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 ].

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Comment by M Izabel on April 5, 2013 at 10:10am

This is a lot, John, and very interesting too.

"Mathematics make strict assumptions and deduce their consequences."


There are non-deductive methods in mathematics such as computer and probability proofs.  Michael also mentioned about experimental mathematics, in which a solution to a mathematical problem makes more sense if it is situated in a real-life situation.  Deductive mathematics, I believe, is applicable as the process of deduction is observable in real life.  Don’t we have cultural/social elements that are deduced from social/cultural rules and norms?    

“Do our data fit the assumptions on which the mathematics are based?”


They do if we consider mathematics as a representation of reality.  Is "a + (b + c) = (a + b) + c" observable in real life?  I think so.  When I was a kid, I saw it in our farm all the time when workers grouped and regrouped themselves for collective efficiency and output.  Imagine if these grouping and regrouping are observable in warfare, political organization, and economic activity.  


"the mathematics of statistical analysis depend on the existence of discrete, randomly sampled data. Such data was rarely to be found in ethnography."


In this case, we should question how an ethnography is done.  Does an ethnographer/anthropologist study only what he observes and participates in? How does he come up with his collective observation/inference about the community he studies?  I don't think ten informants are enough for him to come up with a general conclusion about the community.  


Random survey is an important method ethnographers/anthropologists can use especially if the communities they study are multicultural and the questions they want answered are related to behavior, opinion, and belief that are, most of the time, not visible or observable.  A good example of this is when one studies rape in a rural community in India composed of Christians, Buddhists, Muslims, and Hindus.  One cannot really come up with the community's collective view on rape by just asking informants, who, sometimes, are too shy or scared to express their opinions and beliefs.


"Does China, for example, count as "a culture" to be given the same sampling weight as, for example, "the culture" of  !Kung bushmen?"


This is where set, category, group, and even network theories can help us map out and represent population and mobility.  Maybe in doing so, we will be able to come up with different categories of culture.  Is there a super-culture (Chinese) where a culture (Uyghur) belongs?  Are the culture of !Kung Bushmen "sub-culture" or "co-culture" of that of Botswana?  In my case, the culture of my indigenous community is different to the Philippine culture, which is mostly based on the culture of the majority, the Tagalogs.  It’s possible to categorize an indigenous culture as a co-culture of the national culture where it belongs. 


"Methods textbooks suggested, plausibly I think, that ethnography was a first step in learning about the unknown, ideally reaching  level at which quantitative research would be possible at a later stage, to test explicit hypotheses. But that level was rarely reached, or, if reached, became the foundation for the hypothesis-testing envisioned as the next step."


I saw a TED presentation about technology-based design earlier.  The speaker used the sensing-to-actuating paradigm.  As far as anthropology is concerned, there is no actuating.  What do anthropologists do with their data besides presenting it in a form of a lecture, a book, or an article?  Actuating, in the paradigm, is about doing something after sensing.  In anthropology, it can be solving a problem after gathering and presenting the data about it.  If anthropologists become problem solvers, they will find mathematics useful not only to represent in a numerical language but to calculate.  Can we, for example, predict the demise of a rural community considering many factors such as migration, unemployment, resources, etc. and also considering those factors as numerical and quantitative?  

Comment by John McCreery on April 5, 2013 at 4:29am

Michael, M, here we are three true believers that mathematics has something important to contribute to anthropology. We know that we face an uphill battle to persuade fellow anthropologists to believe as we do. I propose that we need to step back and ask ourselves some basic questions. Thinking about them this morning, I have tentatively identified three areas of particular interest: data, relevance, and accessibility.


Mathematics make strict assumptions and deduce their consequences. Question No. 1 in any application of mathematics to empirical questions is do our data fit the assumptions on which the mathematics are based. Thus, for example, the mathematics of statistical analysis depend on the existence of discrete, randomly sampled data. Such data was rarely to be found in ethnography. Large-scale studies using, for example, the Human Relations Area Files (HRAF) confronted serious coding issues. Does China, for example, count as "a culture" to be given the same sampling weight as, for example, "the culture" of  !Kung bushmen? Methods textbooks suggested, plausibly I think, that ethnography was a first step in learning about the unknown, ideally reaching  level at which quantitative research would be possible at a later stage, to test explicit hypotheses. But that level was rarely reached, or, if reached, became the foundation for the hypothesis-testing envisioned as the next step.


