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Lenin’s theory
knowledge
and mathematical concepts
V. G. Boltyansky,
N. Kh. Rozov
Already in middle school, the uniqueness of mathematics in comparison with, for example, physics becomes clear. All physical laws are valid with some degree of accuracy and are often replaced with progressive measuring technology by new ones, in relation to which personal ones become only a first approximation. It is pointless to ask yourself whether the length of the rod is rational or irrational. For narrowly understood practice, irrational numbers are not needed. In mathematics, the theorem that the diagonal of a square is incommensurate from the side is considered very important. The discovery of this fact in Ancient Greece is considered one of the turning points in the entire development of mathematics.
The published article by V. G. Boltyansky and N. Kh. Rozov contains an attempt to popularize the uniqueness of mathematics from the point of view of the philosophy of dialectical materialism. Perhaps the less experienced among you will master the sections of my article “Matematika” devoted to these issues in volume 26 of the second edition of the “Great Soviet Encyclopedia”.
V.I. Lenin, as you know from the article by I.K. Kikon («Kvant» No. 4), gave a detailed analysis of the main philosophical issues of physics. Lenin does not deal specifically with philosophical issues in mathematics. But his general statements on questions of the theory of knowledge serve as a guide for mathematicians.
A. N. Kolmogorov
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The meaning of the reality surrounding us, the objective laws of nature and society has always been and remains an integral and integral part of human activity, the goal and task of the development of science. The theory of knowledge, one of the branches of philosophical science, is studying the complex and multifaceted process of cognition, identifying the objective laws of the process and the formation of human knowledge about the material world.
Creatively developing the dialectical-materialist theory of knowledge, V.I. Lenin enriched it with new, fundamental provisions, and with exceptional depth, depth and clarity, developed an incredibly wide range of issues related to it. This allows us to speak with good reason about Lenin’s theory of knowledge - one of the branches of philosophical science.
We are surrounded by the material world, the movement and development of which is subject to objective laws that exist independently of man. The properties and mutual connections of the material world are directly or indirectly perceived by people, their senses are reflected in human consciousness. Living contemplation, the immediate sensory connection of a person with objective reality in the process of practical activity, is the starting point of knowledge and allows us to form an idea of the surrounding reality.
The data of experience accumulated by man is addressed and generalized by his abstract-losic thinking. It is thinking that allows a person, pushing away from facts, to cognize what is inaccessible to perception with the help of the senses, makes it possible to penetrate into the beings of objects and phenomena, identify connections between them, discover objective laws of the material world, and reproduce reality.
Naturally, the question arises about whether the picture reproduced by thinking corresponds to this reality itself, whether (and how correctly) our ideas reflect the objective state of things. Only in practice, only by comparing expected results with what actually happens, can a person evaluate the correctness and value of his knowledge. It is practice that serves as the criteria for this or that idea, hypothesis, theory.
V.I. Lenin has a general definition of the process of cognition: “From living contemplation to abstract thinking and from it to practice - this is the dialectical path of cognition of the truth, cognition of objective reality” *).
In the world around us there are no things that are incomprehensible, fundamentally inaccessible to the human mind. However, the meaning of objects and phenomena in their entirety can only be given by an infinite sum of general concepts, laws, etc. And mastering this entire sum of knowledge approximately all the time, i.e., the exhaustive position of the world, is possible, mathematically speaking, only to the limit. At each segment of historical development, our knowledge of reality is only relative and at the same time contains a particle of absolute truth: one aspect of objects and phenomena is reflected by scientific concepts more or less correctly, adequately, others - only approximately or even incorrectly, the third is still completely unknown.
Movement, change, development of the material world occurs in various forms; objects and phenomena of objective reality have appropriate qualities and have various quantitative characteristics.
Various areas of science are studied
*) V.I. Lenin. Full collection soch., vol. 29, pp. 152-153.
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Some specific qualitative features of objects or aspects of a phenomenon, explore one or more theon-connected forms of motion of matter. Thus, biochemistry examines chemical processes in their connection with living organisms, geography describes the picture of our planet and studies the laws of its transformation by people, history has its subject of revealing the laws of development of human society in their specific manifestation, etc.
