States of Distributed Objects in Conceptual Semantic Systems

Abstract

Our classical understanding of objects in spatiotemporal systems is based on the idea that such an object is at each moment at exactly one place. As long as the notions of “moment” and “place” are not made explicit in their granularity the meaning of that idea is not clear. It became clear by the introduction of Conceptual Time Systems with Actual Objects and a Time Relation (CTSOT) using an explicit granularity description for space and time and an object representation such that each object is at each moment in exactly one state – where the states are formal concepts of the CTSOT.

For the purpose of introducing also a granularity tool for the objects the author has defined Conceptual Semantic Systems where relational information is combined with the granularity tool of conceptual scales. That led to a mathematical definition of particles and waves such that the usual notions of particles and waves in physics are covered. Waves and wave packets are “distributed objects” in the sense that they may appear simultaneously at several places.

Now the question arises how to introduce a mathematical notion for the “state of a distributed object”, as for example the state of an electron or the state of an institution, in the general framework of Conceptual Semantic Systems. That question is answered in this paper by the introduction of the notion of the “aspect of a concept $\textbf{c}$ with respect to some view Q”, in short “the Q-aspect of $\textbf{c}$” which is defined as a suitable set of formal concepts. For spatiotemporal Conceptual Semantic Systems the state of an object $\textbf{p}$ at a time granule $\textbf{t}$ is defined as the spatial aspect of the infimum of “realizations” of $\textbf{p}$ and $\textbf{t}$. The one-element states of “actual objects” in a CTSOT are special cases of these states which may have many elements.

The information units (instances) of a Conceptual Semantic System connect the concepts of different semantical scales, for example scales for objects, space, and time. That allows for defining the information distribution of the Q-aspect of a distributed object $\textbf{c}$ which leads to a mathematical definition of the “BORN-frequency”; that is defined as a relative frequency of information units which can be understood as a very meaningful mathematical representation of the famous “probability distribution of a quantum mechanical system”.