The Network Dalect (Figure 13-6 within Multiple Replicas of Variables) is certainly the most computationally tractable. It provides the image of Nodes/Locations/Processors with Links/Paths/Wires between them.
Each link is a `contains` relation, so that the relational structure is explicit.
Networks and other relational dialects have an additional display requirement to specify the direction of nesting, from shallowest to deepest, which requires directed links. The network version often presumes a gravitational metaphor, with deeper nesting shown at lower levels. For a network the level of nesting is defined by counting the nodes lying between beginning and Goal (Input and Output).
Networks are a well-established modeling tool in Computation and in Mathematics. The structure of the generic frame is clearly visible in a network as a link between an upper round node and a lower square node. Multiple contents are multiple lower links. Deletion of structure is just disconnecting a link. Even the inverter diamond fits naturally into the patterns of flow. And networks support any number of types of Container.
Unlike the enclosure dialects, networks do not require Multiple Replicas of Variables. We can use a single node for each variable, and access the variable through multiple links, or pointers. The forms of Arrangement and Replication in Figure 13-6 show this feature clearly. The experience of driving on roads between different cities provides familiarity to the network approach of no replicated objects.
The fluidity of object and reference in networks can be expressed as a Transformation Rule that is unique to this (and similar) dialects, Structure Sharing. In structure sharing, nodes in a network that share the same linking structure can be joined into a single node with multiple links. Entire subnetworks are replaced by links to shared structure. **Structure sharing is not available in textual dialects.** The textual representation of each of the above forms is ([A])([A]). Multiple occurrence of the same variable in a textual dialect emulates shared structure in a network dialect. The absence of structure sharing in symbolic mathematics has lead to rampant replication of symbols and a presumption that replication is free. As noted in Chapter 9, Replication is the source of Complexity.
Not only is Notation not independent of Meaning, it can also actively determine meaning. There is little structure sharing in the physical world since all physical objects are unique.
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*Structure Sharing* is different than structural abstraction. Sharing joins together multiple references to the same form, while abstraction ignores selected features. Human prejudice probably does not apply to structure sharing since sharing is a property of Representation rather than reality. We do have Patent Laws that adjudicate cases of structure sharing in creative works.
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If two sheets of paper have the same sentence written on them, then the two virtual sentences share the same structure. However, the two physical sheets of paper do not share the same structure (i.e. they are not replicas) although they may share abstract properties such as use, color, shape, even content. Representational systems that lack structure sharing risk confusing features of the symbolic model with features of the physical circumstance being modeled.
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William Bricken, Iconic Arithmetic Volume I: The Design of Mathematics for Human Understanding (Unary press, 2019), p. 319–320.