Abstraction (from Latin abstraho – to distract, exclude, separate) is a necessary condition for cognition by forming “secondary images” of reality (its information models), in particular, such as perceptions, concepts, concepts, theories, etc. In the process of abstraction, the choice and processing of information to replace the given empirical image directly with another, not directly given, but implied and conceivable as an abstract object and usually called the same term “abstraction”.

The modern notion of abstraction goes back to Aristotle, according to which abstraction is a method of intentionally unilateral study of reality, a subjective reception of the mental division of the whole and the positing of separate parts of it. In principle, such a postulation does not conclude “no mistake” and is objectively justified by the variety of properties (aspects) of the whole, which are at times so different that they can not become the subject of one science. Science, according to Aristotle, explores the general, and the general is cognized through abstraction. Therefore, abstraction is not only the main prerequisite for scientific knowledge but also “creates science”. In this sense, transient phenomena of experience are important not in themselves, but in so far as they are involved in any abstraction. Aristotle also distinguished empirical abstractions from theoretical ones, believing that the latter is necessary where the thought comprehended and thought itself are inseparable (for example, in mathematics, where knowledge and the subject of knowledge essentially coincide).

This epistemological concept of abstraction, however, did not receive any development either in Hellenic-Roman or medieval philosophy. Scholasticism, including Arabic-speaking Neoplatonism, the theme of abstraction brought essentially to the theme of universals, linking it with the Platonic concept of acide (“invisible,” spiritual principle), which corresponded to philosophical thought oriented to logos, but not to physis. When the medieval “book science” was replaced by the experimental science of modern times, the theological and ontological view of abstraction was replaced by a psychological one: abstraction now seemed to be the forced “action of the soul” to develop common (universally valid) concepts, the necessity of which was due to imperfection of reason, incapable of knowing otherwise (inseparable) “the nature of things.” Both sensationalism and rationalism of the 17th and 18th centuries were almost unanimous in that the “objectification” of abstractions not only obscures the facts of real processes from the eyes of the researcher but also leads to hypostatization of fictitious essences and meaningless representations. A well-known expression of this position was Kant’s requirement of “principle exclusion” for abstractions if they pretend to have any meaning.

The philosophy of the early 19th century changed little in this assessment. In particular, Hegel, recognizing abstraction as the first element of the spiritual assimilation of reality and including it in ordinary and scientific experience (already simple observation, according to Hegel, needs the ability to abstract), at the same time attributed the abstraction to “formal thought”, alien to the philosophical method, and blamed the “abstract” for one-sidedness and emptiness. Only by the mid-19th century, the interpretation of abstraction went beyond the “abstract thought”. Abstraction returns to its scientific Aristotelian meaning. With its help is described not only statics but also the dynamics of the phenomena of nature and social life. In the humanitarian field, this applies primarily to the philosophical method in which the objective dialectic of development is realized through the development of subjective dialectic of concepts, and therefore the principle of abstraction plays a leading role in it (K.Marx). But even in the natural-science methodology of those years, in essence, far from the realized dialectic of concepts, the application of abstract models “achieves striking results in explaining the phenomena of nature.” As a result, the spiritual attitude of the post-scholastic reformation (with its slogan “instead of abstractions – experience”) is gradually replaced by a methodological compromise, when abstract objects are recognized as representatives of the realities necessary to express objective truths. Even positivism accepted this compromise to a certain extent, not only giving abstractions a leading role in scientific research but also recognizing for them a certain “kind of reality” (E.Mach). At the same time, the first classification of abstractions appeared, and the intentional use of definitions through abstraction.

The philosophy of science of the 20th century again returns to the controversy about the objective significance of abstractions. This time, on the one hand, relativistic directions (tendencies) in physics, on the one hand, and transfinite principles of introduction of abstractions in mathematical set theory, which gave rise to a certain “feeling of unease about the dependence of pure logic and mathematics on the ontology of Platonism” served as an excuse (Beth. W. The foundations of mathematics). From the criticism of these tendencies and principles, a deep differentiation of methodological approaches and ways of thinking (according to the type of abstractions used) in modern scientific (especially mathematical) cognition begins, the desire to overcome the emerging “crisis of bases” not only by technical means of improving scientific theories, but also by one way or another solution of epistemological problems of abstraction.

