Abstraction and formalization
Abstraction – This is a method of scientific research based on the fact that when studying a certain object, one is distracted from its non-essential aspects and features in a given situation. This allows us to simplify the picture of the phenomenon under study and consider it in its “pure” form. Abstraction is associated with the idea of the relative independence of phenomena and their aspects, which makes it possible to separate essential aspects from non-essential ones. In this case, as a rule, the original subject of research is replaced by another - equivalent, based on the conditions of the given problem. For example, when studying the operation of a mechanism, a calculation diagram is analyzed that displays the main, essential properties of the mechanism.
The following types of abstraction are distinguished:
– identification (formation of concepts by combining objects related by their properties into a special class). That is, on the basis of the sameness of a certain set of objects that are similar in some respect, an abstract object is constructed. For example, as a result of the generalization of the property of electronic, magnetic, electric machine, relay, hydraulic, pneumatic devices to amplify input signals, such a generalized abstraction (abstract object) as an amplifier arose. It is a representative of the properties of objects of different quality that are equal in a certain respect.
– isolation (isolation of properties inextricably linked with objects). Isolating abstraction is performed to isolate and clearly record the phenomenon under study. An example is the abstraction of the actual total force acting on the boundary of a moving fluid element. The number of these forces, like the number of properties of the liquid element, is infinite. However, from this variety it is possible to isolate the forces of pressure and friction by mentally identifying at the boundary of the flow an element of the surface through which the external medium acts on the flow with some force (in this case the researcher is not interested in the reasons for the occurrence of such a force). Mentally decomposing the force into two components, the pressure force can be defined as a normal component of the external influence, and the friction force as a tangential one.
– idealization corresponds to the goal of replacing a real situation with an idealized scheme to simplify the situation under study and more effectively use research methods and tools. The process of idealization is the mental construction of concepts about objects that are non-existent and impracticable, but have prototypes in the real world. For example, an ideal gas, an absolutely solid body, a material point, etc. As a result of idealization, real objects are deprived of some of their inherent properties and endowed with hypothetical properties.
A modern researcher often, from the very beginning, sets the task of simplifying the phenomenon being studied and constructing its abstract, idealized model. Idealization acts here as the starting point in the construction of theory. The criterion for the fruitfulness of idealization is the satisfactory agreement in many cases between the theoretical and empirical results of the study.
Formalization– a method of studying certain areas of knowledge in formalized systems using artificial languages. These are, for example, the formalized languages of chemistry, mathematics, and logic. Formalized languages allow you to briefly and clearly record knowledge and avoid the ambiguity of natural language terms. Formalization, which is based on abstraction and idealization, can be considered as a type of modeling (sign modeling).
The theoretical level of scientific research is a rational (logical) stage of knowledge. At the theoretical level, with the help of thinking, a transition occurs from a sensory-concrete idea of the object of study to a logical-concrete one. Logical-concrete is a concrete idea of an object in all the richness of its content, theoretically reproduced in the researcher’s thinking. At the theoretical level, the following methods of cognition are used: abstraction, idealization, thought experiment, induction, deduction, analysis, synthesis, analogy, modeling.
Abstraction- this is a mental distraction from some less significant properties, aspects, signs of the object or phenomenon being studied with the simultaneous selection and formation of one or more significant aspects, properties, characteristics. The result obtained during the abstraction process is called abstraction.
Idealization– this is a special type of abstraction, the mental introduction of certain changes to the object being studied in accordance with the goals of the research. Let us give examples of idealization.
Material point- a body devoid of any size. This is an abstract object, the dimensions of which are neglected, and is convenient when describing movement.
Pure black body- is endowed with the property, which does not exist in nature, of absorbing absolutely all radiant energy falling on it, without reflecting or transmitting anything through itself. The blackbody spectrum is an ideal case because it is not affected by the nature of the emitter's substance or the state of its surface.
Thought experiment is a method of theoretical knowledge that involves operating with an ideal object. This is a mental selection of positions and situations that make it possible to detect important features of the object under study. In this it is similar to a real experiment. In addition, it precedes the actual experiment in the form of a planning procedure.
Formalization- this is a method of theoretical knowledge, which consists in the use of special symbols, which allows one to distract from the study of real objects, from the content of the theoretical provisions describing them, and instead operate with a certain set of symbols and signs.
To build any formal system you need:
1. specifying the alphabet, i.e., a specific set of characters;
2. setting the rules by which “words” and “formulas” can be obtained from the initial characters of this alphabet;
3. setting rules according to which one can move from some words and formulas of a given system to other words and formulas.
As a result, a formal sign system is created in the form of a certain artificial language. An important advantage of this system is the possibility of carrying out within its framework the study of any object in a purely formal way (operating with signs) without directly addressing this object.
Another advantage of formalization is to ensure the brevity and clarity of recording scientific information, which opens up great opportunities for operating with it.
Induction– (from Latin induction - guidance, motivation) is a method of cognition based on formal logical inference, which leads to a general conclusion based on particular premises. In other words, this is the movement of our thinking from the particular, individual to the general. By discovering similar signs and properties in many objects of a certain class, the researcher concludes that these signs and properties are inherent in all objects of a given class.
The popularizer of the classical inductive method of cognition was Francis Bacon. But he interpreted induction too broadly; he considered it the most important method of discovering new truths in science, the main means of scientific knowledge of nature. In fact, the above methods of scientific induction serve mainly to find empirical relationships between experimentally observed properties of objects and phenomena. They systematize the simplest formal logical techniques that were spontaneously used by natural scientists in any empirical research.
Deduction- (from Latin deduction - deduction) is the receipt of particular conclusions based on knowledge of some general provisions. In other words, this is the movement of our thinking from the general to the particular.
However, despite attempts in the history of science and philosophy to separate induction from deduction and contrast them, in the real process of scientific knowledge both of these two methods are used at the appropriate stage of the cognitive process. Moreover, in the process of using the inductive method, deduction is often present “in a hidden form.” By generalizing facts in accordance with some ideas, we indirectly derive the generalizations we receive from these ideas, and we are not always aware of this. It seems that our thought moves directly from facts to generalizations, that is, that there is pure induction here. In fact, in accordance with some ideas, implicitly guided by them in the process of generalizing facts, our thought indirectly goes from ideas to these generalizations, and, therefore, deduction also takes place here... We can say that in all cases, when we generalize in accordance with any philosophical principles, our conclusions are not only induction, but also hidden deduction.
