Abstraction

The vertical lines on the scheme of modelling are functions, or mappings, from the states of the world W to representations R (the states of the language L of the model). This mapping is always an abstraction: some aspects of the state of the world are necessarily ignored, abstracted from, in the jump from reality to its description. The role of abstraction is very important: it reduces the amount of information the system S has to process before decision taking. The simplest case of abstraction, mathematically, is when the states of the world w_i are described by a set of variables w₁,w₂,...,w_n, and we can separate those variables, say w₁,W₂,...,w_m, on which M really depends, from the remaining variables w_m+1,...,w_n, on which it does not really depend, so that

M(w₁,w₂,...,w_m,x_{m+1},...,w_n) = M'(w₁,w₂,...,w_m)

We often use the same word to denote both a process and its result. Thus all representations r_i resulting from mapping will be also referred to as abstractions. It should be kept in mind, however, that abstraction is not so much a specific representation (a linguistc object), as the procedure M which defines what is ignored and what is not ignored in the mapping. Obviously, the object chosen to carry the result of abstraction is more or less arbitrary; the essense of the concept is in the transformation of w_i into r_i.

Mathematically, an abstraction r_i can be defined as the set of all those states of the world w which are mapped on r_i. i.e. the set of all w such that M(w) = r_i. The abstraction 'tea-pot' is the set of all those states s in S which are classified as producing the image of a tea-pot on the retina of my eyes.

A cybernetic system, depending on its current purposes, may be interested in different parts, or aspects of reality. Breaking the single all-inclusive state of the world w_i into parts and aspects is one of the jobs done by abstraction. Suppose I see a tea-pot on the table, and I want to grasp it. I can do this because I have in my head a model which allows me to control the movement of my hand as I reach the tea-pot. In this model, only the position and form of the tea-pot is taken into account, but not, say the form of the table, or the presence of other things on it. In another move I may wish to take a sugar-bowl. And there may be a situation where I am aware that there are exactly two things on the table: a tea-pot and a sugar-bowl. But this awareness is a result of my having two distinct abstractions: an isolated tea-pot and an isolated sugar-bowl.

The other aspect of abstraction results from the hierarchical nature of our language. We loosely call the lower-level concepts of the linguistic pyramid concrete, and the higher-level abstract. This is a very imprecise terminology because abstraction alone is not sufficient to create high level concepts. Pure abstraction from specific qualities and properties of things leads ultimately to the lost of contents, to such concepts as `something'. Abstractness of a concept in the language is actually its `constructness', the height of its position in the hierarchy, the degree to which it needs intermediate linguistic objects to have meaning and be used. Thus in algebra, when we say that x is a variable, we abstract ourselves from its value, but the possible values themselves are numbers, which are not `physical' objects but linguistic objects formed by abstraction present in the process of counting. This intermediate linguistic level of numbers must become reality before we use abstraction on the next level. Without it, i.e. by a direct abstraction from countable things, the concept of a variable could not come into being. In the next metasystem transition we deal with abstract algebras, like group theory, where abstraction is done over various operations. As before, it could not appear without the preceding metasystem level, which is now the school algebra.