Mathematics and anthropology

Read, Dwight ANTHRO (Read@ANTHRO.SSCNET.UCLA.EDU)
Tue, 25 Apr 1995 12:42:00 PDT

Yee writes:

"Most of the mathematics I know has no conceivable application to
anthropology (at least directly; the physicists use some weird stuff,
and the chemists use some of that, and...)."

As a mathematician (and anthropologist) let me comment that the question of
"application" is not of concern within pure mathematics; indeed, when an
author does speak of "application" it is usually in the sense of how results
obtained in one area of pure mathematics are applicable to questions in
another area of pure mathematics. Mathematics (and there is a strong
tendency to identify mathematics with pure mathematics) is, today, largely
sui generis. It's questions and goals are defined by mathematics taken as an
abstract system of thought. My area, algebraic structures, in many ways
epitimizes what constitutes pure mathematics. The field of algebraic
structures is concerned with the properties of abstractly defined structures,
regardless of whether they have, or might have, application to real world
phenomena. The Bourbaki school in France has had a program of providing an
axiomatic basis for all of mathematics, which is to say a program of
providing for all of mathematics the framework of: What are the logical
consequences of a specified set of axioms?

Theoretical physics has tended to merge with pure mathematics to the extent
to which it takes an axiomatic approach to theory: What would be the
properties of a universe constituted according to a specified set of laws or
axioms.

Remarkably, pure mathematics has, despite its denial of any need to ground
itself in real world phenomena, provided the intellectual machinery needed in
many areas. Group theory, for example, developed purely as an abstract
mathematical theory, has had enormous impact in theorizing in many areas.
The effective marriage of mathematics and physics has been remarkable--and
lead both physicists and philosophers to ponder why that should be the case
as Yee notes--yet is not likely to be repeated in anthropology.

The reason (in my opinion) relates to an issue that has reared many times
in disucssions on this list, namely our capacity to react intentionally (what
ever that may mean) as sentient organisms that are capable of evaluating
sensory inputs in a complex manner that makes the relationship betweeen
sensory input and behavior output highly complex. In short, we have a brain
between input and output that makes it difficult (to say the least) to model
how inputs are translated into outputs. Further, we not only have the
complexity of the single brain, but the the additional complexity arriving
from our acting as a community capable of both affecting and being affected
by that community. That is, we create culture and we are created by culture.

However, this does not mean that mathematics has limited value in
anthropological theorizing. As Yee comments:

"Despite this, I believe that the most valuable thing mathematicical
training can contribute to students of anthropology is not knowledge
of a small range of methods of practical use (such as how to carry out
chi-squared tests or plot logistic growth curves), but the perspective
which mathematical intution and a training in abstract thinking can
bring."

This perspective is missed entirely when one speaks of mathematics as a
"tool"; mathematics, as the philosoper Benjamin Peirce commented, is the
"drawing of necessary conclusions" and it is also "the study of hypothetical
states of things". It is, in this view, a way of thinking which involves
identification of the structuring properties (axioms) and the logical
consequences of those axioms as what must be true in a universe so
constituted.

Within anthropology, probably the most effective use of this view of
mathematics has come out of structural anthropology--not too surprisingly
since structural anthropology has an intellectual history that traces itself
back (in part) to the work of the Bourbaki schoool of mathematics in France
which was preeminently involved with abstract, algebraic structures. In
effect, Levi-Strauss posed the question: Is culture a structured system of
thought, and if so is it (in effect) an axiomatic system of thought?
(Levi-Strauss never posed the matter in this manner, but in terms of his
inclusion of the work of the mathematician Weil in his monumental
work, "Elementary Structures", it is not an unreasonable reading of what is
at issue.)

Australian kinship systems have been the impetus for a number of papers that
have examined their systems of rules as, in effect, the axioms of an
axiomatic system of thought. That is, what are the logical consequences, in
terms of structure, of those marriage rules? Probably the most extensive
paper in this genre is the one by the mathematician Courrege titled: "A
Mathematical Model of the Structure of Kinship". I quote from the paper:
"The purpose of the present work is to present, using the axiomatic method, a
simple mathematical model permitting the study of the functioning of the
kinship systems in a society where the population is divided into disjoint
matrimonial classes." By functioning is meant the resulting (abstract)
structure.

Within anthropology there have been a number of persons who have
used the language of abstract algebras to examine the logical consequences of
native concepts (hypotheses in Peirce's terms): e.g., Boyd (J), Lehman, Tjon
Sie Fat, H. White, Read, Atkins, Liu, Lucich, Siemens, among others. Topics
taken up, in addition to marriage rules, include kinship terminologies,
genealogical spaces, analogical reasoning, structural analysis, etc.

Yee asks:

"Does anyone know of any
work in the social sciences which uses algebraic topology, functional
analysis, or category theory? "

Probably the most extensive application of category theory (and even
contribution to category theory) in the social sciences is the PhD
dissertation by F. Lorrain at Harvard University: "Social Networks and Social
Classifications: An Essay on the Algebra and Geometry of Social Structure"
(1972). Social networks, as a field, has had a fair amount of mathematical
work spent on it, of which Lorrain's work represents just one of many
approaches that have been taken (see Boyd, J. "Social Semigroups" for another
approach based on algebraic semigroups).

D. Read
READ@ANTHRO.SSCNET.UCLA.EDU