Theory and Program: Isomorphic Structures

Using, Rather Than Merely Alluding To, Theory
©1999 Edward G. Rozycki

RETURN
edited 4/14/12

Related Papers:
Isomorphism Everywhere?
Isormorphism Made Recognizable
Theory Presentation & Evaluation
Using Theory

 

 

 

Introduction
It is common that people use terms belonging to a theory to explain what they are about. They may be be quite mistaken, however, in thinking that they are using the theory from which the terms are borrowed. Using terminology from a theory is not necessarily using the theory itself. Is there a way to determine when a theory is actually being used and to what extent?

A good theory is like a good map. If we plan our itinerary with a good map it gets us to where we want to go. Why? Because the logical structure of the map parallels -- or, as we will say, is isomorphic to -- the spacial structure of the roads we are travelling.

Consider the following:

a. A theory is defined by a set of variables and the set of relations among them. A map generally indicates route intersections, the connections among them and the distances between them.

b. A program is a well-specified procedure. It is defined by a set of operations and the set of relations among them. A program like an itinerary, for example, may specify operations like, "drive north on route 363 for 4.5 miles to route 611." Such operations will generally only work if they are done in a specific sequence.

c. Informally, we can think of an isomorphism as a relationship of perfect correspondence of parts. A program Px is isomorphic to a theory Tz, if and only if for each operation in Px, there corresponds only one variable in Tz. Also, for each relation among operations in Px, there corresponds one and only one relation among variables in Tz. Our itinerary will be isomorphic with our map if and only if for each critical intersection on our trip, there corresponds an operation which takes us to it.

d. To the extent that a program is isomorphic to a theory, those who pursue the program may be said to be using that theory. Mere allusions to a theory will lack many correspondences between operations and variables, and the relationships among them. A person might well, for example, get from one place to another from habit or chance without following a map. Or a teacher may say that he is "reinforcing" student "responses" without actually following operant conditioning theory, as, for example, he writes an "A" on a report card at a time and place long removed from the presence of the student's behavior.

e. Programs pursue goals. An "adequate" theory is one whose isomorphically related program achieves its goal.

Recognizing Isomorphisms

Which of the following diagrams are isomorphic with which? Why?
 

diagram 1

diagram 2

diagram 3

Problem Set A:

1. Can you construct an isomorphism between the English alphabet and the integers from 1 to 26? (Specify matching rules which create this isomorphism.)

2. Is your watch face isomorphic to the hours in the day?

3. In what ways might even a very good map not be isomorphic to the territory it maps?

Answers to the Questions
presented in this paper may be found here.

Comparing Programs and Theories

A program can fail to be isomorphic to a theory in a variety of ways:

1. Program Redundance or Insensitivity. The program can have operations which do not correspond to theory variables uniquely. If the program has extra operations, it is redundant. If one of its operations ambiguously match with more than one theory variable, it is insensitive. Imagine your not knowing that your itinerary has two sets of directions that bring you to the same place. Or worse, imagine your itinerary gives you the same directions for getting to two different places!

2. Bad Program or Bad Theory. The relations among operations may not match uniquely the relations among theory variables. Suppose your itinerary specifications give you different driving distances than your map. Or misdirect you north when the map indicates south! Then your program (itinerary) is badly constructed. However, if you still get where you were intending to go, you might well conclude your map is bad.

Problem Set B:

1. Consider the teacher mentioned above who says he is reinforcing student responses by giving certain grades on the report card. If you consider operant conditioning theory, paying careful attention to the variables it deals with and the relationships among them, especially timing, do you think the teacher is actually following the theory? Why or why not?

2. Is imprisonment necessarily aversive conditioning?

3. If learning by observation alone is possible, what kinds of variables will a theory that purports to explain such learning require?

4. Construct a list of questions you would ask a person who claims his or her research (or training) program is guided by a particular theory.

 

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