Main trends of research in the social and human ... - unesdoc - Unesco

Mathematical models and methods 547
tion.'3 In sociology, several researchers have published applications of process-
es other than the Markov chain type to the analysis of the distribution of
accidents, the distribution of votes cast and the phenomena of spontaneous
concourse: the processes used in such work are usually discrete Poisson dis-
tributions over a continuous period. Coleman's bookI4 gives a good overall
view of the applications to sociology of stochastic processes other than Markov
chains. In addition he introduces contagious and exhaustive versions of the
Poisson distribution, which can be applied to social phenomena of minimal
complexity. As regard Markov chains, wide use is made of them in sociology
as in other disciplines: e.g. for the analysis of attitude change mobility phenom-
ena, etc. In general, however, it must be admitted that the interesting proper-
ties of the homogeneous Markov chain (expressed in particular by the ergodic
theorems) can rarely be applied in this field, since it is only rarely that a social
system can be considered stationary.
Graph theory. Conceived by the German mathematician Koenig and developed
thereafter by C. BergeIS more particularly, graph theory offers a useful means
of solving certain problems under the head of 'operational research'. In social
psychology numerous uses are found for it in the analysis of sociograms
and, more generally, of social structures. It should however be noted that
the most important results in this field are mostly obtained by algebraic
methods. In psychology, there are some applications of graph theory in the
formalization of the theories of Heider, Festinger and others on the coherence
of opinions. Thus, despite its possibilities, applied graph theory is limited, for
the present at least, to a few comparatively specialized fields.
Generally speaking, not only graph theory but also topology is used by the
human sciences, particularly linguistics. In Section IIIA are given some examples
of the application of combinatory topology to problems of grammar. For a
more complete idea of the applications of topology to linguistics the recent
work of Harris should be c~nsulted.~~
It is impossible within the scope of this chapter to expatiate at length on
all the new branches of mathematics with uses in the social and human
sciences. There is no point for instance in dwelling on the value of the recent
branches represented by game theory, information theory and cybernetics,
or the new. applications of certain aspects of mathematics long abandoned by
the Social sciences, such as logic, which is now used in fields as far apart as
genetic psychology, the analysis of court verdicts or the analysis of social
r6les.17 In addition, a review would be needed of the intimate connexions be-
tween certain new branches of mathematics and certain social science research
fields - for instance between coding algebra or the mathematical theory of
automata and linguistics.
However, a few words are appropriate on simulation methods, the importance
of which is growing in most of the social and human sciences. Of course, simu-
lation is not a mathematical theory. It is simply a technique which enables us
to substitute a model or physical maquette, usually constructed with the aid of
a computer, for a theory which one does not know how to express mathemati-