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

544 Raymond Boudon
D. The new mathematics
Another noteworthy circumstance is the demand in the social and human scien-
ces for new mathematical techniques. For many years the mathematics they
used were limited to elementary algebraic procedures, classical mathematical
analysis, or elementary applications of probability calculus and statistical
method. Here again, the example of classical mechanics undoubtedly contribut-
ed greatly to the choice of the mathematical instruments used.
Quite recently, the sciences of man have been using a number of mathe-
matical tools of more recent design. Moreover, the development of the mathe-
matical methodology is sufficiently advanced in some fields to give rise to a
need for new formal instruments and secure their creation. In this section, it
must suffice to cite some instances of branches of mathematics whose application
to the human sciences is in most cases of very recent date.
Advanced algebra, particularly linear algebra, is coming into ever more
general use in the sciences of man. It has uses in economics, where one of the
best known instances of its application consists in the well-known Leontieff
matrices. In psychology, factorial analysis is so called - perhaps in part - be-
cause it consists in factorizing certain matrices, the basis equation of factorial
analysis, as of many methods of dimensional analysis used by psychologists,
being the matrix equation:
R = AA‘
Here R is a matrix of correlation, e.g. between psychometric tests, while A is the
matrix expressing the correlations (saturations) between the tests, and the
unknowns or latent factors responsible for the responses being right or wrong.
Matrix A’ is simply the transposed matrix of A (that is, the matrix obtained by
interchanging the lines and columns of A).
In social psychology, the measurement theory known as latent structure
analysis has- and on some points still does- set the mathematician complex alge-
braic problems. Again, as was stressed in an earlier section sociometric analysis
(analysis of sociograms) frequently relies on matrix algebra to the extent
that any system of relations can always be represented by a matrix.
In anthropology, some research procedures, like the kinship structure analysis
model proposed by Bush, use linear algebra.
In sociology, a very large number of phenomena are ‘naturals’ for represen-
tation in matrix form, followed by some application of linear algebra: social
mobility tables, turnover tables deriving from panel surveys (surveys carried
out more than once on the same sample), etc.
Stochastic process theory. The introduction of stochastic process theory into the
social and human sciences is recent. Even the most mathematicized of these
sciences, economics, still resorts to these processes on a modest scale only. It is
safe to say that currently the only tool from the battery of stochastic processes
which is used on any scale is Markov chain theory, at that almost exclusively
homogeneous Markov chains (chains characterized by the fact that the prob-