Methodological problems of system research
Systems and Structures as a Problem of Modern Science and Technology
The development of modern science is characterized not only by an unusually rapid accumulation of new knowledge but also by the fact that the principles and methods of scientific research have changed essentially and are continuing to change. Among the concepts which reflect this process most intensely, a special place is accorded to the notions of system and structure. Externally this is reflected in the circumstance that the terms designating these notions have become most widespread in scientific and popular literature. A linguist cannot write a few sentences without mentioning the language system; the chemist without mentioning the structure of combinations; the cybernetician without referring to systems of control; the biologist without referring to the structural aspect of organisms; and all of them use these terms with capital letters, as it were, endowing them with special profound meaning.
This is not simply a matter of fashion. The study of entities as structures and systems has become in our time a fundamental problem relevant to all sciences. This direction in methods of understanding, or, one might say, of "perception" of entities began to be formulated already long ago; but it is only in the second quarter of the twentieth century that a real revolution of views occurred and the new direction spread and received universal recognition.
The triumph of this new point of view and the broad transition to system subject matter and system problems resulted not only from the internal development of the sciences themselves but largely also from the development of modern industry. A characteristic feature of our days is the "large scale" technical system, often intricately automated and served by extremely complex electronic computers. Rational regulation of an economy as a whole and of its separate branches requires a holistic concept of a system, including production and its organization, a complex communication net, the organization of supply and distribution, etc. In military matters we encounter systems of gigantic dimensions involving a whole country or even several countries. If systems of this sort are to be controlled, they must be studied. Thus the development of production and of technology influences the formulation of new system objects and new research problems.
The significance of these problems is not only in that they attract attention in many scientific disciplines. More important is the circumstance that they apparently contain some sort of nexus of the development of science as a whole, that is to say, the "technology" of science, methods of scientific investigation. For a linguist it is important, above all, to clarify the meaning of a language system; for a biologist, the nature of a living-being-as-a-whole or of a population; for a sociologist a specific social system. But for science as a whole, something else is considerably more important, namely to try to come to grips with some ways and means which would enable it to investigate entities as systems and structures. For the present, achievements of humanity in solving these problems, for all their practical and theoretical importance, are still quite insignificant.
A famous Austrian biologist, L. Bertalanffy, now living in
To put it in other words: just because we encounter considerable difficulties in the investigation of entities as systems and structures and because the nature of these difficulties, is, in principle, the same in various fields, it is necessary to develop logico-methodological investigations in general. The problem is that of formulating a system of general principles and rules appropriate for building system-structural investigations of specific entities.
In modern science, general methodological analysis has acquired primary significance, inasmuch as the investigator deals, as a rule, with exceedingly complex cognitive constructs, which appear in the role of analytic tools. Here is where one ought to seek the cause of the widespread sharp increase of attention accorded to the methodological problems of science. As is known, the recent broadened session of the Presidium of the
Thus we are naturally led to the necessity of examining this question in greater detail.
What Are the Specifics of a Methodological Approach to the Problems of Science?
The problem facing a representative of a special science is to build up. knowledge of the subject matter investigated; or, put an other way, to describe this subject matter in a symbolic form. In so doing, the scientist makes use of ways and means already worked out in his discipline. As long as these methods "work" without failure and yield knowledge which is internally consistent and which satisfies the problems posed, the scientist need not cogitate on the nature and construction of the methods. A different situation obtains when problems arise which cannot be solved by the old means, or when new entities appear to which the methods cannot be applied. Then the creation of new means becomes a pre-condition for solving the problems.
How is this done? Two polarized approaches are possible. One is by way of an "art." This approach involves combinations of methods already existing in the given discipline by trial and error, which finally leads to a transformation of these means and to the accidental discovery of a solution. The approach also involves the transfer of methods from other sciences, whereby what fails is rejected and what is found most appropriate is adapted and adjusted. The principal factor in this process is the number of attempts and time; in the long run, the required method is found. The main characteristic is the absence of any kind of general knowledge of method which could direct and regulate the search.
Another way of working out new methods of research presupposes a theory of the very methods, that is, a methodology. In this case, the specialist in a discipline tries the combinations not simply of whatever is "at hand," but rather in accordance with his knowledge about all available methods and their relations to the problems. He attempts to borrow from other sciences not just any method but only those which he knows may be appropriate to the problem he confronts and to the description of the entities assigned. If necessary, he creates new methods, knowing in advance, as an engineer does, what sort of methods they ought to be.
But what about the methodology of science itself? Of what sort should it be to insure similar work in the creation of methods of scientific research?
With regard to this there exist two points of view. A representative of one (which may be called the "natural philosophy" point of view) affirms that the subject matter of methodology is nature, that is, the world as such. In this view, the methodol-ogist is not in any way distinguished from a discipline specialist. For example, the physicist analyzes the physical processes in entities, while a scientist working in the field of methodology of physics must also study these processes. The only difference between them is that the physicist will study the physical processes specifically, deriving support from experimental methods on the one hand and from the mathematical apparatus on the other; while the methodologist will study such processes "in general," deriving their general aspects and properties. In the natural philosopher's view the concepts worked out in this "general approach" to physical processes can serve as methods for specific investigations in physics.
The representative of the second point of view (which may be called the "theory of cognition" point of view) affirms that the subject method of methodology is essentially distinct from the subject matter of the concrete sciences. He thinks of the subject matter of methodology as being the activity of cognition, of thought or, more precisely, all human activity, including not only cognition but production as well. One might say that in this view methodology is a theory of human activity. Therefore methodological knowledge can serve as a guide in the search of new methods of scientific research: after all, it describes and even projects in advance the very activity which must be realized for this purpose.
Apparently only the "theory of cognition"point of view regarding methodology justified the singling out of the latter as a genuine science.
With the view of describing the conditions in which specific methodological problems are singled out, let us examine some special situations in schematic form, so-called "antinomies" or "paradoxes," which arise in the course of development of science.
The general logical scheme of such situations can be represented very simply. A certain object A, being a representative of a class, is first analyzed by means of a procedure ∆1 and appears to have the property B. Next the same object is analyzed by means of another procedure ∆2 and appears not to have property B. Upon checking, it turns out that both procedures have been correct and that each can with equal justification be applied to the object in question. Thus at the given stage of a science, it is not possible to reveal the property of that object which underlies such strange results. Thus, both results, "A is B," and "A is not B," obtained respectively by applying procedures ∆1 and ∆2 turn out to be equally-justified and "valid," and this creates a special situation, a "discontinuity" in the development of science.
A number of such situations was established already in the classical Greek period. They were related to quite diverse disciplines – mathematics, physics, and philosophy.
For instance, one wrote down the sequence of natural numbers and a correspondence was made between them and those numbers which were perfect squares, thus:
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16…
1 4 9 16 …
It is obvious that in this sort of correspondence, the further out in the sequence we get, the smaller will be the "weight" of the perfect squares in the sequence of natural numbers. From this it was deduced that the number of perfect squares in a sequence of natural numbers is smaller than the number of all such numbers. But then one proposed another method of correspondence: namely a one-to-one correspondence between the natural numbers and their squares, thus:
1 2 3 4 5 6 7 8
1 4 9 16 25 36 49 64
It is obvious that no matter how far we go out along the sequence, we can always continue the juxtaposition. From this it was deduced that the number of perfect squares is not smaller than the number of all natural numbers.
Thus, applying two different methods of reasoning (both, let us note, correct in the light of the then existing notions) we arrive at two different mutually exclusive assertions.
One might try to object that these assertions were not correct since, from the point of view of modern mathematics, one cannot apply the notions "greater" and "smaller" to infinite sets; that instead one must use the notion of cardinality and the corresponding juxtaposition procedures.4 This is correct. But we know this today. When this question arose and was discussed, apparently from the time of Democritus to the time when the works of G.Cantor appeared, the notion of cardinality did not exist, and it was necessary to use the notions which existed. Moreover, even from this modern point of view one must recognize that both of the assertions regarding the number of perfect squares in the sequence of natural numbers are equally false and equally true. The only important thing in the context of our discussion is that situations arose in which two mutually exclusive results were nevertheless equally valid, and one had to come out of this situation by creating new scientific methods.
In order to remove the impression that paradoxical situations arise because of operations with the "difficult" and somewhat mystical concept of infinity, let us take another example of a physical paradox, singled out by Galileo some 2000 years after the appearance of the mathematical paradox just discussed.
The distinction between uniform and variable motions had been known for a long time. This knowledge, however, was of an intuitive sense-perceived sort, not interpreted in terms of concepts. The direct method of comparing motions governed by the senses, which existed in Aristotle's time, whereby only space intervals traversed in equal time intervals were compared, did not reveal the distinction between uniform and variable motions as a concept.
Next, although in the conceptions of the ancients the notion of velocity was the result and the means of comparing motions in general, regardless of their character, because of this notion's content and because of the way it was constructed, it served as an adequate notion only in comparing uniform motions. Consequently when Galileo took up the investigation of accelerated motions, using the concept of velocity he expressed in the formula
v = s/t (1)
he came up against a logical contradiction, an antinomy. Since the time pieces available to him, in spite of improvements, were still not very suitable for measuring small intervals of time, Galileo undertook to retard the motion of falling bodies by means of inclined planes. This, in turn, brought him to the comparison of rates of fall along vertical and inclined directions. According to Aristotle's definition, of two moving bodies, the one which passes over the larger distance in the same interval of time has the greater velocity. Accordingly, it was assumed that two bodies have equal velocities if they pass equal distances in equal intervals of time.
For Galileo, these definitions were already unsatisfactory. The method of measuring time which he had worked out allowed a conception of velocity as a mathematical ratio of space intervals and time intervals. From this point of view, nothing is changed if one calls two velocities equal if "the intervals passed are proporlional to the times consumed . . . ."5 To the extent that Galileo already subsumed the concept of velocity under the more general concept of the mathematical ratio, this transition was entirely legitimate. The equality of the ratios
S1/t1 = S2/t2 (2)
remains valid both when S1= S2 and when S1 ≠ S2, if t1 and t2 change in the same proportions as the space intervals.
Thus we have two definitions of equal velocities of two moving bodies.
The first definition: the velocities of two bodies are equal if, in equal intervals of time, the bodies pass over equal intervals.
The second definition: the velocities of two bodies are equal if the intervals passed by the one and the other are proportional to the times consumed.
The second definition is a generalization of the first. Equipped with these two definitions, Galileo undertook to compare specific cases of falling bodies. Let two similar bodies fall along CB and CA (see Fig. 1). The velocity of the body falling along CB will be greater than that falling along CA, since it is shown experimentally that during the time it takes the first falling body to travel the distance CB, the second will travel along the inclined plane CA over the segment CD, which is smaller than CB. Hence, one may conclude, in accordance with the first definition, that the velocities of bodies falling along the vertical and along an inclined plane are not equal. At the same time, Galileo's well known postulate to the effect that the velocity of a falling body in any position depends only on the height from which it has fallen, led him to the idea that, since the velocities of the two bodies on the single horizontal line AB are equal, they must also be equal on the segments CA and CB. He tests this hypothesis experimentally and actually finds that the ratios of the times of descent along the entire inclined plane and the entire vertical is equal to the ratio of the lengths of the corresponding paths. Hence, in accordance with the second definition, we may conclude that the velocities of the two bodies are equal.
In this way, following Galileo's reasoning, we obtain two contradictory assertions: 1) the velocities of the bodies falling along CA and CB are equal; 2) the velocities of the bodies falling along CA and CB are not equal.
One cannot seek the source of Galileo's contradiction in his generalization of the conditions of equality of velocities. If we compared the motions of the spheres along CA and CB, taking different segments of CA and CB and using the old definition, we would obtain contradictory results also in accordance with this definition. The velocity of fall along CB would be found to be greater than the velocity along CA in one place, equal to the latter in another, and smaller in still another. Thus, the examined development of the notion of velocity and the generalization of the condition for the equality of velocities were not at the root of the contradiction, but merely an accidental circumstance which facilitated its discovery.
