Concept

A concept is any part of language which has an instrumental meaning. In human language we distinguish neuronal and logical concepts (see Human language).

The vertical lines in the modeling scheme are functions, or mappings R from the states of the world w_i to their representations r_i (the states of the brain of the subject system S). 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 making.

The term abstract concept or simply concept is often used, and will be used here, as asynonym of abstraction.

We often use the same word to denote both a process and its result. Thus all representations r_i resulting from mapping are also referred to as abstractions. We may say, for instance, that triangle is an abstraction from all specific triangles. It should be kept in mind, however, that abstraction, or an abstract concept, is not so much a specific representation (such as a language object), as the procedure 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 procedure that transforms w_i into r_i.

Given a representation function R and a specific representation r, we can construct a specialized function which will recognize the fact that a given state of the world w belongs to the abstraction r. We define this function as follows:

P_r(w) = True if R(w) = w,

False if R(w) =/ w.

Here True and False are two symbols chosen to represent two possible results of the function. Such functions as P_r are known as predicates (see). The predicate P_r(w) takes the value True for those and only those states w which are represented by r.

Whenever we speak of a predicate, we can also speak of a set, namely, the set of all states of the world w for which the predicate is true (takes the symbol True as its value). We shall call this set the scope of the abstract concept. Sometimes it is convenient to define an absraction through its scope, e.g., a member of John Smith's family can be defined by simple enumeration. But the universal way is the definition by property, in which case a reformulation through a set does not clarify the concept. If we "define" abstract triangle as the set of all specific triangles, it still remains to define what a specific triangle is. The real definition of an (abstract) triangle is a predicate, a procedure which can distinguish a triangle from everything else.

A cybernetic system may be interested, depending on its current purposes, 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, as long as they do not interefere with grasping the tea-pot. In another move I may wish to take a sugar-bowl. Also, I may be 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 degree of abstraction can be measured by two parameters: its scope and level; correspondingly, there are two ways to increase abstraction.

The scope was defined above. The wider the scope of a concept, the more abstract it is. The classical example is: cats, predators, mammals, etc. up to animal. With this method, the content of the concept (i.e. the amount of specificity in its procedural definition) decreases as the degree of abstraction increases. We come finally to such universal abstractions as object or process which carry almost no information.

The level of an abstraction is the number of metasystem transitions (see) involved. In the modeling scheme the sibject-of-knowledge system S is a metasystem with respect of the world. Indeed, S {\controls} the world: it takes information from the world as input to create representation using R, processes it using M_a for various a, chooses a certain action a, and executes it at the output, changing thereby the state of the world. The brain of S, as the carrier of world representations, is on the metalevel, a 'metaworld', so to say. Assigning to the world level 0, we define the abstractions implemented in the brain as abstractions of the first level.

However, the creation of models of reality does not necessarily end with the first level abstractions. The brain of the level 1 can become the world for the new brain, the brain of the level 2. Naturally, the necessary machinery comes with it -- the second metasystem transition has taken place. Our human brain is known to have a many-level hierarchical structure. The "brain" of the first level is nothing but our sense organs. The eye contains the retina and the machinery necessary to through light on it and create representations of the world. The retina keeps first level abstractions and serves as the input for the next level of the brain hierarch whose representations will be abstractions of of the second level, etc.

The concept of number can serve as an example of an abstraction of the second level implemented in human language. Specific numbers are first-level abstractions. We derive them directly from our experience. The representation function for this model (arithmetic) is counting. The brain (in our technical sense) includes some physical carrier to keep specific numbers, such as an abacus, or paper and pencil. A specific number, say three is an abstraction from three apples, or three sheep, etc., whatever we are counting. The concept of a number is an abstraction from one, two , three, etc., which all are abstractions themselves which exist only in the first-level brain, and if not that brain, would not be found anywhere.

Another important example of a concept of the second level of abstraction is system (see).

Abstractions of the second and higher metasystem levels will also be called conceptual schemes.

In algebra the second-level abstraction number becomes a material for higher abstractions, and this metasystem transition can be repeated many times. When we say that the concepts used in mathematics or philosophy are very

abstract, we should realize that this high degree of abstraction comes not from the increase in scope, but increase in the level. We constract a language which is, like our brain, a multilevel hierarchical system. High level of abstraction goes in step with the high level of construction.

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!!! Abstraction breaks the world into parts by ignoring something but including something else. This refers to processes, not only to objects. In fact, the only possibility to define what is process (informally -- a part of the world) is to refer to an abstraction mechanism: what is left is a process, what is abstracted away is not this process, but "other processes"

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Our language is a multilevel system. On the lower levels, which are close to our sensual perception, our notions are almost in one-to-one correspondence with some conspicuous elements of perception. In our theories we construct higher levels of language. The concepts of the higher levels do not replace those of the lower levels, as they should if the elements of the language reflected things "as they really are", but constitute a new linguistic reality, a superstructure over the lower levels. We cannot throw away the concepts of the lower levels even if we wished to, because then we would have no means to link theories to observable facts. Predictions produced by the higher levels are formulated in terms of the lower levels. It is a hierarchical system, where the top cannot exist without the bottom.