Analysis and synthesis. Under analysis understand the division of an object into its component particles for the purpose of studying them separately. Such parts may be some material elements of the object or its properties, characteristics, relationships, etc. Analysis is a necessary and important stage in the knowledge of the object. But it constitutes only the first stage of the process of cognition. To comprehend an object as a whole, one cannot limit oneself to studying only its component parts. In the process of cognition, it is necessary to reveal objectively existing connections between them, to consider them together, in unity. To carry out this second stage in the process of cognition - to move from the study of individual components of an object to the study of it as a single connected whole - is possible only if the method of analysis is complemented by another method - synthesis. In progress synthesis the component parts of the object under study, dissected as a result of the analysis, are brought together. On this basis, further study of the object takes place, but as a single whole. At the same time, synthesis does not mean a simple mechanical connection of disconnected elements into a single system. It reveals the place and role of each element in the system of the whole, establishes their interrelation and interdependence.
Analysis and synthesis are also successfully used in the sphere of human mental activity, that is, in theoretical knowledge. But here, as at the empirical level of cognition, analysis and synthesis are not two operations separated from each other. In essence, they are two sides of a single analytical-synthetic method of cognition.
Analogy and modeling. Under analogy refers to the similarity, similarity of some properties, characteristics or relationships of generally different objects. Establishing similarities (or differences) between objects is carried out as a result of comparison. Thus, comparison is the basis of the analogy method.
The analogy method is used in a wide variety of fields of science: mathematics, physics, chemistry, cybernetics, humanities, etc. There are various types of conclusions by analogy. But what they have in common is that in all cases one object is directly examined, and a conclusion is drawn about another object. Therefore, inference by analogy in the most general sense can be defined as the transfer of information from one object to another. In this case, the first object, which is actually subject to research, is called a model, and the other object, to which the information obtained as a result of studying the first object (model) is transferred, is called the original (sometimes a prototype, sample, etc.). Thus, the model always acts as an analogy, that is, the model and the object (original) displayed with its help are in a certain similarity (similarity).
Limits of the scientific method.
The limitations of the scientific method are mainly associated with the presence of a subjective element in cognition and are due to the following reasons.
Human experience, which is the source and means of understanding the world around us, is limited. A person’s senses allow him only limited orientation in the world around him. Man's ability to experience the world around him is limited. Human mental capabilities are great, but also limited.
The dominant paradigm, religion, philosophy, social conditions and other cultural elements inevitably influence the worldview of scientists, and therefore the scientific result.
The Christian worldview proceeds from the fact that the fullness of knowledge is revealed by the Creator and man is given the opportunity to possess it, but the damaged state of human nature limits his ability to know. Nevertheless, a person is capable of knowing God, that is, he can know himself and the world around him, see the manifestation of the Creator’s traits in himself and in the world around him. We should not forget that the scientific method is only a tool of knowledge and, depending on whose hands it is in, it can bring benefit or harm.
Methods of theoretical knowledge are abstraction, analysis and synthesis, induction and deduction, idealization, analogy, formalization, modeling, hypothesis methods and axiomatic, systemic method and approach, etc.
Abstraction . The essence of abstraction consists in mental abstraction from non-essential properties, relationships and connections in an object and between them while simultaneously fixing individual sides, aspects of these objects in accordance with the goals of cognition and the tasks of research, design and transformation. The result of the abstraction process will be abstractions - concepts of natural language and concepts of science.
The abstraction method involves two points. First, the essential is separated from the unimportant, important from unimportant in a cognitive task. Then, various aspects of the object, operating factors, conditions are assessed, the presence of common features is established, membership in certain classes of phenomena, objects, etc. A necessary side of abstraction is the establishment of independence or negligible dependence on certain factors. Next, some object of an ideal or material nature being studied is replaced by another, less rich in properties, having a limited number of parameters and characteristics. The resulting object acts as models first.
It should be noted that the abstraction operation can be applied to both real and abstract objects, which themselves were already the result of a previous abstraction. At the same time, we seem to be moving away from the concreteness and richness of the properties of the original object, impoverishing it, but otherwise we would not be able to cover wide classes of objects and their general essence, interconnection, form, structure, etc. The role of the resulting abstraction is that , that it allows in knowledge to call objects that previously seemed different with one name, to replace complex things with simple ones, to classify diversity according to general characteristics, i.e., to ultimately arrive at a generalization, and therefore at a law.
Analysis - this is the mental division of an object or its aspects of interest to us into separate parts for the purpose of their systematic study. Their role can be played by individual material or ideal elements, properties, relationships, etc.
Synthesis – mental combination of previously studied elements into a single whole.
From the above definitions it is already clear that these are mutually presupposing and complementary methods. Depending on the degree of research, the depth of penetration into the essence of the object or its aspects, analysis and synthesis of various kinds or types are used: direct, or empirical, analysis and synthesis, which are suitable at the stage of the first, still superficial acquaintance with the object of research and its aspects, especially when studying a complex object; recurrent, or elementary theoretical, analysis and synthesis, which are suitable for comprehending moments, sides, aspects of the essence, mastering certain cause-and-effect dependencies; structural genetic analysis and synthesis, which make it possible to identify the most important, central, decisive thing in the object of study, leading to the development of the object into a whole; they cover genetic connections and mediations; their entire chains lead to the completeness of coverage of parts and their content or to a systemic vision and description of the object.
Induction and deduction – the next two methods are, like the previous ones, paired and complementary. They occupy a special position in the system of scientific methods and include the application of purely formal logical rules of inference and inference - deductive and inductive. Let's start by explaining the meaning of induction.
Induction is understood as inference from the particular to the general, when, based on knowledge about some objects, a conclusion is made about the properties of the entire class as a whole. In this case, the following types of induction can be distinguished. Full induction, when a conclusion is made about the properties of a given object based on enumerating all objects of a given class. This is completely reliable knowledge. Every science strives to obtain it and uses it as evidence of the reliability of its conclusions, their irrefutability.
Incomplete induction when a general conclusion is drawn from premises that do not cover all objects or aspects of a given class. Thus, there is a moment of hypothesis in it. Its evidence is weaker than the previous one, because there are no rules without exceptions.