The root of the contradiction is in the circumstance that the concept of velocity constructed on the basis of comparing uniform velocities and characterizing such motions unambiguously is no longer suitable for comparing and characterizing unambiguously non-uniform velocities.
Similar logical contradictions or antinomies are frequently encountered in the history of science. Both assertions which constitute the antimony are equally true and equally false. They are true in the sense that both are valid, if we make our point of departure the then existing specific construction of the underlying concept. They are false in the sense that this construction of the concept can no longer characterize the investigated phenomenon unambiguously.
Paradoxical situations and antinomies occupy a special place in the course of development of science. First of all, it becomes meaningless to ask in those contexts which of the two results is applicable to the object, the first or second. The answer is neither.
Thus, in the light of a comparison of two mutually exclusive findings related to the same object, the object itself becomes separated from the findings and is juxtaposed to them as a third "something," still unfathomed. In Hegel's terms, first we saw the object itself in the concept; now the contcept, like form, becomes separated from the concept. This is the first, and most probably, the fundamental step in the formulation of a theo-retico-cognitive world outlook.
As the object is revealed as something separate from what we see in the findings, and as the findings are compared with each other, we are obliged to make the next step and to ask what underlies the difference in the findings. In answer to this question, the next element in the subject matter of the theory of cognition is revealed: it is in the procedures of obtaining the findings, the procedures of cognitive activity, that we find the cause of the difference in the findings.
The appearance of the theoretico-cognitive point of view makes possible the specifically methodological approach to the tools of science.
The fact of the matter is that in any situation, two different problems can be posed, and correspondingly, both practical and research activity can go along two essentially different lines based on different methods. In the one case, the investigation will be directed toward overcoming this particular single antinomy, toward the working out of new, special concepts which would "remove" the antinomy. In the second case, it can be directed at the clarification of antinomies in general (not only the particular one), on the analysis of ways and means of overcoming them, at the clarification of the structure of the newly acquired knowledge in its relation to the previous antinomic results.
In the first case, we shall remain within the framework of a special science, be it mathematics, physics, or chemistry, and shall be using its specific methods. Then every new antinomy will confront us as the same problem, and we shall be approaching it with the same equipment as before. Our experience in the overcoming of antinomies will not become consciously realized and will have no influence upon our subsequent activity.
In the second case, we must pass beyond the boundaries of a particular science and focus on an entirely different subject, namely the knowledge of things and how such knowledge is achieved and used. Here we shall have to utilize entirely different methods of research and work out concepts essentially different from those used in the special sciences. These will be the concepts of methodology (in the broad sense, including logic and the theory of cognition).
The antinomies or paradoxes arising in the development of science were cited here as examples of situations which necessitate the posing of essentially methodological problems. In such situations a reality is actually formulated which becomes the subject matter of methodology as a science. This reality is activity directed at gaining knowledge.
We can represent the structure of such activity as a block diagram in which the main components are displayed. Special analysis shows that any cognitive act must contain 1) problems (or requirements); 2) objects; 3) means; 4) forms of findings; 5) procedures producing the findings (see Fig. 2).
Such a schema may be viewed as a first approximation to the representation of the subject matter of methodological investigation.
It is important to emphasize that the posing of the question about the object as such, in contrast to the "givenness" of the object in a particular form of a finding, occurs for the first time not, by any means, in the special-scientific investigations, as is commonly thought, but only in methodological ones. In special-scientific investigations, where only one or a few easily combina-ble findings about an object are at hand, the question about the object as such does not arise, and there is no need to juxtapose it to the findings. We are sure that the object is as it is given in these "findings." Only in situations involving antinomies, and analogous ones, we are obliged to focus on the object, to put the question concerning its nature, and to attempt to represent it in a new form different from the findings already at hand. Therefore, it is in fact methodology and the theory of cognition, strange as it seems at first, which turn out to be the theory of objects in the field of objects, i.e., must include within themselves an ontology which constructs models of the world. Therefore it is a mistaken thesis, which now and again appears in philosophical literature, which asserts that the theory of cognition and logic are sciences that have to do with activity and the process of cognition but not with the world. This juxtaposition is not legitimate: the theory of cognition is a theory of activity and ipso facto of the world, including and included in it. This very juxtaposition was a result of an erroneous understanding of objectivity – the famous theory of K. Marx having been forgotten: "The main flaw of all earlier materialism-including that of Feurbach – is in that the subject matter, reality, sensation, was taken only in the form of the object or in the form of perception and not as the sensory activity of man, i.e., practice, not subjectively."
However, this conception of objects in methodology differs essentially from their conception in the special sciences: it is created as a representation of their "higher" objectivity, freed from specific forms of specific problems. For the same reason, methodological ontology has nothing to do with natural philosophy; it exists within the system of methodology and is created not on the basis of physical, chemical, or any other empirics, but on the basis of the analysis of human activity, namely production (practice) and thought.
Thus, as we pass to the area of methodological investigation, we formulate an entirely new discipline, one which does not coincide with any of the disciplines of the special sciences. And we can investigate and describe this discipline only by means of special methods, which are not reducible to the methods of the individual sciences.
That this assertion is not generally accepted, that a controversy goes on around it, can be seen in the widespread and widely accepted thesis of D. Hilbert to the effect that the foundations of mathematics is the business of mathematics itself. Moreover, representatives of many other sciences besides mathematics share and support the thesis that the methodologies of the special sciences should be; developed by the representatives of these same sciences. Therefore, in formulating the position that the methodology of science has its own discipline and utilizes its own specific methods, and that only if this is done will the methodology of science attain the status of a genuine science, we contrast this position with the one which maintains that the methodological problems of every science can be solved by the methods of that science. In our opinion, the triumph of this sort of approach has always led only to the destruction of methodology itself as a science. That this approach was frequently accepted is explained by the fact that it freed scientists from the responsibility of working out specific methods of the methodology of science. In rejecting this thesis, we meet the problem head on: what are, and what ought to be, the fundamental tools of methodology or of a theory of activity. One of these is
The Distinction Between the Object and the "Thing Known"
The object exists independently of knowledge; it existed prior to knowledge. The thing known, on the contrary, is formulated by knowledge itself. As we begin to study or simply to "plug in" some entity into our activity, we consider it from one or from several aspects. These aspects, singled out by us, become a "substitute" or the "representative" of the entire many-faceted entity; they are fixated in the symbolic form of knowledge. To the extent that this is knowledge of something existing objectively, we always objectivise it and it thus determines the "thing." In special-scientific analysis we always view this thing as an adequate representation of the object. And this is correct. However, one should remember that in a methodological investigation this assumption becomes essential, namely that the thing known is not identical with the object: the former is a product of human cognitive activity and, being a special creation of man, obeys certain laws which are not identical to the laws of the object.
Several "things known" may correspond to the same object. This is because the character of the "thing known" depends not only on the object reflected by it but also on the purpose for which the "thing known" had been formulated as a solution to a problem.
In order to clarify these abstract definitions, let us take a simple example. Suppose we have two groups of rams in two villages. These are doubtless objects. People deal with them, use them in various ways, and at some time there arises the problem of counting them. At first one group is counted, say 1, 2, 3, 4, then the other – 1, 2, 3, 4, and finally both numbers are added: 4+4 = 8.
Already in this simple event several very complex and at the same time very interesting moments appear. The objects, namely, the rams, have a number of aspects, and when we count them, we single out one aspect of each group – the number of rams. We express this number by symbols, the digit 4 in the first case, then again the digit 4. Then we perform some strange operation – we add the numbers. If we had five groups instead of two, and if there were four rams in each, we would not have added the numbers but multiplied them: 4x5 = 20; that is, we would have performed another still more strange operation.
Why do I call these operations "strange?" Let us ask whether the operation of addition can be applied to the rams as such. Or, perhaps, multiplication? Or, continuing this line of reasoning, the operations of division, or root extraction, of raising to a power? Certainly not.
But there is still another no less important aspect of this matter. We should ask ourselves: can it be that these operations – addition, multiplication, raising to a power – are applied to the ink marks, the symbols which express the numbers? When we add, we add numbers, not digits. There is a big difference between a number and a digit, because a digit is nothing but a mark, of ink, chalk, or paint, while a number is a thing of an altogether different sort, namely a symbol in which a particular aspect of objects is expressed. We add numbers, not because they are symbols, just as we multiply numbers not because they are digits. We add and multiply, because in these symbols a strictly determined aspect of objects is expressed, namely quantity. This aspect has attained in the objects an independent existence, distinct from the objects; accordingly, when we speak of number as a construct in its own right, distinct from rams as such and of the "quantity" of rams, we have in mind not the object and not aspects of the object, but something created by man, namely a "thing known."
This "thing known" is quite as real as the objects from which it is derived, but it has an entirely different social existence and a special structure different from the structures of objects. The digits themselves are not yet things. The thing arises and starts is existence when the comparing procedure singles out quantity in the group of rams and expresses it by number symbols. We are dealing, therefore, with a connection or a substitution relation between the rams, taken in a definite juxtaposition, and the symbolic expression of numbers. But this relation exists objectively and is expressed only in the form of symbols and in the means of working with that form. The thing known is reality; but the laws of dealing with it as reality are special laws. We deal one way with rams; with numbers we must deal in quite another way. Only after having expressed the "quantity" of rams in a special symbolic form are we enabled to deal with them in a characteristic way, namely, as with quantities. We are not able to do this before; we had to deal with rams as their nature requires; at most we could count them.
The object in itself does not contain any "thing." But the thing can be singled out as special content by way of practical and cognitive actions upon objects. This content can be fixated in symbols. As soon as this is done, the thing appears and stands before man in a reified form as something existing apart from the object from which it has been abstracted. Its reified "given-ness" gives rise to illusions – as if we were dealing with the object itself. This illusory conception of the state of affairs, having arisen already in relatively simple situations, infiltrates next into the higher levels of science and there mixes up everything thoroughly.
The only way to understand the nature of the thing is by clarifying the mechanism of its formation and structure; and this involves an analysis of it as levels of substitution built in successive levels.
The simplest form of the thing can be represented in the following scheme. The first level is formed by operating upon an object X by means of procedures ∆1,∆2, .... The results of these operations are expressed in symbols (A), (B), which fixate and replace the content [X∆1∆2 . . .] singled out in a special activity λ1λ2 – a formal operation on the symbols – and all this constitutes the second level. The results of transformations on the symbol forms of the second level are related to the object X. The original substitution and the reciprocal relation are represented by arrows in Fig. 4.
The symbol constructs (A), (B) and the operations λ1λ2 performed upon them may themselves – give rise to another levels to which new meaningful comparisons are applied; in other words, the symbols themselves become the things operated upon. In this case, the results of the operations on the second level are fixated in symbolic constructs (G) (E) (D), which form the next substitution level upon which the operations on the symbols are performed by means of particular procedures. Other levels may be added, so that in the final analysis we shall obtain a hierarchy of substitution relations, which can be represented by the scheme shown in Fig.5.
Thus we can say that the "thing" is a hierarchical system of substitutions of the object by symbols introduced into particular systems of operations, in which these systems exist actually as objects of a particular kind, reified as scientific literature or as productive activity of society in the creation and utilization of symbol systems. The growing generation is constantly being "socialized" into these substitution systems, masters them, and builds its activity on their bases.
