There are several dozen definitions of this concept. Their analysis shows that the definition of the concept of a system has changed not only in form but also in substance. Let us consider the principal changes that occurred in the definition of a system as systems theory developed and the concept came into practical use.
Elements and relations
The earliest definitions stated, in one form or another, that a system consists of elements (parts, components) aᵢ and connections (relations) rⱼ among them:
$$S = \{a_i,\; r_j\}$$
$$S = \langle A,\; R \rangle$$
$$S = A \cap R$$
In these formalized notations, different set-theoretic representations are used: in the first two, sets are specified without accounting for the interrelationships between the sets of elements and relations; the third reflects the fact that a system is not a simple aggregate of elements and connections of one kind or another, but includes only those elements and connections that lie within the intersection.
Ludwig von Bertalanffy defined a system as "a complex of interacting components" or as "a set of elements standing in certain relations to one another and to the environment." In the Great Soviet Encyclopedia, a system is defined by a direct translation from the Greek systema, meaning "composition," that is, something composed, joined from parts.
The terms "elements" and "components," "connections" and "relations" are usually employed as synonyms (especially in translations of definitions). Strictly speaking, however, "components" is a more general concept than "elements," as it can denote an aggregate of elements. As for the concepts of "connection" and "relation," different viewpoints also exist.
If it is known that the elements are fundamentally heterogeneous, this can be taken into account immediately in the definition by distinguishing different sets of elements:
$$S = \langle A,\; B,\; R \rangle$$
In the definition by M. Mesarovic, a set X of input objects (acting upon the system) and a set Y of output results are distinguished, with a generalized intersection relation established between them, which can be expressed either as in the author's original notation or using other intersection symbols:
$$S \subseteq X \times Y$$
If a certain type of relation applies only to elements of different sets and is not used within each of them, this can be reflected as follows:
$$S = \{a_i \; r_j \; b_k\}$$
- $\{a_i \; r_j,\; b_k\}$ — elements of a new system formed from the elements of the original sets $A$ and $B$.
This form of notation is called a syntagma in mathematical linguistics.
Properties of elements and relations
To refine elements and relations, definitions include properties. Thus, in A. Hall's definition, properties (attributes) Q_A supplement the concept of an element (object):
$$S = \langle A,\; R,\; Q_A \rangle$$
- $Q_A$ — properties (attributes)
A. I. Uyemov, defining a system through the concepts of "things," "properties," and "relations," proposed dual definitions: in one, properties characterize elements (things), while in the other, properties characterize connections (relations):
$$S = \langle a_i,\; r_j,\; q_i \rangle \quad \text{and} \quad S = \langle a_i,\; r_j,\; q_j \rangle$$
- $a_i$ — elements
- $r_j$ — connections (relations)
- $q_i$ — properties of elements
- $q_j$ — properties of connections (relations)
Goal
Subsequently, the concept of a goal appeared in definitions of a system. At first implicitly: in F. E. Temnikov's definition, "a system is an organized set" (where the goal emerges when the concept of "organized" is unpacked); in the philosophical dictionary, a system is "a set of elements standing in relations and connections to one another and forming a certain integral unity." Then it appeared in the form of an end result, a system-forming criterion, or a function (see the definitions of V. I. Vernadsky, R. Gibson, P. K. Anokhin, and M. G. Gaaze-Rapoport), and later with explicit mention of a goal.
Symbolically, this group of definitions can be represented as follows:
$$S = \langle A,\; R,\; Z \rangle$$
- $Z$ — goal, set, or structure of goals.
Some definitions specify conditions of goal formation — the environment, the time interval, that is, the period within which the system and its goals will exist. This is done, for example, in V. N. Sagatovsky's definition, which also forms the basis for one of the techniques of goal structuring: a system is "a finite set of functional elements and relations among them, distinguished from its environment in accordance with a specific goal within a defined time interval":
$$S = \langle A,\; R,\; Z,\; S_R,\; T \rangle$$
- $S_R$ — environment
- $T$ — time of the system's existence
Observer
Further on, definitions of a system began to include, alongside elements, relations, and goals, also an observer N — that is, the person who represents an object or process as a system in the course of investigation or decision-making:
$$S = \langle A,\; R,\; Z,\; N \rangle$$
- $N$ — observer, researcher
W. R. Ashby pointed to the necessity of accounting for the interaction between the system being studied and the researcher. But the first definition that explicitly included an observer was given by Yu. I. Chernyak: "A system is a reflection in the consciousness of a subject (researcher, observer) of the properties of objects and their relationships in the task of research and cognition":
$$S = \langle A,\; R,\; Z,\; N \rangle$$
In subsequent versions of this definition, Yu. I. Chernyak also took into account the observer's language, starting with the formulation: "A system is a representation in the language of the observer (researcher, designer) of objects, relations, and their properties in the task of research and cognition":
$$S = \langle A,\; R,\; Z,\; N,\; L_N \rangle$$
- $L_N$ — the observer's language
Definitions of a system may contain an even greater number of constituents, which is related to the need for differentiating types of elements, relations, and so on under specific conditions.
Evolution of the definition of a system
Comparing the evolution of the definition of a system (elements and relations, then goal, then observer) with the evolution of the use of epistemological categories, one can detect a similarity: initially, models (especially formal ones) were based on accounting only for elements and connections, interactions among them; then attention turned to the goal and the search for methods of its formalized representation (objective function, performance criterion, etc.); and from the 1960s onward, increasing attention was paid to the observer — the person carrying out the modeling or conducting the experiment (even in physics), that is, the decision-maker.
With this in mind, and drawing on a deeper analysis of the essence of the concept of a system, it seems appropriate to treat this concept as a category of epistemology, of the theory of reflection.
