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The Problem with Algebraic Models of Marriage and

Kinship Structure

James M. Cargal ....................................................................................................345

Special Section on ICM

Results of the 2001 Interdisciplinary Contest in Modeling

David C. “Chris” Arney and John H. “Jack” Grubbs ...................355

A Multiple Regression Model to Predict Zebra Mussel

Population Growth

Michael P. Schubmehl, Marcy A. LaViollette, and

Deborah A. Chun . ...............................................................................................367

Identifying Potential Zebra Mussel Colonization

David E. Stier, Marc Alan Leisenring, and

Matthew Glen Kennedy . ................................................................................385

Waging War Against the Zebra Mussel

Nasreen A. Ilias, Marie C. Spong, and James F. Tucker .................399

Judge’s Commentary: The Outstanding Zebra Mussel Papers

Gary Krahn .............................................................................................................415

Author's Commentary: The Outstanding Zebra Mussel Papers

Sandra A. Nierzwicki-Bauer ..........................................................................421

Reviews ..............................................................................................................................427

Annual Index...................................................................................................................431

Acknowledgments........................................................................................................435

Errata ..................................................................................................................................436

Guest Editorial 345

Guest Editorial

The Problem with Algebraic Models

of Marriage and Kinship Structure

James M. Cargal

Mathematics Department

Troy State University Montgomery

P.O. Drawer 4419

Montgomery, AL 36103

jmcargal@sprintmail.com

Introduction

Algebraic models of marriage and kinship systems have been developed

for nearly 50 years, but they continue to be a matter of controversy. Most

anthropologists have found algebraic models abstract and unnecessary, while

mathematical scientists have tended to consider the value of these models selfevident.

I argue here that the anthropologists have been largely correct, and

that the crux of the matter comes down to the question: What constitutes a

mathematical model?

Abstract algebraic modeling of marriage and kinship systems began in 1949

as an addendum to Claude Levi-Strauss’s seminal The Elementary Structures of

Kinship [1969]. The addendum, by the great algebraist Andr磂 Weil, was “at

Levi-Strauss’s request” [1969, 221]. Variations of the same algebraic model

have appeared subsequently in articles and books into the 1990s (for example,

Ascher [1991]). Perhaps the most influential use of the model was in the finite

mathematics textbook by Kemeny et al. [1966], and its most ambitious application

was in the text by Harrison C. White [1963]. I even wrote a paper myself

on the subject [Cargal 1978].

To some extent I am embarrassed by that paper, as I now believe that my

work on algebraic marriage and kinship systems, like the other works, has little

value. Many anthropologists have attacked such models; Korn and Needham

[1970] offer perhaps the best attack. However, though they make some stinging

points, Korn and Needham are caught up too much in mathematical notation,

TheUMAPJournal 22 (4) (2001) 345–353. cCopyright 2001 byCOMAP, Inc. All rights reserved.

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346 The UMAP Journal 22.4 (2001)

as opposed to mathematical substance. The problem with the models does not

lie in the mathematics itself but in what these models do not do.

The rest of this essay is concerned with:

• What is a (good) mathematical model?

• How good are algebraic models of marriage and kinship systems?

What Is a Mathematical Model?

Essays on the nature of a mathematical model abound in texts on operations

research, simulation, probability, and applied mathematics—but are strangely

absent from physics and engineering books. This is not because physicists and

engineers are less philosophical than their counterparts in other mathematical

sciences, but because physicists routinely use models in introductory courses;

there is no need for a transition in later courses to modeling.

Mathematical models are generally defined as mathematical representations

of the subject at hand. They are abstractions that represent ideal assumptions,

and they are supposed to capture the salient features of the subject and

to leave out the irrelevant features. If the model is a good model, predictions

made from the model should be true of the subject that is modeled. The question

of whether the model is a good model is the core philosophical question. Related

questions are:

• Which features of the subject are relevant for the model?

• To what extent can predictions based on the model be used?

• How accurate are the predictions?

Models and Predictions

Prediction is core to the subject of modelsandcore to this critique of algebraic

models of marriage and kinship systems. I suggest that a model is only as good

as the predictions that it accurately makes. What is the purpose of a model if

not to make predictions? The usual answer is that the model can help us

understand the subject. But if the model does not yield predictions, what is

the value of this understanding? I will give two historical examples of models

in physics and astronomy and will then re-examine the algebraic models of

marriage and kinship.

Two Models by Kepler

Although mathematical models of nature seem to have been important to

the Greeks, mathematical models in the modern sense took off around 1600

Guest Editorial 347

with Galileo, Kepler, and others. Kepler constructed two models of interest to

this paper.

• Planetary motion: If we view Kepler’s laws of planetary motion as a mathematical

model, we can see that it is very much a predictive model. The

model not only constructs the path of the planets but also determines the

speed of their revolution, and use of the model enabled more accurate forecasting

of planetary positions. It may be relevant that this model was based

on painstaking analysis of data that were themselves of unprecedented accuracy.

• Planets as platonic solids: In his Harmonice Mundi, Kepler devised a less

well-known model of the planetary system (see Kappraff [1991, 265]). He

showed that the orbits of the six known planets could be inscribed about

the five platonic solids. The five solids are nested inside one another, and

the six planets nest within the solids. Kepler published this model and was

proud of it for his entire life. The model was predictive in only one sense:

It implies that there are no new planets to be discovered. For its time, it is

not a bad model; it is less mystical than prior Greek theories of the universe.

For our time, the model is primitive. We do not reject the model because its

one prediction failed; the prediction could well have turned out to be true.

We reject the model because by our standards it is contrived, and because it

is not predictive enough.

I contend that the algebraic models of marriage and kinship systems are more

in the style of Kepler’s platonic solids model than in the style of his laws of

planetary motion.

