Modeling and Symbolization - History, Philosophy and...

1.6. Modeling and Symbolization

In the previous paragraph, the technical models were considered directly as an integral part of the theory. They correspond in a certain way with mathematical, computer and logical models, about which mountains of books are written, but, unfortunately, this connection does not always get the proper expression. Quite often, technical models are replaced by their isomorphic formal images (interpretations), which is unacceptable. The technologist should clearly understand the uniqueness of each type of model used by him. To do this, first, it is necessary to find out the specifics of the technical models, and secondly, to understand the effectiveness of non-technical models. Let's begin the analysis with consideration of the nature of mathematical models.

A. D. Mouse su belongs, perhaps, a better definition of the mathematical model: "Let's move on to the general definition. Suppose we are going to investigate some set of 5 properties of a real object a using mathematics (here the term object is understood in the broadest sense: , process, etc.). To do this, we select (as they say, build) the "mathematical object" a ' - a system of equations, or arithmetic relations, or geometric figures, or a combination of the two, etc., - the study of which by means of mathematics should answer the questions posed about properties 5. Under these conditions a 'is called the mathematical model of the object a relative to the aggregate of its 5 properties. However, even such a definition raises questions. It is quite obvious that, given the explanation given in parentheses, it can be not only about the real, but also about the imaginary object. It is also very doubtful that the mathematical model must be directly compared with 5.

Many authors believe that the development of a mathematical model is preceded by a pre-model: "The construction of a model begins with a verbal-semantic description of an object or phenomenon. & lt; ... & gt; This stage can be called the formulation of the pre-model & quot ;. It is not difficult to understand that in the context of interest to us the premodel is nothing but a technical model. But we believe that the technical model is not a premodel, but a full-fledged conceptual image, and therefore it is not enough to say that it is verbally -sense description of an object or phenomenon, for such is the mathematical model. So, the subject of discord is quite obvious. We will present it in a technical context.

One part of the authors argues within the framework of the conceptual framework: (a) technical object - & gt; mathematical model. They eliminate technical science, while the importance of both technical and mathematics is distorted, and the mentioned elimination is a sign of panmathematicalism.

Another part of the authors uses three coordinates: (b) technical object - & gt; predmodel - & gt; mathematical model. Compared to the first version, an intermediate appears, but its characteristic is superficial. Nevertheless, the conceptual progress is obvious, and it is not difficult to understand the possibility of its growth.

In this case, the framework appears: (c) technical object - & gt; technical model - & gt; mathematical model. By its potential, the conceptual framework (c) outperforms its competitors - the frameworks (a) and (b). But, strictly speaking, it is not enough. The fact is that all three frames are built on the illusion that you can go to models directly from the objects being studied. We do not tire of repeating that scientific knowledge is realized as a conceptual transduction, one of the stages of which is modeling. The model correlates not only with the object under study, but also with principles, laws, facts, etc. This does not negate the fact of correlation in the context of the technical model - & gt; mathematical model & quot ;. What is the nature of this correlation?

The mathematical model is an isomorphic image of the technical model. This means that it is possible to establish a correspondence between the technical and mathematical concepts. We give the simplest example. Let us write three widely known linear functions:

Equation (1) is taken from mathematics. Equation (2) is a record of Hooke's law from the theory of the resistance of materials. Equation (3) expresses Ohm's electrical law. The correspondence between the components of these equations, for example between y, uy , is literally striking, and the piquancy of the situation is that they belong to different theories. The isomorphism of theories points to their known similarity, but in conceptual terms, theories are fundamentally different from each other. A mathematician is competent in mathematics, a physicist in physics, a technologist in technical science, etc.

It is worth noting that the question of the correlation between the informal sciences and mathematics has always caused some perplexity. With the light hand of the famous American physicist Eugene Wigner, even talk about incomprehensible the effectiveness of mathematics. Indeed, it seems incomprehensible why an appeal to mathematics that is not part of technical science is so fruitful. But the fact is that the division of scientific labor has made it possible to accumulate in the mathematics a huge amount of knowledge, which, by virtue of its isomorphism to other sciences, can effectively use in them. That's why mathematical modeling is necessary.

Mathematical modeling in is carried out in its interests. The term "mathematical model" is misleading, for it gives the impression of a transition to the field of mathematics. But in mathematical modeling, mathematics plays an auxiliary role. The mathematical model is nothing more than an improved technical model. Thus, instead of the expression mathematical modeling it is better to use the term "technical and mathematical modeling", avoiding the emphasis on the word "mathematics".

However, the mathematization of the technical model of its improvement does not end. In modern studies, the mathematical model, as a rule, is transformed into a computer model. The creation or choice of an algorithm and the writing of a program for calculating a mathematical model are of decisive importance in the matter of computer modeling. The computer model allows you to make the necessary calculations as quickly as possible. The speed of computers makes it possible to vary the parameters of a mathematical model, which is especially important in conducting mental experiments. Finally, computer models have one more advantage over mathematical models: they allow the most efficient visualization of the conceptual process.

Computer visualization is often associated only with the development of a visual image of the phenomenon, but such an object-oriented interpretation does not correspond to the true state of affairs. In fact, a visual image is created not only of directly studied phenomena, but of the whole process of conceptual transduction. Computer models allow you to see the invisible - concepts. It seems that the fantasy dream of many centuries has come true.

In connection with the consideration of mathematical and computer models, it is necessary to mention also logical modeling.