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, was abstract ourselves from its value, but the possible values themselves are numbers, i.e. linguistic objects formed by abstraction 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.

There is another parameter to describe logical concepts. This is the degree to which the language embedding the concept or concepts is formalized. A language is formal, or formalized, if the rules of manipulation of linguistic objects depend only on the `form' of the objects, and not on their `meanings'. The `form' here is simply the material carrier of the concept, i.e. a word, an expression. The `meaning' is the sum of association it evokes in the human brain. While `forms' are all open for examination and discrimination, i.e. objective, `meanings' are subjective, and are communicated indirectly. Operations in formal languages can be delegated to mechanical devices, machines. A machine of that kind becomes an objective model of reality, independent from the human brain which created it. This makes it possible to construct hierarchies of formalized languages, in which each level deals with a well-defined, objective reality of the previous levels. Exact sciences operate using such hierarchies, and mathematics makes them its object of study.

Classification of languages by these two parameters leads to the following four types of language-related activities (I take the table from by book The Phenomenon of Science):

{\begtt

~ Concrete language Abstract language ~

~

Unformalized Art Philosophy language

~

~

Formalized Descriptive Theoretical language sciences sciences, ~ mathematics \endtt}

Art is characterized by unformalized and concrete language. Words and language elements of other types are important only as symbols which evoke definite complexes of mental images and emotions. Philosophy is characterized by abstract informal thinking. The combination of high-level abstract constructs used in philosophy with a low degree of formalization requires great effort by the intuition and makes philosophical language the most difficult type of the four. Philosophy borders with art when it uses artistic images to stimulate the intuition. It borders with theoretical science when it develops conceptual frameworks to be used in construction of formal scientific theories. The language of descriptive science must be concrete and precise; formalization of syntax by itself does not play a large part, but rather acts as a criterion of the precision of semantics.

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A piece of knowledge is true if the predictions made by the subject (the user) of knowledge on the basis of this knowledge are true. A prediction that test T is successful (denoted as T!) is true if the test T is, in actual fact, successful.

What this definition says is, essentially, that for every statement S the statement 'S is true' is true if and only if S itself is true. We call such statements equivalent. But equivalency is not a tautology; the meanings of these two statements are different, even though they always are either both true or both false. To see this clearly we want to make more formal the system aspect of our definition.

Statements of a language represent some pieces of knowledge which can be reconstructed from statements by the recipient of the message. (which may be the sender of the message itself). These pieces of knowledge produce, ultimately, some predictions: T_1!, T_2!, ... etc. that certain tests are successful. Knowledge about the world constitutes a metasystem with respect to the world in which the tests T_1, T_2, ... etc. take place. When we speak of statements and determine their various properties, we create one more metasystem level; this is a metasystem transition. While the first level is the level of the use of knowledge, the second level is that of the analysis and evaluation of knowledge.

Generally, the construction of a new level of analysis and evaluation may be extremely fruitful and all-important. What we usually want are efficient procedures to decide whether statements are true or false. But as long as the semantics of truth statements goes, the results are more modest. The informal meaning of the statement 'S is true' is that some correct process of analysis of S concludes that S is true. Obviously, an attmept to define what is correct will lead us back to the concept of truth. Our definition avoids this loop by reducing 'S is true' to S. But if somebody insists on defining a correct process of evaluation, then the definition will be as follows. The evaluation process, when applied to a statement, is correct if it results in the value 'true' if and only if it results in 'true' when evaluating all predictions generated by this statement. When applied to a prediction T!, the correct evaluation process must result in 'true' if and only if the test T is successful. This definition is a reformulation of our original definition of truth.

The metasystem staircase does not, of course, end with the second level. We can create the third metasystem level to decide whether the statement 'S is true' is true, an so on infinitely. But if we only mean the semantics of it, then it is not worthwhile to do. Even though formally different, all these statements will be equivalent to each other.

We shall go along with the common understanding of the true and the false according to which what is not true is false, and vice versa. Now we put the question: What does it mean that a statement is false? To be meaningful, the statement of falsehood must be, as any other statement, interpreted as a prediction or a recursive generator of predictions. Let us start with the case of prediction.

Consider a prediction T!. If it is not true, there are two possibilities: that test T ends in failure, and that it never ends at all. The statement of the end in failure is an ordinary prediction: T runs, stops, we compare the result with Failure and this comparison succeeds. But what about the case when T never ends, i.e. is the statement that T is infinite?

The statement that a certain process is infinite is not a prediction. But it is a generator of predictions. Our definition of a prediction T! includes a procedure, let it be denoted as P(t_i), which is applied to the states t_i, i= 1,2,... etc. of the test T and determines, in a finite time, whether each given state is Success, Failure, or neither of the two, which we shall denote as Not-end. Therefore, the statement that the state t_i is Not-end is a prediction, namely, the success of the procedure that runs P(t_i) and compares the result with Not-end. Now we can define what is the meaning of the statement that T is infinite: it is a generator which produces the following row of predictions:

t_1 is Not-end

t_2 is Not-end

t_3 is Not-end

...

etc., infinitely

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We easily agree that test is either finite or infinite. Now that we have the exact meaning of being infinite, we can formulate this assumption in its exact form.