Thus, drawing a distinction between the object and the thing known permits us to introduce another most important concept of methodology:
The Ontological Representation of the Content of Knowledge
A most important result of the above analysis is the view to the effect that the application of juxtaposition to objects created new content. We shall express this content by the symbols [X∆1∆2…]. The content is fixated, expressed in symbolic form (A) (B) and in the methods of operating on this symbolic form X1X2- Applying different comparative operations to the symbols (A) (B), we obtain a new content, which we express in symbols (D) (E) (F) and quite frequently relate to the object X. For instance, we measure the values of volume and pressure of a mass which regularly correspond to each other (the first level), obtain a series of values p1, p2, p3,…..V1,V2, V3…(the second level), then juxtapose them thus P1V1 ↔P2V2 ↔ P3V3 ↔…and find the mathematical formula of their dependence pV = constant (which must be placed on the third level). The content of this form we consider to be a "law," obeyed by the gas and consequently something directly related to our object.
Not infrequently, however, such a direct reference cannot be carried out, since the content revealed through the symbolic operations does not correspond to the properties of the object empirically observed or revealed. In that case, one constructs a special symbolic representation which is "interposed," as it were, between the symbolic form of knowledge and the empirically given objects.
To turn to the schematic diagram of the "thing" discussed above, the situation in which a newly obtained finding cannot be related to the object can be represented as in Fig.6
Here the rectangle marked by the dotted line encloses the representation of the "discontinuity," the result of our inability to relate the results of operating on the fourth level of knowledge directly to the object X. In order to eliminate this dis continuity, we' build a special symbolic construct represented by the little rectangle with the letter O), which is to represent the thing "as such" in a certain way. On the basis of this specific function, we can speak of representations of this sort as ontological representations of the content of knowledge. This expresses precisely the specific cognitive role of such symbolic constructs: they are to represent the object so as to insure its connection to the newly obtained findings. The so-called "ideal entities" appear in just this way, e.g., the point mass, the ideal level, the perfectly elastic body, the mathematicalpendulum, etc.
As an example, let us take the "mathematical pendulum." The equation of its oscillations contains the symbols M (the mass of the pendulum) and Z (its length). Suppose we have at hand a real pendulum clock with a massive disc and a long rod. Can we apply the mathematical equation in-describing the real motion of the clock pendulum? We shall find that if we measure its actual parameters, i.e., the length of the rod and the mass of the disc, we shall obtain incorrect results. The mathematical equation can be related directly only to a particular ideal entity – the "mathematical pendulum," which, in turn, can be represented only by symbols. In order to apply the mathematical equation to a real pendulum, we must first reduce the latter to a mathematical pendulum, called "reduced length" and "reduced mass" of the real pendulum (cf. Fig.7).
A similar picture involving ontological representations appears in all other scientific results.
An ontological representation endows the thing known with a second form of existence: it "collapses" its many-leveled structure in a single representation.
Now, having introduced these general methodological concepts, we can return to the problems of system-and-structure investigation, and ask:
What Is a System?
The term "system" is defined by means of terms like "connection" (or "interconnection"), "element," "whole," "unity." In purely verbal formulations, it is still possible to find agreement. However, the representatives of various sciences endow these words with meanings so diverse that the agreement is only apparent. For some, "connection" is only a geometric inter-relation among the parts; for others it is the dependence among the parts of the aspects of the whole. Some use "structure" to mean a geometric inter-relation; others reduce the term to mean the "set" of elements. Frequently the theoretical definitions are at variance with empirical material. Thus, for example, the well known British cybernetician St. Beer [evidently Stafford Beer – Translator] takes system to mean the inter-connection of the most diverse elements, as an example offers a billiard ball, where, as a matter of fact, there are no interconnections but only a functional unity of the whole. Therefore it would probably be most accurate to say that at the present time there are no satisfactory, widely accepted conceptions of system and structure. Nor was the Society for General Systems Research able to offer such concepts. G. H. Goode and R. E. Macall, in their analysis of "large scale" systems have declined to undertake any attempt to define the boundaries of the range of systems examined.
"As it usually happens in any discipline," they remark, "these boundaries pass over broad vague territories and a search for their precise position would elicit large but fruitless controversies." Indeed, the position they have taken is the only widespread one among those who investigate particular systems and structures.
It is senseless to challenge this position if we know in advance that our opponents do not have the means to solve this problem. But this is not all. It is hardly worthwhile to prove explicitly that the unrestricted and diffuse character of the basic concepts makes scientific research difficult and not very productive.
It seems useful to put the following question in this situation: why have all the attempts to single out specific criteria of systems failed to give positive results for such a long time? The answer is, in a sense, a trivial one: it seems one tries to unify into a single category too many diverse phenomena. Thereby the formal, and also the merely intuitively understood, characteristics of entities and of the elements which compose them are seized upon (to justify the unification),
while the more essential and profound differences which actually determine the nature and the life of "systems" remain unseen.
There is also a methodological dogma which plays a considerable role in the determination to investigate the "systems in general," namely that one should always strive to single out from a mass of diverse phenomena some general invariants at the expense of an empirical approach to details and to the limitations which they impose.
Apparently the only effective path of theoretical working out of the problem should be that of dividing the empirical material encompassed by the investigations into narrower fields which are defined by some other "non-system" characteristics more relevant to the objects.
In particular, it is necessary to make the exceedingly important essential distinction between
"Organizations" and "Structures"
Even if they possess all the usually listed characteristics of systems, complex objects may have nothing to do with each other from the "systemic" point of view. Let us show this in simple models.
Imagine a wooden base with little pits containing little marbles (cf. Fig. 8).
The marbles are firmly in their position and together they form a strictly determined configuration. This configuration may be described with the aid of some "holistic" characteristics or other. For example, its rhomboid character can be established (Fig. 8). Each marble has its strictly determined position, rigidly defined both in relation to the whole and to the relation of the other marbles. If we remove one of the marbles, then the whole will doubtless be changed. Instead of a rhomboid configuration we shall obtain a triangular one. Therefore these four little marbles constitute some unified whole: a change of position in at least one of them, or its disappearance, changes the whole. But there is an important peculiarity: a change of position of one marble in no way is reflected in the positions of the others. As the position of one marble changes, there is a change in the whole, even though the remaining elements remain the same. These properties define a class of systems of one of the simplest kind, namely an organization with relations. In this type of system connections are absent.
Let us take another example. Using the same wooden base, let us place the marbles in the same rhomboid configuration. However, in contrast to the previous example, let us connect them to each other by springs (cf. Fig. 9).
When the springs are balanced, the system is at rest, and the four marbles form a rhomboid configuration. Let us change the position of one of them. The configuration of the system will, of course, change, but not in the same ways as in the previous example: the system of springs, disturbed from its equilibrium position as a result of a change of position of one of the marbles, will be set in motion. All the marbles will be displaced, and an entirely new configuration will appear. As in the first case, a change in the whole was brought about; but now this has taken place with respect to the positions of all the elements. This is the peculiarity of the systems of the second type: the elements are not only related to each other but also connected to each other. Just for this reason the change of position of one of the marbles results in the changes of the positions of others. Thus we are dealing in this case with an essentially different type of system – the system of structural connections.
However, this distinction between types of systems, namely according to whether they are based on "relations" or "connections," is by no means the only one. No less important is the distinction between
Cognitive and Object Systems
In an overwhelming majority of modern works on the problems of structure and system, or on the methodology of structure investigation, this fundamental distinction is not made. The system of knowledge about an object is usually identified with the system of the object itself, while the system of the thing known is then mechanically overlaid upon the system of the object. Further, this essentially single system is treated in different ways. At times it appears as a system of cognition, at times as a system of the object reflected in the cognition. And even the continually resulting antinomies cannot shake the faith in the unity and identity of these three systems.
In linguistics, for example, the following phrases are constantly used as synonyms: "the system of findings about a language," "a language system," and "a language system as the subject matter of language science"; and it is shown that the system of the thing (of knowledge) cannot be distinguished from the object.
The same position is revealed clearly in cybernetics. W. Ross Ashby in the fifties could not imagine a system otherwise than a set of mutually connected parameters. Beer, in contrast to Ashby it would seem, speaks all the time about object systems, but upon closer examination it appears that they are the same as the "systems of ontological representations of things known, which are interpreted as object systems only on the verbal level. Nevertheless, the systems of knowledge of the thing known and of the object are evidently entirely distinct and cannot under any circumstances be identified with each other.
Scientific knowledge is always systemic. Even the simplest forms of knowledge, such as "Birches are white," "Metals are conductors of electricity," etc., are examples of systems. Their form consists of connected elements, while their content is both segmented and integrated into a whole. Whatever more complex forms of knowledge we might take, whether separate propositions or entire theories, they will always be systemic. The difference is only in the form and complexity of the systems themselves.
Let us now turn to objects. If we' have in mind the world of matter, every real object – that is, if we examine it as such aside from connections with the various problems of research – constitutes a complex whole and has a definite structure. Depending on the problems of research, however, it can be and is examined in different ways: first as a simple body, from the point of view of its "external" properties, if one may say so [which, in turn can be (a) attributes, (b) functions]; second as a complex body from the point of view of composition, i.e., as a set or complex of elements [these, in turn may be considered (a) as heterogeneous, in which case the composition is characterized only "qualitatively," or (b) as homogeneous in a particular relationship, in which case the composition is endowed also with a quantitative characterization]; finally, third, as a "network" of mutually connected elements. In this last case it is not the elements that come to the forefront, not even the relations among them, but rather the connections among the elements. It is important to note here that these are the objective connections, not connections among the elements of knowledge about the object (these would take us back to the systemic character of knowledge), but connections among the elements of the object itself and in the object itself. These are connections not as products of mental activity but as that which is investigated and must be reproduced in a particular way in symbolic form.
Let us now examine the relationship between the system of the object and the system of our knowledge about it.
Very frequently, even if knowledge is systemic, the system of the object is in no way reflected in it. To see this, it is sufficient to analyze a few simple examples.
Let us imagine a simple model: three little spheres are placed at a certain distance from each other and form a certain organized entity (cf. Fig. 10). Let us further suppose that our task is to describe this system, including, naturally the relations among its elements. This is easy to do with the help of coordinates. Suppose the axis passes horizontally through the center of the first sphere. Then we can describe the relations of the other two spheres by means of two characteristics, namely their distances from it and the angles relative to the axis. Let these be respectively l1a1. . . and l2a2. Let us examine the relation between this system of description and the objective system of the organization. Let us first suppose that all the spheres are identical. Then in several cases their specific identification as the elements of the objective system is not necessary. Only the relations themselves are identified in the characteristics of the description. Moreover, these characteristics appear as elements of knowledge, and they are grouped in a particular way. Among the characteristics a1 and l1 there are essentially no relations or connections; we may interchange them at will, although there is a definite sequence which fixates the order of obtaining these characteristics. There is a certain correspondence between the characteristics α1 and l1 expressed in the proximity of their notations, be it through a comma or through an "and." There is also a proximity relation between the groups l1a1 and l2a2 expressed by an "and."
To the extent that there is a relation and a particular subordination between the first and the second "and" (the first connects characteristics related to one element of the objective system; the second connects characteristics related to different elements), we can, in principle, speak of the systemic character of the description. There is hardly any sense of objecting against this, but it is also important to emphasize that the relations in the objects are in no way reflected (or represented)' in such descriptions and in the symbols of the logical connections. The system of the object is one thing, and the system of the description quite another. There is neither an isomorphism nor a representational relation between the two.
It is also important to emphasize the considerable arbitrariness of such systems of description relative to the system of the object. The arbitrariness results not so much from the objective peculiarities of the system described (although this, too, is the case) as from the very method of description. If, for example, we take the axis of coordinates as the vertical instead of the horizontal line, that is, the line which connects the first two spheres, our description, as. it is easily seen, will change considerably. The position of the second sphere will now be characterized not by two coordinates but only by one. At the same time, the first symbol of the logical connection will disappear. The position of the third sphere will still be characterized by two coordinates, but these characteristics will be different, since the magnitude of the angle will change. Evidently, this signifies an essential change in the entire descriptive system.
