Boundaries

Yesterday we talked about how objects relate to each other across scales via abstraction. We took it as a given that we had an object, and asked about its properties, uses, and constituent objects. Today, I want to examine our assumption that an object is a discrete entity.

Let's consider a boundary to be an object-defining distinction. Take our putting green, for example:



Any reasonable person viewing the above picture can easily distinguish the putting green from the fairway. The boundary between these two objects is clear, that is, uncontroversial.

But now consider this picture:



[NB: I need to get a better picture, one that zooms in even more so you are really looking at the blades of grass and can't see a boundary at all.]

This is a boundary between a green and a fairway, but it is no longer possible to perceive it as such. [UPDATE THIS WITH NEW PICTURE To which object--green or fairway--does the blade of grass with its roots in the green but its stalk in the fairway belong? Or vice versa? And what of the roots that surely cross from one side to the other below ground?] In this picture, our boundary is unclear or controversial.

Scale and abstraction explain why the boundary is clear in one picture but unclear in the other. At the "hole" scale, the putting green and the fairway are seen in the abstract: only certain of their properties are in view, e.g., their smoothness, location, etc. However, at the "blade of grass" scale, we have to account for certain other properties of blades of grass that we otherwise ignore when we collect them into the abstraction "putting green," namely, their growth habits and their own shape and structure.

It appears, then, that a boundary between two objects that is clear at the next-higher scale may be unclear at the next-lower scale. A boundary is fuzzy if it is dependent upon scale for its clarity. The fuzziness or width of a boundary is a function of the properties of the objects participating in the boundary, and it becomes a property of the higher-scale object that the boundaries between its constituents be of such-and-such an acceptable fuzziness.

Armed with these additional terms, we may now also ask the following when presented with an object:

• What are the boundaries of this object's constituents?
• What is the actual and acceptable fuzziness of these boundaries?
  
• How do its constituents' constituents' properties affect the fuzziness of these boundaries?
• What boundaries does our object participate in?
• How do its properties affect the fuzziness of these boundaries?