Logic, as we know, stood at the origins of the birth of formal sciences. It appeared even before mathematics. Formal logic was created by Aristotle in the IV. BC, and Euclidean geometry as the first mathematical discipline, constructed in accordance with the axiomatic method, appeared about a century later.

The organic connection between logic and mathematics became especially evident after the creation of mathematical logic by Gottlob Frege and Bertrand Russell. Both of them as logicians were trying to reduce mathematics to logic. The program of logicism turned out to be too pretentious, but in proving the correctness of mathematical theorems, very few people do without mathematical logic.

The organic connection of logic with informatics and, consequently, with the theory of computer technology was discovered very early. From the very beginning it was revealed that the bit can be considered as a logical judgment with the values ​​1 (true) and 0 (confusion) with which Buhl's logic deals. Bitwise operations are easily described in the language of Boolean logic. In the subsequent connection of logic with computer science did not weaken, but, on the contrary, it was strengthened. Logic was especially effective in proving the correctness of algorithms and programs, as well as descriptions of knowledge bases and logical procedures for inference and decision making. In the modeling, the potential of continuous, fuzzy and interval logic and related formal systems is often in demand. So, fuzzy logic has numerous applications in the field of modeling the functioning of technical systems in conditions of uncertainty. Many authors believe that logical modeling is by no means inferior in its productive capabilities to mathematical modeling.

So, in the simplest form, the technologic modeling can be represented as follows:

However, taking into account the above explanations, the presented line of model transduction makes sense to correct:

In all four cases, it is a question of technical models. Of course, it is advisable to distinguish between their variations, but always remember that the substantiation of the corresponding adjectives can lead to significant cognitive errors. For example, transforming the term "mathematical mathematical model" into the concept of a mathematical model is fraught with distortion or even loss of the technical component. But technical science never goes into mathematics, logic or computer science.

Above we have discussed in some detail the need for various types of modeling in technical science, but there are certain limitations in this matter. Both logic and mathematics in their modern form, and computer sciences are represented by a whole spectrum of different theories, and the pluralism of theories does not decrease, but, on the contrary, increases. Since all theories are interrelated, then without special evidence it is clear that technical science needs all concepts less or more, although their relevance for technical science is different. In any technical and logical science only selected logical, mathematical and computer theories (including programming languages) are used. This circumstance seems to contradict the syntactic nature of the formal sciences, in which, by definition, the qualitative features of those phenomena that are the object of pragmatic disciplines are not taken into account. But, as it turns out, syntax to syntax is different. Presumably, it is this phenomenon that leads to selectivity in the case of technical modeling when using the achievements of formal sciences.

This section does not consider the internal structure of technical science, but its relations with other branches of science, i.e. inter-scientific relations of technical science. If analyzes the relationship of technology with logic, mathematics and informatics, then they talk about modeling. However, technical science is also related to all other branches of science, for example, physics, chemistry, biology, geology, economics, politics . Historically, it so happened that these ties were not given a specific name, and yet they deserve it. Of course, if desired, we could generalize the phenomenon of internship modeling. In this case, we would have to introduce a number of its types: technical-physical, technical-chemical, technical-geological, technical-biological, technical-economic, technical-political-political, etc.

The discussed terminological difficulty can be eliminated by another way: introduce a special term that would be used to characterize the types of internusional links. The term "Symbolization", deserves special attention in this sense, since it quite successfully expresses the essence of inter-scientific relations, including those that are characteristic for technical science. Indeed, the essence of inter-scientific ties is that the concepts of the two sciences are put in mutual correspondence. If interna-tional communications are conducted for the development of technical science, then the concepts of other sciences are considered symbols of technical concepts.


In energy, as well as in physics, the concept of energy is widely used. It even gives the impression that the technical energy is no different from physical energy. But this impression is deceptive: the concept of physical energy is organically combined with physical concepts, whereas the concept of technical energy is in the conceptual framework of the technical theory. Thus, the two concepts of energy are related to different sciences and therefore do not coincide with each other. The subject of interest of the technologist is not physical energy, but useful energy, which can be effectively used by man. Useful energy is not a part of physics. But for all the difference in physical and technically useful energy, they obviously are not separated by an impenetrable moat. In the absence of physical energy, there is no technical energy, i.e. physical energy is a condition for the availability of technical energy. Demonstrating this circumstance, the technologist considers physical energy as a symbol, a representative of technical energy. The same is done with all other concepts, including laws and principles, non-technical sciences.

Any internship connection, including modeling, acts as a symbolization, and it is always carried out in a certain direction. In the case of technical-physical symbolization, the originals are the concepts of technical science, and the symbols of physics are the symbols.

In the case of physicotechnical symbolization, the originals are the concepts of physics. Thus, conceptual symbolization is the operation that allows us to express an inter-scientific relationship, establish coherence (coherence) of the sciences. In the absence of coherence of sciences, conceptual symbolization can not, in principle, take place. Therefore, conceptual symbolization is an interna-tional method.


1. Since technical science is woven into a wide network of inter-scientific relations, it has a transdisciplinary character. Accordingly, conceptual symbolization is transdisciplinary.

2. Any simulation relevant to technical science is done in its interests.

3. With the technical-mathematical, technical-logical and technical-computer simulation, the conceptual device is not of mathematics, logic and informatics, but of the technology itself.

4. An inter-scientific method is conceptual symbolization, including interna- tional modeling.

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