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Pythagoras and the Pythagoreans. Doctrine and school of Pythagoras

15.09.2020

The teachings of Pythagoras are one of the most interesting phenomena in Greek philosophy. It has an independent meaning and is important as one of the elements in Plato's philosophy, if it cannot be considered its main element. The direction of Pythagoras runs through the entire history of Greek philosophy: it originates simultaneously with the Milesian school and is transformed during its last period into neopythagoreanism.

Pythagoras. Bust in the Capitoline Museum, Rome

It is remarkable that the further we move away from Pythagoreanism, or the later some commentator lived, the more abundant his information, the more he knows about the teachings of Pythagoras and about himself, the more eloquent is his narration - and no less eloquent are those who could speak, because they are not separated by centuries. Poetic fiction adorned the figure of the Croton sage with semi-legendary features - the divine Pythagoras remembers his previous lives, he is surrounded by universal adoration, the Roman king Numa Pompilius is none other than his disciple. This very fiction took on a historical form, and on its basis that incorrect information grew, which was repeated for a long time and took deep roots, namely, that Pythagoras was the first to call himself a philosopher. The latest research has shown that this term came into use only with the disciples of Socrates: only in the IV century. BC words wisdom (Sofia)and sophistry are beginning to be replaced by a new expression "philosophy", that is, "love of wisdom" .

There are historians of philosophy who deny any scientific significance of the teachings of Pythagoras and see its entire meaning in religious beliefs, in the salvation of the soul. But can Pythagoreanism be considered only a sect? What is its actual meaning?

The religious beliefs of the Pythagoreans are nothing more than threads that connect this teaching with the East. These threads begin and end in knots, and these knots are difficult to untangle, if not impossible. Did Pythagoras really penetrate the secrets of the Egyptian priests and from there did he take out his conviction that the body is the grave of the soul, as well as the belief in the immortality of souls, in their judgment and their migration? Was the founder of the great Greek teaching in Babylon, and was it not under the influence of Zend-Avesta that he transferred bloodless sacrifices to Greece? Did he penetrate into India and did he borrow the theory of vision from the brahmanas? The travels of Pythagoras are one of the hobbies of Eastern explorers and the subject of attacks for all those who deny the originality of Greek philosophy. Wanting to deny borrowing, these researchers usually deny travel itself.

It is not impossible that the trading affairs of his father could have forced Pythagoras to take trips to Egypt, Babylon, and even India, but he could have taken his religious beliefs from another source. Namely: the doctrine of the immortality of the soul, attributed to Pythagoras, is already found in Hesiod, and Orphic theogonies are captured by other features that characterize his beliefs. Herodotus mentions the Egyptian origin of the Orphic and Pythagorean mysteries (II, 49, 81, 123). But whether these elements were brought into Pythagoreanism directly or through the Orphic is difficult and immaterial to decide. Equally difficult and insignificant is the question of whether Pythagoras was a student of Pherekides, the author of one of the theogonies, and whether he borrowed from there the doctrine of the transmigration of souls into demons. It is incredible that he was a student of the Milesian philosopher Anaximander, although there is a certain connection between these teachings.

But the importance of the teachings of Pythagoras lies not in religious beliefs. Its meaning is a deep philosophical worldview.

Pythagoras, among other (almost 20) works, is also credited with the Golden Poems, where there are many proverbial thoughts and other deeper, but less well-known, such as “help the one who carries his burden, and not to the one who is going to throw it off "," the value of the statue lies in its form, the dignity of a person in his actions. " The ideal of Pythagoras was the likeness of God and, according to his teaching, in order to become God, one had to become a man first. The teachings of Pythagoras had all the features of a vivid ethical theory.

The personality of the Croton sage is charming. In the stories about him, Pythagoras is surrounded by an aura of beauty, eloquence and profundity. According to sources, "he never laughed." His biography is covered in a hazy haze: born between 580 and 570. BC, resettlement from the island of Samos (off the coast of Asia Minor) to the southern Italian colony of Croton between 540 and 530, then flight to neighboring Metapont and death in his advanced years. This is all we know about the positive Pythagoras.

When order and harmony are the ideal, then nothing can be higher numbers ... According to the teachings of Pythagoras, order and harmony are realized in number. The number is therefore the essence of the world, the secret of things, the soul of the universe. The number is not a symbol because it is much larger than a symbol. And without the number, everything would merge in boundless indifference. Since a thing is a number, it is good: a lie never penetrates into a number, because a lie is disgusting and hateful to its nature, and truth is inherent in number. Pythagoras reduces virtue to numbers, and such ethics is an important part of his entire philosophical teaching.