The divergence between the system of description and the system of the object can be revealed by other lines of comparison. An exceptional role in the whole field of system-structural investigation is played also by the distinction between
The System of the "Thing" and the System of the Object
The distinction appears most clearly when we compare so-called "empirical" and "abstract-logical" systems of describing a complex object.
In order to carry out this comparison, we shall use the scheme of so-called "dual finding." Suppose we have an object which is, with respect to its inner structure, a "black box" to use the expression of cybernetics. This object, however, can be known with any desired thoroughness and precision with regard to its "external" or empirical properties. Suppose for simplicity that it has three inputs and putputs, A, B, and C, and we can in accordance with our purposes change the values of A, B, and C within certain limits (cf. Fig. 11). Suppose, further, that in another situation we have a complete, virtually absolute representation of the internal structure of this object. We shall compare these findings in the two situations, passing from one to another, attempting to clarify the relation of the actual structure of the object (that is, the black box) to the empirical findings which we obtain about it.
To fix ideas, suppose the object at hand has a very simple structure consisting of elements A, B3 and C naturally linked by two-way connections. For simplicity, suppose also that each of these elements yields one datum, respectively A, B, and C. We can vary these values at will in the "input" and measure the corresponding changes of value at the "output" of the other elements. In other words, in our discussion, the structure elements of the object will not differ from its aspects empirically revealed. This is a very drastic simplification and we thus avoid one of the fundamental problems of structural analysis; but this will considerably facilitate our discussion and will not impede the clarification of the fundamental thing which we wish to achieve. Suppose further, that in studying the object we use an empirical procedure, accepted in all the natural sciences. We fix one of the values, say C, so as to hold it constant during the entire experiment and, while changing the other value, say A, we shall determine the resulting changes in the third value B. We shall obtain a pair of sequences of corresponding values
This will be a tabular expression of the dependence which exists in the given object between A and B. In order to express this dependence, we shall have to establish particular correspondences between the values found, and discover a mathematical formula which will fit all the values put into the table. Let this formula be β = f1(α). This mathematical expression will give us a particular representation of the object examined, namely, an empirical representation of the dependence of B upon A, C being constant.
Let us, however, put the following question: To what extent is this mathematical function a representation of the connection between A and B in the structure of the object? A simple argument shows that this is in no way the case. Surely the changes in B stimulated by the changes impressed upon A were the results not only of the direct connection between A and B, but equally of the indirect connection A → C → B (the fact that C has remained constant during the experiment is by no means evidence that the connection was ab sent or that it did not "work"). But this is not all. One of the components in the changes in B was the feedback upon B from A via C. Thus, one may say that the function β = f1(α) represents not the connection of B and A as such, but a sum maryaction of a whole series of connections, essentially all the connections in the structure of the object, including A ↔ B and B ↔ A and A ↔C ↔ B and B ↔ C ↔ A. In other words, the function β = f1(α). represents the action of the connection A ↔B, modified by the presence of all the other connections of the object. It represents the connection A ↔ B as it acts and is evidenced in the structure of all the other connections. This means, among other things, that the function β = f1(α) already tacitly contains the actions of all the other connections of the object, and does so in a concealed unexpressed way. Therefore the function β = f1(α) is not a representation of the connection A ↔B, but rather a representation of the entire object under investigation from a particular aspect.
Evidently we can repeat the argument with regard to the dependence of C upon A while B is kept constant, then with regard to the dependence of C upon B with A constant, then with regard to the dependence of A upon B with C constant, etc. We shall have six functions altogether:
Each of these will fix the dependence be tween two aspects of the object and thereby the connection between these aspect-elements. But they will not fix the connections as such, only as they are revealed in the presence and in the action of the other connections of this structure. Each will be an expression of an empirical finding about the object as a whole, and will not inform us of the corresponding connection in its pure form. No matter how we try to exhibit this connection by means of empirical analysis, we cannot do it; we shall always get the summary action of all the connections of the structure.
Let us make a special note of the fact that using the apparatus of a function of two variables will not be of help. We make use of this device when, in investigating the dependence between two aspects of the object, we cannot keep the third constant. But the use of this device does not bring us any closer to the goal of revealing structural connections of the object as such.
In examining the values of two aspects as independent variables, we evidently unite the actions of two connections in one expression, say A ↔ B and C ↔ B, ignoring the third. Having obtained the system of equations
we arrive at a situation analogous to the one discussed above.
This example shows clearly the distinction between the empirical description, represented in this case by means of mathematical functions, and an abstract-logical description of the structure of an object.
In this way, no matter how far we go in the empirical sphere, no matter how we change the direction of approach, the real structure of the object remains unknown. The "black box" remains a "black box." In order to exhibit each of the connections in a pure form, other approaches and methods of analysis are acquired than empirically established correspondences. In science, the development of these approaches and methods began apparently during the Renaissance, namely with Galileo, although he had some precursors, of course.
Already Aristotle began to study freely falling bodies, and he studied them purely empirically. He took bodies of different weights and measured the time of the fall from the same height. The means of measuring time in those days were quite crude, and within the limits of attainable precision, the monotone reduction of time of fall with the increasing weight of the bodies was clearly observed. Qualitative laws were formulated: "the heavier the body, the less time it takes to fall from a given height; or, "the heavier the body the more rapidly it falls." The juxtaposition of a series of values of weight and of time gave the formulas of dependence, approximately correct and of considerable range of validity: t = k/W. These formulae were checked and made more precise during some two thousand years, but they all remained essentially unchanged. In the writings of Leonardo da Vinci we find very clever schemes of experiments, directed toward checking this law, but they, as all the others, could demonstrate at best the lack of precision of the law so as to give finally a very complex formula expressing the dependence of the speed of falling upon the weight of the falling body; but they never could lead us to the modern theoretical formula, given by Galileo: "All bodies fall in the same way independently of their weight." And we must note that if we were to test this universally recognized and entirely valid formula empirically in our natural conditions, that is, where it ought to be naturally applied, we would be convinced only of one thing, namely that the formula does not correspond to empirical reality. This knowledge is of the abstract-logical kind.
The manners and methods of empirical working out of problems of this sort were intensively developed in several sciences, but failed to yield any substantial results. That is why in some stage of the development of science the problem was inverted. The fundamental method of investigation became that of
Constructing Structural Models
Whereas the movement had been from the empirically demonstrated dependences of the aspects of objects toward the structural connections which determine them, whereby the given object was analyzed and dismembered into abstractions, now one begins to build at the very start another object, a structured one, which is examined as a facsimile or model of the object under investigation, for which purpose it was specifically created. To the extent that the structure of the model is created by the investigator himself, it is known; and to the extent that it is studied as a model of the object under investigation, the structure of the latter is considered known.
Such were already the first investigations of structures in mechanics (J. Bernoulli, T. D'Alembert). Their method was later carried over to the study of the structure of matter (the so-called "molecular-kinetic" and "electronic" theories, etc.) and in recent times has spread to all the other sciences. Essentially this inversion of the problem is evidently the only productive means known to us today for investigating the structures of objects and for conceiving them in thought.
At the same time (and this aspect of the matter must be clearly realized) the circumstance that the structures of object models are being built and formulated does not remove the problem of empirical analysis of the structures of the original objects. In the dominant currents of modern positivistic methodology or the "logic of science," the problem of constructing system models has attained a specifically mathematical flavor and is taken extremely one-sidedly. The question of the correspondence of the model to the original object or, in other words, the question of the "adequacy" of the model (naturally with regard to a given problem), is relegated to the background or is abandoned altogether. In consequence it appears that we must first build the structure (a "formal" one, as they say) and only then decide whether it can be considered a model of the object studied. All this has to do with the solution of the first problem, namely with pure "mathematics," i.e., a "formal" discipline engaged in the construction (within arbitrary limits) of possible structures; and this construction is essentially independent of the problem of investigating any particular object. However, in empirical investigation we are always interested in only one particular structure, which gives the "correct" representation of the given object. This means that the "mathematical" theory of structure building, although it is quite a natural idea and quite fruitful as a theory in certain respects, nevertheless can never replace or crowd out completely the problem of investigating empirically particular structured objects. The mathematical theory only stands side by side with the latter and gives it specific formal tools, which, if they are to be the logic of empirical investigation, must be supplemented by special methods of empirical analysis. These methods, as we have already said, remain for the time being mostly uninvestigated.
In one way the matter is aggravated even more, but in another way it is somewhat alleviated in certain respects when among the most important problem the following one is singled out namely,
The Analysis of Historically Developing Systems
The methods of studying the structures of developing objects are more complex than those of studying non-developing objects, because the former type of objects always contains actually two systems of connections at the same time – those of function and those of genesis. These two systems are, on the one hand, essentially distinct and must be distinguished; but on the other hand, they cannot be separated from one another. If we put the problem, say, of studying and depicting the connections of functioning of an organic object separately from the connections of genesis, this is often simply impossible to do: at each moment of time, at each synchronic "cross section" of the object, the genetic connections continue to act and to exert influence on the functional connections and even, in addition, to define the structure of the latter. Therefore it is either impossible to set apart the functional connections or, if they are somehow fixated, they cannot be explained. They appear to be unreal, mysterious.
This was discovered already a long time ago, and Hegel and Marx showed in their works that the solution of this problem is in working out "historical theories" of such objects. However, if one accepts this thesis, one accepts the following formulation of the question: in order to investigate and to depict the functioning structure of an object, it is necessary first to investigate and to depict its genetic structure (perhaps not the entire structure but at least those parts upon which the character of the functioning structure depends). In order to analyze one structure, the functioning one, we must first analyze the other – the genetic one. Here a paradox arises. The understanding of the functioning structure depends upon the understanding of the structure of genesis. But the converse is also true: the degree of understanding the structure of genesis depends on how deeply and thoroughly we have analyzed the already developed state of the object before us. K. Marx pointed out the necessity of investigating the developed states of organic objects from the point of view of their development; but it is also he who said the famous words, that the key to the understanding of the anatomy of an ape is in the anatomy of man. The overcoming of this antinomy is in the working out of a method of investigation in which both functional and genetic analysis would be combined and in which investigation of the "older" state of the object would be a means of depicting its genesis, while the knowledge of the genetic laws would serve as a tool for analyzing and for achieving a more profound understanding of the functioning system in the most developed state. Herein lies the complexity of the problem as we pass to the investigation of organic objects.
However, the problem is facilitated by the same circumstance. It is not difficult to see that knowledge of the regularities of genesis can be used in such a way that they yield complementary, extremely important data on the ways and means of depicting the functioning structure of the given object, data which cannot arise in depicting a usual non-organic object. Namely, we may suppose that this construction must depict the developmental history of the object before us from its earliest structural state to the latest most complex one. More generally this can be expressed so: we may suppose that the method and the sequence of depicting the functional structure of an organic object must correspond to the regularities of its development.
Then the problem of finding the structure of a given organic object will be reduced to three more special problems: 1) to carry out an empirical "non-structural" analysis of the "final" most developed state (although this analysis is to be oriented toward the singling out of particular structural moments); 2) to find in some way a structure which could be considered as the simplest one, the genetically original one; Hegel, and Marx after him, called such a structure a "cell" of the object investigated; 3) to find the regularities of development or, more precisely, of the unfurling of this structure into more complex ones, regularities which would lead us finally to the structure characterized by all the manifestations which were singled out in the empirical "non-structural" analysis of the state which "has become." The solution of these problems will then be the solution of the fundamental original problem, namely that of revealing the functioning structure of the given object.
Each of these problems has its own difficulties. Empirical analysis, as has already been made clear, gives us no opportunity for singling out the structural connections of the object. That is why we are obliged to carry out an investigation which we have called a "non-structural" one. At the same time, this investigation is oriented toward singling out those moments in the empirical material which depend on the structure of the object, and therefore the investigation must derive from some particular structural analysis, from some general notions about structures and their manifestations.