Historically, the first was the so-called enumerative (or popular) induction. It is used when some kind of regularity or repeatability is noticed in experience, about which a judgment is formulated. If there are no refuting examples, then a general conclusion is drawn in the form of an inference. This type of induction is considered complete. Complete induction is otherwise called scientific, since it gives not only a formal result, but also a proof of the non-randomness of the found regularity. Such induction also makes it possible to capture cause-and-effect relationships. An example of complete induction: successively tested metals - one, another, a third, etc. - have electrical conductivity, from which it follows that all metals are electrically conductive, etc. An example of incomplete induction: each even number is divided by two, and although they There is an infinitely large set of all of them, we still conclude that all even numbers are a multiple of two, etc.
Deductive inference is an inference in which a conclusion about the properties of an object and about itself is made on the basis of knowledge of the general properties and characteristics of the entire set. The role of deduction in modern scientific knowledge and knowledge has increased dramatically. This is due to the fact that modern science and engineering practice are faced with objects that are inaccessible to ordinary sensory perception (the microworld, the Universe, the past of humanity, its future, very complex systems of various kinds, etc.), so increasingly we have to turn to thoughts, rather than observation and experimentation. Deduction is of particular importance for the formalization and axiomatization of knowledge, the construction of hypotheses in mathematics, theoretical physics, management theory and decision making, economics, computer science, ecology, etc. Classical mathematics is a typically deductive science. Deduction differs from other methods in that if the initial knowledge is true, it gives true inferential knowledge. However, the power of deduction cannot be overestimated. Before applying it, it is necessary to obtain true initial knowledge, general premises, and therefore special significance remains for the methods of obtaining such knowledge, which were discussed above.
Idealization . For the purposes of scientific knowledge, construction, design and transformation, so-called “ideal objects” are widely used. They do not exist in reality and are fundamentally not implemented in practice, but without them theoretical knowledge and its applications are impossible. These include a point, a line, a number, an absolutely rigid body, a point electric charge, a charge in general, an ideal gas, an absolutely black body and many others. Science cannot be imagined without them. The mental construction of such objects is called idealization.
For idealization to proceed successfully, the subject's abstracting activity is necessary, as well as other mental operations: induction, synthesis, etc. At the same time, we set ourselves the following tasks: mentally deprive real objects of certain properties; We mentally endow these objects with certain unreal ultimate properties; we name the resulting object. To accomplish these tasks, multi-stage abstraction is used. For example, abstracting from the thickness of a real object, a plane is obtained; depriving the plane of one dimension, they get a line; depriving a line of its only dimension, they get a point, etc. But how to move to the limiting property? Let us, for example, arrange the bodies known to us in a row in accordance with the increase in their hardness. Then, in the limit, we get an absolutely rigid body. The examples can easily be continued. An ideal object such as incompressibility is constructed theoretically when the compressibility property is taken to be zero. We get an absolutely black body if we attribute to it the complete absorption of incoming energy.
Note that abstraction from any of the properties is necessarily the attribution of the opposite property to it, and the previous one is discarded, otherwise we will not obtain an ideal object.
Analogy . This is one of the methods of cognition when, from the similarity of some features and aspects of two or more objects, a conclusion is drawn about the similarity of other features and properties of these objects.
Let's build an analogy. It is known that the Sun is an ordinary star in our Galaxy, which contains about 100 billion such stars. These luminaries have a lot in common: huge masses, high temperatures, a certain luminosity, radiation spectrum, etc. They have satellites - planets. By analogy with our Solar system, scientists conclude that besides ours, there are also inhabited worlds in the galaxy, that we are not alone in the Universe. An analogy does not provide absolute certainty for a conclusion: it always has an element of conjecture, assumption, and only experience and practice can make a final verdict on this or that analogy.
Formalization . The term itself is ambiguous and is used in different meanings. The first is as a method for solving special problems in mathematics and logic. For example, proof of the consistency of mathematical theories, the independence of axioms, etc. Questions of this kind are solved by using special symbols, which makes it possible to operate not with the statements of the theory in their meaningful form, but with a set of symbols and formulas of various kinds. Secondly, in a broad sense, formalization is understood as a method of studying various problems by displaying their content, structure, relationships and functions using various artificial languages: mathematics, formal logic and other sciences.
What is the role of formalization in science? First of all, formalization ensures a complete overview of certain problems and a generalized approach to them. Further, thanks to symbolism, with which formalization is inevitably associated, polysemy (polysemy) and vagueness of terms in ordinary language are eliminated, as a result of which reasoning becomes clear and strict, and conclusions are evidential. And finally, formalization ensures the simplification of the objects being studied, replacing their study with the study of models: a kind of modeling arises based on symbolism and formalisms. This helps to more successfully solve various cognitive, design, engineering and other tasks. From the above it is clear that formalization is associated with modeling; it is also associated with abstraction, idealization and other methods.
Modeling . Modeling, as a powerful and effective method, is used empirically in the form of mock-ups and at the theoretical level in the form of symbolic constructions. There is a distinction between analogue modeling, when the original and the model are described by the same mathematical equations, formulas, diagrams, etc. Sign modeling is more complicated. Here, the role of models - substitutes for real objects - are numbers, diagrams, symbols, etc. Actually, a significant part of the technical project is expressed in exactly this way. But this type of modeling is further developed thanks to mathematics and logic in the form of logical-mathematical modeling. Here operations, actions with things, processes, phenomena, properties and relationships are replaced by sign constructions, the structure of their relationships, and the expression on this basis of the dynamics of objects and their functions.
Another step forward was the development of model representation of information on computers: computer modeling. The models constructed in this case are based on a discrete representation of information about objects. The opportunity opens up to simulate in real time and build virtual reality.
Axiomatic method – it is a method of organizing existing knowledge into a deductive system. It is widely used in mathematics and mathematized disciplines. When using this method, a number of simple ideas, previously proven or obvious, are introduced into the foundations of the theory in the form of initial provisions. In mathematics they are called axioms, in theoretical physics and chemistry - “beginnings” or principles. All other knowledge - all theorems, all laws and their consequences - are derived from them according to certain logical rules, i.e. deductively.
