The paradigms of OLAP, multidimensional modeling and data mining have first emerged in the areas of market analysis and finance to address various needs of people working in these areas. Does this mean that they are useful and applicable in these areas only? Or, can they also be applicable in the other more traditional areas of science and engineering? What characterize the systems for which these paradigms are suitable? What are the goals of these paradigms? How do they relate to the traditional body of knowledge that has been developed throughout the centuries in the areas of mathematics, statistics, systems science and engineering? Where, how and to what extent can we leverage the conventional wisdom that has been accumulated in the aforementioned disciplines to develop a foundational basis for the above paradigms? The goal of this paper is to address these questions at the foundational level. We argue that the paradigms of OLAP, multidimensional modeling and data mining can also be applied successfully to complex engineering systems, such as membrane-based water/wastewater treatment plants, for example. We develop mathematically-based axiomatic definition of the concepts of 'dimension,' 'dimension level,' 'dimension hierarchy' and 'measure' using set theory and equivalence relations.