A most complicated task, which requires especially refined methods of research, is the construction of the "cell" of the theoretical representation of an organic object. In analyzing the logical structure of Marx's Das Kapital, A.A. Zinoviev has described a series of general criteria of a "cell," the knowledge of which makes it possible to answer the question of whether this or that structure is a "cell" of the given object or not. But these criteria do not suffice for depicting the structure of the "cell" itself. In order to depict it, we need some additional procedures.
Likewise, special ways and means of analysis are required for determining the methods of reasoning, which will disclose to us the mechanism and the rules of unfurling of the "cell" into more complex structures, which represent the object before us in greater detail and more concretely.
Whichever of these directions of research we take, everywhere the main problem and, one may say, the "nexus" of all problems will turn out to be the discovery and the depiction of the connections of the object. However, the solution of the problem is made very difficult by the continual confusion of the concepts with one another.
"Relation" and "Connection"
The concept of connection seems to be intuitively clear, especially when we think of it in terms of concrete examples, as, say, the connection between a cause and its effects or as two bodies connected by a shaft. It is in this intuitive sense that the concept was used in the logical tradition of Francis Bacon and J.S. Mill without eliciting any particular objections. Today, however, it becomes increasingly clear that this intuitive understanding can no longer satisfy us. It is no longer sufficient, since there is a continual mixing of structural connections of objects on the one hand with formal connections of sequential discussion and on the other with relations. When we are dealing with the assertion "Peter the First was taller than Napoleon," it is still comparatively easy to guess that its content is a relation, not a connection. However, when we take slightly more complex assertions, as, for example, "Ivan is Peter's brother" or "A is a part of B," it is no longer a simple matter to decide whether we are dealing with a relation or with a connection. For this reason, in the logical tradition of the second half of the nineteenth century and the first half of the twentieth, assertions like "A is the cause of B" and "A is the brother of B" were considered together as indistinguishable assertions about relations. This naturally led to a corresponding theoretical recognition of the categories "relation" and "connection" themselves. The former was considered as a generic concept of the latter and the intuitively conjectured distinction between them was held to be outside the scope of logic.
Actual attempts to draw a typical distinction between "relation" and "connection" began comparatively recently. These attempts were stimulated on the one hand by the paradoxes associated with the formal link of a syllogism (the so-called "implication" in its various forms) and on the other hand by attempts to explain the nature of so-called "nomological assertions" and "causal implications," to which many "laws of nature" were referred, as well as complex assertions concerning every day life. Many foreign scientists, among them H. Reichenbach, A. Burks, M. Schlick, N. Good man have in effect clarified a special structure of content of such assertions and strived to construct specific theoretical explanations for them. These attempts, however, did not, after all, yield any really essential results on the one hand; and on the other, even if they had been successful, they could not in any case lead to a singled out general concept of connection, since they were directed from the start at special cases.
The first fundamental attempt to derive general criteria for distinguishing information about relations and information about connections and, correspondingly, the relations and connections themselves was undertaken from 1955 to 1960 by A.A. Zinoviev.
In Zinoviev's view, the solution of this problem could not be obtained by a direct attempt to define the specifics of the connection itself: along this line we do not move beyond tautological assertions, such as "a connection is a connectivity or mutual dependence," or "assertions about connections are those in which connections are fixated," etc. Therefore Zinoviev began his investigation from the other end, namely from an analysis of the logical structure of the findings about connections and of the rules of their formal transformation in reasoning. Having derived, on the one hand, indisputable assertions about connections in the different scientific contexts, such as, "As A changes, B changes," "A is a cause of B," etc., and on the other hand, typical assertions about relations, such as "A is larger than B," he compared the methods of formal transposition of the ones and the others into other contexts and discovered a basic difference. It turned out that the assertions about connections singled out by him obey logical rules of derivation different from those obeyed by assertions about relations. If we simplify the matter somewhat, we may state it thus: In assertions about relations, the transitive -relation rule holds, "If A > B, B > C, then A > C," while this rule does not hold for connections. From the premises "A elicits B" and "B elicits C," it does not necessarily follow that "A elicits C," although in some cases this may be so.
Having thus shown a specific logical property of knowledge about connections, Zinoviev next tried to characterize the connections themselves as a special content of this knowledge. "Having defined assertions about connections as a special type of assertions," he wrote, "we can define the connections themselves as that which reflects knowledge of this sort." However, in order to overcome the usual tautology (connections are what is expressed in the knowledge about connections), it was evidently necessary to construct special representations for the connections themselves, distinct from the form of their fixation and expressions. On this basis, we could next examine and describe procedures and methods of constructing the assertions themselves.
If we turn to the material of modern science, we shall see that there are in science several different methods of representing connections. The best known and, one may say, popular are representations by means of line segments which connect the symbols representing the elements, as, for example, in the structural formulas of chemistry. Another form of representing connections is by "lines" depicting the channels of signal transmission in the block diagrams of information processing or other machines. Special forms of representation are graphs, tables, and certain elements in physical and engineering models.
However, all of these forms of graphical representation widely used in modern science have all had a common shortcoming, namely that they do not express their kinship with the assertions about connections and do not show the procedures of analysis and of the construction of the assertions themselves, which have to be derived in order to realize the program of logical investigation described above. Therefore, it became necessary to reject these representations and to look for those kinds which might in some way disclose the secret of singling out the connection as a specific objective entity. Fortunately it turned out that such a form had already been found in the preceding development of logic and was even represented schematically in a table. These were the schemes of the so called inductive or experimental derivation of causal connections of Bacon, Herschel, and Mill.
These methods came into logic together with the science of the new era and were a generalization of the methods of practical laboratory research of the seventeenth and eighteenth centuries. These were methods of observation aimed at determining causal connections and dependence.
One of the most important among them was the so-called method of "unique distinction." John Stuart Mill and his followers expressed it as follows. If, when a certain factor is introduced an event occurs, and after the factor is removed the event disappears, whereby we neither introduce nor remove any other circumstance which might exert an influence in the case at hand, nor make any changes in the initial conditions of the event, then the factor mentioned constitutes a cause of the event. Later this principle came to be rep resented as a syllogistic scheme:
Conclusion: The circumstance A is a cause of a.
This scheme is imposed upon actual investigated situations: if any two objects, factors, or events behave as represented in the scheme, we could assert that a causal connection exists between them. In this scheme of inductive derivation, Zinoviev found what he needed: it satisfied all the requirements posited above. It was a special representation of the content of findings about connections, distinct from the form of the findings themselves, and at the same time, unlike all other types of representation, it showed the very method of constructing findings.
It remained only to introduce a slight refinement and generalization of concepts and a change of notation.
In the principles and schemes of traditional logic, the concepts "event," "circumstance," and "factor" had been very vague. There was no precise identification of the boundaries of those factors which must remain constant in the course of an experiment or of an observation. The order of analysis and the taking into account of the objects or events A and a were in no way fixated, etc. In order to eliminate all these shortcomings of traditional schemes, Zinoviev introduced several new, formally precisely defined concepts with corresponding symbolic representations. The symbolic representations of the "objects of juxtaposition" became fundamental. Here both the symbols representing actual things, a, b, c, entered together with the symbols of the properties or criteria singled out, Q, R, P. As a whole, the "object" was represented by groups of symbols in the form (Qa), (Rb), (Pab), etc. The absence of the "object" was also considered as an object, and was represented symbolically by (-Qa). The fixation of the object in a corresponding knowledge was represented, symbolically by "Qa" or "-Qa." After these symbolic representations and the corresponding concepts have been singled out, the above-mentioned scheme of inductive juxtaposition could be represented in the form of a table:
The first row was to represent one situation in juxtaposing the "objects" Qa and Rb; the second row was to represent the other situation of juxtaposition, in which the absence of Qa was "accompanied" by the absence of Rb. The juxtaposition of these two situations permitted the conclusion that a connection was present between Qa and Rb and to assert "if Qa, then Rb." As a whole the table was called a "set." The order of the juxtaposition of "objects" in situations and of situations in the sets determined the type of the derived connection. Thus, according to Zinoviev, different sets of situations can serve as a generalized model of all those contents of findings which we call "connections."
On the basis of these notions and symbolic representations, Zinoviev constructed a mathematico-logical enumeration of connections, defined the conditions of logical validity for different complex assertions constructed from simpler ones according to certain rules. This work, we repeat, is the most basic and fundamental of all done up to the present on the problem of logical definition of connections.
However, in spite of its virtues, this work has one essential shortcoming. It cannot embrace all of the notions of connection existing at the present time and widely used in science. One could put this even stronger: the notion of connection introduced in this work does not in general correspond to most of those uses, in particular to all we know about connections of objects and elements as a whole, all of the kinematic and mechanical notions of connection, etc. And evidently Zinoviev himself felt this in the course of further development of his theory, since he wrote in a later work that the proffered method
. . . gives only an approximate, simplified description of the field of our interest. In particular, it will encompass only a limited sphere of assertions which does not exhaust the entire field of assertions, defined by the current usage of the word "connection" and which perhaps partly falls outside of it. Be sides, the linguistic means of expressing these assertions (special words and constructions of sentences, tables, graphs, charts, etc.) cannot be examined on this course literally in the form in which they are encountered in the contexts of every day and scientific languages.
From the context in which these remarks were made, one could form an impression that the limitations of the proffered method are explained by its deductive nature and that probably such will be any other deductive method, no matter how content-rich its foundations might be. In our view, however, it is not a matter of the deductive character of the method but of its foundations. These turned out to be too narrow and, perhaps, even simply incorrect. In justification of this assertion, we should like to examine
The Basic Contradictions in the Existing Notion of Connection
Among the sorts of findings about connections encountered in modern scientific literature, two polar types stand out. One fixates the dependences or the connections among the properties or the indications of objects; the other, the connections among the objects themselves, considered as elements of a whole. A characteristic example of a finding of the first kind is the analytic form of an expression of some "law," say, Boyle's Law concerning the dependence of volume upon pres sure: pV = constant. As an example of a finding of the second kind we may take a description of a structural formula of some chemical compound, say Ca(OH)2. If we take instances of findings of the second kind, it turns out that neither the methods of their construction nor the means of formal operations with them correspond to what Zinoviev has described in the notion of the "objects of juxtaposition" of situations and of sets.
It might seem at first that this assertion is not substantiated and that it is not difficult, in principle, to express and to describe the connection, of objects in situation tables. In particular, attempts are made to do this in the following manner.
One of the most natural and self evident models of connection is an image of two spheres joined by a shaft, as shown in Fig. 12.
Suppose the spheres a and b are characterized by their "positions" relative to an origin of coordinates. The object a has the characteristic L, while object b has the characteristic K. Suppose also that we do not see the rod which joins the spheres together, and hence do not know whether they are joined or not. In order to find out, it is evidently necessary to "pull" the sphere a to change its position from L1 to L2, and to see what will happen to sphere b. If the position of b remains unchanged, we can assert that objects a and b are not joined (in any case in the range of position changes L2 – L1); if the position of b changes, say to
And it would seem that we could draw a conclusion from these tables concerning the connection of the objects in question.
Such or similar reasoning serves to substantiate the identity of the content described in charts of juxtaposed situations with that represented in a model of two joined spheres (let us represent it symbolically by aSb). At first glance this reasoning appears to be correct and sufficiently substantiated.
Still, these arguments contain one not very evident point in which there occurs a substitution for the analysis of the object and where an error enters. In the model of the two spheres joined by a rod, the connection existed between the spheres a and b themselves, that is, directly between the objects or, considered otherwise, between the elements of the mutual connection aSb. This connection between the objects a and b was revealed and fixated only in force of the characteristics of the positions of the spheres, L and K. The associated changes in the positions spoke for the existence of a connection between objects. When we passed to the table representing the set of juxtaposition situations, objects of essentially different nature appeared in it, namely the "objects of juxtaposition," La and Kb; and the conclusion concerning the existence of the connection, which we had to make on the basis of this set, could be referred only to those conventional "objects," not at all to the objects of the model, a and b. In this way the idea that the content of the model and that of the table of juxtaposition are identical turned out to have been unsubstantiated.