The main thesis of Pythagoras and his followers: "Everything is number." However, what they understood by "number" is radically different from the concept of number in modern mathematics. From modern positions, a number is an abstract expression of a certain number of elements of a set, which is discrete or continuous.

In this sense, the differences between numbers are only quantitative, and all the same numbers are indistinguishable from each other.

According to Pythagoras, number is a certain even though a formal principle of the existence of a thing, but the very concept of number does not break away from the source of its origin - specific things of the world around us. In the conceptual form of number, the initial intuition that served as the basis for the appearance of this concept is clearly expressed in this sense.

Pythagoras's "number" in its arithmetic aspect is inseparable from the geometric and physical aspects. In this sense, each number takes on its own individual and unique face. Even in modern times, we distinguish numbers by quantity, distinguish such classes of numbers as even and odd numbers, and among odd numbers - prime. From the point of view of Pythagoras, each number has its own special figured structure - so, for example, the same number can correspond to a different structural arrangement of elements within this number. There are "triangular", "rectangular", "pentagonal", etc. "Numbers".

The "geometrism" of a number is a number taken taking into account the peculiarities of its internal structure. The internal structure of a number is determined, firstly, by decomposing it into factors and, secondly, by representing it as a sum of numbers. In number theory, there is a so far unproven Goldbach's theorem, according to which any even number can be represented as the sum of two primes. The difficulty of this theorem is not least due to the fact that the additive properties of numbers (associated with the operation of addition) are considered as a function of their multiplicative properties (associated with the operation of multiplication - since it is through this operation that primes are defined).

From a "geometric" standpoint, we have the question of whether it is possible to represent any "rectangle" w * n in the form of two "linear" parts, each of which no longer admits of being represented as a rectangle. If the introduction of a new dimension is considered as a transition to a new quality, then the statement of the theorem is tantamount to the statement about the internal quantitative characteristic of any quality, due to which it is possible to represent a plane object in the form of a connection of linear elements. That is, that a quality related to an integral object can be represented in the form of a structural relationship between its two main qualities, already related to parts of this whole, each of which has an internally integral character. Mathematically, this idea is expressed among the Pythagoreans in the form of the so-called "golden section". We see a similar situation in the idea of \u200b\u200bthe atom in Democritus.

Let us assume that the theorem is true, i.e. that for Vn 3 the primes pk and pi are such that 2n \u003d pk + pi. Then pk + 1 \u003d 2t \u003d pr + ps, pi-1 \u003d Pm + Pt 2n \u003d pr + ps + Pm + Pt, etc.

Thus, we see that any number of a natural series, composed of units identical to each other, is broken down into the simplest, then indivisible constituent elements - atoms with uniqueness - prime numbers.

Clarification of the structure can be meaningfully described as a transition from the general characteristics of the existence of any thing as One to the unique and inimitable specifics of the existence of a given thing. And in this idea, we also see similarities with the idea of \u200b\u200bthe atom.

In addition, the number is also inseparable from the thing that served as the basis for the initial intuition of the given number. In this sense, the structure of the elements of a given number reflects the structural structure of the thing's being, i.e. the structure of its essence, and not its form externally observed by us.

The figured structure of a number expresses not the connection between the individual parts of the body to which the given number belongs, but the connection between its most important, essential qualities. This is the structure of the being of a thing, not the structure of the visible appearance of a thing. The idea of \u200b\u200bdividing the whole into parts, each of which expresses, in turn, an integral quality, is expressed by the principle of the golden ratio: the whole refers to its larger part as this large part refers to the smaller part: x / a \u003d a / (x-a).

In this sense, the greater part turns out to be similar to the whole, and due to this similarity, the property of integrity is transferred to it. Arithmetically, this ratio is expressed in an approximate form by the ratio of two consecutive Fibonacci numbers, and the limit of this ratio with increasing n is exactly equal to the "golden ratio": lim un / un + 1 \u003d a.

Two adjacent Fibonacci numbers are at the same time relatively prime, i.e. have no common divisors. However, these numbers themselves may have divisors. When we assert that any even number can be made up of the sum of two primes, we put forward a stronger requirement - that not only these numbers, which together give us the original even number, should be mutually prime, but that these numbers themselves should be as follows. that any two numbers that give us a sum of either the first or the second numbers must be coprime. That is, each of the divided parts of the original integer can only be decomposed into a sum of mutually prime numbers. And this just means that each of the divided parts must be a prime number.

