Abstract Measurement Theory
This book presents a theory of measurement, one that is "abstract" in that it is concerned with highly general axiomatizations of empirical and qualitative settings and how these can be represented quantitatively.
The need for quantitative measurement represents a unifying bond that links all the physical, biological, and social sciences. Measurements of such disparate phenomena as subatomic masses, uncertainty, information, and human values share common features whose explication is central to the achievement of foundational work in any particular mathematical science as well as for the development of a coherent philosophy of science. This book presents a theory of measurement, one that is "abstract" in that it is concerned with highly general axiomatizations of empirical and qualitative settings and how these can be represented quantitatively. It was inspired by, and represents a generalization and extension of, the last major research work in this field, Foundations of Measurement Vol. I, by Krantz, Luce, Suppes, and Tversky published in 1971.
Abstract Measurement Theory presents an overview of the subject with a high degree of generality; it explores several new directions of development; and it introduces a number of significant recent results. One of its major new directions is the extension of measurement to non-Archimedean situations through the use of nonstandard analysis. Among the other topics discussed are the classification and axiomatization of the possible scale types that can occur in science, the theory of numerical representations for ordered relational structures, the generalization of extensive measurement to situations where concatenation operations need be neither associative nor commutative, and the measurement of "conjoint" ordered situations-ones that can be factored into separate, ordered components. Throughout the book, emphasis is placed on attaining a deeper and more exact understanding of the role of axiomatization in the theory of measurement.