Social position and behaviour: a formalization

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 methodologies.