The establishment of the axiomatic method in science is associated with the appearance of the famous “Principles” of Euclid. The main requirements for this method are as follows: consistency of the axioms, that is, in the system of axioms or principles there should not be simultaneously a certain statement and its negation; completeness, that is, there should be no axioms without consequences, and their number should give us all the consequences or their negations; independence, when any axiom should not be deduced from others. There is nothing to add to this system.
The advantages of the axiomatic method are that axiomatization requires a precise definition of the concepts used and rigor of reasoning. It organizes knowledge, excludes unnecessary elements from it, eliminates ambiguity and contradictions, and allows us to take a fresh look at previously achieved knowledge within the framework of a certain theoretical system. True, the application of this method is limited, and within the framework of mathematics it also has certain boundaries. In clarifying this issue, an outstanding role was played by the theorem proven by Kurt Gödel about the fundamental incompleteness of developed formal knowledge systems. Its essence is that within the framework of this system it is possible to formulate statements that can neither be proven nor disproved without leaving this axiomatized system into a metatheory. For all mathematics, arithmetic plays this role. Gödel's result led to the collapse of mathematicians' illusion about the universal axiomatization of mathematics.
Logic and philosophy
The second group is methods for constructing and justifying theoretical knowledge which is given in the form of a hypothesis, which as a result acquires the status of a theory. Modern hypothetico-deductive theory is based on a certain empirical basis - a set of facts that need explanation and make the creation of a theory necessary. It is the idealized object that makes the creation of a theory possible. Scientific theories are primarily distinguished by the idealized objects they are based on.
QUESTION No. 25
Formalization, idealization and the role of modeling
According to Radugin (p. 123)
Methods for constructing and studying an idealized object
The discovery of stable connections and dependencies is only the first stage in the process of scientific knowledge of the phenomena of reality. It is necessary to explain their foundations and causes, to identify the essence of phenomena and processes. And this is possible only at the theoretical level of scientific knowledge. The theoretical level includes all those forms of knowledge in which laws and other universal and necessary connections of the objective world are formulated in a logical form, as well as conclusions obtained using logical means and consequences arising from theoretical premises. The theoretical level represents various forms, techniques and stages of indirect cognition of reality.
Methods and forms of cognition at the theoretical level, depending on the functions they perform, can be divided into two groups. The first group includes methods and forms of cognition, with the help of which an idealized object is created and studied, representing the basic, defining relationships and properties, as if in a “pure” form. The second group is methods of constructing and justifying theoretical knowledge, which is given in the form of a hypothesis, which as a result acquires the status of a theory.
Methods for constructing and studying an idealized object include: abstraction, idealization, formalization, thought experiment, mathematical modeling.
a) Abstraction and idealization. The concept of an idealized object
It is known that every scientific theory studies either a certain fragment of reality, a certain subject area, or a certain side, one of the aspects of real things and processes. At the same time, theory is forced to abstract itself from those aspects of the subjects it studies that do not interest it. In addition, theory is often forced to abstract from some differences in the objects it studies in certain respects. From a psychological point of viewthe process of mental abstraction from certain aspects, properties of the objects being studied, from certain relationships between them is called abstraction.Mentally identified properties and relationships appear in the foreground, appear as necessary for solving problems, and act as a subject of study.
The process of abstraction in scientific knowledge is not arbitrary. He obeys certain rules. One of these rules is complianceinterval of abstractions.The interval of abstractions is the limits of rational validity of a particular abstraction, the conditions of its “objective truth” and the limits of applicability, established on the basis of information obtained by empirical or logical means. The abstraction interval depends, firstly, onthe assigned cognitive task;secondly, what one is distracted from in the process of comprehending an object must be to outsiders (according to clearly defined criteria) for a specific object being abstracted; thirdly, the researcher must know to what extent a given abstraction has legal force.
The abstraction method involves performing a conceptual development and conceptual assembly of objects when studying complex objects.Conceptual developmentmeans displaying the same initial object of study in different mental planes (projections) and, accordingly, finding many intervals of abstractions for it. So, for example, in quantum mechanics, one and the same object (an elementary particle) can be alternately represented within two projections: one as a corpuscle (in some experimental conditions), then as a wave (in other conditions). These projections are logically incompatible with each other, but only taken together they exhaust all the necessary information about the behavior of particles.
Conceptual buildrepresentation of an object in a multidimensional cognitive space by establishing logical connections and transitions between different intervals that form a single semantic configuration. Thus, in classical mechanics, the same physical event can be reflected by an observer in different systems in the form of a corresponding set of experimental truths. These different projections, however, can form a conceptual whole thanks to "Galileo's rules of transformation" governing the ways of moving from one group of statements to another.
Abstraction as the most important technique of human cognitive activity is widely used at all stages of scientific and cognitive activity, including at the level of empirical knowledge. On its basis, empirical objects are created. As V.S. Stepin noted, empirical objects are abstractions that capture the characteristics of real objects of experience. They are certain schematizations of fragments of the real world. Any feature, the “carrier” of which is an empirical object, can be found in the corresponding real objects (but not vice versa, since an empirical object represents not all, but only some signs of real objects, abstracted from reality in accordance with the tasks of cognition and practice) . Empirical objects constitute the meaning of such terms in empirical language as “Earth,” “current-carrying wire,” “distance between the Earth and the Moon,” etc.
Theoretical objects, unlike empirical ones, are not just abstractions, but idealizations, “logical reconstructions of reality.” They can be endowed not only with features that correspond to the properties and relationships of real objects, but also with features that no such object possesses. Theoretical objects form the meaning of such terms as “point”, “ideal gas”, “absolute black body”, etc.
In logical and methodological research, theoretical objects are sometimes called theoretical constructs, as well as abstract objects. Objects of this kind serve as the most important means of understanding real objects and the relationships between them.They are called idealized objects, and the process of their creation is called idealization. Thus, idealization is the process of creating mental objects, conditions, situations that do not exist in reality through mental abstraction from some properties of real objects and relationships between them or endowing objects and situations with those properties that they do not actually possess or cannot possess, with the goal of a deeper and more accurate knowledge of reality.
The creation of an idealized object necessarily includes abstraction abstraction from a number of aspects and properties of the specific objects being studied. But if we limit ourselves to only this, then we will not yet receive any integral object, but will simply destroy a real object or situation. After abstraction, we still need to highlight the properties that interest us, strengthen or weaken them, combine and present them as properties of some independent object that exists, functions and develops according to its own laws. And this is achieved as a result of usingidealization method.
