Look at the next two lines.

2+3=5

@@+@@@=@@@@@

Do they mean the same thing?

Many of you will be tempted to say that they represent the same thing, a simple addition problem.  But they are very different in their external form but the underlying properties (namely, two things added to three things makes five things).  The two patterns are said to be isomorphic (Hofstadter, 1980).  That is a bit of a misnomer, because they are not the same shape, but the conceptual level is the same.  Superficial different, but behaving the same way in some abstract sense.  In this case, and most cases, we identify the similarity by the behavior.  Do they behave the same way in some identifiable sense.  That is an isomorphism.  In the above example, something representing a two (2 and @@) are combined to something representing a three (3 and @@@) to give something representing a five (5 and @@@@@).  This discussion of the example necessarily requires some use of isomorphism to make clear.  Numbers are abstract and thus require use to use some concrete isomorphism to talk about (numerals being the most common).

This use of isomorphisms is the same way that a computer program is tested to see if it works.  Does is behave in a desired way?  This criterion is the basis of programs that are used to model some aspect of the brain or mental function.  Take, for example, the letter and word identification model of McClelland and Rumelhart (1981).  If you are not familiar with the model, it might help to go to these pages which have an interactive illustration of a very simple version of the model that is visually presented.  The specifics are not really that important.  However, they claim that this is a good model, simple as it is, because its behavior is like that of a research subject.

In their book on these types of models, they are very clear about the nature of isomorphic proof:

The model has been applied in detail to the role of familiarity in the perception of letters in visually presented words, and has proved to provide a very close account of the results of a large number of experiments (Rumelhart & McClelland, 1986, p. 20).

Here, in their textbook on the PDP approach, they make their philosophic point clear.  Isomorphism, similarity in behavior as it were, is the goal.  In a sense, this has been the goal at least since the Turing test was proposed by British mathematician Alan Turing (1950).  The Turing test, also called the imitation game, was designed as a way of trying to see if a machine had intelligence.  It is a widely debated task, but the principles are simple.  Turing sets up a situation: say a there are two people in a separate room and we can communicate only by text terminal.  One is a man and the other a woman.  Our task is to figure which is which by asking questions of the two people.  One, say the woman, is trying to help us, and the other, say the man is trying to make us guess incorrectly.  This rules out simple questions like "who is the woman?"  We can ask any question we like.  You can try this version of the game online here.  I have run this task in my cognition class and it is not as easy as it sounds.  Now, Turing, using this base, changes the question, remove one of the people and replace it with a machine.  If we are not able to be any more correct in guessing which of the entities is the machine than we are which person was the woman, then the machine must be able to think.  It can think because it imitates, successfully, human verbal interaction.  Imitation! Isomorphism!

External behavior is taken as an indication of internal properties and it is up those asking the questions, either us in the Turing test or the researcher in psychology, to ask the most discriminating questions so that we can tell when we have a fraud in our midst.  Even if we do not accept this as a criterion of a thinking machine, there is much greater acceptance of this criterion for evaluation of a model of some brain or cognitive function.  This criterion would be the "weak AI" of John Searle (1980).

How does this apply to this model of the visual system or even the visual system itself?  Well, I don't have a better criterion at this time than that of isomorphism for testing the model.  I will compare its outputs against data in the literature.  More substantively, the model needs to indicate how the brain achieves a sort of isomorphism with the physical world so that it allows us to move and behave in this space.  By isomorphism, I am not indicating that the brain action is in any physical sense the same as the object.  The isomorphism is on a more abstract level.  This is not the field theory of Köhler (1920) where isomorphism became literally associated with the electromagnetic fields of the world.  At this point I am not ready to specify the nature of the isomorphism, but the search for this isomorphism will guide this effort.