THE NEED TO USE MATHEMATICAL METHODS FOR PHILOSOPHICAL PROGRESS
THE NEED TO USE MATHEMATICAL METHODS FOR PHILOSOPHICAL PROGRESS
Gulhan Nuradin
Ph.D., associate professor, Taraz Regional University named after Mohammed Haidar Dulaty,
Kazakhstan, Taraz
Shakhzoda Mirzabayevа
first-year master student, Taraz Regional University named after Mohammed Haidar Dulaty,
Kazakhstan, Taraz
Gozel Ahmedova
first-year master student, Taraz Regional University named after Mohammed Haidar Dulaty,
Kazakhstan, Taraz
The unity of controllability and uncontrollability is clearly considered in the course of the development of physical and mathematical sciences. It can be explained like this. Most of the concepts of physics and mathematics, being an idealized copy of reality, are explained by the following features: accuracy, provability, inference, clarity, etc. These features and characteristics require researchers that formal logic preserve laws without any violations and contribute to the rejection of eclecticism, sophistry and relativism, which sometimes take place in the study of philosophy.
Mathematical approaches make it possible to accurately describe and study complex systems, thereby becoming a real structure of knowledge. Mathematics allows us to abstract certain instrumental parts of observed objects and the relationships between them, as well as to logically find new non-control components and relationships. Through this, mathematical approaches make it possible to form knowledge about fundamental artistic, uncontrolled beings, objects, and parameters. The heuristic essence of mathematical structures lies in the fact that it is a method of scientific discoveries, in predicting new facts, a means of forming new scientific ideals and concepts. How can one generalize the situation by the unity of the controlled and the uncontrolled into one general form? Cognition begins with facts (phenomenon, trend, event, components, relationships, etc.) perceived by the senses of the observed. It should be noted here that we include sensory control in a broad sense, that is, experimental technical structures and devices. In this sense, the concept of “controllability” does not coincide with the concept of "sense perception" in our traditional sense.
Based on the logical processing of incoming data, abstract concepts that cannot be controlled (the concept of matter, continuity, the sum of the angles of a triangle, etc.) are formed. The receiving subject can create entire scientific theories based on the concepts of abstractness, idealism. This is evidenced, for example, by the systems of numbers of matter and the complex, non-Euclidean geometries (N.I. Lobachevsky, Ya. Boliam, B. Gimana).
At the same time, in science, inside numbers, very often in mathematics, they set themselves the goal of indirectly perceiving these abstractions in a sensual way: this is the reduction of the latter to the connection of other sensually perceived features of concepts with the reflection of specific forms in reality, their interpretation in the language of geometry (i.e. e. spatial forms). Subsequently, giving these idealizations the nature of a material and technical system, there are several interpretations of the hyperbolic geometry of N. I. Lobachevsky, proposed by E. Beltras, F. Klein, A. Poincaro. They testify that it is possible to move from the uncontrollability of abstraction and idealization of science to the sensory perception of the object in question in a spatial model.
It should be noted that in the cases described, sensory control is not a return to the first level of fine control.
In fact, this is an observation of a higher order, rich in the content of the loss of the achieved achievements that underlie abstract thinking.
However, it should be borne in mind that the objective non-transitional-uncontrollable features of reality can be made indirectly controllable. In this regard, we should not forget about the existence of a fundamental uncontrollability of the features of the objective world and homometric bodies (continuity, infinity, the ratio of length to diameter). But this does not mean that this is due to their recognition.
We can draw the following conclusion: understanding, starting with the study of the sensory controllability of objective reality, ultimately reveals uncontrollable features in it, trying to give the reverse side the character of indirect controllability. Let us dwell on the illustration presented in the development of mathematical understanding, including in the process of formation and on the example of a fundamental concept, such as number.
If integer, fractional, rational numbers are introduced from the practice of a specific calculus and a request for measurement, then under them lie real geometric and arithmetic objects that can be observed directly (for example, 2-2=4). Negative, irrational, complex, transcendental numbers are not caused by the influence of observed facts of reality.
Thus, mathematical representations and representations of various numbers arise, on the one hand, from practical inquiry, and on the other hand, from inquiries within the topic itself.
