Models, congruence, and commutivity

D. C. Mikulecky

Professor of Physiology

Medical College of Virginia Commonwealth University

http://views.vcu.edu/~mikuleck/

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, E_{1} and E_{2}. 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 t_{1} and t_{2 }on their equations E_{1}(x,y) and
E_{2}(x,y) such that

t_{1} E_{1}(x,y) = t_{2} E_{2}(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,

E_{1}(x,y) = t_{1}^{-1· }t_{2}
E_{2}(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 E_{1}(x,y) (counterclockwise) or you can go
clockwise.

E_{2}

(x,y) -------
> E_{2}(x,y)

E_{1} |
| t_{2}

V V

E_{1}(x,y)
<-------- t_{2} E_{2}(x,y

( t_{1 })^{-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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