Idealization helps the researcher to isolate in its pure form the aspects of reality that interest him. As a result of idealization, an object acquires properties that are not required in empirical experience. Unlike ordinary abstraction, idealization places emphasis not on the operations of abstraction, but on the mechanism replenishment . Idealization gives an absolutely accurate construct,mental construct, in which this or that property, state is presented in extreme, most expressed form . Creative constructs, abstract objects act asideal model.
Why is it necessary to use abstract objects (theoretical constructs) in cognition? The fact is that a real object is always complex; significant and secondary properties for a given researcher are intertwined in it; necessary regular relationships are obscured by random ones. Constructs, ideal models, are objects endowed with a small number of specific and essential properties and having a relatively simple structure.
Researcher , based on a relatively simple idealized object, give a deeper and more complete description of these aspects. Cognition moves from concrete objects to theirabstract, ideal models, which, becoming more and more accurate, perfect and numerous, gradually give us an increasingly adequate image of specific objects. This widespread use of idealized objects is one of the most characteristic features of human cognition.
It should be noted that idealization is used both at the empirical and theoretical levels. The objects to which scientific statements refer are always idealized objects. Even in cases where we use empirical methods of cognition - observation, measurement, experiment, the results of these procedures directly relate to idealized objects, and only due to the fact that idealized objects at this level are abstract models of real things, the data of empirical procedures can be attributed to actual items.
However, the role of idealization increases sharply in the transition from the empirical to the theoretical level of scientific knowledge. Modern hypothetico-deductive theory is based on a certain empirical basis - a set of facts that need explanation and make the creation of a theory necessary. But a theory is not a simple generalization of facts and cannot be logically deduced from them. In order to make it possible to create a special system of concepts and statements, called a theory, we first introducean idealized object, which is an abstract model of reality, endowed with a small amountproperties and having a relatively simple structure. This idealized object expresses the specificity and essential features of the field of phenomena being studied. It is the idealized object that makes the creation of a theory possible. Scientific theories are, first of all, distinguished by the idealized objects they are based on. In the special theory of relativity, the idealized object is an abstract pseudo-Euclidean four-dimensional set of coordinates and instants of time, provided that there is no gravitational field. Quantum mechanics is characterized by an idealized object, represented in the case of a collection of n particles by a wave in n-dimensional configuration space, the properties of which are associated with the quantum of action.
The concepts and statements of a theory are introduced and formulated precisely as characteristics of its idealized object. The basic properties of an idealized object are described by a system of fundamental equations of the theory. The difference in idealized objects of theories leads to the fact that each hypothetico-deductive theory has its own specific system of fundamental equations. In classical mechanics we deal with Newton's equations, in electrodynamics with Maxwell's equations, in the theory of relativity with Einstein's equations, etc. The idealized object provides an interpretation of the concepts and equations of the theory. Clarification of the theory equations, their experimental confirmation and correction lead to clarification of the idealized object or even to its change. Replacing the idealized object of a theory means reinterpreting the basic equations of the theory. No scientific theory can be guaranteed that its equations will not sooner or later be subject to reinterpretation. In some cases this happens relatively quickly, in others after a long time. So, for example, in the doctrine of heat, the original idealized object - caloric - was replaced by another - a collection of randomly moving material points. Sometimes modification or replacement of the idealized object of a theory does not significantly change the form of its fundamental equations. In this case, it is often said that the theory remains the same, but its interpretation changes. It is clear that this can be said only with a formalistic understanding of scientific theory. If by theory we mean not only certain mathematical formulas, but also a certain interpretation of these formulas, then a change in the idealized object should be considered as a transition to a new theory.
b) ways to construct an idealized object A
What are the ways of forming an idealized object. In the methodology of scientific research, there are at least three of them:
1. It is possible to abstract from some properties of real objects, while at the same time retaining their other properties and introducing an object that has only these remaining properties. So, for example, in Newtonian celestial mechanics we abstract from all the properties of the Sun and planets and imagine them as moving material points that have only gravitational mass. We are not interested in their size, structure, chemical composition, etc. The sun and planets act here only as carriers of certain gravitational masses, i.e. in the form of idealized objects.
2. Sometimes it turns out to be useful to abstract from some relationships of the objects being studied to each other. With the help of such abstraction, for example, the concept of an ideal gas is formed. In real gases there is always a certain interaction between molecules. Abstracting from this interaction and considering gas particles as having only kinetic energy and interacting only upon collision, we obtain an idealized object - an ideal gas. In the social sciences, when studying individual aspects of social life, individual social phenomena and institutions, social groups, etc. we can abstract from the relationships of these parties, phenomena, groups with other elements of social life.
3. We can also attribute properties to real objects that they do not have, or we can think of the properties inherent in them in some limiting value. Thus, for example, in optics special idealized objects are formed - an absolutely black body and an ideal mirror. It is known that all bodies, to a greater or lesser extent, have both the property of reflecting some part of the energy incident on its surface and the property of absorbing part of this energy. When we enhance the reflection property to its maximum value, we get an ideal mirror - an idealized object whose surface reflects all the energy incident on it. By enhancing the absorption property, in the limiting case we obtain an absolutely black body - an idealized object that absorbs all the energy incident on it.
An idealized object can be any real object that is conceived in non-existent, ideal conditions. This is how the concept of inertia arises. Let's say that we are pushing a cart along the road. The cart moves for some time after the push and then stops. There are many ways to lengthen the path covered by a cart after a push, for example, lubricating the wheels, creating a smoother road, etc. The easier the wheels turn and the smoother the road, the longer the cart will move. Through experiments, it is established that the smaller the external influences on a moving body (in this case, friction), the longer the path traversed by this body. It is clear that it is impossible to eliminate all external influences on the moving body. In real situations, a moving body will inevitably be subject to some kind of influence from other bodies. However, it is not difficult to imagine a situation in which all influences are excluded. We can conclude that under such ideal conditions a moving body will move indefinitely and at the same time uniformly and rectilinearly.
c) Formalization and mathematical modeling
The most important means of constructing and researching an idealized theoretical object is formalization Formalization in the broad sense of the word is understood as a method of studying a wide variety of objects by displaying their content and structure in a symbolic form, using a wide variety of artificial languages.