I wonder how European mathematicians treated positive and negative numbers? A number of European mathematicians, such as the French mathematician Nicolas Schuquet (1445-1500), the German mathematician Michael Stiefel, the French mathematician Francois Viet, were categorically against this. They called it pointless. French mathematician, physicist, philosopher Blaise Pascal (1623-1662) says that the operation of subtracting 4 from 0 loses all its meaning. Therefore, negative numbers thus had no geometric meaning at the time. From about the middle of the 18th century, many mathematicians began to make extensive use of negative numbers, no matter how much they opposed the fact that reciprocals actually exist.
Other mathematicians such as René Descartes, Gottfried Leibniz, Isaac Newton followed the right course because of reciprocals. They clearly showed the opposite of algebraic roots with positive and negative roots, depending on their magnitude.
Newton, one of the founders of theoretical natural science, also used this direction. He used geometric movement to expand the integration of positive and negative numbers. He called forward motion positive and backward motion negative. That is, if the first movement increases the distance between the rows, the second decreases. If in geometry a line drawn in a certain direction is considered positive, then a line opposing that is considered linearly positive, then a line opposing that is considered negative. Newtonian thought develops because of the volumetric objectivity of positive and negative numbers and their relationship. The interpretation of such numbers determines the objective reality. In addition, it should be noted that the rational numbers were a measure of the volumetric function. They help us to know whether the part of the vertical line taken as a measure of length is productive of the length of the other part of the vertical line. As a result of the measurement, a full or fractional number is obtained. “Quote” [1]. This is the level of measurement created using the first numbers in mathematics.
It is interesting to contrast real numbers from the control and non-control side. In modern mathematics, real numbers are obtained by adding irrational numbers to rational numbers. In the Pythagorean scientific school, the countable klemg is represented by whole numbers.
But the Pythagorean school established the fact that the ratios of two parts are not always computable, since they are not expressed in terms of a ratio of integers. For example, the Pythagoreans state the fact that the diagonal and vertical quadrangular sides are not measurable, that is, they are equal to the number of irrational ones .... Such a discovery led to the collapse of the foundations of mathematics, changing the existing ideas of philosophers and mathematicians about continuity and superiority, as a final and endless. This discovery was based on theoretical, deductive-logical thinking using positive integers with empirically known even, odd, prime, and quadratic numbers. The number is incalculable... cannot be created by any means, by geometric experimental methods.
French specialists Am Daan-Dalmedico and Jeanne Peiffer, working in the field of the history of mathematics, write: “What is true from a geometric point of view turned out to be non-existent when it was shown through the ratio of integers. Such arithmetically immeasurable missing parts came out of the theory of Pythagorean relations”. “Quote” [2].
The discovery of infinity influenced the development of mathematics and philosophy, bringing to the fore questions of a philosophical and methodological nature.
Firstly, this discovery led to the study of a deep and general theory of finite and infinite volume, to the problem of understanding the controlled and the uncontrolled.
Secondly, the discovery of infinity gave importance to the relationship between geometry and arithmetic in Greek mathematics. Therefore, he proposed to explain the concept of volume from the point of view of the geometric method. “Recourse to geometry for help is perfectly understandable,” writes Mark Klein, a well-known American historian of mathematics. If we consider the numbers 1 and ... as lengths, then the difference between 1 and ... is unknown. “Quote” [3].
Thirdly, the problem of measuring the finite and the infinite has acquired a new meaning. Actual measurements are made with limited accuracy compared to mathematical ones. The Czech philosopher, specialist in the field of scientific logic and methodology K. Berko notes: “Physical and non-physical volumes should be understood in geometric terms in accordance with their actual measured value. It comes from objects and objects of various geometric nature”. “Quote” [4].
Real and unreal human knowledge, especially mathematical knowledge, has the character of world science.
Fourth, some geometric quantities of rational numbers, such as the sides of a rectangle, cannot be measured diagnostically.
The phenomenon of finiteness and infinity is an integral part of the concept of number. “In general, to obtain a measure of volume, we need not only natural and rational numbers, but also irrational numbers,” says V.B. Biryukov and V.I. Mikheev K. In the last words of Berka's book.
