Mathematical tradition in Anthropology. An Introduction 1. Edmund R.Leach


Anthropologist sees the world  as a world of extreme complexity or as a series of Big Data ( NP hard ) problems , hence, some field complexities could be described as“ botanic rarities of the most exotic kind “ by literary forms , whereas another complexities are ready for scientific computational analysis.

As is known the first attempts to introduce systematic scientific analysis of culture as “ a set of mechanical devices “ ( Malinowski ) or  as a  sort of “computer software “( Leach )  were made by functionalists . In 1933  White émigré  Russian functionalist S.M. Shirokogoroff also used equations of statistical physics in order to describe self-organization effect of his “ethnoses theory “( used by Soviet ethnography, later ). In 1940s Levi-Strauss and Andre Weil attempted to use elements of modular algebra and becoming category mathematics in kinship classifications  in the terms of structuralism. At the same time Levi-Strauss had found simplification of this mathematics in the form of Jakobson ‘s binary arithmetic ( “system of phonological distinctive features “), generalized the first by “functionalist-structuralist” Edmund Leach.

Probably, the best expression of becoming mathematical tradition in anthropology belongs Edmund R. Leach, Cambridge’ applied mathematician having engineering background. Some passages by Leach in this context are very impressive, indeed :   


“ I tend to think of social systems as machines for the ordering of social relations or as buildings that are likely to collapse if the stresses and strains of the roof structure are not properly in balance. When I was engaged in fieldwork

I saw my problem as trying to understand "just how the system works" or "why it held together."


“ In my own mind these were not just metaphors but problems of mechanical insight; nor was it just make-believe. To this day, in quite practical matters, I remain an unusually competent amateur mechanic and retain an interest in

architecture which is much more concerned with structural features of design than with aesthetics “


“ I had learned to work with binary arithmetic before I had ever heard of computing or of Saussurean linguistics. I recall that when, in 1961, I first encountered Jakobson' s system of phonological distinctive features my inner reaction was: "Ah! I have been here before!" “


“ My engineering background also effected the way I reacted to Marxism “.


“ My concern with design stability does not mean that I am unmoved by the aesthetics of great architecture, but it adds a dimension which less numerate observers probably miss. My private use of the concept of "structure" in social

anthropology is thus different both from the usage developed by Radcliffe - Brown and Fortes (where it simply refers to the skeletal framework of society without any consideration of design features) and from Levi -Strauss's transformational usage, which borrows from Jakobson's phonology, though my engineer's viewpoint is much closer to the latter than to the former.”


 “ In terms of my engineering metaphor, Fortes describes the social machinery and its component parts but is unconvincing when he tries to explain how the system works . Firth gives us an instruction manual for operating the machinery,but he does not tell us what the bits and pieces would look like if we took it apart. Or to pursue my art and architecture model: it is wholly appropriate that Firth should be entranced by the highly decorated solidity of the Romanesque Cathedral at Conques and that Fortes should have been overawed by the symmetrical Gothic fragilities of King's College Chapel “


“ anthropologists are engaged in a scientific discipline which is capable of revealing facts of (social) nature in much the same way as experiments in physics reveal the facts of physical nature”


“ I never had the makings of a true mathematician, but I was mathematically literate. I learned about "transformational" theory (in the form of advanced algebra and the nineteenth century developments of projective geometry) several years before I entered Cambridge as an undergraduate. If some of my anthropological work is"structuralist" in style, it is for that reason”.


“ Another key point, about which I was also quite explicit, was that my use of "function" derived from mathematics and not from biology or psychology, as was the case with the followers of Radcliffe-Brown and Malinowski. Consequently, from my point of view, there was no inconsistency between " functionalism" and "structuralism" (in its then novel continental sense) “.


“ Human society was made by man, so man should be able to understand society, in an engineering sense, e.g. why it holds together and does not collapse. Behind this there is the further perception that all the artifacts (including human society) which man thus "makes" must necessarily be projective transformations of what the human brain already "knows." This implies, to use computer terminology, that social products are generated by "software programs," operating through but limited by the computer-like machinery of the human brain. The "software" comes from our cultural environment; the "hardware" derives from our genetic inheritance.”