Operations with formalized objects mean operations with symbols. As a result of formalization, symbols can be treated as concrete physical objects. The use of symbolism ensures a complete overview of a certain area of problems, brevity and clarity of knowledge recording, and avoids the ambiguity of terms.
The cognitive value of formalization lies in the fact that it is a means of systematizing and clarifying the logical structure of a theory. Reconstruction of a scientific theory in a formalized language makes it possible to trace the logical relationship between various provisions of the theory, to identify the entire set of prerequisites and foundations on the basis of which it is developed, which makes it possible to clarify ambiguities and uncertainties, and to prevent paradoxical situations. The formalization of a theory also performs a kind of unifying and generalizing function, allowing a number of theoretical provisions to be extrapolated to entire classes of scientific theories and to use a formal apparatus for the synthesis of previously unrelated theories. One of the most valuable advantages of formalization is its heuristic capabilities, in particular the ability to detect and prove previously unknown properties of the objects being studied.
There are two types of formalized theories: fully formalized and partially formalizedtheories. Fully formalized theories are constructed in an axiomatically deductive form with an explicit indication of the formalization language and the use of clear logical means. In partially formalized theories, the language and logical means used to develop a given scientific discipline are not explicitly fixed. At the present stage of development of science, partially formalized theories predominate in it.
The formalization method contains great heuristic possibilities. In the process of formalization, through the reconstruction of the language of scientific theory, a new type of conceptual construction is created, which opens up opportunities for obtaining new, sometimes the most unexpected consequences, through purely formalized actions. The formalization process is creative. Starting from a certain level of generalization of scientific facts, formalization transforms them, reveals in them such features that were not recorded at the content-intuitive level. Y.L. Ershov, in his works devoted to the use of formalized languages, provides a number of criteria confirming that with the help of formalizing a theory, non-trivial consequences can be obtained, which were not even suspected until they were limited to the content-intuitive formulation of the theory in natural language. Thus, the formulation of the axiom of choice was initially not in doubt. And only its use (in conjunction with other axioms) in a formal system that claims to axiomatize and formalize set theory revealed that it leads to a number of paradoxical consequences, which cast doubt on the possibilities of its use. In physics, when trying to axiomatize field theory, isolating certain statements about the quality of its axioms led to obtaining a large number of consequences suitable for explaining experimental data.
The creation of formalized descriptions has not only cognitive value itself, but is a condition for use at the theoretical levelmathematical modeling. Mathematical modeling is a theoretical method for studying quantitative patterns based on the creation of a sign system consisting of a set of abstract objects (mathematical quantities, relationships) thatallow for different interpretations. Mathematical modeling as a theoretical method found its wide application in the late 40s of the twentieth century. in individual sciences and in interdisciplinary research. The basis of the mathematical modeling method is the constructionmathematical model. A mathematical model is a formal structure consisting of a set of mathematical objects. The significance of the mathematical method in developing a theory is determined by the fact that, while displaying certain quantitative properties and relationships of the original, it replaces it in a certain way, and manipulation with this model provides deeper and more complete information about the original.
In the simplest case, the model is a separatemathematical object, that is, such a formal structure with the help of which it is possible to move from empirically obtained values of some parameters of the material object under study to the values of others without resorting to experiment. For example, by measuring the circumference of a spherical object, use the formula to calculate the volume of this object.
Researchers have established that in order for an object to be successfully studied using mathematical models, it must have a number of special properties. First, the relationships within it must be well known; secondly, the properties essential for the object must be quantified (and their number should not be too large); and, thirdly, depending on the purpose of the study, the forms of behavior of the object (which is determined by laws, for example, physical, biological, social) must be known for a given set of relations.
Essentially, any mathematical structure (or abstract system) acquires the status of a model only when it is possible to establish the fact of an analogy of a structural, substrate or functional nature between it and the object (or system) under study. In other words, there must be a certain consistency obtained as a result of the selection and “mutual adjustment” of the model and the corresponding “fragment of reality.” This consistency exists only within a certain interval of abstraction. In most cases, the analogy between an abstract and a real system is associated with the relation of isomorphism between them, defined within the framework of fixing the interval of abstraction. In order to study a real system, the researcher replaces it (up to isomorphism) with an abstract system with the same relations. Thus, the research problem becomes purely mathematical. For example, a drawing can serve as a model for displaying the geometric properties of a bridge, and a set of formulas that form the basis for calculating the size of the bridge, its strength, the stresses arising in it, etc. can serve as a model for displaying the physical properties of the bridge.
The use of mathematical models is an effective way of learning. Just the translation of any qualitative problem into a clear, unambiguous and rich in its capabilities language of mathematics allows us to see the research problem in a new light and clarify its content. However, mathematics also reveals something more. Characteristic of mathematical knowledge is the use of the deductive method, i.e. manipulating objects according to certain rules and thus obtaining new results.
According to Tarasov (pp. 91-94)
Idealization, abstraction- replacement of individual properties of an object or an entire object with a symbol or sign, mental distraction from something in order to highlight something else. Ideal objects in science reflect stable connections and properties of objects: mass, speed, force, etc. But ideal objects may not have real prototypes in the objective world, i.e. As scientific knowledge develops, some abstractions can be formed from others without recourse to practice. Therefore, a distinction is made between empirical and ideal theoretical objects.
Idealization is a necessary precondition for constructing a theory, since the system of idealized, abstract images determines the specifics of a given theory. The theory system distinguishes between basic and derived idealized concepts. For example, in classical mechanics the main idealized object is a mechanical system as the interaction of material points.
In general, idealization allows you to accurately outline the characteristics of an object and abstract from unimportant and vague properties. This provides a huge capacity for expressing thoughts. In this regard, special languages of science are formed, which contributes to the construction of complex abstract theories and the process of cognition in general.
Formalization - operating with signs reduced to generalized models, abstract mathematical formulas. The derivation of some formulas from others is carried out according to the strict rules of logic and mathematics, which is a formal study of the main structural characteristics of the object being studied.
Modeling . A model is a mental or material replacement of the most significant aspects of the object being studied. A model is a specially created human object or system, a device that in a certain respect imitates and reproduces real-life objects or systems that are the object of scientific research.
Modeling relies on analogies of properties and relationships between the original and the model. Having studied the relationships that exist between the quantities describing the model, they are then transferred to the original and thus make a plausible conclusion about the behavior of the latter.
