Social Position and Behaviour: A Formalization

Modern society has witnessed anever-increasing development in the social sciences;partly due to changes in mentality, and partly due tothe growing requirements of the economic and political world – requirements which frequently takeon the guise of necessity, as for instance in thecases of market research and electoral opinion polls.Such development has produced an increasing paralleldemand for mathematical accuracy and exactitude inthese fields.The intention here is to rationalize the basic logicand methodology of the sociological procedure; anintention, it is hoped, which could be influential inimproving practical sociological work by rendering itmore comprehensible. This requires the employment offairly sophisticated mathematical and statisticalnotions.Section 1 outlines the definitions considered essential for the rationalization of the basiclogic framework previously described.Section 2 discusses the general concept of statisticalvariance. 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 theimmersion of metric spaces in Banach spaces, andshows how the theorem can be used to constructtheoretically satisfactory immersion algorithms.Section 7 together with the material, which ispreviously introduced in Section 4, outlines thecluster analysis and the principal components methodologies.