Divisions create conceptual difficulties. At its simplest one space becomes two, separated by a boundary, in this case a line. Three things are then present: the two spaces and the boundary itself. If the boundary has any width at all, then it also occupies space, with a boundary around its edge, which has no width and so occupies no space. If it occupies no space, then we cannot say where it is. Is the boundary part of the line or the space, or both? It cannot be part of the space because if the line is removed then the boundary disappears, so it must belong to the line. But what proportion of the line's width does the boundary occupy? None, since the boundary has no width. So even though the line itself is a boundary between the two spaces, it actually contains no boundaries in itself — at least none that occupy any of its surface area.
Saturday 30 June 2007
Boundaries
Divisions create conceptual difficulties. At its simplest one space becomes two, separated by a boundary, in this case a line. Three things are then present: the two spaces and the boundary itself. If the boundary has any width at all, then it also occupies space, with a boundary around its edge, which has no width and so occupies no space. If it occupies no space, then we cannot say where it is. Is the boundary part of the line or the space, or both? It cannot be part of the space because if the line is removed then the boundary disappears, so it must belong to the line. But what proportion of the line's width does the boundary occupy? None, since the boundary has no width. So even though the line itself is a boundary between the two spaces, it actually contains no boundaries in itself — at least none that occupy any of its surface area.
