Algebraic models of marriage systems have been in use for nearly fifty years. These models continue to be a matter of controversy. Most anthropologists have found algebraic models abstract and unnecessary. Mathematical scientists have tended to consider the value of these models self-evident. I argue here that the anthropologists have been in large correct, and that the crux of the matter comes down to the question of what constitutes a mathematical model. Different mathematical cultures answer this question in different ways. Although there is some formal mathematical content to this note, it is written with the intent that the non-mathematical reader can easily absorb the gist.

Abstract algebraic models of marriage in anthropology began in 1949 as an addendum to Claude Levi-Strauss's seminal The Elementary Structures of Kinship [1969]. The addendum was written by the great algebraist Andre Weil "at Levi-Strauss's request" [1969, p. 221]. Subsequently variations of the same algebraic model have appeared in articles and books into the 1990's (for example, Ascher [1991]). Perhaps the most influential use of the model was in the textbook by Kemeny, Snell, and Thompson [1966]. The most ambitious application of the model was in the text by Harrison C. White [1963]. I will make reference in this article to my own paper on the subject [1978]. My paper does have a few analytical features worth mentioning. However, the purpose of this article is not to try to bring attention to my earlier work. The purpose of this paper is nearly the opposite. Whereas I feel that I did a number of things beyond what else I have seen in this area; to some extent I am embarrassed by the paper. I believe that my work on algebraic marriage systems, like the others, has little substantial value. Whereas many anthropologists have said as much quite forcefully, they generally use the wrong arguments. The best paper attacking the algebraic models is the paper by Korn and Needham [1970]. However, though their paper makes some stinging points, they are too much caught up in mathematical notation as opposed to mathematical substance, and the answer does not lie in the mathematics itself. The problem with algebraic models of marriage systems is in what these models do not do. The rest of this essay is concerned with two topics. First: what is a mathematical model? Second: how good are algebraic models of marriage systems?

What is a Mathematical Model?

The question of what a mathematical model is turns up in the first chapter of nearly every text on operations research and every text on simulation. It also occurs frequently in probability texts and applied math texts. On one hand, essays on mathematical models appear to be almost ubiquitous, but on the other hand it is amazing that in physics books and engineering books this topic is generally not treated explicitly. This is not because physicists are less philosophical than their counterparts in other areas of the mathematical sciences, but because physicists use models in their introductory courses and there is no need for a transition to modeling. Mathematical models are generally defined to be 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 and valid model is the core philosophical question at the heart of much that is written about models. Other 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

The matter of predictions based upon models is core to the subject of models and to this critique of algebraic models of marriage 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 answer is always the same: The purpose of the model is to help us understand the subject. But if the model does not yield predictions, what is the value of this understanding? I am going to give two historical examples of models in physics and astronomy and then we will examine the various models of the algebraic models of marriage.

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 with Galileo, Kepler and others. Kepler constructed two models of interest to this paper. He was very much a man of his own time and ahead of his time. The model for which he is so famous is his laws of planetary motion. If we view these laws 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 the speed of their revolution. It enabled more accurate forecasting of planetary positions. It may be relevant that this model was based upon painstaking analysis of data that was itself of unprecedented accuracy. Kepler constructed another model of the planetary system that is also well known but not nearly as well known as his laws of motion (see Kappraff [1991, p.265]). In his Harmonics Mundi, he showed that the orbits of the six planets could be inscribed about the five platonic solids. In this model, 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 only predictive in one sense; it implies that there are no new planets to be discovered. Two things need to be said about this model. 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 systems are more in the style of the Kepler's platonic solids model than in the style of his laws of planetary motion.

The Quantum Physics Model

For many people, if there is one thing they know about quantum physics, it is the Heisenberg uncertainty principle. In fact, the Heisenberg uncertainty principle is so important that in von Neuman's formalization of quantum physics, the Heisenberg uncertainty principle is an axiom (von Neuman [1955]). Quantum physics also has the characteristics that it is difficult to understand and it is completely unintuitive. However, 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 characteristics. First it is a mathematically consistent theory and second it has been the source of thousands of predictions, and in all testable cases, the predictions have been correct. The most famous single such prediction might be Bell's theorem (an elementary account of Bell's theorem is given by Peat [1990]). Hence a mathematical model that is best known for its statement of what cannot be predicted 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 systems for the criteria of how well it predicts.

