Geometry of Concepts

Abstract

Every study organizes data according to a specific conceptualization scheme, which is defined by the purpose and method of exploration. Co-processing data from diverse studies requires concept mapping. The model presented in this work places all elements of knowledge into a topological space. Conceptualization schemes subdivide this space into subspaces of lower dimensionality where every element has well-defined coordinates. Allocating semantics to the conceptualization scheme enables the use of abstract mathematical approaches, such as category theory and geometry, for concept and data mapping. Relative coordinates of concepts, models and data are defined via morphisms that represent complex relationships among these elements. Data models for implementing morphisms in a database are presented here. This work provides a framework for data and knowledge integration illustrated with practical examples. It addresses several important challenges in interdisciplinary data integration and ontology building, such as defining complex relationships and unambiguous data interpretation. The geometrical interpretation enables visualization of the intangible world of data and knowledge and facilitates interactive and meaningful discussions of the subject.