Let us assume for the sake of argument that the data are sufficiently precise for mathematical analysis. The relevance of that analysis to any particular case remains problematic. Consider, for example, the relation between the mathematical models constructed by economists and the management of the firms whose behavior they purport to describe. Kyoto University Business School Professor Koichi Hioki observes that economic models may work fairly well as descriptions of market behavior in the aggregate; but that is of little comfort to executives managing particular firms, composed of particular individuals, with organizational structures and corporate cultures that reflect distinct histories and pose specific problems under changing market conditions. A recent example comes to mind. A friend is the president of the Japan branch of a Fortune 500 manufacturer of large, industrial systems. He is also in charge of the company's business in Korea. There the company's best salesman suddenly died, taking with him a vast network of personal connections and the customer goodwill that reflected his meticulous attention to their needs and rapid response to whatever problems arose with the systems he was selling. There is no one ready to be his successor. Even those with the requisite technical knowledge will need months or years to establish their own networks and credibility with customers. Awareness of this sort of issue is why Hioki has been working with Chirohika Nakamaki for over two decades to develop what they call the "Anthropology of Administration." It is, as Hioki sees it, the very particularity of ethnographic methods that make them valuable to management. 


Here is an area in which, thanks to computers and the Internet, massive progress has been made. The sort of research that I am currently conducting, based on social network analysis of archival data, would have been impossible when I was in graduate school in the 1960s. The task of transferring the ad contest credits data I use to Hollerith cards would have been daunting. Writing my own software in FORTRAN, delivering the cards with the programs and data on them to the computer center and waiting at least a day to discover the first bug in what I had done—no way. Plus, I can now work with software developed by people who have been thinking about network analysis for the decades intervening between then and now. Just the last three years have seen a spurt of new introductory and intermediate to advanced level textbooks. Software has improved dramatically. There are courses and other resources on line. I don't have to invent social network analysis. The tools exist, ready to hand. And, as I mentioned in a previous message, I don't have to have the makings of a mathematician to use them. The math in terms of which I think about my data rarely rises above the level of everyday arithmetic and use of Venn diagrams. I can count on others to deal with the deep math that underlies the tools I use. All I have to know is enough to recognize when they have produced something useful for moving my own research forward. Since I despise my own ignorance, I have recently begun to relearn calculus; but that is for the sake of keeping my grey cells exercised. I don't expect to catch up with the people doing the advanced modeling work that is way beyond my current horizon. 

If I turn now to the bibliography of Leach's publications to which Michael has pointed us, I can now imagine a project that would involve

  1. Scanning, OCR, and editorial cleanup of all of the publications in question.
  2. Pushing the resulting text through a semantic network analysis program to map what Leach was thinking about and how it changed over time. It would also be fascinating to do a citation analysis of the people he cites and compare the semantic and social networks.
  3. At the end of the day, however, I would still need to go to the relevant pages where "mathematics" or other math-related terms appear to explore their usage in more detail—turning from positioning in an abstract mathematical space to thick description and interpretation in an effort to better understand what Leach was thinking about. 

Could be an interesting bit of research for a Ph. D. in the history of social anthropology. The fact that I can even conceive it in these terms shows how far we have come in the last few decades. 

Comment by M Izabel on April 4, 2013 at 8:42pm

As long as anthropology remains descriptive (not prescriptive), the application of mathematics will only be representational such as the use of set, group, category, and network theories as models for social and cultural realities.  Mathematics is about solving problems.  Does the current anthropology solve sociocultural problems?  I don't think so.  I went back to the real life applicability of quadratic equation/formula.  It is obvious to me that it is applicable in the study of society and culture.  It can help us calculate space-time related problems such as the birth of inner cities or ghettos or the depopulation of rural areas or even overpopulation.  It has some economic applications too that we can use in the study of household and even informal economies.  Lastly, mathematics is peppered with variables and anthropology is with factors.  From that alone, we can see the connection between the two.