The Quantum Physics Model

Quantum physics is difficult to understand and completely unintuitive,

but it is has been perhaps the most successful model of modern physics. The

quantum theory of physics has been totally dominant in its domain (atomic

mechanics) because of two features:

• It is mathematically consistent.

• It has been the source of thousands of predictions, which in all testable cases

have been correct. The most famous such prediction might be Bell’s theorem

(for an elementary account, see Peat [1990]).

Hence, a mathematical model that is bestknownfor its statement of what cannot

be predicted (i.e., the Heisenberg uncertainty principle) has been as successful

as any model in science history, because it is the exemplary predictive model.

Following the example of quantum physics, we should evaluate algebraic

models of marriage and kinship systems according to the criterion of how well

they predict.

348 The UMAP Journal 22.4 (2001)

Algebraic Models of Marriage and Kinship

The Core of the Algebraic Model

    Algebraic models of marriage and kinship are based on clans and their relationships. In some cases—specifically, the Kariera, the Aranda, perhapsthe Tarau, and perhaps the Murngin—the structure of the kinship system is an algebraic group. This single observation, which is apparently due to Andr´eWeil, is the heart of 40-plus years of writing on mathematical aspects of marriage and kinship systems.
    However, it is not in any way a remarkable observation. Clan systems are either hierarchical or they are not. If, as in the above cases, the clan relationships are not hierarchical, they are likely to be symmetric. Group theory could be described as the mathematics of symmetry (see Armstrong [1988] and Stewart [1992, Chapter 9: The Duellist and the Monster, 115–129]). It would be a remarkable discovery indeed if there were symmetric clan structures that could
not be described in the language of mathematical groups!
    If we use groups to describe symmetrical clan structures, there should be a payoff or return to anthropologists: There needs to be some reason to bring group theory into anthropology. This may seem obvious, but it is not. Anyone who has worked in certain areas of industry has seen mathematical models (and simulation models in particular) that seem to have no purpose. On the
subject of simulation modeling, E.C. Russell says: “The goal of a simulation project should never be ‘To model the . . . .’ Modeling itself is not a goal; it is a means of achieving a goal” [1983, 16]. That models without a purpose exist is a consequence, I believe, of the divergence of mathematics and physics that has been going on for 200 years but which has greatly accelerated in the last 30 years.
    The purpose of a mathematical model is to predict. What do algebraic models of marriage and kinship systems predict? The answer is, nothing.
    Let us ask a simpler question: What information do algebraic models of marriage and kinship systems provide? In his genesis of algebraic models, Andr´e Weil [1969, 221] says “I propose to show how a certain type of marriage and kinship laws can be interpreted algebraically, and how algebra and the study of groups . . . can facilitate its study and classification.” Given the fortyplus years since Weil’s essay, it is reasonable to ask whether this has been accomplished: Has anyone aided the study and classification of marriage and kinship systems through the use of group theory (or any use of abstract algebra whatsoever)?
    Another approach is this: How are the traditional anthropological methods for classifying marriage and kinship structures deficient? How does group theory make up for such purported deficiency?

                                                                          Guest Editorial 349

Prediction and Falsifiability
    A key element in the twentieth-century view of scientific theories is falsifiability,a key doctrine in the work of the eminent philosopher of science Karl R. Popper (see, for example,
Popper [1959]). A scientific theory must be falsifiable, that is, there must potentially be observations or experimental results that would force abandonment of the theory.
    Falsifiability seems nearly equivalent to predictability. A scientific theory must make predictions so that the theory itself can be tested. This seems like a reasonable criterion for scientific theories and for models, but it certainly is not a criterion met by algebraic models of marriage and kinship systems. What predictions do the algebraic models make?
    If algebraic models of marriage and kinship systems are not predictive models, perhaps they are so-called explanatory models. This begs several questions:
• Can a model that makes no predictions be explanatory?
• What exactly does an explanatory model explain?
• How do we judge the merits of an explanatory model?
    Explanatory models don’t predict but perhaps they give us insights. Let us take an example from my own paper [Cargal 1978]. The Kariera have four clans, and their clan system can be represented by the graph in Figure 1.

 


Figure 1. Diagram of the Kariera kinship system. Solid arcs represent a clan of marriage and kinship and the dotted lines represent the clan of a child.


This representation, the traditional algebraic view of the Kariera, is a valid way to introduce algebra, since the graph depicted is also the Cayley graph of
                                                                   350 The UMAP Journal 22.4 (2001)
the Klein four-group Z2 × Z2. In my paper, I suggest looking at the subclans of each sex: “[T]here is sex differentiation among the clans referred to. Natives think in terms of men of Clan A or women of Clan A . . . ” [1978, 161]. I give a graph of the eight subclans and then relabel that graph to realize a Cayley graph of a group. Figure 2 shows the group of subclans of the Kariera.

 
Figure 2. A1 denotes men of Clan A and C0 denotes women of Clan C. The relations are S and O.An S means subclan of children of same sex and an O means subclan of children of opposite sex.

    The group of Figure 2 happens not to be homomorphic to the group of       Figure 1. This is a formal mathematical statement to the effect that the two groups are fundamentally different. The one group, of order eight, is not merely a refinement of the other group, of order four. To my knowledge, the group of order eight does not appear anywhere else in the literature; but it is as valid as the standard group (to whatever extent that group can be said to be valid). 
    We have two algebraic representations of marriage and kinship relations in the Kariera people, that is, two fundamentally distinct models. If these models have anthropological content, they should give conflicting information, or at least the second model should provide new information. My paper gives a great deal of analysis of this and other groups. It is rather enthusiastic analysis and to some extent I find myself a little impressed as I look at it now. But with the distance of more than 20 years since doing the research, I can ask, as anthropologists asked: What significance does this have to anthropology?

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