Algebraic Models of Marriage

The Core of the Algebraic Models of Marriage

Algebraic models of marriage are based upon viewing clans and the relationships between the clans. In some cases, specifically the Kariera, the Aranda, perhaps the Tarau, and perhaps the Murngin, the structure is an algebraic group. This single observation, which is apparently due to AndrĒ Weil, is the heart of forty-plus years of writing on marriage systems and mathematics. However, it is not in any way a remarkable observation. Clan systems are either hierarchial or they are not. If as in the above cases, the clan relationships are not hierarchial, they are likely to be symmetric. Group theory is a major part of abstract algebra. It could be described as the mathematics of symmetry. Although group theory has many facets and applications, the study of symmetry is a principal application. One of the best recent introductions to group theory is Groups and Symmetry [Armstrong, 1988]. An elementary essay introducing groups and symmetry is Chapter 9 (The Duellist and the Monster) in Stewart [1992]. Otherwise, there are hundreds of books on group theory, abstract algebra, linear algebra, and geometry that discuss the connection between groups and symmetry. (Also there are cases, such as in linear algebra texts, where it is not stated that the algebraic structure is a group.) It would be a remarkable discovery indeed if there were symmetric clan structures that could not be described in the language of algebraic groups. Even a theoretical example of such a structure would be quite interesting. If we are going to use algebraic groups to describe symmetrical clan structures, there should be a pay-off or return: there needs to be some reason to bring group theory into anthropology. This may seem obvious, or elementary, 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 [1983, p. 1 6] 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.

That models without a purpose exist I believe is a consequence of the divergence of mathematics and physics that has been going on for two-hundred years but which has greatly accelerated in the last thirty years. Again, the most obvious purpose of a mathematical model is to predict. Then it is logical to ask what algebraic models of marriage systems predict. The answer is nothing. Then let us ask a simpler question: What information do algebraic models of marriage systems provide? In his genesis of algebraic models AndrĒ Weil [1969, p. 221] says . . . I propose to show how a certain type of marriage laws can be interpreted algebraically, and how algebra and the study of groups . . . can facilitate its study and classification. Given that there have been forty-plus years since Weil's essay it is reasonable to ask whether this has been done. Has anyone aided the study and classification of marriage systems through the use of group theory or through any use of abstract algebra whatsoever? Another approach is this: How are the traditional anthropological methods for classifying marriage structures deficient?

Prediction and Falsifiability

A key element in the twentieth century view of scientific theories is falsifiability. A scientific theory must be falsifiable. The notion of falsifiability seems nearly equivalent to predictability. The criterion of falsifiability was a key doctrine in the work of the eminent philosopher of science Karl R. Popper (see for example Popper [1959]). In particular a scientific theory must make predictions in order that these predictions can be tested and so that the theory itself is 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 systems. What predictions do the algebraic models make? If algebraic models of marriage systems are not predictive models 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 it seems don't predict but they give us insights. Let us take an example from my own paper [Cargal, 1978]. The Kariera have four clans. This system can be represented by the graph in Figure 1.

Figure 1

In this representation, the solid arcs represent clan of marriage and the dotted lines represent clan of child. This representation is a valid way to introduce algebra since the graph is also the Cayley graph of the Klein-four group: Z2 Z2. This is the traditional algebraic view of the Kariera. In my paper, I suggested looking at the subclans of each sex: ... there is sex differentiation among the clans referred to. Natives think in terms of men of Clan A or women of Clan A ... [Cargal, 1978, p. 161]. I then gave a graph of the eight subclans. Then, I relabeled that graph to achieve a Cayley graph of a group. Figure 2 shows the group of subclans of the Kariera.

Figure 2

In this representation A1 denotes men of Clan A and C0 denotes women of Clan C. The relations are S and O. S means subclan of children of same sex and 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 group of order eight (eight elements) is not merely a refinement of the group of order four. The group of order eight that I developed in my paper does not appear anywhere else to my knowledge. It is as valid as the standard group (to whatever extent the first group can be said to be valid). We have two algebraic representations of the Kariera tribe. These are two fundamentally distinct models of the Kariera. If these algebraic models have any anthropological content then two different models should give conflicting information. At least the existence of a second model should provide new information given the first model imparted 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 twenty years (since doing the research) I can ask, as anthropologists asked, What significance does this have to anthropology? A second question is now rather insistent: What happened in the last twenty years that I should now have developed an attitude far more sympathetic to anthropologists than when I did my own work in marriage theory? (This is answered in the addendum to this paper.)

