 In the mid-twentieth century the two main lines of inquiry in measurement theory, the one dedicated to the empirical conditions of quantification and the one concerning the classification of scales, converged in the work of Patrick Suppes (1951; Scott and Suppes 1958; for historical surveys see Savage and Ehrlich 1992; Diez 1997a,b). Suppes’ work laid the basis for the Representational Theory of Measurement (RTM), which remains the most influential mathematical theory of measurement to date (Krantz et al. 1971; Suppes et al. 1989; Luce et al. 1990). RTM defines measurement as the construction of mappings from empirical relational structures into numerical relational structures (Krantz et al. 1971: 9). An empirical relational structure consists of a set of empirical objects (e.g., rigid rods) along with certain qualitative relations among them (e.g., ordering, concatenation), while a numerical relational structure consists of a set of numbers (e.g., real numbers) and specific mathematical relations among them (e.g., “equal to or bigger than”, addition). Simply put, a measurement scale is a many-to-one mapping—a homomorphism—from an empirical to a numerical relational structure, and measurement is the construction of  scales.[6]  RTM goes into great detail in clarifying the assumptions underlying the construction of different types of measurement scales. Each type of scale is associated with a set of assumptions about the qualitative relations obtaining among objects represented on that type of scale. From these assumptions, or axioms, the authors of RTM derive the representational adequacy of each scale type, as well as the family of permissible transformations making that type of scale unique. In this way RTM provides a conceptual link between the empirical basis of measurement and the typology of  scales.[7]