrative figures that demonstrate the sensitivity of his simple cylindrical matrix model to initial conditions. What kind of implications could this have for the evolution of kinship systems over time? In the section ‘Local rules and emergent proper- ties’ (pp. 246-249) he presents an interesting example: let a,, be a matrix of OS and 1s. In his example there are 40 columns, but the rows are unbounded since they represent time. One of the transition rules he considers, F22, defines the sequence

1 a f+l,j =

if u,,~_~ + u,,~ + u,,~+~ = 3

0 otherwise,

where the second subscript is reduced modulo 40, so that it wraps around in a circle. In order to be well-defined, the first row has to be given. Subsequent rows are given by the rule that an entry is 1 iff there is exactly one 1 in the three closest entries in the row above. This is similar to a one-dimensional ‘game of life’. Fat claims that this rule may be ‘chaotic or aperiodic’, because it does not repeat after 38 time periods. We have three comments. First, since rule F22 is a function on a finite set (having 240 elements), we know a priori that it must repeat eventually. Second, we could not resist writing a small program that found the first repeat: a134 = a,ro, so it has a period of 24, assuming our program is correct. Last, his matrix has an error in row 1: the 41st column should be a 1; which suggests that any ‘chaos’ comes from a bug in his program.

We also have some reservations on the empirical basis for these complex systems. Is it possible that he is over-interpreting the lists of kin types that anthropologists have casually put into footnotes? Delet- ing or inserting one kin type would completely destroy one of the helical models depending on prime numbers. His models are implausi-

Book review 189

ble, and since these societies are almost all extinct, there seems little hope of (dis)confirmation. Finally, is there not more hope in looking at statistical models that would give predictions on the relative fre- quencies of marriages based on simple moiety or marriage class systems together with the male bias toward younger wives?

Who should read this book? We would recommend it to social scientists who have the mathematical sophistication to fill in the gaps and lapses in a rather difficult exposition. However, Representing Kinship may very well prove to be a source book that inspires future endeavors.

John Boyd, Carl Hirschman,

Timothy Brazil School of Social Sciences

University of California Irvine, CA 92217

USA

References

Boyd, J.P.

1969 “The algebra of group kinship”. JoumnZ of Mathematical PsychoZogy 6: 139-167. 1991 Social Semigroups. Fairfax, VA: George Mason University Press.

Brink, C. and J. Pretorius

1992 “Boolean circulants, groups, and relation algebras”. American Mathematical Monthly 99: 146-152.

Epling, P.J., J.P. Boyd, and J. Kirk

1973 “Genetic relations of Polynesian sibling terminologies”. American Anthropologist 5: 1596-1625.

Fat, F.E. Tjon Sie

1990 Representing Kinship: Simple Models of Elementary Structures. Leiden, The Netherlands: Faculty of Social Sciences, Leiden University.

White, H.C.

1963 An Anatomy of Kinship. Englewood Cliffs, NJ: Prentice-Hall.