Working definition of a system
Considering the various definitions of a system and their evolution, and without singling out any one of them as the principal definition, one can not only observe how difficult it is to briefly define such (usually intuitively grasped) concepts as "system," but also recognize the fact that at different stages of representing an object as a system, in various specific situations, different definitions may be employed. Moreover, as one's understanding of the system is refined or one moves to a different stratum of its investigation, the definition of a system not only may but should be refined.
A definition that includes elements, relations, the goal, and the observer — and sometimes the observer's "language" for representing the system — helps to formulate the problem and outline the main stages of a systems analysis methodology. In organizational systems, for example, if the person competent to make decisions is not identified, the goal for which the system is being created may not be achieved. But there are systems for which the observer is obvious. Sometimes there is no need to even explicitly use the concept of a goal.
For instance, the version of systems theory developed by Yu. A. Urmantsev for studying relatively low-level biological objects such as plants does not include the concept of a goal, as it is not characteristic of this class of objects; instead, the notion of purposefulness and development is expressed in the form of a special kind of relations — laws of composition.
Thus, when conducting systems analysis, one should first represent the situation using the most complete possible definition of a system, and then, having identified the most essential components influencing decision-making, formulate a "working" definition that can be refined, expanded, or narrowed depending on the course of the analysis.
The "working" definition of a system helps the researcher (developer) begin describing it. In order to correctly select the necessary elements, relations, their properties, and other constituents included in the adopted "working" definition of a system, the persons who formulate this initial, verbal representation of the system must use these concepts in the same sense.
The choice of a definition of a system reflects the adopted conceptual framework and is essentially the beginning of modeling. Therefore, from the very outset, it is advisable to present definitions in symbolic form, which facilitates a more unambiguous understanding by all participants in the development or investigation of the system.
Viewing the definition of a system as a means of beginning its investigation, and striving to preserve integrity when transforming or designing a system, led the author of this section to propose a definition in which the system is not decomposed into the most elementary particles (as is done in the definitions already cited), but is instead represented as an aggregate of high-level components that are fundamentally necessary for the existence and functioning of the system under investigation or being created:
$$S = \langle Z,\; \text{Str},\; \text{Tech},\; \text{Cond} \rangle$$
- $\{Z\}$ — set or structure of goals
- $\{\text{Str}\}$ — set of structures (production, organizational, etc.) that implement the goals
- $\{\text{Tech}\}$ — set of technologies (methods, tools, algorithms, etc.) that implement the system
- $\{\text{Cond}\}$ — conditions of the system's existence, i.e., factors affecting its creation, functioning, and development
This definition makes it possible not to destroy the system under investigation, but to preserve its main structures within it, developing them in accordance with the stated goals; when creating a new system, it helps to obtain a holistic design concept and to realize a goal-oriented approach to the system's creation.
Materiality and immateriality of a system
During the formative period of systems research in the 1960s and 1970s, debates about whether systems are material or immaterial arose quite frequently. Not everyone finds this problem clear even today.
On the one hand, seeking to emphasize the materiality of systems, some researchers replaced the term "element" in their definitions with the terms "thing," "object," or "item"; although the latter can also be interpreted as abstract objects or subjects of investigation, the authors of these definitions clearly intended to draw attention to the tangibility and materiality of the system.
On the other hand, in Yu. I. Chernyak's definition cited above, and especially in S. Optner's definition, a system can only be interpreted as a representation — that is, as something existing solely in the consciousness of the researcher or designer. Any specialist who understands the regularities of the theory of reflection would, it seems, object: surely the concept (the ideal representation of the system) will subsequently exist in material embodiment, and for decision-making problems it is important to emphasize that the concept of a system can serve as a means of investigating a problem and solving a task. Nevertheless, the aforementioned definitions were subject to criticism during that period from proponents of the materiality of systems, particularly philosophers.
V. G. Afanasyev demonstrated the futility of the debate about the materiality or immateriality of a system: "…objectively existing systems — and the concept of a system; the concept of a system used as a tool for cognizing the system — and once again the real system, knowledge of which has been enriched by our systems-based representations — such is the dialectics of the objective and the subjective in a system…".
In connection with the issue under discussion, let us note that in the Great Soviet Encyclopedia, alongside the definition cited above, the following is also given: a system is "an objective unity of entities, phenomena, and knowledge about nature and society that are linked to one another by laws" — that is, it is emphasized that the concept of an element (and consequently of a system) can be applied both to existing, materially realized objects and to knowledge about those objects or their future implementations.
Thus, in the concept of a system (as in any other category of cognition), the objective and the subjective constitute a dialectical unity, and one should speak not of the materiality or immateriality of a system, but of approaching the objects of investigation as systems, of representing them differently at different stages of cognition or creation.
Strata of system consideration
For example, Yu. I. Chernyak shows that the same object at different stages of its examination can be represented in various aspects, and accordingly proposes to depict the same system at different levels of existence: philosophical (epistemological), scientific-research, design, engineering, and so on, through to material embodiment.
In other words, at different stages of a system's consideration, different concepts can be embedded in the term "system"; one can speak, as it were, of the system's existence in different forms. M. Mesarovic proposes distinguishing strata of system consideration.
Analogous strata may exist not only during the creation but also during the cognition of an object — that is, in representing actually existing objects as systems abstractly conceived in consciousness (or in models), which can then help create new objects and develop recommendations for the transformation (restructuring, reconstruction) of existing ones.
A systems analysis methodology (or a model of systems research) need not necessarily encompass the entire process of cognition or design of a system; it may be developed for one of its strata (which, as a rule, is the case in practice). To prevent terminological or other disagreements among the researchers or developers of a system, one must first clearly specify which particular stratum of system consideration is being discussed.