Modeling as a method of scientific knowledge is based on a person’s ability to abstract the studied characteristics or properties of various objects and phenomena and establish certain relationships between them.
Although scientists have long used this method, it was only from the middle of the 19th century. modeling is gaining strong recognition among scientists and engineers. In connection with the development of electronics and cybernetics, modeling is becoming an extremely effective research method.
Thanks to the use of modeling the patterns of reality, which in the original could only be studied through observation, they become accessible to experimental research. The possibility arises of repeated repetition in the model of phenomena corresponding to unique processes of nature or social life.
If we consider the history of science and technology from the point of view of the use of certain models, then we can state that in the early stages of the development of science and technology, material, visual models were used. Subsequently, they gradually lost, one after another, the concrete features of the original, and their correspondence with the original acquired an increasingly abstract character. Currently, the search for models based on logical foundations is becoming increasingly important. There are many options for classifying models. In our opinion, the most convincing option is the following:
a) natural models (existing in nature in their natural form). So far, none of the structures created by man can compete with natural structures in terms of the complexity of the problems they solve. There is science bionics , the purpose of which is to study unique natural models with the aim of further using the acquired knowledge in the creation of artificial devices. It is known, for example, that the creators of the model of the shape of a submarine took the body shape of a dolphin as an analogue; when designing the first aircraft, a model of the wingspan of birds was used, etc.;
b) material-technical models (in a reduced or enlarged form, completely reproducing the original). At the same time, experts distinguish (88. P. 24-25): a) models created in order to reproduce the spatial properties of the object being studied (models of houses, district buildings, etc.); b) models that reproduce the dynamics of the objects being studied, regular relationships, quantities, parameters (models of airplanes, ships, plane trees, etc.).
Finally, there is a third type of models - c) symbolic models, including mathematical ones. Sign modeling makes it possible to simplify the subject being studied and to highlight in it those structural relationships that most interest the researcher. While losing to material-technical models in terms of clarity, iconic models gain due to deeper penetration into the structure of the fragment of objective reality being studied.
Thus, with the help of sign systems it is possible to understand the essence of such complex phenomena as the structure of the atomic nucleus, elementary particles, and the Universe. Therefore, the use of symbolic models is especially important in those areas of science and technology where they deal with the study of extremely general connections, relationships, and structures.
The possibilities of symbolic modeling have especially expanded due to the advent of computers. Options have emerged for constructing complex sign-mathematical models that make it possible to select the most optimal values of the quantities of complex real processes under study and carry out long-term experiments on them.
In the course of research, the need often arises to construct various models of the processes being studied, ranging from real ones to conceptual and mathematical models.
In general, “the construction of not only visual, but also conceptual and mathematical models accompanies the process of scientific research from its beginning to the end, making it possible to cover the main features of the processes under study in a single system of visual and abstract images” (70. P. 96).
Method of historical and logical : the first reproduces the development of an object, taking into account all the factors acting on it, the second reproduces only the general, the main thing in the subject in the process of development. The logical method reproduces the history of the origin, formation and development of an object, so to speak, in its “pure form,” essentially, without considering the circumstances that contribute to it. That is, the logical method is a straightened, simplified (without losing the essence) version of the historical method.
In the process of cognition, one should be guided by the principle of the unity of historical and logical methods: one must begin the study of an object from those aspects, relationships that historically preceded others. Then, with the help of logical concepts, it is as if to repeat the history of the development of this cognizable phenomenon.
Extrapolation - continuation into the future of trends, the patterns of which in the past and present are quite well known. It has always been believed that lessons for the future can be learned from the past, because the evolution of inanimate, living and social matter is based on well-defined rhythmic processes.
Modeling - representation of the object under study in a simplified, schematic form, convenient for obtaining predictive conclusions. An example is the periodic system of Mendeleev (for more details about modeling, see above).
Expertise - forecasting based on an assessment of the opinions of specialists - (individuals, groups, organizations), based on an objective statement of the prospects of the corresponding phenomenon.
The three listed methods seem to complement each other. Any extrapolation is to a certain extent a model and an estimate. Any predictive model is an estimate plus extrapolation. Any predictive assessment implies extrapolation and mental simulation.
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The discovery of stable connections and dependencies is only the first stage in the process of scientific knowledge of the phenomena of reality. It is necessary to explain their foundations and causes, to identify the essence of phenomena and processes. And this is possible only at the theoretical level of scientific knowledge. The theoretical level includes all those forms of knowledge in which laws and other universal and necessary connections of the objective world are formulated in a logical form, as well as conclusions obtained using logical means and consequences arising from theoretical premises. The theoretical level represents various forms, techniques and stages of indirect cognition of reality.
Methods and forms of cognition at the theoretical level, depending on the functions they perform, can be divided into two groups. The first group is methods and forms of cognition, with the help of which an idealized object is created and studied, representing the basic, defining relationships and properties as if in a “pure” form. The second group is methods of constructing and justifying theoretical knowledge, which is given in the form of a hypothesis, which as a result acquires the status of a theory.
Methods for constructing and studying an idealized object include: abstraction, idealization, formalization, thought experiment, mathematical modeling.
A) Abstraction and idealization. The concept of an idealized object
It is known that every scientific theory studies either a certain fragment of reality, a certain subject area, or a certain side, one of the aspects of real things and processes. At the same time, theory is forced to abstract itself from those aspects of the subjects it studies that do not interest it. In addition, theory is often forced to abstract from some differences in the objects it studies in certain respects. From the point of view of psychology, the process of mental abstraction from certain aspects, properties of the objects being studied, from certain relationships between them is called abstraction. Mentally identified properties and relationships appear in the foreground, appear as necessary for solving problems, and act as a subject of study.
The process of abstraction in scientific knowledge is not arbitrary. He obeys certain rules. One of these rules is compliance interval of abstractions. The interval of abstractions is the limits of rational validity of a particular abstraction, the conditions of its “substantive truth” and the limits of applicability, established on the basis of information obtained by empirical or logical means. The abstraction interval depends, firstly, on the assigned cognitive task; secondly, what one is distracted from in the process of comprehending an object must be to outsiders(according to clearly defined criteria) for a specific object being abstracted; thirdly, the researcher must know to what extent a given abstraction has legal force.
