Forms are specific Patterns of Containment.
All the possible containment patterns constitute the language of James forms. The mathematical abstraction that comes closest to describing James forms is a partial ordering.
A partial ordering is a Graph consisting of nodes and links. The nodes are containers that are delineated by a boundary. The links are containment relations. A physical, or finite, strict partial ordering has these defining characteristics:
There is a top node and a bottom node.
There is a direction, every node is on a Path between top and bottom.
Links identify specific directional relations between nodes.
Nodes bound links.
Within the theory of relations, a partial ordering has three characteristics
irreflexive: no node is linked to itself
antisymmetric: no node is above or below itself
transitive: you can travel to distant nodes
One objective of Iconic Arithmetic Volume II is to look at these rather strange notions in detail. They fail to convey the intent of a containment pattern. Consider contains to be parent-of. You cannot be your own parent (irreflexive), and you can’t be your parent’s parent (antisymmetric). As well, your parents are not the parent-of your children (intransitive). Technically then containment is not a partial ordering because it is not transitive. There is a transitive concept that we might call contained-at-any-depth. In our example it would be the ancestor relation. But the deeper point is that using conventional abstractions based on sets does not particularly help us to think differently about the formal structure of distinction.
# Interpretation
Figure 6-3 provides a quick introduction to the interpretation of boundary configurations that we will use. These patterns unify counting, adding, multiplying,
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raising to a power, and the assortment of inverse operations. However, numbers and numeric operations are but one interpretation of what a container form might mean. When translating from one language to another (here, for example, between configurations of containers and conventional arithmetic), the more primitive, less redundant and therefore foundational language will have multiple alternative interpretations within the more sophisticated and complex language. Thus, the single boundary configuration A <B> can be read both as A + –B and as A – +B. Interpretation from a simpler foundation is one-to-many.