Quality & Quantity 34: 275–297, 2000.
© 2000 Kluwer Academic Publishers. Printed in the Netherlands.
Social Position and Behaviour: A Formalization
MASSIMO CLAVELLI and ANNA ATTIAS
Università di Roma “La Sapienza”, Facoltà di Economia. Dipartimento di Matematica per le
Decisioni Economiche, Finanziarie e Assicurative, e-mail: firstname.lastname@example.org or
Abstract. Modern society has witnessed an ever-increasing development in the social sciences;
partly due to changes in mentality, and partly due to the growing requirements of the economic and
political world – requirements which frequently take on the guise of necessity, as for instance in the
cases of market research and electoral opinion polls. Such development has produced an increasing
parallel demand for mathematical accuracy and exactitude in these ﬁelds.
The intention here is to rationalize the basic logic and methodology of the sociological procedure;
an intention, it is hoped, which could be inﬂuential in improving practical sociological work by
rendering it more comprehensible. This requires the employment of fairly sophisticated mathematical
and statistical notions.
Section 1 outlines the deﬁnitions considered essential for the rationalization of the basic logic
framework previously described. Section 2 discusses the general concept of statistical variance.
Section 3 introduces the notion of “heterogeneity” and offers several propositions linked to this
concept. Section 5 revolves around the notion of “inaccuracy”. Section 6 presents a theorem relating
to the immersion of metric spaces in Banach spaces, and shows how the theorem can be used to
construct theoretically satisfactory immersion algorithms. Section 7 together with the material, which
is previously introduced in Section 4, outlines the cluster analysis and the principal components
Key words: behaviour, social position, metric spaces, Banach spaces, variance, cluster analysis,
When dealing with sociological problems, that is with reference to the science of
social fact, the statistical problems which arise, and therefore the statistical science
employed, need a more sophisticated treatment. Studying the stratiﬁcation of social
body and its interacting phenomena is totally different from evaluating the effect-
iveness of a batch of light bulbs or the average length of bolts regularly used in a
factory, largely because one is dealing with a mass in movement whose changes in
state may be inﬂuenced by a very large number of circumstances, not all of which
Even if the work is the result of the authors’ common discussion, we declare that Sections 3,
5, and 6 and related proofs in appendices must be attributed to Massimo Clavelli, while Sections 2,
4, and 7 and related proofs in appendices must be attributed to Anna Attias; Introduction, Premise,
Section 1 and References must be attributed to both authors.