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- any representation of one phenomenon by another, e.g., ANALOGY OR METAPHOR.
- any formal (i.e. mathematical or logically formal) representation of a set of relationships.
- a physical or a pictorial or diagrammatic representation (including maps) of a set of relationships.
- computer models, which can allow the simulation of real world processes.
Models vary in the degree to which they are regarded as approximating reality (their degree ofisomorphism with the reality). Their functions also vary, and may be heuristic as well as explanatory, including:
- the proposal of new hypotheses for exploration by suggesting comparisons between unfamiliar phenomena and those better known or better explained (e.g. between cultural and biological evolution);
- the simplification of complex reality for analytical purposes by the provision of an unambiguous general concept (Weber's ideal type of BUREAUCRACY) or to highlight fundamental explanatory causal mechanisms in isolation from complicating factors (e.g. Marx's model of CAPITALISM AND CAPITALIST MODE OF PRODUCTION);
- comparisons between the ‘ideal’ model and the real world (as in both Marx's and Weber's models, or the THEORY OF GAMES), intended to increase awareness of real world processes. Ultimately no clear-cut distinction exists between the terms ‘model’ and ‘theory’, since both of these terms imply some simplification of reality, necessary in order to achieve generality.
a three-dimensional representation (made of plaster of paris, wood, plastic, cardboard, or another material) of an architectural complex or city that already exists or is to be built. Models vary in scale. A model reproduces either all of the original’s details or approximates the original. It can be used for testing and improving architectural compositions. Models are also important elements in museum (primarily architectural) expositions.
(1) A sample that serves as a standard for series or mass reproduction (such as a model of an automobile or a model of clothing); also the type or design of a product or structure.
(2) A product (made, for example, from wood, clay, wax, or plaster) from which a mold is taken for reproduction in another material (such as metal, plaster, or stone).
(3) A person who poses for an artist, and in general, objects being represented (“real life”).
(4) A device that reproduces and simulates (usually on a reduced, “toy” scale) the design and operation of some other device (the “real” device) for scientific purposes (see below), practical purposes (for example, production tests), or sports.
In the broad sense of the word, a model is an image—including an arbitrary or imaginary image, such as a picture, description, diagram, blueprint, graph, plan, or map—or prototype (sample) of some object or system of objects (the “original”) that is used under certain conditions as a “substitute” or “representative” for the original. For example, a globe is a model of the earth, and the screen of a planetarium is a model of various parts of the universe (or, more accurately, of the stellar sky). In the same way it may be said that a stuffed animal is a model of the animal, and a passport photograph (or a list of items and, in general, any list of data from a passport or personal record form) is a model of the passport holder, although, by contrast, a painter calls the person he draws the “model.” In mathematics and logic, the aggregate of objects whose properties and interrelations satisfy the given axioms in terms of which the objects are described is usually called the model of the system of axioms.
All the above examples fall naturally into two main groups: examples of the first group express the idea of the “imitation” (description) of something “real” (some reality or “real life” that is the original for the model); in the remaining examples, by contrast, the principle of “real embodiment,” the realization of some abstract concept, is manifest (and here the model itself acts as the original concept). In other words, a model may be a system either on a higher level of abstraction than its original (as in the first case) or on a lower level (in the second). In various mathematical and logical refinements of the concept of “model,” systems of abstract objects for which the question of relative “seniority” is usually meaningless act as the models and originals. (For a more detailed discussion of possible classifications of models, based in particular on the nature of the means of their construction, see.)
In the natural sciences, such as physics and chemistry, the first interpretation of the term is usually followed. Here a description of a system in the language of a scientific theory (for example, a chemical or mathematical formula, an equation or system of equations, a fragment of a theory, or even an entire theory) is called the model of the system. One speaks of “models of a language” in the same sense, although the second interpretation is currently used more widely. In this case some linguistic reality is called a model, in contrast to the description of the reality (a linguistic theory). However, both interpretations may coexist. For example, relay-switching circuits are used as “experimental” models of the two-valued formulas (Boolean functions), and the latter, in turn, are used as “theoretical” models of the former.