Abstract theory

The simplest version of abstraction is the act of abstraction, more precisely, the act of selective reflection or interpretation of data. With the same data in different situations, various acts of abstraction are possible. Although the arbitrariness of distractions is undeniable, they are usually justified to the extent that abstraction leads to success in cognition or practice. An arbitrary act of abstraction can only by chance give such a result. For example, identifying, as a rule, choose only such grounds for identification, which would give the abstraction of identification a certain epistemological sense. Usually, this is determined by the purpose, or task, or some other setting. In general, the installation essentially depends on the structure of the abstract image (an abstract object) and its reorganization (when the installation is changed). At the same time, abstraction can be conscious, reflected at the level of thinking, or unconscious realized at the level of functional properties of receptors (sense organs, devices). However, in any case, the abstraction should give a certain “partial image” from an almost infinite set of possibilities (external data flow).

The interpretation of abstraction as a diversion presupposes either a transitive or intransitive form of the verb “to distract”. Although the positions of these forms in the language itself are equal, their semantic roles are not the same. Usually (but not always) they express additional aspects of abstraction: the transitional form fixes attention to the part selected from the whole, intransitive, on the contrary, on the whole, devoid of the part. The first (positive) aspect introduces the information (abstract) image directly, while the second (negative) – only indirectly, through the incompleteness of the basis, leaving the completion (painting) of the image to the share of idealization or imagination. That is why abstract is often characterized as negative, “only as a moment of something real” (Hegel). This division of aspects of abstraction is, generally speaking, conditional, but the choice of one or the other of them had a noticeable effect on the value attitude toward abstraction. Thus, Aristotle saw the epistemological value of abstraction in the solution of the positive problem of cognition, and Kant, on the contrary, recognized for abstraction only negative work, referring to the solution of the positive problem at the expense of reflection. These polar points of view emphasize the importance of understanding abstractions in the context of modern scientific practice since the habit of singling out the negative (negative) aspect of abstraction still dominates its vocabulary definitions: the common meaning of the term “abstraction” is a literal translation from Latin.

Of course, a pure act of distraction in itself is not capable of providing a useful, meaningful image. It is necessary to analyze the sufficient grounds for abstraction-subjective, on the one hand, and objective-on the other hand, in which information “captured” by the abstraction process and included in its result could be considered virtually independent of other data and therefore of outsiders for this abstraction. The search for an objective outsider, or rather, clarifying which characteristics of the whole (or environment) are extraneous to the information image, is one of the main issues of abstraction. In part, it coincides with the notorious question of essential properties, but only in a strictly scientific formulation of it, when by essential they mean such definable properties of the object that can completely represent (replace) this object in a certain epistemological situation. This confirms the relativity of the “essence of the matter,” represented by abstraction because the properties of the objects themselves are neither significant nor extraneous and can be so only for something and with something. Also, the abstract image is realized with a fullness, not exceeding the completeness of the available data. And this is not enough to generate abstract high-order objects created specifically ad usum theoreticae. Thus, the first empirical concepts of the figures of material bodies in the observable space – the “abstraction of the sensory figure” – are created inductively, being abstracted from all the properties of these bodies, except for shapes and sizes. But geometrical images in the proper sense are obtained by logical reconstruction of inductive concepts, replenishing empirical properties with the theoretical “point” (in the set-theoretical sense) “arrangement” of figures, the possibility of their continuous (congruent, affine, topological) transformations, in general, all the properties that are necessary for the formulation or the proof of geometric theorems. Obviously, abstract objects of this order are only genetically related to distraction. Their content is not exhausted by experience data. Here we are talking about a certain interpretation of reality, about its understanding “in-laws”, which in itself is impossible without generating new semantics, without adding to the data of experience new information that does not follow logically from these data. But since abstraction is declared as scientific, it is limited in its arbitrariness not so much by the correspondence of facts, but rather by the fact that it “can not introduce any logical contradictions” (F. Klein). Observance of this restriction on the application of abstractions essentially distinguishes the norm of science from the norm of art, where it is permissible not only to “depart from the fact”, but also to go to the internal contradiction in depicting facts for the sake of solving a certain artistic problem.

It’s no secret that in the system of scientific representations, abstraction does not always obey the logic of empirical facts. Moreover, dogma can serve as the basis for adopting one or another abstraction. This, in particular, is the postulate of ancient science on the perfection of circular motion (the “dogma of the circle”), which set abstractions on the facts of astronomical observations and for a long time determined not only the nature of the first theories of celestial mechanics, but also the approach to the mathematical description of physical phenomena by means of exponentials. And yet, in general, scientifically, abstraction is dominated by the ideology of empiricism. For theoretical natural science, this is obvious. But the mathematician, when there is a need to justify an abstraction, does not neglect the possibility of presenting this abstraction “from the visual side,” to find its prototype in sensory experience. This does not mean, of course, the actual exclusion of abstraction, but it allows us to understand the genesis of abstraction, its connection with what can already be accurately “tied” to an empirical fact.