We postulate and consider it self-evident that a test is either finite or infinite. This excludes any third, or middle, possibility, which gave this principle the name of the law of the excluded middle. From this we derive that a prediction may be only true or false. We further derive the law of the excluded middle for arbitrary statements. Consider some statement S which is a generator of predictions. Let us run this generator and define the following refutation process. Each time that a prediction if produced it it replaced by the symbol True if it is true and False if it is false. The first time the symbol False appears in the refutation process, the process ends and succeeds. This process is either finite, or infinite. If it is infinite, the statements S is true, otherwise it is false. No other outcome is possible.

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Now that we know that a statement is either true or false, we want to classify them correspondingly. This is done through predicates.

We call predicate an agent which examines a statement and the world about which the statement is, i.e. the world of the tests T_1, T_2, ... etc., and comes with one of the two outputs: True if the statement is true, or False if false. The symbols True and False are referred to as truth values. Predicates are abstractions of the second level (see abstraction. In some cases we can construct predicates as machine procedures, in other cases a person or a group of persons may work for a long time in order to come with an answer.

In a wider sense, a predicate is any function which takes one of the two truth values. Such a function is understood as taking the input values, constructing a statement from them, and evaluating the truth value of this statement.

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Up to know we have been speaking of truth in abstraction from the actual means we have to establish the truth of a statement. The most general term for establishing the truth of a statement is proof.

By proof of a statement S we mean any process which the subject of knowledge accepts as the evidence that S is true and is ready to use S for predictions on the basis of which to make decisions.

There are two cases of proof which can never arise any doubt because they do not base on any assumptions: verification and direct refutation.

Verification is a direct use of the definition of truth to decide if the statement is true.

Predictions are, in principle, verifiable. You only have to initiate the test process that the prediction is about and wait until it comes to the final state. Thus, if a prediction is correct, there is, in principle, a way to verify this, even though it may require such a huge time, that in practice it is impossible.

Now consider the situation when you want to verify a prediction T!, but the test process T is infinite. This means that the prediction is false, but you will never be able to verify this directly. To sum up, the following three cases are possible when we try to verify a prediction:

Testing process Prediction is

-------------------- ----------------- 1. ends in success verifyably true 2. ends in failure verifyably false 3. never ends false, but this is not directly verifyable

As to the verififcation of general statements (generators of predictions), it may be possible only if the number of prediction produced is finite. In the more usual case, with an infinite number of prediction produced, the direct verification is impossible.

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A general statement which may produce an unlimited number of predictions can never by directly verified. But it may be directly refuted, i.e. found to b false. This happens when it produces a prediction T!, but in fact the test T ends in failure.

What we call theories are general statements which may produce infinite number of specific predictions. Theories are not directly verifiable, but they may be refuted.

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When a statement cannot be directly verified or refuted, it is still possible that we take is as true relying on our intuition. For example, consider a test process T the stages of which can be represented by single symbols in some language. Let further the process T develop in such a manner that if the current stage is represented by the symbol A, then the next stage is also A, and the process comes to a successful end when the current stage is a different symbol, say B. This definition leaves us absolutely certain that T is infinite, even though this cannot be either verified, or refuted. In logic we call some most immediate of such statements axioms and use as a basis for establishing the truth of other statements. In natural sciences some of the self-evident truths serve as the very beginning of the construction of theories.

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We distinguish between factual statements and theories. If the path from a statement to verifiable predictions is short and uncontroversial, we call it factual. A theory is a statement which can generate a wide scope of predictions, but only through some intermediate steps, such as reasoning, computation, the use of other statements. Thus the path from a theory to predictions may not be unique and often becomes debatable. Between the extreme cases of statements that are clearly facts and those which are clearly theories there is a whole spectrum of intermediate cases.

Top-level theories of science are not deduced from observable facts; they are constructed by a creative act, and their usefulness can be demonstrated only afterwards. Einstein wrote: "Physics is a developing logical system of thinking whose foundations cannot be obtained by extraction from past experience according to some inductive methods, but come only by free fantasy".

The statement of the truth of a theory has essentially the same meaning as that of a simple factual judgment: we refer to some experience which justifies, or will justify, the decision-making on the basis of this statement. When this experience is in the past we say that the truth is established. When it is expected in the future we say it is hypothetical. There is no difference of principle between factual statements and theories: both are varieties of models of reality which we use to make decisions. A fact may turn out to be an illusion, or hallucination, or a fraud, or a misconception. On the other side, a well-established theory can be taken for a fact. And we should accept critically both facts and theories, and re-examine them whenever necessary. The differences between facts and theories are only quantitative: the length of the path from the statement to the production of predictions.

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