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

550 Raymond Boudon
just influenced by mathematics - that is too obvious to state - but strictly
dependent on the solution of a problem in pure mathematics. The most striking
example of this is perhaps the aporia which stultified the Walras-Pareto theory
of economic equilibrium as long as certain results in the realm of pure mathematical
analysis, like the ked-point theorems, remained unknown. It can be
reckoned that the Walras theory of general equilibrium was blocked by our
ignorance of mathematics from the 1870s to the 1960s~ and it is noteworthy
that those chapters of books on mathematical economics which dealt with
general equilibrium are out of date wherever the books antedate 1960: an
example is Allen’s Mathematical
To take another example already cited, there is no doubt that, if the theory of
stochastic processes is still very rarely applied to economic and social phenomena,
it is largely that it needs further development by mathematicians before it can
be used in connection with the problems of the sociologist and the economist.
Indeed, one thing which is quite certain is that in the years to come the social
and human sciences will have occasion to use increasingly complex mathematics.
Today we are still in the preliminary phase of development of a mathematical
methodology in the human sciences and the mathematicians of real substance
who have decided to devote themselves to this field are still infrequent. For
the rest, while some disciplines like linguistics and economics have already
reached the stage of posing complex mathematical problems to the mathematician,
it is otherwise in psychology and in sociology, disciplines characterized
by a high proportion of banal mathematical researches. However everything
points to the probability that there will be rapid progress in this field over the
next few decades.
E. The new epistemology
The last trait which must definitely not go unmarked in a chapter on social
science methodology, consists in the epistemological changes which have follow-
ed the development of mathematical methods and models.
Considering the situation in its least specific and most superficial aspect, that
of opinions on the usefulness or otherwise of mathematics in the social sciences,
it is roughly true to say that there is an increasing tendency to exhibit one or
other of two forms of nafvet6. The iirst consists in the belief that human phe-
nomena, for some obscure reason, cannot be susceptible of mathematical
treatment.
This dogmatic opinion dates from a period when mathematics was looked on
as a quantitative science, and when mathematical approaches to human phe-
nomena were most commonly ill-judged. Nowadays, when there are whole
branches of mathematics which do not use either the notion of number or the
familiar operations of arithmetic, it is very difficult to demonstrate a priori, by
philosophical arguments, that the social sciences cannot depend on mathematics.
There is a growing feeling that the possibilities of applied mathematics are not
restricted a priori by the nature of the subject to be treated and that the main
function of mathematics is to clarify. We should, however, note that this evolu-