The specificity of mathematics is that it does not consider specific materials of the body, does not analyze any specific form of movement of matter. “Pure mathematics,” wrote F. Engels, “has as its object spatial forms and quantitative relations of the real world…” *. In other words, mathematics studies the material world from a special point of view: the subject of its study is the multi-network relations themselves (in the general philosophical understanding of this term), which are inherent in all objects of real reality without exception and which “are characterized, in contrast to qualitative ones, only by their indifferent attitude to the specific nature of those objects that they connect…” **).
This specificity of the subject determines the wide applicability of mathematics in other sciences. The internal inseparability of the qualitative and quantitative properties of objects or phenomena of real reality creates objective proposals for the use of mathematical methods in the study of many of the most important problems of biology, geology, language, etc. In this sense, K. Marx said that “…science only reaches perfection when it manages to use mathematics” ***).
*) K. Marx and F. Engels, Works, vol. 20, p. 37.
**) A. N. Kolmogorov, Mathematics, TSB, vol. 26, p. 476.
***) Sat. “Memories of Markey and Engels”, M., 1956, p. 66.
Mathematics is generally considered an abstract science, and this is undoubtedly true. It is the specificity of mathematics, which studies spatial forms and quantitative relationships, that determines its special abstractness. F. Engels drew attention to this: “…in order to be able to explore these forms and relationships in their pure form, it is necessary to improve them from their content, to leave this latter aside as something indifferent…” *).
In other words, we are talking about research, the results are not with the help of a microscope or chemical reagents, but only by the power of abstraction.
Each object or phenomenon of objective reality has an infinite number of properties, qualities and are inextricably linked with other objects and phenomena, with the entire material world. Our thinking cannot immediately embrace all these properties and all the diversity of connections. Therefore, when carrying out scientific research, we are forced to focus our attention on only one or several properties of the object being studied, only on some of its connections with other objects. We do not pay attention to all other properties and connections, that is, we are distracted, abstracted from them.
In other words, all knowledge is necessarily associated with processes of abstraction, with abstractors. “Thinking, ascending from the concrete to the abstract, does not hunt - if it is correct … - from the truth, but approaches it … all scientific (correct, gray, not nonsense) abstractions reflect nature deeper, more accurately, more fully,” notes V. I. Lenin **).
No abstraction also has its weakness, which can be seen first
*) K. Marx and F. Engels, Works, vol. 20, p. 37.
**) V.I. Lenin. Popl. collection soch., vol. 29, p. 152.
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produces larger or smaller sizes. Thus, “houses” really exist, but “material points” do not.
The same can be said about the geometric concept of “segment”. In nature, we do not observe “in their pure form” objects that are long and “straight,” but “devoid of thickness.” This concept, therefore, does not arise as a result of the abstraction of identification, but is an idealization.
Idealization in general terms is as follows. The real object is replaced in our minds by an abstract model, which is obtained not only by abstracting from non-essential properties, but also by endowing the model with idealized properties, which in an exaggerated form reflect the essential properties of the object. For example, if we study the movement of a celestial body distant from us at a distance many times greater than its size, we can build an exaggerated model, considering this body to be a material point. In the same way, the concept of “touch” in an exaggerated form reflects the possibility in reality of the relative position of objects.
Of course, this does not mean that the concept that arose as a result of idealization is completely divorced from the objective world and devoid of real meaning. Allowing us to especially deeply and clearly trace the most essential properties of an object, these concepts have a well-known significance in practical applications of mathematics.
A special type of abstraction, idealization, is the abstraction of local feasibility. It is characterized by a distraction from the real boundaries of our constructive capabilities, conditioned by our limited life in space and time.
This form of abstraction allows us to reason, for example, about any large number (say, about the number ) and assume that we seem to have the opportunity to reach it, starting with 1, 2, 3, 4 … etc. Obviously, this possibility is purely potential: we will not have enough for this throughout our entire life.
However, we abstract from the practical impracticability of this task, from the limitations of our capabilities, and consider it permissible to consider any numbers, thereby arriving at the concept of “an infinite series of natural numbers.”
As a result of the use of this type of abstraction, concepts such as “straight” (extending unlimitedly in both directions), “plane” and others also arise.