Comment by Michael Alexeevich Popov on April 4, 2013 at 11:28am

M.Izabel and John, thank you for comments

Unreasonable effectiveness of mathematical applications in some sense is puzzling topic for pure mathematicians as well. Own experience and archive investigations can probably help to understand some aspect of this problem. For example : Leach archive -

The Royal Institute of Anthropology has published a bibliography of Leach's publications.

The catalogue was edited for publication on the internet by Elizabeth Pridmore.

Index Terms
Leach, Edmund Ronald (1901-1989) anthropologist
King's/PP/ERL contains:
1 Published work.
12 boxes; paper.
2 Unpublished work.
9.5 boxes; paper.
3 Papers from colleagues.
1.5 boxes; paper.
4 Overseas research papers.
10.5 boxes; paper.
5 Academic Career.
10 boxes; paper.
6 Correspondence.
41.5 boxes; paper.
7 Personal and family papers.
1 box; paper.
X Extra material.
Comment by John McCreery on April 4, 2013 at 9:21am

Yes, Wittgenstein did. But to mention it in an ad hominem manner is snark and deserves to be treated as such.

But that aside, this tempest in a teacup points to a major gap in anthropological theory. We tend to divide the world into natives assumed to share "a culture" and experts who share an expertise in some aspect of culture. We don't do at all well, in fact have little or nothing to say, about middle-brow thinkers, local intellectuals/amateurs who develop their own idiosyncratic or collective versions of traditions that mingle low and high-brow thinking.

In my experience both Daoist priests in China and advertising creatives in Japan fall work in this middle ground, where neither mindless obedience to cultural rules or clearly articulated philosophies account for what they are doing (Bourdieu's habitus may be relevant here). 

I can readily appreciate why someone eager to promote more involvement with mathematics by anthropologists would point to venerated ancestors who seemed to have mathematical leanings. I wholeheartedly support the cause in question, since without mathematical literacy anthropological debates are, at best, either purely rhetorical exercises or medieval logic chopping. But I still see it as a perfectly fair question to ask how mathematics actually figured in the revered ancestors' thinking. 

Comment by M Izabel on April 4, 2013 at 7:32am
Comment by M Izabel on April 4, 2013 at 7:28am

Let me play neutral here.

I think Michael was too fast in his thinking and was too economical as far as detailed language is concerned.  I don't know if this is how mathematicians think--ignoring the easy ones and expecting people to know them.  Maybe we can ask Michael for elaboration next time. 

Your two initial questions, John, can be interpreted in many ways. I read your questions as about language and reality in relation to mathematics.  Sure enough, Wittgenstein talked about the same stuff in his philosophy of mathematics. 

Comment by John McCreery on April 4, 2013 at 2:53am

Michael, with all due respect, you almost certainly know a great deal more math than I do, but your fieldwork technique and historical analysis are crap. Where do you get the "later Wittgenstein's taste" in what I wrote (I haven't mentioned Wittgenstein)? And if I were writing in that vein, the last thing that I would assume is that there is an unchangeable mathematics as a whole. After all, if mathematics is just another language game, it will evolve historically like other language games. These errors are most unfortunate, since they lead one to wonder if, to use a parallel example, one is talking with a genuine physicist or a New Age groupie who has read The Dao of Physics.

But let's put aside this sort of ad hominem on both sides, which adds nothing to what could be an extremely interesting discussion. Let us begin, instead, with one of your quotes from Leach,

“ 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”.

When I read Leach describing himself as being mathematically literate with having the makings of a true mathematician, this description resonates strongly with my personal experience. Having scored higher on the math than the verbal section of the SAT, I entered the Honors College at Michigan State University assigned to an honors calculus class. There I learned the difference between having a certain level of mathematical ability and being a talented mathematician, a totally different category to which several of my classmates belonged. 

