City Research Online

Abstract

This thesis represents a study in measurement theory extended to non-classical systems characterized by an uncertain-ty relation which is finer than the known uncertainty relations of physical and technical sciences. It suggests corresponding changes to the classical measurement theory.

Classical measurement is simple, objective and determinative, but its formalism is limited to countable sets and systems, to Boolean logic and algebra with underlying Euclidean spaces and it excludes the human experimenter. Its language satisfies the requirements of objectivity to the extent that it admits hidden parameters accounting for this objectivity. These factors limit the representational theory of measurement in several respects. Difficulties arise whenever the axioms of distributivity and of tertium-non-datur fail in the systems to be measured, which is the case in complex engineering, frequently in systems at the verge to catastrophy, in complex problems of astrophysics and cryogenics and in most human sciences, mainly in psychology and in social science. In fact, the situation that presents itself resembles problems encountered in the transition from classical mechanics to complex quantum mechanics, but at an even higher mathematical level.

Complex empirical systems are shown to obey Tarski's calculus of systems; they comply with Brouwerian lattices with unity and underlying co-dimensional Banach spaces containing a so-called negligible set H (the dual to Planck's constant h) which enters a non-complementarity condition and an uncertainty relation for complex fuzzy systems corresponding to the non-commutativity condition and to the Heisenberg uncertainty relation of quantum mechanics, respectively. The incompatible conjugates of the measuremental uncertainty relation are - under these conditions and conform to the ’’part and the whole doctrine” - precision and relevance or significance. This is where human subjectivity (knowledge and will of the experimenter) replaces the classical objectivity. The experimenter can give preference either only to precision or only to relevance.

The uncertainty of measurement is clearly of mathematical origin (the negligible set), much finer and higher allocated than any technical or physical uncertainty. Its existence is proven beyond doubt.

Having determined this uncertainty, we establish a classical measurement channel penetrating into the complex fuzzy regime of the system by constructing a quotient space (or algebra) modulo uncertainty. This step restores Boolean conditions and the classical objectivity of measurement.

Finally, using some topological theorems due to M.H. Stone, it is shown that the homomorphic representation of measurement is equivalent to a monotone homeomorphic representation and that the space of measurement is a Stone space. In this way a combinatorial measurement on classical and non- classical systems is possible. The essence of non-classical measurement is the determination of a negligible set and the corresponding uncertainty relation; without them no such measurement of the state of a complex fuzzy system is possible. And the impossibility to measure entails the impossibility to calculate.

Publication Type:	Thesis (Doctoral)
Subjects:	T Technology
Departments:	School of Science & Technology > School of Science & Technology Doctoral Theses Doctoral Theses

Export

Downloads