Abstracting from empirical data, receive abstractions of the first order. They are also called real. Each subsequent step from these abstractions generates abstractions of a higher order than the first. They are called ideal. This scale of orders is not, of course, absolute, nor is it an absolute criterion that allows one to distinguish abstractions from non-abstractions once and for all. At least in the sphere of scientific knowledge, “empirical” (concrete) and “theoretical” (abstract) are correlative concepts, and an alternative between them is possible only through abstraction. True, in the process of cognition, the concrete always has an exoteric meaning. It is considered and explained in the aspect of “abstract reality”, since any “fact only in abstraction can be known by thought”. In its turn, the abstract, on the contrary, is always esoteric. It is an attribute of thinking that represents the ideal moment of reality as the content of concepts. The objectivity of this ideality is found, as a rule, in applications, i.e. in general, where abstractions are applied. Then the epistemological relation turns around: the inductive way “experience – abstraction” is replaced by a deductive way “abstraction – experience”. That is why, going back to abstract objects of a high order, it is necessary to take care that the return path of their “exclusion” is somehow ensured. For scientific knowledge, the ability to wrap the relation, to make abstraction an independent starting point for research, whether its empirical equivalent is found or not, is the most important condition for development. The same can be said about logic, which only feels in the sphere of abstractions on “native soil”. This possibility makes it possible not only to combine observation and experiment with logical deduction but also to compensate for the absence of an experimental basis in principle, which “then the force of abstraction must replace” (K.Marks). And from this is already a direct path to the axiomatic method in science, which in turn becomes an instrument of abstraction and analysis and as meaningful axiomatics that retains an explicit link with empirical experience, and as axiomatics of a formal one that does not preserve such a connection. In the latter case, the value of the method is particularly obvious, because the transition from contentive axiomatics to a formal one is a far-reaching generalization, which, as a rule, requires abstractions of a higher order than those dispensed by a meaningful interpretation of concepts. Therefore only formal axiomatics reveals the difference between the intuitive sense of abstraction implicit in the language of the researcher and their generalized meaning encoded in the language of the formal theory. Hence, as a consequence, the ambiguity of abstraction, elusive on the inductive path.

The philosophical idea of ​​the intentional incompleteness of knowledge due to abstraction, when the abstract point of view is consistently carried out, is supplemented by the requirement of its completeness to the domain of the meaning of abstraction. The question of the content of this area, its depth or its boundaries is not always, of course, possible to find an a priori answer. But it is precisely the problem of the completeness of abstraction that naturally leads to the epistemological concept of the interval of abstraction as a characteristic of freedom (admissibility) of abstraction or as a measure of the information capacity of abstraction expressing a kind of “abstraction concept”, the conditions for its “model realizability”. In this sense, the interval of abstraction does not depend on “external” (empirical) determination but is determined by the intrinsic logic of abstraction as “the theme that forms the basis for fulfillment” (Hegel).

The elaboration and analysis of abstractions is a special purpose and task of science, at least since “every science explores the common” (Aristotle). The desire for generality is consonant with the desire for order. And if one of the tasks of science is to “open” facts, then another, no less important, is to bring facts to order. Therefore, the search for generalized points of view begins with the search for patterns that are fixed in the abstraction of the “scientific law”, which gives “a kind of natural coordinate system with respect to which we can order phenomena” (V.Geyzenberg). A simplified image of reality without “side effects” or “a mass of details” is only the initial work of abstraction, which in its truly scientific manifestation goes far beyond what can be extracted from the data of experience. The thesis that cognition abstracts (coarsens) reality through abstraction, encounters an objection that the genuine interests of cognition tend, as a rule, “beyond” the available experience to the invariant “essence of the matter” presented in abstraction. By itself, a pure act of abstraction only precedes the search for such invariants, masking the further non-trivial process of a mental analysis of the relationship between abstraction and reality.

Apparently, there is no field of knowledge where abstraction does not serve as a rational basis for cognition, although in various fields the abstractions used and the features of their use are, of course, different. The most developed system of abstractions is mathematics, which is essentially a science of abstractions. Natural science, to the extent that it uses mathematics, borrows from its abstractions, adding its own to borrow. But at the same time, there are general scientific abstractions that are necessary both at the first steps in the formation of concepts and at all levels of the formation of knowledge about natural and social life. That is why abstractions are not “scaffolding”, which after the construction of any branch of knowledge can and even need to be discarded. This is not only a form but the very essence of science.

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