“… being a functionalist and being a structuralist; I have quite consistently been both at once. But both my functionalism and my structuralism derive from my grounding in mathematics and engineering “.


“ Furthermore, I have an engineer's interest in design, in how local regions of complex unbounded systems "work ." Indeed, I have consistently maintained that the social systems with which anthropologists have to deal are not, in any empirical sense, bounded at all. To discuss the plurality of cultures is for me nonsense…”


[ “ Glimpes of the unmentionable in the history of British social anthropology “  Ann. Rev. Anthropol. 1984. 13:1-23 ].

Please see also :



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Comment by John McCreery on April 11, 2013 at 4:46am

Here is something a bit more serious, courtesy of Jacob Lee. 

Comment by Michael Alexeevich Popov on April 8, 2013 at 11:41am

Thank You.It is a good joke. Indeed, "Anthropology is the science of the sense of humour " ( Malinowski,1937)


Comment by John McCreery on April 6, 2013 at 3:44pm

Michael, you are making a moral argument—"have no right"—to a scientific discussion.

I agree totally that there is no easy way to do math as mathematicians do. But, to my mind, that puts the onus on those who insist that anthropologists should do math to show the way a bit, in a way that demonstrates—or at least provides a taste—of the benefits. 

In my own case, I can demonstrate how I use mathematical tools, albeit crude and simple ones, to do research impossible without them.  The analogy is a simple one. The network analysis software I use supplies the social scientist with a tool that works like a biologist's microscope, rendering visible structures of interest invisible without it, thus posing questions for further inquiry—origins, change over time, the impact of external factors. 

I would like to see something similar from you. Perhaps something like this.

Comment by Michael Alexeevich Popov on April 6, 2013 at 2:04pm

John and M.Izabel,

Let me to make some sort of generalization.

1a. Malinowski, Shirokogoroff, Levi-Strauss and Leach made their best to introduce scientific standard (of 1920-30s ) in anthropology. We cannot forget about it, we have no right to ignore their contribution.

1b. Unfortunately there is no special easy way to do math and science even for anthropologists.

2a. There is natural way to understand mathematics and mathematics of classical anthropologists. I agree it is hard.

2b. Mathematics is access to the world of science and into the world of the right questions - unfortunately some efforts are needed, indeed.

3a. Following mathematicians- anthropologists I believe that we can develop advanced platform for scientific anthropology as well as today's anthropologist is able to compete with another scientists .

3b. Unfortunately I agree with Tiger (2006) that impact of literary tradition in anthropology ( as " confused form of literary scholarship " ) is " fundamentally unfortunate " for anthropology...

P.S. I love arts as well, in 2003 at Space Congress I developed mathematical choreography for space ballet.  


Comment by John McCreery on April 6, 2013 at 8:00am

I am also interested in what Michael is up to. I wish he would tell us more about it, instead of assuming that we know what he is talking about.

Comment by M Izabel on April 6, 2013 at 3:10am

I won't call it enthusiasm.  I have partly given up on anthropology a long time ago.  My life now revolves around arts.   Culinary is my bread and butter; sculpture, my hobby that relieves stress; and creative writing, my challenge to myself.  If I still do anthropology, it's because I'm still hopeful a little bit that maybe in the future I can go back to what I really love doing-- anthropologizing, if that's a word.

I read somewhere that mathematics makes a science legitimate.  It helped Physics break away from Philosophy.  Since the last discussion I posted asks whether anthropology is a science, I find Michael's focus on mathematical anthropology interesting and relevant.

Yes,  (a+b)+c=a+(b+c)=(a+b+c) is basic, but it is a foundational rule or assumption in mathematics.  Didn't you ask about anthropological data fitting mathematical assumptions?  Yes, that equation looks trivial and easy, but its existence in real life tells us that mathematics can be applied in the study of society and culture.  If the foundational assumptions in mathematics are observable in the field, is it not possible to calculate, map, or mathematically represent what we see and observe? 

Comment by John McCreery on April 6, 2013 at 2:12am

Michael, M, I admire your enthusiasm. I do wonder, however, what you are hoping to accomplish here? 

Michael, why do you care about anthropology? As you have observed there are lots of people working in fields like computational sociology, big data, and brain science, doing the sorts of things that you are interested in. Why aren't you one of the physicists instead of the kids taking physics for poets and still not liking physics very much?