Modern mathematics operates with a huge number of concepts that cannot be obtained as very strong, immediate abstractions of the properties of an object in the mathematical world accessible to animal contemplation. These concepts are formed in the process of multi-stage layering of abstraction.
The original, primary mathematical concepts arise from objective reality, reflecting the abstraction of quantitative relations and spatial forms of the object of action. But when these concepts arise, they seem to live their own life, forming their own world in the human mind, and they themselves give rise to further reflections and new abstract concepts. These abstract concepts, in turn, again give rise to new abstractions, so to speak, of a higher order, etc.
In order to understand what has been said, let us turn to the concept of “numbers”. Number, which initially acts as a result of counting objects (identification abstraction), is then generalized to the concept of a natural number (potential feasibility abstraction). Then the concept of a fractional number (identification abstraction) is introduced.
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deductions *). The next abstraction is rational numbers (positive and negative), and then real numbers.
The multi-stage nature of abstraction is clearly visible here. In the same way, the concepts of “number”, “vector”, “geometric transformation” and a number of other abstractions (referring, very indo, to spatial forms and quantitative relations of the real world) give rise to the concept of “group”, which is an abstraction of a higher order.
However, one should not think that the multi-stage nature of mathematical abstractions, the presence of abstractions of different orders, means that mathematics is separated from the real world. What is very instructive in this regard is that many of the “refined” abstract concepts and theories of modern mathematics are already finding important applications. Thus, the concept of “group” and subtle issues of group theory find important applications in particle physics.
The specific character of mathematical theories leaves a specific imprint on all mathematics.
Physicists study, for example, the image of gas processes, that is, they study the properties of a gas provided that its temperature does not change. Another example from physics is the study of supercooled liquids, i.e., the study of the properties of substances provided that it has melting points, but (due to special experimental conditions) is not crystallized, but remains in a liquid state. The laws discovered under these conditions will be the only ones under certain hypotheses.
However, in physics, laws also make sense only when the corresponding conditions and hypotheses are really feasible. In fact,
*) Dividing the whole, the unit into parts, we then identify the symbols , , , , etc., considering them to represent the same fractional number.
If someone created a “theory” about the behavior of substances at temperatures below absolute zero, then physicists would hardly accept it, or it uses a hypothesis that contradicts the properties of the real world known to us.
In mathematics, the situation (at least externally) is different. The objects around us have three dimensions, but mathematicians have built theories of -dimensional (and even infinite-dimensional) spaces. We know that multiplication of numbers is commutative: , but in mathematics they also study number systems in which . Other examples of theories that contradict “common sense” can be given.
But it is characteristic that these seemingly “wild” theories have important applications in the study of reality! Suffice it to recall, for example, that the relative theory makes extensive use of four-dimensional space.
The most striking example of a hypothetical theory influencing the whole of mathematics was Lobachevsky’s geometry. N.I. Lobachevsky replaced Euclid’s theorem about parallel others with his own theorem (through a given point in a given plane one can check the free set of lines that do not intersect a given line), and then on the basis of this axiom (and other axioms coinciding with Clide’s εν) he built a logical, structured and complete geometry.
The discovery of N. I. Lobachevsky largely suggested the complete triumph of the axiomatic method of constructing mathematics. The essence of this method is that all theorems are derived purely logically from a relatively small number of unprovable axioms. In other words, a mathematical theory is developed and built on the assumption that certain, pre-fixed axioms hold.
Mathematician in the process of his creativity
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Chestava deals with abstract concepts. Each time in his reasoning he does not turn to objective reality; he directly explores not it, but precisely this “understood world”. To prove a particular theorem, there is no need to describe and systematize facts and phenomena, there is no need to build cumbersome and expensive installations, and there is no need to conduct jewelry-precise experiments.
Mathematical research occurs in a purely speculative way, obeying the laws of logic. A mathematician in the process of his work represents only the goal of carrying out logical, rigorous, exhaustive proofs of theorems, and constructing consistent theories.
But, in addition to the purely logical activity of thinking, a large role in the development of mathematics (as, indeed, of any science) is also played by human creative capabilities, such as imagination and intuition. Emphasizing the importance of creative imagination, intuitive and then the importance of fantasy in the process of cognition, V.I. Lenin wrote: “It is in vain to think that only pose needs it. This is a deep prejudice! Even in mathematics it is necessary; even the discovery of differential and integral calculus would have been impossible without fantasy” *).