Cultures of Mathematics

Until the turn of the century applied mathematics meant primarily mathematical physics. Probability may have been serious in the eighteenth century, but statistics is very much a twentieth century development (with merely a pre-history before 1900). Computer science started in the 1940's with a pre-history going back to maybe 1930 or even to the nineteenth century. Operations research began in World War I and took off with World War II and the development of computers. Models in physics are different than models in many mathematical disciplines. Mathematical logic, for example, is concerned a great deal with models that are quite different than the models that interest physicists [See Bell and Slomson, 1977]. Models in statistics are different still [see Hinkelmann and Kempthorne, 1994]. In much of mathematical culture a model must be a mathematical system that is formally specified and which is internally consistent. There is absolutely no requirement that it be useful. It is true that operations researchers and statisticians are interested in models they can use to solve problems. But the importance of making predictions is not emphasized enough (I don't recall it being emphasized at all). This is why E. C. Russell made his edict I quoted earlier:

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, p. 1 6].

Like all of us that have worked in industry, Russell has seen many computer simulations with no stated goal but to model the system in question. These projects can go on endlessly sucking up money, keeping (some) researchers quite happy and accomplishing nothing else. In anthropologists, I have seen two kinds of attitude. Generally, to the very few anthropologists who can understand the mathematics, the (positive) worth of algebraic models of marriage is self-evident. The remaining anthropologists are skeptical. They tend to ask: What good is this model? What will it do for me? Mathematicians tend to respond condescendingly knowing that the anthropologists are mystified by the mathematics. Their reply is generally of the form: It will help us understand the structure of the marriage system. Unfortunately, I do not remember any anthropologist being prescient enough to ask What will this model predict? C. P. Snow's The Two Cultures [1969] is a justifiably celebrated work. He discusses the rift in academia between the humanities and the sciences. (Incidentally my personal experience is that people in the humanities refer to the book a lot, but the only people I have met who actually read the book are in the sciences.) C. P. Snow's analysis is still correct, however like all large cultures, the two cultures he mentions have many subcultures (and this of course was known to Snow). It appears that in the sciences that one manifestation of different subcultures is the understanding of what a model is. One consequence of this is that in the case of what constitutes a mathematical model, there may be an unlikely alliance between anthropologists and physicists versus mathematicians.

Mathematics is Not a Science

Mathematicians who work in physics and engineering are scientists. However, though it is said to be the language of science mathematics is generally considered outside of science. This is because scientific facts must be verified empirically. Mathematical truths must only be true to their axioms. The axioms must be consistent.

Cultural Anthropology: Is it a Science?

Anthropology is supposed to be a science. Anthropology has its subcultures too. Physical anthropology can be quite different from cultural anthropology. For one thing, physical anthropology has more of a hard science orientation. However, the models of this paper concern cultural anthropology. Cultural anthropology is now being influenced by people such as Derrida and Foucault who have pervaded literature for the last two decades. Hence cultural anthropology may not be too hospitable to a traditional concept of science (see Fox [1992]). This is a further reason that anthropologists may not care about mathematical models of marriage. Nonetheless mathematicians are to apply their models to an area such a anthropology, the paradigm that they should use is the one of the scientific method and of predictions, testing and falsifiability.

There is good reason why mathematicians define their models the way they do. However, in applying their models outside of mathematics, they might be wise to use the standards of physics. Lastly, when mathematicians apply mathematics to previously non-mathematical areas, the people who fail to understand the mathematics may nonetheless be quite right in their criticism. It is important that the needs of the applied area be recognized, that in applying mathematics, mathematicians should look outside their own needs and their own culture.

Addendum

I would like to explain why I came to reject an application of mathematics that I had published. Since this is not germane to the arguments of the paper I am making this merely an addendum. Until recent times almost all math majors studied physics. Math majors are still encouraged to take physics, and physics provides much motivation for calculus not to mention most examples in differential equations and vector calculus. However, it is becoming common for math majors to get around the physics requirement as I did, and there is a smaller proportion of mathematicians with physics training than there used to be. When I did my work in anthropology, my background was a master's in mathematics with emphasis on logic and abstract algebra. At the time I published my paper, I was working in aerospace and had made it my business to be comfortable in statistics and computer science. Between 1983 and 1987 I did my Ph.D. in operations research (which was a sub-department of industrial engineering at my school). Operations research is a mathematical discipline with strong ties to statistics and computer science but not physics. In 1988 I was again working in military aerospace, essentially as an applied mathematician. A problem came up in classical Newtonian physics. I merely needed to use computer models constructed for the problems that I faced. However, I felt that I should understand the underlying physics. Therefore I began the study of physics not realizing that it is addictive. Exposure to physics changed my appreciation of the meaning of mathematical models.

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