Such multiple use of the term becomes understandable if one takes into account the fact that in specific sciences models are related in various ways to the use of simulation—that is, to the elucidation (or reproduction) of the properties of an object, process, or phenomenon by means of another object, process, or phenomenon (its “model”). Typical examples are the “planetary” model of the atom and the concept of “electron gas,” which appeal to more graphic—or, more accurately, more customary—mechanical concepts. Therefore, the first requirement that naturally arises for a model is complete identity of the structure of the model and the original. This requirement is realized in the condition of the isomorphism of the model and the “modeled” object with respect to the properties of interest to the researcher: two systems of objects (in the case here in question, the model and the original), with certain sets of predicates (sets of properties and relations), are said to be isomorphic if a one-to-one correspondence has been established between them (that is, each element of either of them has a unique “counterpart” among the elements of the other system) such that corresponding objects have corresponding properties and exist within each system in corresponding interrelationships. However, fulfillment of this condition may prove difficult or unnecessary. Indeed, it is unreasonable to insist on it, since the use of isomorphic models alone does not simplify the task of research, which is the most important incentive for the use of models.
Thus, at the next level one reaches the concept of a model as a simplified image of an object—that is, the requirement of homomorphism of the model with the original. (Homomorphism, like isomorphism, “preserves” all properties and relations defined in the initial system, but in contrast to isomorphism this representation is, generally speaking, unambiguous in only one direction: the images of certain elements of the original in the model are “glued together,” just as the images of sections of an object that are close to each other merge into a single point on the retina of the eye or in a photograph.) However, even this interpretation of the term “model” is not definitive and indisputable: if we pursue the goal of simplifying the studied object by modeling it in certain relations, there is no reason to require that the model be simpler than the original in all regards—on the contrary, the use of any set of methods, no matter how complex, to construct the model is rational, as long as the methods simplify the solution of the problems posed in the specific instance. Therefore, the most general definition of the concept of a model can be reached by requiring only that, for models and originals, no matter how complex, the structure of certain “simplified versions” of each system be identical. In other words, we shall now call two systems of objects A and B the models of one another (or say that they simulate one another) if some homomorphic transform of A and homomorphic transform of B are isomorphic with respect to one another. According to this definition, the relation of “being a model” has the properties of reflexivity (that is, any system is its own intrinsic model), symmetry (any system is a model of each of its own models—that is, the original and the model can change roles), and transitivity (that is, the model of a model is a model of the original system). Thus, “modeling” (in the sense of the last definition of the concept of a model) is a relation of the equality (identity or equivalency) type that expresses the “identity” of the given systems with respect to those properties that are preserved in given homomorphisms and isomorphisms. Of course, the same also applies to the first definition of a model as an isomorphic image of an original, whereas the homomorphic relation, which underlies the second definition given above, is transitive and antisymmetric (the model and the original are not equivalent!), thus giving rise to a hierarchy of models (beginning with the original) in descending order of complexity.
The models used in modern research were first used in explicit form in mathematics to prove the consistency of the non-Euclidean geometry of Lobachevskii with respect to Euclidean geometry. The method of interpretation that was developed in these proofs later found particularly broad application in axiomatic set theory. Model theory, a special discipline within whose framework a model (or “algebraic system”) is understood to be an arbitrary set with defined groups of predicates and/or operations, regardless of whether such a model can be described by axiomatic means (to find such descriptions is one of the fundamental tasks of model theory), was formed at the boundary between algebra and mathematical logic. This concept of model was further detailed within the framework of logical semantics. As a result of algebraic (logical) and semantic refinements of the concept of the model, it was found advisable to introduce the concept independently of that of isomorphism (since axiomatic theories also allow nonisomorphic models).