Here we see different types of proportional relationship, i.e. harmony. The simplest case is when the whole is divided into two equal parts. More complex - when this whole is divided according to the principle of the "golden ratio". An even more subtle kind of harmony is the division of the whole into two prime numbers. In the latter case, we have in full measure wholes - indivisible and unique unity, atoms - which together make up the original whole and at the same time characterize its uniqueness. Further restrictions on the separated parts no longer make sense, since we have already come to indivisible "atoms" as a result of such a separation.

In fact, taken together, these parts give us the magic number 7 - which just characterizes the structure of the relationship between the existence of the whole and its main qualities. It is no coincidence that Plato, as one of the brightest followers of Pythagoreanism, has the number of inhabitants of an ideal state equal to N \u003d 5040 \u003d 7! After all, the idea is internal

it is precisely this structure that is expressed by the number of all permutations An \u003d n! given n elements, while maintaining the identity of the geometric structure itself. This number at the same time expresses the sum of all possible sums - which give us the original number as a holistic formation. But the idea of \u200b\u200bthis sum is expressed through the introduction of a determinant.

Thus, the Pythagorean idea of \u200b\u200bthe figured arrangement of a number is realized in modern mathematics in the theory of matrices and determinants. In turn, matrix theory is a part of group theory, which acts in mathematics as a way of conceptualizing the idea of \u200b\u200bsymmetry. As applied to the physics of elementary particles, this problem is considered as the problem of obtaining a nonlinear wave equation for field operators. This wave equation is equivalent to a system of integral equations. The proper solutions of these equations are just elementary particles. "Therefore, they are mathematical forms that replace the regular bodies of the Pythagoreans." Interestingly, the eigen-solutions, for example, of the differential equation of a stretched string lead to numbers expressing harmonic vibrations of the string among the Pythagoreans.

Strictly speaking, the field equation is a mathematical representation of a whole class of symmetry types. In modern physics, in addition, symmetries associated with space and time are revealed and which are expressed in the group-theoretical properties of the basic equation. For example, this is the Lorentz group, which plays an important role in the theory of relativity. There are also other groups expressing, for example, the quantum numbers of elementary particles. It is surprising that various aspects of symmetry, which are expressed in the group structure of the field equation, very closely correspond to the experimentally observed properties of elementary particles.

As V. Heisenberg points out in this regard, "modern physics is moving forward along the same path that Plato and the Pythagoreans followed. This development of physics looks as if at the end of it a very simple formulation of the law of nature will be established, as simple as it is. hoped to see more Plato. It is difficult to indicate any solid basis for this hope of simplicity, other than the fact that until now the basic equations of physics were written in simple mathematical formulas. A similar fact is consistent with the religion of the Pythagoreans, and many physicists in this regard share their belief but so far no one has ever given really convincing evidence that this should be the case. "

Natural numbers, as AF Losev says, "are the successive potentiations of the moment of spreading multiplicity." That is, according to the ideas of the Pythagoreans, there is individuality, semantic uniqueness and irreducibility to each other of any member of the natural series. Each of its members is a kind of "atom" in the sense of Democritus. Each new number of the natural series is formed not simply by mechanical addition of the next unit, but it is a completely new whole with its own unique face.

The unit is the Cosmos as a whole, and the plurality of numbers express the hierarchy of its structure, down to individual things and their parts. The foundation of being

The cosmos is the beginning for all existing things. The number characterizes not just the visible appearance of things, it is the harmonies that permeate the entire Cosmos and every single thing in it.

Number is a combination of the limit and the infinite. That is, the thing, as it were, carves itself against the background of infinity, thereby forming a special and unique set of qualities. The idea of \u200b\u200bnumber initially presupposes the idea of \u200b\u200ba closed, integral quality. Each number is a kind of surmounted and transformed infinity, infinity captured and held in a finite form. The principle of quantitative change expresses only one, the most schematic and abstract kind of difference. The difference between one quality and another is another kind of difference. Each new kind of distinction is more subtle than the previous one. For example, if good and evil from the point of view of logic are equal opposites, then from the point of view of ethics we have here an asymmetry.

Similarly, in physics, we discover more and more complex and subtle types of symmetry. Group theory is a purely mathematical way of defining invariants - quantities that remain constant under various kinds of group transformations. This means that such an object to some extent turns out to be independent of the choice of the corresponding theoretical scheme or method of description (for example, the choice of the coordinate system). Thus, within mathematics itself, there are ways to determine the truth of its statements, the principles of selection among possible solutions of those that correspond to reality. V. Heisenberg asserts, in particular, that modern theoretical physics actually stands on the ideas of a kind of Platonism. “Plato, - said W. Heisenberg, - perceived the essential elements of the doctrine of atoms. Four elements - earth, water, air and fire - corresponded to four types of the smallest particles. These elementary particles were, according to Plato, the basic mathematical structures of the highest symmetry. The smallest particles of the earth element were represented by cubes, the water element - icosahedrons, the air element - octahedrons, and, finally, the smallest particles of the fire elements were represented in the form of tetrahedrons.But these elementary particles, according to Plato, were not indivisible. They could decompose into triangles and again Thus, for example, from two elementary particles of air and one elementary particle of fire, an elementary particle of water was built. The triangles themselves were not matter, they were only a mathematical form. Consequently, in Plato, elementary particles were not just something given, unchanging and indivisible; they demanded more explanation, and the question of elementary particles was reduced to Plato to mathematics. The last basis of phenomena was not matter, but a mathematical law, symmetry, mathematical form. "

