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Mathematical models and methods 551
tion of opinion is not yet complete, as can be seen from P. Sorokin’s rather
ill-informed book, Fads and Foibles in Modern Sociology, and Don Martindale’s
recent essay, ‘Limits to the Use of Mathematics in the Study of Sociology’.z3 The
first of these publications is a sort of pamphlet written against the quantification
maniacs (quantophrenes). But one gets the impression that the adversaries the
author is combating were manufactured for the needs of the cause. In any case,
no one today would even think of claiming hegemony for the quantitative
methods and forgetting the considerable importance of qualitative methods in
sociology. Martindale’s essay is an attempt to demonstrate that sociology’s
subject matter and the special relationship between observer and observed
limit theappiicability of mathematics. Thisisadoubtful argument since sociology
does not deal with a particular subject matter, but with a variety of subject
matters, extremely diverse in logical character, resulting in forms of analysis and
definitions of the observer-observed which in turn are extremely diverse; to
appreciate this one need only recall that the mathematical approach is the
natural one for the analysis of ‘polls’ while ‘community studies’ have never yet
drawn on mathematics, the distinction resulting from the form of the data
collected in either case.
It should be noted, however, that sociology is today the only discipline in
which, at long intervals, doubts of the utility of mathematical methods are
still expressed.
The second form of nayvet6 consists in believing that the sciences of man can
be handled with a mathematical language and models shaped for the use of the
natural sciences. As was pointed out in Section I, a high proportion of the applied
mathematics in the human sciences dating from the ‘pre-modern’ period consists
of comparatively crude imitations of models borrowed from physics and biology.
The mathematics used on the phenomena of social diffusion and migration
are typical in this respect : throughout the ‘pre-modern’ period, mathematical
methodology in these areas consisted of pure transpositions of the law of
gravitation or the logistic curve. Nowadays, on the one hand it is more clearly
appreciated, that it is more often a matter of inventing new mathematical methods
than applying existing ones, and on the other it is acknowledged that the application
of mathematics to the social and human sciences cannot so easily attain
to logical integration and effectiveness as in the natural sciences.
But these phenomena of opinion are of only superficial interest. It is more
worthwhile to dwell on a certain number of concepts which have become central
in the methodology of the human and social sciences and are intimately tied in
with the development of mathematical methods: these are the concepts comprehended
in the terms model and structure. The definition of these two terms
poses major problems, and there is an abundant philosophical and epistemological
literature which endeavours to delimit their signification.”4 This fact
alone demonstrates that, despite the importance and widespread use of the two
concepts, these definitions and meanings are manifold. Despite this, it is possible,
eschewing any idea of hard and fast definition, to trace the notable features
of the history of the social sciences in recent times which explain the importance
of the blanket notions of model and structure.