To be sure, we could realize such an identity if we introduced the model of joined objects itself as a convention, moving, so to say, from the table and fixating the symbols La and Kb themselves in the representations of the objects. But then this model will evidently no longer be a representation of the objects themselves and of their connection but a special artificial representation of the content of findings about the connection and about corresponding ideal things.
From several extremely important and essential points of view such a representation would correspond to the essence of the matter, since it is just with respect to position that the spheres are connected to each other; if we wished to check whether they had some other forms of connection, it is quite possible that the characteristics of position would turn out "inoperative" here, and we would have to search for some other manifestation of this connection. In this way, there exists an essential unity between the object as such and those of its properties which reveal its connection with other objects. In this plan it is correct to introduce a special construct, designated by Qa and Rb to express the unity of the object and its manifest property. At the same time, however, it is indisputable that in models like the two joined spheres we express connections as if between the objects themselves, unrelated to any of their properties. In precisely the same way there exist connections expressed in the form of dependences among the properties alone, unrelated to the determination of their objects.
If we examine all these cases from the point of view of the distinctions introduced above, the systems of findings, the system of the thing and the structure of the object, we shall easily see that the content expressed in the tables of juxtapositions is really related not to the object but to the thing. Exactly in the same way, the constructs of the type Qa ahd Rb are not objects in the precise meaning of this word but findings about the "thing." But this means that also the connections, which we introduce as the tables of juxtaposition of these things, are connections not of the objects but of the things known. The fact that the ones and the others obey entirely different structural laws and have different systemic segmentation has already been pointed out above.
At this juncture we see the first fundamental contradiction in the concept of connection introduced by Zinoviev. Constructed in this way, it cannot encompass and express connections among the elements of the real structure of objects, elements obtained in the process of decomposing this structure. But then another question follows from this assertion: in what manner do we reveal the structure connections of the objects themselves?
The second contradiction in the present concept of connection is associated with the same distinction between "object" and "thing known." If the "objects" in the situation juxtaposition tables are actually not objects but things known and, if further, the "things" are nothing but bundles of substitution in symbols revealed operationally in the objects of contents, then they should be viewed in just this way, that is, taking into account many levels of symbol substitution and analyzing the new elements which each level adds to the process of revealing the content of findings. The failure to take into account this aspect of the matter is a most important defect in the present concept of connection. For just this reason it is not able to "grasp" the real linguistic tools and the peculiarities of content of the various scientific assertions about connections.
The fact of the matter is that the content of knowledge about connections is given not only by what juxtapositions are realized on the level of the original objects, but also by what symbolic means the derivation (and therefore the content) are fixated, and by what precisely becomes correspondingly the object of the juxtapositions that follow. The overwhelming portion of contemporary findings about connections has for its content juxtapositions in which, besides the objects (decomposed into parts and synthesized from them), also symbols of various kinds play a part. These symbols lie on different levels of substitution and rep resent different contents. For instance, in modern chemistry we have besides the reacting substances themselves and their descriptions also formulas of compounds, structural formulas, physico-chemical and physical models of atoms and of molecules. And the juxtaposition which singles out new con tent of objects (their structure in particular) goes on always in place of transitions from some symbolic tools and levels of substitution to others. Schematically and, so to say, "in cross section," this whole system of substitutions and of juxtapositions based on them can be represented as in Fig. 13.
We must say, in particular, that the appearance of new representations of the composition and of structure of chemical compounds or of physico-chemical and physical models of substances changes radically the character of reasoning and of deductions in chemistry. The very logic of thinking changes, as well as the logical rules of concrete and formal solutions of problems. In particular, the methods of constructing assertions about connections also change. In order to obtain information about connections on the basis of structural formulas already obtained, we need entirely different schemes of juxtaposition and of procedures in general, entirely different from those which we had to use in obtaining information on the basis of formulas of compounds. The same is taking place in other sciences as their symbolic tools develop and new levels of substitution appear.
Therefore, it is quite natural that a logical theory of knowledge about connections, which fails to take into account these instances, turns out to be very limited and is unable to encompass even the most important types of knowledge, let alone all. In order to construct a really general logical theory of assertions about connections, an essentially different approach to the problem is required, as well as different logical bases, in particular such as take into account the empirical distinctions between objects and connections between properties, on the one hand, and on the other, the multi-level structures of all knowledge.
As we realize this principle we wish to examine
The "Logical Encirclement" of the Notions of Connection
An analysis of the history of thought shows that all original notions about connections arise at the intersection of several methods of analysis of objects, and for this reason they unite in themselves different groups of thought processes. In order to demonstrate the method of reasoning it self while analyzing the notions, we shall examine a simplified combination of some such thought processes.
The first is a purely empirical derivation initially of a correlation, later of dependence of two properties (parameters) of some object or event.
The simplest illustration of this line of investigation is the discovery of a dependence between the pressure and the volume of a gas in the polemic of R. Boyle against Linus.25 Boyle had to convince Linus of the existence of the elasticity of air. He took a bent glass tube (like a syphon tube) with its shorter arm sealed and filled it through the longer (open) arm with mercury. As he added more mercury, the air in the shorter arm became compressed but it nevertheless balanced the ever taller column of mercury. To characterize the "elasticity" of air, Boyle had to juxtapose the diminishing volumes of air to the corresponding excesses of pressure in the longer arm. The most "natural" formula for fixating the correlations of volumes and the excesses of pressure was the following table:
Only later, a student of Boyle's, Richard Townley, noted that the product of pressure and volume remained approximately constant and so discovered an invariant. This permitted the establishment of an analytical formula (and function), the dependence between pressure and volume.
The mathematical theory of proportions gave an operational symbolic form of expressing the empirically discovered dependence between two properties of an object.
A second thought process can be called an "explanation" of the category of dependence. After the simplest dependence among properties of objects were established and expressed in mathematical
form, a long period of search and explanation began. We are not now discussing the question of the reasons why such explanations became necessary, or the conditions which made their appearance possible. We accept them as a historical fact. The means of such explanations were given by "engineering constructs," i.e., "artificial" objects somehow connected with one another. These could be, for example, two spheres connected by a rod, a string, or a spring.
Notions about such "artificial" objects were essentially "superimposed" upon the empirically disclosed dependences among the properties of the "natural objects" and served as means of understanding them. One property of an object changes as a result of a change in another, or, in other words, one depends upon another, because they are somehow connected with each other. The investigator begins to "see" the table of changing values of the properties a and b through the image of spheres tied to each other. He imagines it as the result of changes in the state of connections
and at the same time as a manifestation of this connection. If, for example, we change the "position" a, then correspondingly position b will change, and this will be expressed in a table, which can later be expressed in some analytically expressed dependence.
"Artificial" engineering constructs in the form of two objects tied to each other became an explanatory model of dependences. But if we take the construct as such, there will be no connection in it; the rod and the spring will remain just a rod and a spring. They become a "connection" only thanks to the role of the construct as an "explanatory model" of the dependence of properties, which had been disclosed empirically in the object studied. In other words, particular elements of the "engineering constructs" (rods, springs, conveyor belts, transmission mechanisms, and such) appear as "connections" only because they are accepted as the "basis" of the various dependences among the properties of objects.
It is especially important to emphasize one more circumstance. Connections are objects of a kind, but in their direct givenness as objects they exist only in the explanatory models. In the "natural" objects there is nothing like them; there are no rods nor springs. Essentially they are ascribed to objects, to the extent that "engineering constructs" appear as representations or as models of those objects. Therefore, when we speak of the existence of "connections" in objects, we must understand that in reality they exist only either in empirically disclosed dependences of properties or in those internal mechanisms which, at the basis of these dependences, are specific to the object in question, the "internal mechanisms," disclosed empirically and fixated in tables and functions.
In this way a mental construct is put together which includes two different processes of investigation: 1) the empirical disclosure of dependence between two properties of the object investigated, and 2) an explanation of this dependence by means of relating it to another object, constructed by man in the form of elements connected to each other. The mental construct can be represented by a diagram as in Fig. 14.
It is this construct taken as in whole which corresponds to the first forms of scientific conception of connections.
The mathematical form of functions expresses the dependence of two properties of an object; but it contains no expression of the connection, and there is no basis for introducing this concept. In other words, the concept of connection cannot appear so long as we use only the tables and the mathematical formulas in expressing dependences. The connection appears and can be singled out as something apart and independent only when we introduce artificial constructs of objects connected to one another. It is in these constructs that the connection gets its real material existence in the form of a rod, a spring, or a string; i.e., becomes a particular, one may say, material object. But an object by definition cannot be an object of empirical analysis. The only reality brought out empirically are the dependences. But any disclosed dependence is immediately understood by the investigator as a definite "connection." This occurs at the expense of the mental construct described above. Every dependence is understood as a connection, and every connection exists in reality and is manifested in some empirically brought out dependence. Thus, on this level of understanding connections, there is always a duality: its one side is the inter-dependence of the properties of an object; the other is the connectedness of the elements of the model. Such duality is harmless as long as a one-to-one correspondence is established between the two.
This correspondence, however, begins to be violated as soon as we pass in our empirical analysis from the dependence among two properties (parameters) to a dependence among many parameters. We have already examined the simplest example, using the method of "dual findings." We supposed that the structure of the object in question consists of three elements, A, B, and C, and two-way connections among them. With the help of empirical procedures, it is possible to bring out dependences among any two parameters and to express them in the form of mathematical functions. There will be six expressions in all [Cf. Eq. (2)].
For each of these we can choose a model of a corresponding mechanism of connection so that we shall have six different models for representing a single structure of an object. Since, however, the object is only one, there will naturally arise the problem of uniting them all into a single synthesized model. To do this mechanically is impossible: the fact is that in each function there "resides" the whole structure, as we have already explained, and therefore each of the six models of the connection mechanism is a particular functional analogue of the whole structure. If, however, the synthesis will be conducted not mechanically, this only means that some new model will be constructed with new elements and connections, and these connections will be such that not one of them separately will correspond to the mechanisms which simulate the empirically disclosed dependences. Only when taken together in interdependence with each other within the framework of a single mechanism will these connections yield a basis for explaining all the enumerated functions. This is represented schematically in Fig.15. After the synthetic model of the mechanisms has been built, there appears a perfectly obvious discontinuity between the structure connections of the object and the functional representations of the dependence of its properties. In order to overcome this rupture, another special task must be undertaken, namely to derive the dependences from the mechanical model. This is that very substitution of structure analysis by means of hypothetical models of which we spoke above. Thereby different manifestations are derived from the analysis of the mechanisms, including possible dependences among properties. They are juxtaposed to the set of properties and dependences which we were able to bring out by way of empirical analysis of the object itself. If the properties brought out from the model coincide with the properties of the object, we assume that the model of the mechanism has been constructed correctly.
In the process of bringing out the properties from the mechanical model, it becomes necessary to operate in a special way upon the different components of the model itself. Perhaps the most important and widespread method of operation is the mechanical combination of elements and of the fragments of the model with each other, the addition of some to others, and the decomposition of complex models into simpler ones.
Because of this there appears on the level of the mechanical models themselves still an additional content. In the image of two spheres connected by a rod, both the spheres and the rod were identical material components of the model. The functional distinction of the rod arose from its being related to a special symbol representing a function in the expression of dependence. Now, when the mechanism itself is decomposed into elements, another entity must make its appearance, namely that which connects and fixes the parts into the whole, parts which were obtained when the object was decomposed or fractured. If the whole (which consists of the two spheres connected by a rod) is fractured in such a way that only the spheres are taken into account, then the rod ceases to be a material component of the mechanism and becomes a complementary formal instrument, taken in from the outside, as it were, and having no analogues in the mechanism itself. If, in decomposing the whole, we take into account not only the spheres but the rods as well, then all of them appear as different but, from the formal point of view, equal elements of the whole. Then something else must appear in the role of whatever binds the elements together.