For example, if from the point of view of one type of symmetry we have identity and indistinguishability, then from the point of view of a more subtle and deeper symmetry we get inequality - and thus the desired principle of selection turns out to be in our hands: correct, i.e. corresponding to reality, there are solutions corresponding to symmetries of a more fundamental type. Mathematics in this sense turns out to be not just a language for describing reality or a method used to cognize it, which is indifferent to the nature of the objects being cognized.

It has ontological content. Number is not only a characteristic of certain things, but also an internal characteristic of a similarly arranged human soul.

But the soul acquires this inner structure under the influence of harmonies that permeate the whole world. As Plato says, “since day and night, the cycles of months and years, the equinox and the solstice are visible, our eyes opened the number to us, gave us the concept of time and prompted us to explore the nature of the universe, so that we, observing the circulation of the mind in the sky, would benefit from the rotation of our thinking ". It is due to the unity of man and the world that the foundations of knowledge and the foundations of being are inseparable. The original Pythagorean idea of \u200b\u200bsymmetry as a criterion of truth is thus in fact the concretization of the idea of \u200b\u200bthe interconnection of all the phenomena of the world and the existence of some unified universal principles connecting these phenomena into a harmonious Cosmos. The principles of symmetry arise as a result of the clarification of the original sensory intuitions of truth. And therefore, the mere clarification of the mathematical ideas we use allows us to obtain a description of the real, and not only the possible world.

It is indicative that the principle of the golden ratio can be taken as the basis of the so-called irrational number system - that is, that any natural number can be represented as the sum of a finite number of integer powers of a. For example, 2 \u003d a1 + a-2, 5 \u003d a3 + a-1 + a-4.

That is, we get an irrational basis of the natural series. However, an irrational number is not pure chaos as a lack of order. It's just a very complicated order. For example, the formula V2 immediately sets the rule for obtaining all digits of a number in its decimal value. The same is true for transcendental numbers. It is not for nothing that the number n, expressing the idea of \u200b\u200ba circle, was endowed with divine properties, and the world as a whole was understood as a ball, expressing the idea of \u200b\u200bperfection to the maximum extent.

So, on the one hand, we see the idea of \u200b\u200bvarious degrees and ways of expressing the very idea of \u200b\u200bDifference, and on the other, the idea of \u200b\u200ba synthesis of the heterogeneous. Addition is the simplest kind of connection, the synthesis of two wholes - within one dimension, one quality. Multiplication already expresses the idea of \u200b\u200ba synthesis of individual qualities. You can also talk about the combination of the spiritual and the physical in general - both in the being of a person and in the being of the world. Finally, the idea of \u200b\u200bthe unity of man and the world expresses the idea of \u200b\u200ba synthesis of the most heterogeneous. Universal discrimination and universal synthesis is the most complete expression of the Pythagorean idea of \u200b\u200bNumber.

In this sense, the Pythagorean idea of \u200b\u200bthe qualitative irreducibility of one number to another sets the boundaries of any possible mathematics, establishes an unattainable limit for mathematics of its development and change, associated with the isolation of the individual characteristics of each number, as well as with the deepening of the very understanding of what a number is. This is precisely what its enduring significance and eternal relevance consists in.

The philosophy of Pythagoras (2nd half of the 6th - early 5th centuries BC) developed as esoteric knowledge. Pythagorean circles were brotherhoods, moreover secret, associated with numerous taboos and prescriptions. The main provisions of this philosophy were attributed to the Teacher himself. Although it is still difficult to establish what belongs to Pythagoras and what to his students. What allows Pythagoras to be attributed to the first philosophers? This is undoubtedly his teaching about number as the substance of all things. Number has a material and substantial character, it is observable, spatially, corporeal and at the same time retains all the properties of intelligible principles.

The numbers of the Pythagorean tradition are both mathematical quantities, and physical bodies, and living beings. Each number is the essence of the substance of our real world. Each number brings something into the world: a monad (one) brings order, certainty, a dyad (two) - uncertainty, bifurcation.