It is worth noting the internal factors in the development of science. The internal logic of the development of mathematics contributes to the expansion of the field of numbers, the emergence of irrational, real and complex numbers.
Irrational numbers arose as a result of theoretical analysis. The problem of control and uncontrollability of irrational numbers is a complex philosophical problem. After all, no measurement can define unmeasurable lines.
Irrational numbers come from the mathematical calculation of the right angle, the hypotenuse of a triangle with its sides. Therefore, the hypotenuse and sides of a right triangle cannot be equated to positive and negative arithmetically real numbers. In other words, r/d is not defined by rational numbers.
For a deep understanding of irrational numbers, the famous philosopher A.F. Losev wrote the book The Problem of Symbols and Realistic Art. This is a geometric figure of an irrational order, the vertical diagonal of which is equal to one.
There is another type of invariant, both theoretical and practical, that results in transcendental numbers. One of the basic concepts in mathematics is that in order to determine the place and origin of transcendental numbers, one must look into their genesis. For example, if we take π, the first of the transcendental numbers, it has deep roots in human history. It is also closely related to the activity of people in life. There is an important rule: the diameter of a circle is approximately 1/3 of its length (or 3 times less than its length). This decision was made as a result of experimental measurement of things in circular objectivity. However, the Egyptians also proved that “the diameter is not equal to 1/3 of the circumference.”
How did the Egyptians determine this fact? Obviously, this result was achieved primarily through more accurate measurements. An even greater role is played by the contribution of theoretical ideas about the length of polygons written and illustrated inside the circle. First of all, it was Archimedes who determined this fact. He concluded that “the length of a circle is not less than 1/7 of its triple diameter and not more than 1/71 of its diameter” (the length of a circle is less than 1/7 of its triple diameter, but greater than 1/71 of its diameter). It is worth noting that in early times, the solution to the problem of measuring the circumference of a circle was one of the largest discoveries in the field of mathematics that Archimedes made on the scientific path. Archimedes also fought against the indestructible lock between theoretical and applied mathematics. Archimedes was able to boldly combine his open theoretical research with experience. He showed how to determine the area and volume of bodies.
In mathematics, there were several types of this approach. Many mathematicians, called “quadraturists”, opposed this decision. They tried to draw a small square equal to a given circle. If, for example, a and d are the same side of a square and the diameter of a circle, then the equation ... and the square of the circle must equal finding the value of π. From equality it is easy to determine that the side of the square, equal to the radius of the circle, is ... In this case, the quadrature of the circle is built into the structure of the complex of its length.
For three thousand years, the main property of the number π was determined. According to Florica Cimpan, historian of mathematics and researcher of the number π, “by means of this number one can capture the thoughts and feelings not only of scientists, but also of philosophers and artists”. “Quote” [5]. Only in 1882, the German mathematician Ferdinand Lindemann proved the transcendence of the number π, i.e., as a result of determining the squaring of the circle, there is no, i.e., the circumference of a circle cannot be accurately measured by a geometric section using a compass and ruler.
Transcendental numbers are the result of a theoretical measurement of a curve by a straight line. If the result of such a measurement is treated only rationally, it cannot be marked with any irrational or other algebraic numbers.
The complexity of the problem is that any transcendental number is an invariant value of the length of the curve (determined) by the length of the straight line segment. For example, the ratio of the circumference of a circle to the length of its diameter, the property of the number π, is not observed in geometric existence. In this regard, it should be noted that it is impossible to find a geometric explanation by observing transcendental numbers. Although the number π, equal to 3.14, is the circumference as a geometric body and its diameter, the ratio of the circumference to the diameter is not absolutely exact, but only approximate. The circumference of a circle can be determined, although not in absolute terms. The transcendence of cognition is determined by an approximate value, not an exact one.
In any case, the value of the stability in question cannot be determined by an exact number. Because it cannot be designated by any specific number, therefore it is a transcendental number. This is the circumference. It is impossible to imagine or see that there is a certain number between the length and diameter of a circle, which means that there is no common measure. It is also completely false that there is a number that determines the length of a circle by its diameter. π, e, etc., it is also impossible to show the last decimal value of transcendental numbers.
