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Onto-Med >> Theories >> GFO Part I Basic Principles

 
   


7.3 Boundaries of Material Structures

Let $x$ be a material structure which occupies a topoid $T$ and let $b$ be the spatial boundary of $T$. Does a material structure $y$ exist which occupies the boundary $b$? This seems to be impossible because material structures occupy three-dimensional space regions, while $b$ is two-dimensional. Nevertheless, we assume that such material entities exist, and we call them material boundaries. These are dependent entities that are divided into material surfaces, material lines and material points. Every material surface is the boundary of a material structure, every material line is the boundary of a material surface, and every material point is the boundary of a material line. We introduce the basic relation $\Gmatb(x,y)$ with the meaning $x$ is a material boundary of the material structure $y$. One may ask whether a material boundary of a body $B$ is a kind of ``skin'', a very thin layer that is a part of $B$. We do not assume this and consider material boundaries as particular dependent entities.

In contrast to spatial and temporal boundaries, material boundaries cannot coincide. Instead, in order to explain the notion of two material boundaries touching each other, their spatial locations must be considered. Two material structures (or their material boundaries) touch if their occupied space regions have spatial boundaries with coincident parts. One has to take into consideration here that the spatial boundary which is occupied by a material boundary depends on granularity and context. Cognitive aspects may refine this dependency. For example, the spatial boundary occupied by a material boundary may depend on an observer's distance from the considered objects.

Our notion of material structure is very general; almost every space-region may be understood as the location of a material structure. Without an elaborated account of unity, we single out material objects as material structures with natural material boundaries. A body is a connected material object that consists of an amount of solid substrate. An organism is an example of a body. The notion of a natural material boundary depends on granularity, context and view. This notion can be precisely as defined. Let us consider a material structure $S$, which occupies a topoid $t$ and let $B$ the material boundary of $S$ which occupies the boundary $b$ of $t$. A part $A$ of the boundary $B$ is considered to be natural if two conditions asre met (1) there is a material structure $P(A)$ outside of $S$ such that $P(A)$ and $S$ touch at $A$ (the spatial boundary occupied by $A$) and (2) $P(A)$ and $S$ (or a tangential part of $S$ with boundary $A$) can be distinguished by a property. Examples of such properties are fluid, solid, gaseous. As an example, let us consider a river. A river (at a time point of its existence, i.e., considered as a presential) is a material structure which consists of fluid substrate and has natural material boundaries at all places, with exception of the region of the river's mouth. The solid river bed may be distinguished from the river fluid and the river fluid may be distinguished from the air above the river. Within our framework certain puzzles can be easily solved. In Leonardo's notebooks there is mentioned:

What is it ... that divides the atmosphere from the water? It is necessary that there should be a common boundary which is neither air nor water but is without substance, because a body interposed between two bodies prevents their contact, and this does not happen in water with air. (15)

How can two things - the water and the air - be in contact and yet be separated? Leonardo's problem can be analysed as follows. There are two material structures $W$ and $A$ (water and air), $W$ consists of liquid substrate, $A$ consists of gaseous substrate. $W$ and $A$ have natural boundaries because at the ``touching area'' we may distinguish $W$ and $A$ by the properties ``fluid'' and ``gaseous''. These natural boundaries touch because their occupied space-boundaries coincide. The touching phenomenon is explained by the property described in the Brentano-space theory that pure space boundaries may coincide; they may be at the ``same place'' but, nevertheless, different. What is ``interposed'' between the two natural boundaries are two coinciding space-boundaries which do not occupy any space.

Robert Hoehndorf 2006-10-18
 
       
     
     
     

   
     
     
       
 

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