Models, congruence, and commutivity
D. C. Mikulecky
Professor of Physiology
Medical College of Virginia Commonwealth University
Modeling is the art of bringing entailment structures into congruence.
Congruence of mathematical objects:
Imagine a collection of ellipses of different sizes, eccentricity, and orientation in the plane. Take any two of them, E1 and E2. There will be a coordinate transformation on one of them, T, which will rotate it, translate it and scale its major and minor axes so that falls on top of the other. These two ellipses are then clearly congruent. They differ only under coordinate transformations. They are not equivalent since it takes a coordinate transformation to bring about this congruence.
In this manner, the notion of congruence can be extended to objects under transformation. In general, objects are congruent if there are two coordinate transformations t1 and t2 on their equations E1(x,y) and E2(x,y) such that
t1 E1(x,y) = t2 E2(x,y)
The equation for an ellipse, E(x,y) is a relation on the Cartesian product of the Real numbers with themselves. This Cartesian product, R x R, simply means that in each pair (x,y) the x comes from the first set in the product and the y from the second. Thus the equation for an ellipse is a particular locus or a particular subset of all the points in the plane.
We can express these relations as mappings as
E: (x,y) à E(x,y).
From the congruence equation above,
E1(x,y) = t1-1· t2 E2(x,y)
Which can be represented by a diagram of mappings which is said to commute, that is you can go directly from (x,y) to E1(x,y) (counterclockwise) or you can go clockwise.
(x,y) ------- > E2(x,y)
E1 | | t2
E1(x,y) <-------- t2 E2(x,y
( t1 )-1
Modeling relations: transductions one kind of entailment converted to another in an invariant way:
Causal Entailment à Inferential Entailment
In other words, we simple require that these be brought into a kind of congruence.
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