Thus, because of the rupture between the connection proper and the mathematical expressions of functions on the one hand, and on the other, the appearance of the formal activity of decomposing the model into parts and synthesizing the parts, there arise "connections" in the proper sense of the word. Because of their special mental roles, they are freed essentially of all material properties and are endowed with a purely formal operational content. After this it becomes possible to introduce special symbolic representations of connections – most frequently as line segments – which no longer represent material elements but rather the connection itself in its limiting abstract sense.
It is here that we first enter the sphere of specifically structural investigation, and here the connections become finally formulated as specific components of structural models and as a special content of findings about objects, distinct from the dependences among properties.
However, this very process results in the circumstance that structural representations of objects composed of symbols of elements and symbols of connections are distinguished from the representations of the "life" mechanisms of the objects and begin to exist as independent entities in the general system of findings about objects.
One of the most important results of this separation consists in that the symbols for the elements and the symbols for the connections exist for a comparatively short time as simple elements of the structural representation of objects. They are soon organized into operational systems and began to be used in accordance with special rules. The operational system of such symbols represents essentially a special "mathematics": the investigator gets the opportunity to move about on that level in a purely formal way, relating only the final result to the object, again in accordance with particular rules. Evidently the fundamental meaning of this movement described above consists in just this, if we examine it on the level of logic and the methodology of research. The formalization of research, its transfer to the level of operating with symbols, radically changes the type of research work itself, immensely simplifying and accelerating it and freeing it from the necessity of conducting a long chain of empirical observations and procedures, at the same time improving the quality of the results.
This whole process can be followed, in particular, in the historical appearance and utilization of structural formulas in chemistry, which already for a long time represent an operational system. Its elements are line segments, representations of connections, which obey rigidly fixed rules of operation. It is not difficult to see that the very appearance of these formulas is entirely consistent with our scheme given above. Evidently essentially the same role is now being assumed by elements of engineering constructs in many branches of modern science, constructs which initially appeared as means of representing objects of investigation but now are organized more and more into systems with particular rules of operation. However, this process cannot at this time be considered as completed, although its tendency is being outlined rather clearly.
Examining in this manner the findings about connections, we discover a particular structure, which can be called
The "Organism" of a Concept
The main thing in the characteristic of a scientific concept is that it exists by no means in the head of this or that individual, but is an objective entity, fixated in symbols and having a hard hierarchical structure. From the above exposition it is clear that such a structure can be represented in the form of a series of levels or elements connected in a particular way, whereby symbols of various types may be constantly introduced into it to perform certain functions. Nevertheless, all the levels of this structure constitute a single whole. Therefore, any concept or finding can be considered as an objective organism possessing its own logic of motion and its potentialities of development. And only this organism as a whole constitutes that which creates the content of any concept. It is not difficult to see that such an approach to the structure of a scientific concept departs from what we usually encounter in formal logic. But this is just what we tried to show, namely that the formal-logical approach does not disclose and cannot disclose either the content of concepts, or the objective structure of a complex cognitive organism, or its specific functions in cognitive activity.
In our analysis, the structure of the organism of concept was represented in the form of the totality of rigidly connected blocks or levels. Not one of these blocks constitutes the concept proper, although in each of them there are specific rules of operation; and not one of these blocks, taken separately, can single out the content of a concept. All this is accomplished by means of the unity of all the blocks.
But what is the purpose of this unity? In defining its nature, we can say that a concept is a special sort of machine. The block scheme, by means of which we represent a concept, is included in a particular system of utilization, so that each of the blocks "works" in its particular way. Tables, analytical formulas, different types of models, are all "plugged in" to systems of action upon objects and a "net" of connections is super imposed by means of symbols. The meaning of the net is in that it makes it possible to pass from one activity to another. It is this availability of a method of passing between the blocks or, what is the same thing, between the different types of activity, which allows us to speak about the structure of the whole, of a single organism of a concept. Every layer and level live in accordance with particular laws and are fragments of some particular type of activity. However, they are also connected among themselves and this permits the substitution of one type of activity for another. Because of this, the separate fragments of the different activities are "fused" into a single structure, constitute a single organism, obeying specific laws of existence.
This "organismic" nature of a concept is particularly important in system-structural investigations. The fact is that the cognitive constructs in this type of investigation are especially complex. The more complex the structure the more necessary is a consciously methodological approach to the building of the constructs. Besides, the concepts of a system-structural investigation are characterized by a complex many-sided content, which appears on several different levels. That is why a clear definition of each level is required, as well as of the rules of transition among them. Otherwise, as the content is worked on, the machine may fail in some nexus, and this will lead to errors in the investigation.
We have examined some problems of methodology of system-structural investigation and have indicated only some of the difficulties with which the investigator is confronted. But perhaps this will permit to imagine the novelty and the complexity of the problems which arise in this field, new in human thought.
It will be no exaggeration to say that system-structural investigations disclose a new, unusually important field of scientific creativity. As we enter this field, we enter a country of wonderful discoveries which promise to humanity more than can be now imagined. The construction of a theory of life and the regulation of large economic systems, the rational organization of education, and a wide planning of scientific investigations, modern city planning, and the creation of most complex cybernetic devices—all these and many other problems cannot be successfully solved without the tools of system-structural investigation.
However, in order to create such means, much hard and tedious work is required in a special field of cognition, namely that of methodology. This work will be successful only under one condition, namely if it is able to overcome the hypnotic power of the old forms of thought. That which the scientist encounters in the investigation of a complex object will usually seem at first strange and contrary to common sense.
Often in the investigation of systems and structures we deal with a contradiction to the usual intuitive logic. This was disclosed in its time in the evaluation of some of Hegel's positions. For instance, his assertion that the whole is equal to a part still elicits protests and irritation in many. But if we analyze the works of Hegel in detail, we find that such an assertion has a strict sense in the methods of analysis created by Hegel. An equally astonishing failure of understanding is often found in the analysis of the problem of contradiction which, by its very nature, cannot be made consistent with common sense and with the intuitive logic upon which it is based. When K. Marx speaks of the "splitting" of commodities into use value and exchange value, many understand this as a splitting of the actual commodity—such as a bible or a bottle of wine. It is often quite difficult to convince people that when simple forms of commodities are converted into complex ones, we do not have a literal splitting of the commodity into commodity and money. Marx applies here a special method of structural-functional investigation, which does not correspond to the process of actual development of commodities, and he makes a special note of it. Usually, however, this disclaimer is not noticed.
Essentially the same thing happens and will continue to happen with many if not all the concepts specific to structure-system investigations. This is understandable: in order to analyze systems and structures, essentially new forms of thought must be created which are not only not customary but often contradict common sense and intuition. To accomplish this, the fetichism of old notions must be overcome. Without this work of clearing up old notions and creating new forms of scientific thinking, we cannot count on a construction of a methodology for system-structural research.
SOVIET LITERATURE ON GENERAL SYSTEM THEORY, THEORY OF COGNITION, AND RELATED TOPICS
I. The Development of Concepts in the Methodology of Science
Bransky, V. P. Filosofskoie znachenie nagliadnosti v sovriemionnoi fizike. (The Philosophical Significance of Graphic Demonstration in Modern Physics)
Davydov, G.A. Vopros o prirodie poniatia v "Filosofskikh tietradiakh" V. I. Lenina. Sb. "Dialektika – teoria poznania. Istorichesko-filosofskie ocherki." (The problem of the nature of concept in the Philosophical Notebooks of V. I. Lenin. Compendium Dialectics – The Theory of Knowledge. Historico-philosophical Sketches).
_______. Iedinstvo filosofskogo i iestiestvienno-nauchnogo podkhodov ê priedmietu issliedovania. Sb. "Dialektika – teoria poznania. Problemy nauchnogo metoda." (The unity of the philosophical and the natural-scientific approaches to the object of investigation. Compendium Dialectics – The Theory of Knowledge. Problems of Scientific Method.)
Il'ienkov,E. V. Dialektika abstraknogo i konkretnogo v "Kapitalie" K. Marksa. (The Dialectics of the Abstract and the Concrete in Marx's "Capital.")
_______. Ideal'noie. "Filosofskaia Entziklopedia" t. 2. (The ideal.) Philosophical Encyclopedia.
_______.Vopros î tozhdiestvie myshlienia i bytia v domarksistkoi filosofii. Sb. "Dialektika –theoria poznania. Istorichesko-filosofskie ocherki." (The question of the identity of thought and being in pre-Marxist philosophy. Compendium Dialectics—The Theory of Knowledge. Historico-philosophical Sketches.)
Ladenko, I. S. Istoria nauki v svietie teorii myshlienia. (The history of science in the light of the theory of thought.) Voprosy Filosofii, 1964, No. 9.
_______ . Problemy obosnovania matematiki i logicheskiy empirizm. Sb. "Dialekticheskiy materializm i sovriemionnyi pozitivizm." (The problem of the foundations of mathematics and logical empiricism. Compendium Dialectical Materialism and Modern Positivism.)
Lektorky, V. A. and Sadovsky, V. N. Osnovnyie idiei "geneticheskoi epistemologii" J. Piaget. (The basic ideas of the "developmental epistemology" of J. Piaget.) Voprosy Filosofii, 1961, No. 4.
Mamardashvili, M. K. Protzessy analiza i sinteza. (The processes of analysis and synthesis.) Voprosy Filosofii, 1958, No. 2.
_______. Niekotoryie voprosy issliedovania istorii filosofii êàê istorii poznania. (Some problems of research in the history of philosophy as a history of cognition.) Voprosy Filosofii, 1959, No. 12.
Ovchinnikov, N. F. Î razrabotkie teorii nauchnogo znania. (On the development of the theory of scientific knowledge.) Voprosy Filosofii, 1964, No. 2.
Sadovsky, V.N. Krizis neopozitivistskoi kontzeptzii "logiki nauki" i antipozitivistkie tiechenia v sovriemionnoi zarubezhnoi logikie i metodologii nauk. Sb. "Filosofia marksizma i neopozitivizm." (The crisis of the neopositivist conception of the "logic of science" and anti-positivist currents in modern logic and methodology abroad.) Compendium The Philosophy of Marxism and Neopositivism.
Sazonov, B.V. Ê kritikie neopozitivistskogo analiza "iestiestviennogo" iazyka nauki. (Contribution to the critique of neopositivist analysis of the "natural" language of science.) Ibid.
Shchedrovitzky, G. P. Î niekotorykh momentakh v razvitii poniatiy. (On some features of the development of concepts.) Voprosy Filosofii, 1958, No. 6.
_______. O vzaimootnoshenii formal'noi logiki i neopozitivistskoi "logiki nauki." Sb. "Dialekticheskiy materilizm i sovriemionnyi pozitizm." (On the interrelation between formal logic and neopositivist "logic of science." Compendium Dialectical Materialism and Modern Positivism.)
_______. Î razlichii iskhodynykh poniatiy "formal'noi" i "sodierzhatel'noi" logik. Sb. "Problemy metodologii i logiki nauk." (On the distinction between the fundamental notions of "formal" and "applied" logics. Compendium Problems of Methodology and of the Logic of Science.)
Shchedrovitzky, G.P. and Sadovsky, N. Ê kharakteristikie osnovnykh napravlieniy issliedonvania znaka v logikie, psikhologii i iazykoznanii. Soobshchenia I-Ø "Novyie issliedovania v pedagogicheskikh naukakh." (On the characterization of basic directions of research on the symbol in logic, psychology, and linguistics. Communications I-Ø, New Investigations in the Pedagogical Sciences, vol. 2, 1964, vols. 4 and 5, 1965.)