In transcendental numbers, there is not only no common measure between quantities whose value is known through each other, but there is also no real number that accurately gives their ratio. Therefore, the number π is a strictly controlled object in mathematical, more precisely, geometric reality.
Thus, it is necessary to be able to distinguish between three types of numbers: irrational, belonging to an algebraic group and forming, according to L. Euler, numbers “higher than algebraic” - transcendental. Therefore, transcendental numbers are structurally much more complex than irrational numbers.
I wonder if it is possible to manage transcendental numbers and designate them as you like?
Taking into account the observability and visibility of transcendental numbers, A.F. Losyev asked: “Is it possible to visually see a circle (turn or circle) and its area, composed by the outer radius? Of course it is possible. So don't tell me that irrational and transcendent volumes are invisible. No matter how your thoughts are against it, you clearly see them”. “Quote” [6].
Revealing the objective property of transcendence, A.F. Losyev argues that the consciousness and scientific understanding of a person are able to bring any incomparable and immeasurable elements into a clear and legitimate order.
Criticizing the simple understanding of mathematics (especially in the combination of transcendence and irrationality), he said: “Therefore, if someone is afraid of irrationality, he does not know and does not understand mathematics and that irrationality and transcendence are visible to the naked eye. Then where did this mysticism go?
A.F. Los'ev, studying the diagonal of a square and the length of a circle, combines the meaning and controllability of irrational and transcendental numbers into one concept. This is not an easy problem.
Accordingly, mathematical methods, measured on the basis of rational, irrational and transcendental numbers, help to conduct this process more accurately and accurately.
So, in physics and other sciences, it is determined by experience and observation, and according to this method, the experiment is carried out mentally. That is, transcendental and irrational numbers have two different immeasurable values. Although volumes can be exactly determined by squares of each other's numbers, in the case of irrational numbers, the measured volumes do not have a common measure. In this case, the incommensurability of volumes means that these volumes cannot be exactly determined by rational numbers, and we can only represent them with square numbers. This is evidenced by the fundamental nature of the Pythagorean theorem a2+b2=c2, which is important in all mathematics.
In the case of transcendental numbers, there is not only a common measure between quantities known through each other, but there is also no real number representing their ratio.
If we look at the history of the development of the concept of numbers, we will see that its apex is the introduction of complex numbers. Where are the roots of complex numbers? Their genesis is closely related to the internal factors in the development of mathematical science, namely: how were complex numbers formed and how do they develop?
The solution of an equation of the third degree was first found (in 1535) by the Italian mathematician Nicollo Tartaglia. He achieved this success by identifying the relationship between the root of the equation and the coefficient. Moreover, mathematicians have established that the equation has 1 positive and 2 imaginary roots. Tartaglia's achievements were widely reported in the work of another Italian mathematician, Gerolallo Cardano, called The Great Art (1545). For the first time, Cordano used in calculations the concept of imaginary volume, which was a new object in mathematics, and emphasized that although these volumes are useful, they are useless. Although Cordano's formula produced complex numbers when using the algebraic method to solve a cubic equation, it only chose real roots. Instead of considering the disassembled volumes as true volumes, Cordano recognized that these sephistic or false numbers have a place in mathematics.
It is possible to explain the operation of finding the square root of any number and solving algebraic radical equations of degree 3 and higher. This purely formal algebraic procedure requires one more new (ordinal) level, i.e., the formation of imaginary numbers.
The latter (territorial) is an extension and logical generalization of the territory of numbers and represents a much higher rank. Thus, complex numbers are a high-level (level) result of human activity as a product of the development of mathematics.
The appearance of complex numbers is a sign of the development of mathematics and has made a significant contribution to the progress of mathematics. Complex numbers are a powerful tool for solving problems in (mathematics), physics, mechanics, electrical engineering, geodesy, cartography.
The cognitive process has the following (natural) property. The fact that the solution to a problem is unmanageable at one level (for example, through the geometric interpretation of negative numbers) creates a problem at a higher level (in this case, complex numbers). After that, it is necessary to find new connections of objective reality.
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- Stroyk D.Ya. Brief outline of the history of mathematics. Moscow, 1984.