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

Mathematical models and methods 549
of their use is in the analysis of the mechanisms characterizing groups or organizations
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Finally, in psychology, there are several examples of particular interest in
which simulation has slightly different connotations from those it has in the
preceding examples. The outstanding instance is the case study by Newell,
Shaw and Simon,z1 describing their attempts to simulate the human process of
deductive reasoning on a computer by programming the machine with a diversity
of items of ‘knowledge’ and behaviour choices.
To close this section, a point which needs stressing is the profound interaction
already observed with regard to several examples, between the development of
mathematics and that of the sciences of man. It is not going too far to say at
this date that the social sciences have had a real influence, however modest, on
the development of some branches of mathematics. Take the example of graph
theory, due, as was said earlier, to the researches of Koenig published in 1936.
This theory remained comparatively unnoticed until the beginning of the 1950s
when a first practical use was found for it in the sociometric studies of Ross and
Harary. In addition, several instances can be cited of theories, first begotten of
encounters with problems posed by the human and social sciences, which have
since become full-fledged branches of mathematics: one example of this is tbe
mathematical theory of grammars, which had its start in research directed to
analysing the grammars of natural languages. It very soon became apparent
that this problem was of infinite complexity and this led on to the consideration
of artificial languages: for instance, the language composed of two ‘words’ a
and b, in which all sentences are structured on the basis of m repetitions of a and
n repetitions of b. At this point it becomes possible to formulate mathematical
problems, sometimes complex, about the grammars capable of generating such
a language. This example suffices to show how it was possible for the theory
of formal grammars to achieve a degree of independent existence to the point
of becoming a special branch of mathematics. Naturally it is to be hoped that
applications wil be found for this theory. However to a certain extent this consideration
has ceased to be envisaged with any immediacy.
Another example of a theory born of the concerns of the human and social
sciences but which has since become a sort of branch of mathematics, is game
theory. Nevertheless, the position in this case is rather different from that of
formal grammars. Game theory raises and propounds numerous mathematical
problems, and to this extent is the starting point for results belonging to the
nature of pure mathematics. However, it cannot be deemed to be a theory of
pure mathematics inasmuch as it has retained a close connexion with the problems
of the human and social sciences, and especially with those of economics.
Indeed, the most important problems which game theory is now seeking to
solve are those posed by defining the notion of rational action in the complex
cases of non-competitive non-zero-sum games with n participants : these are
unquestionably the games which express most realistically the situations encountered
by the economist.
However if there is some effect by the social and human sciences on mathematics,
there are also cases where the development of the social sciences is not