The purely speculative, mental path of mathematical research, which allows for the widespread use of fantasy and hypothetical constructions, creates a sense of independence of mathematical concepts and theory from the objective world, a lack of real content in abstractions of a higher order that arise in the process of “free creativity” of the mind.
In fact, formally the following possibilities open up: a mathematician puts forward arbitrary axioms he has invented and deduces from them using the laws of the logic of reason.
*) V.I. Lenin. Full collection cit., vol. 45, p. 125.
personal consequences; if he is “lucky” and the result is a consistent theory, then we can talk about a mathematical discovery. In other words, a mathematician, not being consistent with any real objects, nor with any formulation of the continent, can, it would seem, be guided only by his imagination, make any hypotheses and develop theories on their basis, just to get something logical consistent and “mathematically interesting.”
Developing similar thoughts, one of the greatest French mathematicians, Henri Poincaré (1854-1912), argued that the basis of mathematics (and even science in general) is free, arbitrary to accept an agreement, invented to explore and convenient for him. He believed that mathematics is a product of free “pure reason” and does not depend on the existence of materials, objects, and is not related to the practical activities of people. As for the possibility of practical application of the achievements of mathematics, this is only a partial function, which we are not able to explain *).
Paying tribute to the scientific merits of A. Poincaré, V. I. Lenin sharply criticized his philosophical concepts, describing him as “a very confused philosopher.” It should be borne in mind that with all the inaccessibility and openness of the idealistic
*) Similar views can sometimes be found even now. For example, a group of famous foreign mathematicians, speaking under the pseudonym Nikolai Bourbaki, wrote, touching on the relationship between the “experimental world and the mathematical world”: “The fact that there is a theory of connection between experimental phenomena and mathematical structures - this, it seems, has been confirmed in a completely unexpected way by the recent discoveries of modern physics, but the global reasons for this are completely unexpected to us (if any meaning can be attributed to these words), and perhaps we we will never know them” (N. Burbak, Essays on the History of Mathematics, M., 1963, p. 258).
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Due to the nature of such views, it is impossible to refute them by purely logical arguments. The correct answer can only be given by a consistent dialectical-materialist point of view, recognizing the practice as criteria that are true.
Logic is a working tool of mathematics: a statement derived from premises, axioms and definitions according to the rules of logic is considered true in mathematics. But “obviousness,” “clarity,” and “convincingness” of these steps of logical reasoning do not arise by themselves; the validity of logical constructions is confirmed in the centuries-old experience of practical human activity.
It is this experience that confirms that the internal laws of development of logical thinking are identical with the law of development of objective reality. F. Engels wrote on this occasion: “All of our theoretical thinking is dominated with absolute force by the fact that our subjective thinking and the objective world are subject to the same law and that therefore they cannot contradict each other in their results, but must harmonize with each other”*).
Thus, the requirement of formal mathematical rigor, i.e., accuracy of definition, complete proof of theorems, consistency of theory, etc., is one of the aspects of the specific manifestation of the criterion of practice in mathematics.
*) K. Marxy F. Engels, Works, vol. 20, p. 581.
However, mathematical rigor itself cannot completely replace the criterion of practice and is not an exhaustive guarantee of the truth of mathematical knowledge, that is, the correctness of a person’s reflection of objective reality.
For the question of compliance with the premises of real action accepted in the theory is also significant. This question already goes beyond the competence of logic and can only be resolved in the course of public voluntary (including scientific) activity of people. Only the experience of this activity can distinguish strictly logical “theories” built on unrealistic, fantastic hypotheses and axioms, from truly scientific theories that bring us closer to the knowledge of objective truths.
Scientific and historical practice shows that primary mathematics and the axioms describing them are taken from life and are a reflection of the objective properties of real objects. Since understood logic reflects the objective logic of things, that is, the objective connections of objects in the mathematical world, new concepts and theories are technically connected with the real world and therefore naturally, and not at all by chance, find more and more new applications in the scientific and practical activities of people.
The current rapid process of “mathematization” of knowledge, even more stimulated by the development of computer technology, fully confirms the power of mathematical methods and their organizational connection with life.