In accordance with the various functions of modeling methods, the concept of “model” is used not so much for the purpose of obtaining explanations of various phenomena as for predicting phenomena that are of interest to the research. Both of these aspects of the use of models are especially productive if complete formalization of this concept is rejected. The “explanatory” function of models is manifested in their use in teaching, and the “predictive” function is manifested in their use in heuristics (such as in “probing” new ideas and drawing “conclusions by analogy”). For all the diversity of these aspects, they are united by the concept of the model primarily as an implement of knowledge—that is, as one of the most important philosophical categories. Significant expansion of the variety of models used is characteristic of the concept in all its diverse aspects at the current stage of scientific development. The inclusion of time characteristics among the parameters that describe changing (developing) systems (or the use of functions in the mathematical sense as the primary elements of the model) makes possible expansion of the concept of isomorphism to isofunctionalism and, using it, the representation (simulation) not only of “rigorously defined,” invariant systems but also of various processes (such as physical, chemical, production, economic, social, and biological processes). This opens up broad prospects for using as models programs for digital computers, whose languages may be considered as “universal modeling systems.” Of course, this also applies to normal (natural) languages, but with regard to linguistic models, claims of their unconditional isomorphism with the situations described are found to be untenable and unnecessary. Also, preliminary consideration of all parameters to be modeled, which is necessary for a literal interpretation of the term “model” as introduced by any precise definition, is frequently impossible (this accounts, incidentally, for the need for modeling). Because of this, the broad interpretation of the term “model,” based on intuitive concepts of “modeling,” once again proves to be particularly productive. This applies to all types of “probabilistic” models of teaching, “behavioral models” in psychology, and the models of self-organizing (self-adapting) systems typical of cybernetics. The requirement for unconditional formalization as a prerequisite for construction of a model would only restrict the potential of scientific research. The introduction of various dilutions into the formal definitions of the concept of a “model” is also a highly promising way of surmounting the difficulties that arise here. This results in the emergence of “approximate,” “fuzzy” concepts, such as “quasi model” and “near-model.” In this case the concept of a “model” is used in both senses mentioned above, and often simultaneously, for all its modifications at all levels of abstraction. For example, the “recording” of genetic information in chromosomes models the parent organisms and at the same time is modeled in the organism of the progeny.
REFERENCESKleene, S. C. Vvedenie v metamatematiku. Moscow, 1957. Section 15. (Translated from English.)
Ashby, W. R. Vvedenie v kibernetiku. Moscow, 1959. Chapter 6. (Translated from English.)
Lakhuti, D. G., I. I. Revzin, and V. K. Finn. “Ob odnom podkhode k semantike.” Filosofskie nauki, 1959, no. 1.
Modelirovanie v biologii [collection of articles]. Moscow, 1963. (Translated from English.)
Beer, S. Kibernetika i upravlenie proizvodstvom. Moscow, 1963. (Translated from English.)
Chao, Yiian-jen. “Modeli v lingvistike i modeli voobshche.” In the collection Matematicheskaia logika i ee primeneniia. Moscow, 1965. Pages 281–92. (Translated from English.)
Miller, G., E. Galanter, and K. Pribram. Plany i struktura povedeniia. Moscow, 1965. (Translated from English.)
Gastev, Iu. A. “O gnoseologicheskikh aspektakh modelirovaniia.” In the collection Logika i metodologiia nauki. Moscow, 1967. Pages 211–18.
Curry, H. B. Osnovaniia matematicheskoi logiki. Moscow, 1969. Chapters 2 and 7. (Translated from English.)
Chomsky, N. Iazyk i myshlenie. Moscow, 1972. (Translated from English.)
Carnap, R. The Logical Syntax of Language. London, 1937.
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IU. A. GASTEV
["A Manual for the MODEL Programming Language", J.B. Morris, Los Alamos 1976].
Note: British spelling: "modelling", US: "modeling".
model(1) A particular unit of hardware, known by its style or type.
(2) A graphical representation of an object.
(3) A mathematical representation of a device or process used for analysis and planning. See data model, data administration, financial planning system and scientific application.