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to conceptual tools. From the Rosen modeling relation (Figure 14.6), it appears that metaphor
is a one-to-many expansion. A formal model, which is a set of scaling relationships, can be
encoded into and decoded from a material system as a metaphor. In our scheme, a single
metaphor can apply to more than one material thing or situation. Metaphor is associated with
a coded representation (Figure 14.6). It is in the one-to-many quadrant of Figure 14.7. If two
material systems can be represented by one formal model (Figure 14.6), then the two material
systems become analog models of each other. Analogies are compressions down to only what
the analogs have in common. They therefore belong in the many-to-one quadrant, diagonally
opposite to metaphor in Figure 14.7.
There are various usages to the term metaphor. Certainly some metaphors go only one
way, from representation to what is represented. The metaphor “time is money” does not
reverse to “money is time.” At the least, the reverse of time and money will not be the same
metaphor. We understand that some things called metaphor by some scholars go both ways
symmetrically, the first to another and the other back to the first. If there are two items with
links going both ways, like Aristotle’s plow and ship’s prow, in our parlance that would be an
analog compression down to structures both able to cut through fluid material (water or soil).
We wish to make the distinction between metaphor and analogy. For us, metaphor only
works in one direction of model to material object, as distinct from analogy, which is a
mutual compression down to what the two or many have in common (Figure 14.6).
The one-to-one quadrant of Figure 14.7 yields a mathematical function, where a point
on one variable plots to just one point on another. It is sometimes taken as a model that
means that one thing determines another. The many-to-many quadrant yields what
mathematicians call relations. Each variable value can map to many points on the other
variable. There is slack in a relation. When scatter diagrams between a pair of variables
produce an oval cluster of points, the width of the oval, or the thickness of the sausage, shows
the slack in the relationship. That is quantified by r
for correlation coefficients. Function
means a specific relationship, while relation pertains to a more general condition.
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Figure 14.7
A two-by-two lattice that spells out many-to-one and one-to-many
relationships in combination.
Figure 14.7 invites comparison between the cells. Those comparisons are entered in a new
lattice in Figure 14.8. The margins of the new lattice are “model” and “narrative.” The cells
are situated at one of four transitions in moving through the modeling process: model to
model; model to narrative; narrative to model; and narrative to narrative. The function
version of metaphor leads to a model, in that models are local and explicit, relative to
narratives. Alternatively, metaphor to relation is more open, and suggests narrative, not a
model. There is slack in a narrative as there is in a relation.