I, too, would describe myself as mathematically literate but not having the makings of a mathematician. But what does this mean in practice? The biggest practical difference is that whenever I hear someone distinguish and X from a Y, I do not automatically assume that X and Y are distinct categories. I begin by asking myself, is the relationship between X and Y a nominal, ordinal, integer, or real number scale (does it put them in two separate boxes, rank one box higher than the other, allow me to assign a score (1 vs 3, for example), or do arithmetic with the scores). I am also aware of the existence of limits, of asymptotes that approach but never reach limits, and discontinuous functions—so I may be a bit less confused that the mathematically illiterate who try to read a book like Melanie Mitchell's Complexity: A Guided Tour (though in most cases they will never even try). I have been intrigued, ever since that calculus class (and the course in symbolic logic, mostly Quine, I took at the same time) by the idea of a formal system in which an infinite set of results can be generated from a small, finite set of primitives and a handful of rules for their combination (an idea that makes both Noam Chomsky and Claude Lévi-Strauss more plausible than they may seem on other grounds). I am constantly aware, however, of the limits of what I know. I can, for example, work out how to use a series of commands in a network analysis program called Pajek to determine the overlap in membership between people in a network of ad contest winners working on projects with different attributes. Do I really understand the matrix algebra on which Pajek's algorithms are based? No. Could I write the program from scratch? No way. What my mathematical literacy amounts to in practice is a handful  of very powerful ideas, which contribute metaphors to the way I think about things, and a handful of rudimentary arithmetic and algebraic techniques that let me do a few interesting things while analyzing my data. 

This, not some vague Wittgensteinian assumption, is what lies behind my historical questions. I wonder what Leach's mathematical literacy really amounted to. What I have read of Leach suggests that he, too, found mathematical ideas a fertile source of metaphor. I cannot recall ever seeing either mathematically informed analysis of data or rigorous mathematical proof in anything I have read. In a similar vein, I wonder about the relationship of Lévi-Strauss and Weil. Again the dim impressions in my memories suggest that L-S use of mathematics was, like Leach's, largely metaphorical. Certainly, by the time he wrote the Mythologiques, he was thinking more in terms of music and Hegelian dialectic than making any explicit use of mathematics. 

I hasten to add that I do not discount the thinking of either Leach of L-S because their use of mathematics was largely metaphorical. Like the philosopher Max Black to whom Victor Turner often alluded, I have cheerfully aware of the role of metaphor in stimulating mathematical and other forms of scientific thinking. I do not, however, mistake metaphor for the rigorous proof or empirical demonstration required to convince me that the scholars in question have moved beyond metaphor. 

Comment by Michael Alexeevich Popov on April 3, 2013 at 12:48pm


Your philosophical questions in later Wittgenstein's taste are based on assumption that [1] there is unchangeable mathematics as a whole ,and [2] we always must know where such sort of mathematics has the beginning and where such mathematics has the end... Because mathematics is rapidly evolving and changeable entity and because our initial assumptions are uncertain, we can say always everything about everything.

Concrete example: in 1940s Levi-Strauss believed that he uses  a kind of "Set theory " + " Abstract algebra", however, it was quite different category mathematics ( Weil was a founder of such math ). Hence Levi-Strauss 's "canonical formula " could be understood today as Levi-Strauss 's lemma on the existence of a nontrivial anti-automorphism of the quaternion group of order eight , etc

P.S. About 30 y. ago  ( after " Poetics and Politics ... ") social and cultural  anthropologists made, probably, wrong conclusion that they are merely "artists", "field philosophizing walkers" and future "poets". As a consequence, the two generations of today's anthropologists believe that to make taking science seriously in anthropology is impossible...

I suppose that mathematicians - anthropologists were more optimistic. Today anthropologist equipped as Clarendon lab of the 1960s can make different big things in the field ( from advanced computation, data cubes, simulation and even animation) ,our informants became data-makers and our future field work could be associated with advanced mining data indeed.

I suppose that  Mathematics of Computation is the best competitive philosophy in the field.  

Comment by John McCreery on April 1, 2013 at 4:54am

I hadn't known that Leach had a background in engineering or that Malinowski had conceived of culture as "a set of mechanical devices." The historian in me has two questions: First, has anyone explored the relation between mathematics as metaphor and mathematics as mathematics in the history of anthropology? Second, considering both these topics, has anyone considered the relation between anthropological thinking and the mathematics available at the time the thinking was done?


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