M., why do you think that anthropologists should be interested in (a+b)+c=a+(b+c)=(a+b+c), when the arithmetic is trivial and the fact that people work together in different combinations also seems trivially obvious? What difference does the arithmetic make, as opposed to knowing, for example, that a, b and c are men or women, brothers or sisters, a man and two women (who might be a husband with two wives) or a woman with two men (who could be her brothers or her lovers, rivals for her attention), cousins, members of a clan or lineage, co-residents in the same village, co-owners of shared property or participants in a formalized system of labor exchange to make sure that everyone's fields get planted and everyone's harvest gets harvested?

Michael, you throw around a lot of big-sounding words, "big data," "quantum-computing algorithms," that sort of thing. You appear to be unaware that when talking to people ignorant of the technical meaning of the terms you throw at them and likely to be math phobic, it would be a good idea to walk them through some examples of the sorts of things they are interested in and show how the new mathematical toys would improve their understanding of them. 

Personally, I have a theory why mathematicians still know what Lévi-Strauss did with group theory and anthropologists simply don't care. When L-S wrote The Elementary Structures of Kinship kinship was a big topic in anthropology. Why? Anthropologists were still mainly into people who lived in stateless societies where, it was, said, kinship provided the framework for most if not all of social life. Getting the kinship system right was job No. 1 for the fieldworker studying a new people. There was also a lot of theorizing about the nature of prehistoric societies, where kinship was presumed to be important. Along comes L-S and says, look at this, some simple mathematics explains kinship. That was, for its time, an exciting idea. The problem was it never went anywhere. You had to assume things like moieties, four-section systems, or  some form of circulating exchange in what was, ultimately, a local closed system. Try doing this sort of thing with say, 30-generation deep Hakka lineages whose ancestors have wandered over a whole lot of China during several ups and downs in the dynastic cycle. You don't get very far. At the end of the day L-S didn't have a lot to say about kinship in the sorts of places that anthropologists were increasingly studying, peasant villages and urban neighborhoods in Asia or Latin America. Nowadays, of course, kinship is rarely taught at all. L-S is back there in the dustbin of history with folks who obsessed over the differences between Crow and Omaha kinship terms and what happens when you cross patrilineal and matrilineal rules of inheritance (for different kinds of goods) with virilocal marriage. The mathematicians remember him fondly as that famous anthropologist who took group theory seriously. The anthropologists? They have moved on to other things, and hauling out his name as if it were a sacred totem doesn't have a lot of persuasive power.

So here's the rub. If we want to persuade people that what we are talking about is important to them, we have to start where they are and lead them in small steps to where we want them to go. Shouting, in effect, "You idiots! Don't you know?!" is no way to make friends for mathematical anthropology, let alone win new converts.

Comment by M Izabel on April 5, 2013 at 9:59pm
Comment by M Izabel on April 5, 2013 at 9:52pm

Yes, Michael, the mathematicalization of anthropology and sociology has been done already, but mostly representational (map, diagram, model) and metaphorical (analogy, one-to-one correspondence, substitution).  Can you direct me to some recent anthropology articles that are formulaic and calculative in their application of mathematics like how mathematics is employed in economics?

Apparently,  Harvey Goldberg already developed a probability measure in his investigation of the FBD (parallel cousin) marriage among Tripolitanian Jews in 1967.  I think that was a good start.

Comment by Michael Alexeevich Popov on April 5, 2013 at 3:55pm

M Izabel and John,

Current confusing litarary tradition of social anthropology is based on total simplification ( please, see -  Mark Moberg. Engaging anthropological theory.Routledge 2013  with zero knowledge on mathematical tradition at all ), hence,some obvious facts are usually rejected ( by anthropologists - as- self - made- philosophical -skepticists ) for example:

1. Levi-Strauss's mathematical applications are already known in applied mathematics and there are some developments here;

2. Cultures and subcultures are already objects of Big Data mathematics ( Google used already quantum like algorithms in 2012 );

3. There are also serious developments in Computational sociology and Brain Data science, associated with anthropology as a science.

Thus, because we lost already two generations of anthropologists ( for poetry ), we need to find Competetive Platform for recovery of serious anthropology in conditions of very high competition in science. 


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