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Mathematical models and methods 539
problem of constructing types where n objects are characterized in terms of rn
quantitative characters (where, for instance, m psychometric tests are given to
n subjects) can be obtained by seeking the eigenvectors of the correlation
matrix of the m characters. But to calculate these is so complicated, as soon as
the correlation matrix reaches a quite small dimension, that the application of
this method became possible only when computers became available.
b. While the contribution of mathematical methodology is important in the
analysis of large quantities of data, there is another logical field in which mathematical
methods and models have proved their effectiveness: that of the synchronic
analysis of systems.
Two examples will be considered. The first is the analysis of kinship structures,
the second the mathematical theory of grammars.
In kinship structure theory, the anthropologist finds himself faced with facts
which, at a level of unsophisticated apprehension, seem peculiar and incoherent :
thus, given two archaic societies, he observes that in society A, a man may
marry his father’s sister’s daughter and his mother’s brother’s daughter, but not
his mother’s sister’s daughter or his father’s brother’s daughter. In society B,
the rules are identical, except that here it is forbidden to marry the father’s
sister’s daughter as well. In this kind of situation, it is impossible to resort to
the classical explanations of the incest taboo, such as those which ascribed it to
concern for eugenics. Thus, we are faced with an assemblage of facts which at
first sight are strange and irrelevant and the temptation may come to imitate
certain ethnographers prior to Murdock and Lkvi-Strauss and record these facts
purely and simply.
One of the novel factors of mathematical methodology which has evolved
over the last few years in the field of anthropology has been to produce theories
whereby it can be demonstrated that these marriage rules on permitted and
prohibited degrees of kindred and affinity, despite their oddity, form coherent
systems to the extent that each of the rules can be deduced from a set of axioms.
A further example completely comparable to the first one is the mathematical
linguistics associated more particularly with the name of Chomsky. Mere again,
the starting point of analysis is the apparent oddness of the rules governing the
formation of grammatically correct utterances in a natural language. Take for
example one of the rules for the pronunciation of the English language: if one
compares the words ‘compensation’ and ‘condensation’, which are closely
paronymic, one is surprised at the apparently arbitrary difference in pronunciation.3
In pronouncing ‘compensation’ the e-vowel of the second syllable is
elided, while the corsesponding vowel in ‘condensation’ is audible. On meeting
this kind of thing - and examples are legion - one may 5e content to assume
that the rules of a language consist of a collection of arbitrary items which it
would be enough to note. But this attitude is in flagrant defiance of certain
basic facts: for, if the grammar of a language must be deemed no more than a
hodge-podge of rules associated with a multitude of exceptions, how are we to
explain the ease with which adults of English mother tongue - since English is
the language being considered in the above examples - and even tiny children
apply these rules with perfect assurance, even to vocables which they have never