An object is a representation of a relation between actions.
In the simplest case it is a subset of A_{1}x A_{2}, where A_{1}
and A_{2} are some sets of actions. Such an object is a set
of ordered pairs a_{1}, a_{2}> with a_{i} \in A_{i}.
Such a pair tells that if action a_{1} takes place, then
it is followed by a_{2}.
An egg can be defined by pairs of actions which include:

(look at it and perceive a white spot)

(throw it on the floorand perceive a small pool of liquid)

(Keep it at the temperature of the motherhen's body for a certain
number of days and a chicken hatches), etc.
A more precise definition of an object can be achieved through relations
including more variables, including those which stand for other
representations: we allow here metasystem transitions which,
in the last analysis take us back to actions.
Suppose I am aware of a teapot on the table in front of me.
This is a result of my having
the mechanism of abstraction in the brain. I recognize the image on my
retina as belonging to a certain set of images, the abstraction `teapot'.
But there is more to it. I perceive the teapot as an object.
The object `teapot' is certainly not a definite image on the retina
of my eyes; not even a definite part of it.
For when I turn my head, or walk around the table, this image changes all
the time, but I still perceive the teapot as the same object. The teapot
as an object must, rather, be associated with the transformation of the image
on my retina which results from the changing position of my eyes.
This is, of course, a purely visual concept.
We can add to it a transformation which produces my tactile sensations
given the position and movements of my fingers.
The general definition of an object suggested by this example consists
of three parts.
 First we define a set R_{{}ob of representations which are said
to represent the same object;
in our example this set consists of all images of the
teapot when I look at it from different viewpoints, and possibly, my
sensations of touching and holding it.
 Then from the set of all possible actions we separate
a subset A_{cogn} of actions
which will be referred to as cognitive; in our case
A_{cogn} includes such actions as looking at the teapot, turning my head,
going around the table, touching the teapot etc.  all those actions
which are associated with the registration of the fact that a teapot
is there.
 Finally, we define a family of functions f_{a}(r), where for every
cognitive action a \in A_{{cogn}, the function
f_{a}: R_{{ob} \to R_{{ob }
transforms a representation r \in R_{{ob} into f_{a}(r) = r'
which is expected as a result of action a.
The most important part here is the third; the first two can be subsumed
by it. We define an object b as a family of functions f_{a}:
b = {f_{a}: a \in A_{cogn}}
It also can be seen as a subset of
A_{cogn} x R_{ob}x R_{ob}.
The set A_{cogn} is the domain of the index a; the set R_{ob}
is the domain and codomain of the functions of the family.
When I perceive an object b, I have a representation r
which belongs to the set R_{ob}; I then execute some cognitive actions,
and for each such action a I run my mental model, i.e. perform
the transformation f_{a} on r. If this anticipated representation
f_{a}(r) matches the actual representation r' after the action a:
f_{a}(r) = r'
then my perception of the object b is confirmed;
otherwise I may not be sure about what is going on. Observing
a teapot I check my actual experience against what
I anticipate as the result of the movements of my head and eyeballs.
If the two match, I perceive the teapot
as an object. If I travel in a desert and see on the horizon castles and
minarets which disappear or turn topsyturvy as I get closer,
I say that this is a mirage, an illusion, and not a real object.
The concept of an object is naturally (one is tempted to say, inevitably)
arises in the process of evolution. It is simply the first stage in
the construction of the world's models. Indeed,
since the sense organs of cybernetic animals
are constantly moving in the environment, these actions are the first
to be modeled. In the huge flow of sensations a line must be drawn between
what is the result of the animal's own movements, and other changes
which do not depend on the movements, are objective.
Description of the world in terms of objects factors out
certain cognitive actions.
Function f_{a} factors out the action a by predicting
what should be observed when the only change in the world is
the subject's taking of the action a. If the prediction comes true,
we interpret this as the same kind of stability as when nothing changes
at all. The concept of object btings in a certain invariance, or stability,
in the perception of a cybernetic system that actively explores
its environment.
Copyright© 1997 Principia Cybernetica 
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