The abstraction method involves performing a conceptual development and conceptual assembly of objects when studying complex objects. Conceptual development means displaying the same initial object of study in different mental planes (projections) and, accordingly, finding many intervals of abstractions for it. So, for example, in quantum mechanics, one and the same object (an elementary particle) can be alternately represented within two projections: one as a corpuscle (in some experimental conditions), then as a wave (in other conditions). These projections are logically incompatible with each other, but only taken together they exhaust all the necessary information about the behavior of particles.
Conceptual build– representation of an object in a multidimensional cognitive space by establishing logical connections and transitions between different intervals that form a single semantic configuration. Thus, in classical mechanics, the same physical event can be reflected by an observer in different systems in the form of a corresponding set of experimental truths. These different projections, however, can form a conceptual whole thanks to "Galileo's rules of transformation" governing the ways of moving from one group of statements to another.
Abstraction as the most important technique of human cognitive activity is widely used at all stages of scientific and cognitive activity, including at the level of empirical knowledge. On its basis, empirical objects are created. As V.S. Stepin noted, empirical objects are abstractions that capture the characteristics of real objects of experience. They are certain schematizations of fragments of the real world. Any feature, the “carrier” of which is an empirical object, can be found in the corresponding real objects (but not vice versa, since an empirical object represents not all, but only some signs of real objects, abstracted from reality in accordance with the tasks of cognition and practice) . Empirical objects constitute the meaning of such terms in empirical language as “Earth,” “current-carrying wire,” “distance between the Earth and the Moon,” etc.
Theoretical objects, unlike empirical ones, are not just abstractions, but idealizations, “logical reconstructions of reality.” They can be endowed not only with features that correspond to the properties and relationships of real objects, but also with features that no such object possesses. Theoretical objects form the meaning of such terms as “point”, “ideal gas”, “absolute black body”, etc.
In logical and methodological research, theoretical objects are sometimes called theoretical constructs, as well as abstract objects. Objects of this kind serve as the most important means of understanding real objects and the relationships between them. They are called idealized objects, and the process of creating them is called idealization. Thus, idealization is the process of creating mental objects, conditions, situations that do not exist in reality through mental abstraction from some properties of real objects and relationships between them or endowing objects and situations with those properties that they do not actually possess or cannot possess, with the goal of a deeper and more accurate knowledge of reality.
The creation of an idealized object necessarily includes abstraction - abstraction from a number of aspects and properties of the specific objects being studied. But if we limit ourselves to only this, then we will not yet receive any integral object, but will simply destroy a real object or situation. After abstraction, we still need to highlight the properties that interest us, strengthen or weaken them, combine and present them as properties of some independent object that exists, functions and develops according to its own laws. And this is achieved as a result of using idealization method.
Idealization helps the researcher to isolate in its pure form the aspects of reality that interest him. As a result of idealization, an object acquires properties that are not required in empirical experience. Unlike ordinary abstraction, idealization places emphasis not on the operations of abstraction, but on the mechanism replenishment. Idealization gives an absolutely accurate construct, mental construct, in which this or that property, state is presented in extreme, most expressed form. Creative constructs, abstract objects act as ideal model.
Why is it necessary to use abstract objects (theoretical constructs) in cognition? The fact is that a real object is always complex; significant and secondary properties for a given researcher are intertwined in it; necessary regular relationships are obscured by random ones. Constructs, ideal models, are objects endowed with a small number of specific and essential properties and having a relatively simple structure.
The researcher, relying on a relatively simple idealized object, gives a deeper and more complete description of these aspects. Cognition moves from concrete objects to their abstract, ideal models, which, becoming more and more accurate, perfect and numerous, gradually give us an increasingly adequate image of specific objects. This widespread use of idealized objects is one of the most characteristic features of human cognition.
It should be noted that idealization is used both at the empirical and theoretical levels. The objects to which scientific statements refer are always idealized objects. Even in cases where we use empirical methods of cognition - observation, measurement, experiment, the results of these procedures directly relate to idealized objects, and only due to the fact that idealized objects at this level are abstract models of real things, the data of empirical procedures can be attributed to actual items.
However, the role of idealization increases sharply in the transition from the empirical to the theoretical level of scientific knowledge. Modern hypothetico-deductive theory is based on a certain empirical basis - a set of facts that need explanation and make the creation of a theory necessary. But a theory is not a simple generalization of facts and cannot be logically deduced from them. In order to make it possible to create a special system of concepts and statements, called a theory, an idealized object is first introduced, which is an abstract model of reality, endowed with a small amount properties and having a relatively simple structure. This idealized object expresses the specificity and essential features of the field of phenomena being studied. It is the idealized object that makes the creation of a theory possible. Scientific theories are, first of all, distinguished by the idealized objects they are based on. In the special theory of relativity, the idealized object is an abstract pseudo-Euclidean four-dimensional set of coordinates and instants of time, provided that there is no gravitational field. Quantum mechanics is characterized by an idealized object, represented in the case of a collection of n particles by a wave in n-dimensional configuration space, the properties of which are associated with the quantum of action.
The concepts and statements of a theory are introduced and formulated precisely as characteristics of its idealized object. The basic properties of an idealized object are described by a system of fundamental equations of the theory. The difference in idealized objects of theories leads to the fact that each hypothetico-deductive theory has its own specific system of fundamental equations. In classical mechanics we deal with Newton's equations, in electrodynamics - with Maxwell's equations, in the theory of relativity - with Einstein's equations, etc. The idealized object provides an interpretation of the concepts and equations of the theory. Clarification of the theory equations, their experimental confirmation and correction lead to clarification of the idealized object or even to its change. Replacing the idealized object of a theory means reinterpreting the basic equations of the theory. No scientific theory can be guaranteed that its equations will not sooner or later be subject to reinterpretation. In some cases this happens relatively quickly, in others - after a long time. So, for example, in the doctrine of heat, the original idealized object - caloric - was replaced by another - a set of randomly moving material points. Sometimes modification or replacement of the idealized object of a theory does not significantly change the form of its fundamental equations. In this case, it is often said that the theory remains the same, but its interpretation changes. It is clear that this can be said only with a formalistic understanding of scientific theory. If by theory we mean not only certain mathematical formulas, but also a certain interpretation of these formulas, then a change in the idealized object should be considered as a transition to a new theory.