Shchedrovitzky, G. P. Metodologichskie zamiechania ê problemie tipologicheskoi klassifikatzii iazykov. Sb. "Lingvisticheskaia tipologia i vostochnyie iazyki." (Methodological comments on the problem of typological classification of languages. Compendium Linguistic Typology and Oriental Languages.)
Shvyrev, V.S. Î neopozitivistskoi kontzeptzii logicheskogo analiza nauki. Sb. "Dialektichesky materializm i sovriemionnyi pozitivizm." (On the neopositivist conception of the logical analysis of science. Compendium Dialectical Materialism and Modern Positivism.)
Zinoviev, À. À. Î razrabotkie dialektiki êàê logiki. (On the working out of dialectics as logic.) Voprosy Filosofii, 1957, No. 4.
_______. Ê problemu abstractniho a konkretniho poznatku. (On the problem of abstract and concrete cognition.) Filosofsky Casopis, 1958, No. 2.
_______. Problema stroienia nauki v logikie i dialektikie. Sb. "Dialektika i logika. Formy myshlienia." (The problem of the structure of science in logic and dialectics. Compendium Dialectics and Logic. Forms-of Thought.)
_______. Dva urovnia v nauchnom issliedovanii. Sb. "Dialektika – teoria poznania. Problemy nauchnogo metoda." (Two levels of scientific research. Compendium Dialectics – The Theory of Cognition. Problems of Scientific Method.)
_______. Logika vyskazaniy i teoria vyvoda. (The Logic of Propositions and the Theory of Inference.)
_______. Logicheskoie i fizicheskoie sledovanie. Sb. "Problemy logiki nauchnogo poznania." (The logical and the physical consequence. Compendium Problems of the Logic of Scientific Cognition.)
Ï. Theory of Mental Activity
Ladenko, I. S. Ob otnoshenii ekvivalentnosti i iego roli v niekotorykh protzessakh myshlienia. (On the equivalence relation and its role in some mental processes.)
_______. Î protzessakh myshlienia, sviazannykh s ustanovlieniem otnoshenia ekvivalentnosti. (On the mental processes involved in establishing the equivalence relation.)
Le Fevre, V. A. and Dubovskaya, V. I. Sposob "rieshenia zadach" êàê sodierzhanie obuchenia. "Novyie issliedovania v pedagogicheskikh naukakh." (Problem solving as the content of education. New Research in the Pedagogical Sciences.)
Niepomniaschaya, N. I. Logika i psikhologia v contzeptzii J. Piaget. (Logic and psychology in the conception of J. Piaget.) Voprosy Filosofii, 1965, No. 4.
Pantina, N.S. Issliedovania umstvennogo razivitia dietiei v protzessie dieiatel'nosti s didakticheskimi igrushkami. Sb. "Razvitie poznavatiel'nykh i volievykh protzessov u doshkol'nikov." (Investigation of mental development of children in the process of activity with didactic [educational?] toys.) Compendium The Development of Cognitive and Will-directed Processes in Pre-school Children.
Rozin, V. M. Analiz znakovykh sriedstv geometrii. (The analysis of the symbolic tools of geometry.) Voprosy Filosofii, 1964, No. 6.
Shchedrovitzky, G. P. "Iazykovoie myshlienie" i iego analiz. ("Language thinking" and its analysis.) Voprosy Iazykoznania, 1957, No. 1.
Shchedrovitzky, G. P., Alekseyev, N. G., and Kostelovsky, V. A. Printzip "parallelizma formy i sodierzhania myshlienia" i iego znachenie dlia traditzionnykh logicheskikh i psikhologicheskikh isslie-dovaniy. Soobshchenia I-IV. (The principle of "parallelism of form and content of thought" and its significance for traditional logical and psychological investigations. Communications I-IV.) Doklady Academy of Pedagogical Sciences RSFSR, Nos. 4 and 5, 1960.
Shchedrovitzky, G. P. and Ladenko,
_______.Ê analizu protzessov rieshenia zadach. (Contribution to the analysis of the problem-solving process.)
_______. Î printzipakh analiza ob'iektivnoi struktury myslitiel'noi dieiatiel'nosti na osnovie poniatiy sodierzhatiel'no-geneticheskoi logiki. (On the principles of analysis of the objective structure of mental activity on the basis of the notions of content-genetic logic.) Voprosy Psikhologii, 1964, No. 2.
Ø. Methodology of Investigations of Structures and Systems
Afanas'iev, V. G. Problema tzelostnosti v filosofii i biologii. (The Problem of Wholeness in Philosophy and Biology.)
Blauberg, I. V. Problema tzelostnosti v marksistskoi filosofii. (The Problem of Wholeness in Marxist Philosophy.)
Grushin, B. A. Ocherki logiki istoricheskogo issliedovania. (Outlines of the Logic of Historical Research.)
Khailov, Ê. Ì. Problema sistiemnoi organizovannosti v teoreticheskoi biologii. (The problem of systemic organization in theoretical biology.) Zhurnal Obshchei Biologii, vol. xxiv, 1964, No. 5.
Kremiansky,"V. I. Niekotoryie osobiennosti organizmov êàê "sistiem" s tochki zrenia fiziki, kibernetiki, i biologii. (Some peculiarities of organisms as "systems" from the point of view of physics, cybernetics, and biology.) Voprosy Filosofii, 1958, No. 5.
Le Fevre, V.À. Î sposobobakh priedstavlienia ob'iektov êàê sistiem. Tezisy dokladov simpoziuma "Logika nauchnogo issliedovania" i seminara logikov. (On methods of representing objects as systems. Theses of reports at the symposium The Logic of Scientific Research and logicians' seminar.)
Lektorsky, V. A. and Sadovsky, V. N. Î printzipakh issliedovania sistiem. (On the principles of system research.) Voprosy Filosofii, 1960, No. 8.*
Moskaieva, A.S. and Rozin, V.M. Ê analizu stroienia znania tipa "Nachal" Evklida. Tezisy dokladov simposiuma "Logika nauchnogo issliedovania" i seminara logikov. (Contribution to the analysis of the construction of cognitive systems of
Problemy issliedovania strukrur i sistiem. Materialy i konferentzii. (Problems of Research on Structures and Systems. Materials and Conferences.)
Sadovsky, V.N. Ê voprosu î metologicheskikh printzipakh issliedovania priedmietov priedstavliaiushchikh soboi sistiemy. "Problemy metologii i logiki nauk." (Contribution to the problem of methodological principles of research on objects which are systems.) Problems of Methodology and of Logic of Science.
Shchedrovitzky, G. P. Metologicheskie zamiechania ê problemie proiskhozhdienia iazyka. (Methodological comments on the problem of the origin of language. Nauchnyie Doklady Vysshei Shkoly, Filologickeskie Nanki, 1963, No. 2.)
_______. Miesto logicheskikh i psikhologicheskikh metodov v pedagogicheskoi naukie. (The role of logical and psychological methods of pedagogical science.) Voprosy Filosofii, 1964, No. 7.
Shvyrev, V. S. Ê voprosu î kazual'noi implikatzii. Sb. "Logicheskie issliedovania." (Contribution to the problem of causal implication. Compendium Logical Investigations.)
Spirkin, A. G. and Sazonov, B. V. Obsuzhdienia metologicheskikh problem issliedovania sistiem i struktur. (A discussion of the methodological problems of research on systems and structures.) Voprosy Filosofii, 1964, No. 1.
Zinoviev, A. A. Logicheskoie stroienie vyskazaniy of sviaziakh, "Logicheskie issliedovania." (The logical structure of propositions about connections. Logical Investigations.)
_______. Ê opriedielieniu poniatia sviazi. (Toward a definition of the notion of connection.) Voprosy Filosofii, 1960, No. 8.
_______. Dieduktivnyi metod v issliedovanii vyskazaniyî sviaziakh, "Primienienie logiki v naukie I tiekhnikie." (The deductive method in the investigation of propositions about connections. The Application of Logic in Science and Technology.)
_______. Ê voprosu î metodie issliedovania znaniy (vyskazyvania î sviaziakh.) (On the problem of the method of research on findings [on propositions about findings].)
_______. Logika vyskazaniy i teoria vyvoda. (The logic of propositions and the theory of deduction.)
*This article appeared in English translation in General Systems, Vol. 5 (1960).
* Translation by Anatol Rapoport of Problemy Metodologii Sistemnogo Issliedovania.
 Cf. V. A. Lektorsky and V. N. Sadvosky. O printsipakh issliedovania sistiem. Voprosy Filosoffi, 1960, No. 8.
 Metodologieheskie Problemy Nauki.
 "Art" in the sense which this word had in the Middle Ages: artful meaning expert, perfect performance, based
on rich experience. Today cyberneticians often use this word in this sense, e.g. L. Kuffinial (Cf. his article in the
Compendium Nauka i Cheloviechestvo, Moscow, 1963.)
 The concept of cardinality was introduced by the famous German mathematician G. Cantor (in this connection Cf. Novyie Idiei v Matematikie, St. Petersburg, 1914, vol. 6).
 5. Galileo Galilei. Dialog o Dvukh Glavnieishikh Sistlemakh Mira – Ptolomeevoi i Kopernikovoi.
 Here we must note that besides antinomies there exist a number of other situations, in which the problem of investigating cognitive activity is put in exactly the same way and the components of this activity are singled out. We do not analyze those situations since from the point of view of our interest, they do not differ in any way from antinomies.
 Cf. G.P.Shchedrovitzky and V.N. Sadovsky. K kharaktieristikie osnovnykh napravlieniy issliedovania znaka
v logikie, psikhologii i iazykoznanii. Soobshchenia 1. Novyie Issliedovania v Pedagogicheskikh Naukakh, vol. 2,
 By ontology we mean in this case the construction of special representations of objects as such.
 K. Marx and F. Engels. Works, vol. 3, p. 1 (The reference is to the Soviet edition.)
 D. Hilbert. Osnovania Geometrii.
 In this connection see our articles "K analizu protsessov rieshenia zadach," (Doklady, Academy of Pedagogical Sciences, RSFSR, 1960, No. 5); "O printzipakh analiza ob'iektivnoi struktury myslitiel'noi dieiatel'nosti na osnovie poniatiy sodierzhatiel'no-geneticheskoi logiki" (Voprosy Psikhologii, 1964, No. 2).
 Cf. S. E. Khaikin. Mekhanika. Moscow, 1947.
 Cf. St. Beer. Kibernetika i Upravlienie Proizvodstrom. Moscow, 1963, p. 22.
 G.H. Goode and R.E. Macall. Sistiemotekhnika: Vvedienie v Proiektirovania Bol'shikh Sistiem.
 Cf. e.g., O Sootnoshenii Sinkhronnogo Analiza i Istoricheskogo Izuchenia Iazykov. Academy of Sciences U.S.S.R., 1960, pp. 95, 102, 125, and elsewhere.
 Cf. B. A. Grushin. Ocherki Logiki Istoricheskogo Issliedovania. Moscow, 1961.
 Cf. A.A. Zinoviev. Voskhozhdienie ot Abstratnogo k Konkretnomu (na Materialie K. Marksa). Dissertation,
 Cf. S.I. Povarin. Logika. Petrograd, 1916.
A.A. Zinoviev. Logicheskoie stroienie znaniy o sviaziakh. Sb.Logicheskie Issliedovania.
 A.A. Zinoviev. K opriedielieniu poniatia sviazi. Voprosy Filosofii, 1960, No. 8, p. 59.
 Cf. V.F. Asmus. Logika.
 V. Minto. op. cit.. pp. 268-269.
 Cf. V. F. Asmus, op. cit., pp. 268-269.
 A. A. Zinoviev. Logicheskoie stroienie znanii o sviaziakh. Cf. Logicheskie Issliedovania, pp. 113-138.
 Cf. F. Rosenberger. Istoria Fiziki.
 Cf. "The System of the 'Thing' and the System of the Object"
 Cf. "Constructing Structural Models" above.