The main meaning of numbers is that they are in the human soul. Number preserves the objectivity of the world, it is the world itself and what forms the basis of our mind, our thinking abilities. In Pythagoreanism, it is not the external world that comes to the fore, although it is inevitably present, but the internal world of the human soul. A number is primarily a state of mind. Number is what is born and lives in the soul. Hence follows an interest in the numerical justification of the external world, in cosmology and cosmogony, but the doctrine of the soul is also connected with this.

This side of the teaching originates in Orphism. The presence of two principles in a person is recognized: light and dark. The light beginning is the soul, the dark one is the human body itself. The body is the prison of the soul. It is it, the body, that interferes with the natural state of the soul, according to the teachings of Pythagoras, the path to the salvation of the soul lies through the achievement of harmony, which is inherent in the whole world and must be restored in the individual human soul. Therefore, it is necessary to achieve the elimination of the affects of anger, despondency, rage and learn to master your feelings, giving preference to reason.

Philosophical views of Heraclitus.

The figure of Heraclitus of Ephesus (born c. 544 BC - unknown year of death) is one of the most significant in world philosophy. Heraclitus is the greatest dialectician. His substance is fire. The world is an eternally living fire. We are talking about a mobile, dynamic element. Heraclitus emphasizes that the essence of the world is active, that the nature of being is mobile. How is the idea of \u200b\u200bsubstance combined with the idea of \u200b\u200bmobility? Fire is a symbol of mobility, variability, disappearance and birth of the world.

The philosophy of Heraclitus is undoubtedly dialectical: the world "governed" by the Logos is one and changeable, nothing in the world repeats itself, everything is transient and one-time, and the main law of the universe is struggle, everything is born through struggle and by necessity, Heraclitus says.

Philosophy of the Eleats.

The philosophy of Parmenides (early 6th century BC) breaks with the (physical) physiological tradition, depriving it of the status of truth. Truth is achieved on the paths of cognition corresponding to the sum of requirements that are realized in the process of thinking, or rather, the true path of cognition is one, and acts of sensory perception that do not fit into its framework form a foggy world of opinions. Parmenides was the first to express the simple idea that with all the many opinions, there is one truth. At the same time, he focuses on the thought process itself: in the center of his attention is the neutral thinkable, and not the thinking subject. But he does not turn away from a person, moreover, he substantiates the orientation of a person to being and, for this purpose, points out some requirements for his thinking: if a person can think, then what does it mean to “think”. At the same time, Parmenides places special emphasis on the fact that it is necessary to think, highlighting the sum of the rules of thinking, and speaks of the need as a mediation of true thinking, that is, its dependence on something third, which allows you to stay on the path of truth and thereby avoid ways of lying.

So, thinking (as the ability to contemplate an object, to speak out about it) is subject to the sum of requirements, the main of which looks like a tautology: in order to think, it is necessary to remain in the field of pure thought, to solve the problem of being by reason, without resorting to the usual experience of the senses.

What is being for Parmenides? The most important definition of being is its comprehensibility by reason: that which can be cognized only by reason is being, but being is inaccessible to feelings. Therefore, "the same thing is thought and that about which thought exists" - in this position of Parmenides, the identity of being and thinking is affirmed. Being is what is always, what is one and indivisible, what is motionless and consistent, "like the thought of it." Thinking is the ability to comprehend unity in non-contradictory forms, the result of thinking is knowledge (episteme). Sensory perception deals with a plurality of different things and signs, and about the world of sensually perceived single objects surrounding a person. A person can only have an opinion (doxa) - an ordinary, everyday idea, opposed to knowledge as a result of the comprehension of the one.

Zeno of Elea, defending and justifying the views of his teacher and mentor Parmenides, rejected the plurality of things and their movement. Zeno strove to show that multiplicity and movement cannot be thought without contradiction, therefore they are not the essence of being, which is one and motionless. Zeno's reasoning was called "aporia" (literally "difficulty", "hopeless situation").

Aporia is an intractable problem in the contradiction between the data of experience and mental analysis. The most famous are 4 Zeno's aporias against the movement: "Dichotomy", "Achilles and the Turtle", "Arrow" and "Stadium".

The first aporia says that movement cannot start, because a moving object must first reach half of the path before it reaches the end, but to reach half, it must reach half of half ("dichotomy" - literally "halving" ), and so on - ad infinitum; that is, to get from one point to another, you need to go through an infinite number of points, and this is absurd.

The second aporia says that the movement can never end: Achilles will never catch up with the turtle, because when he comes to the point, the turtle will move away from its "start" for such a part of the initial distance between Achilles and itself, so much its speed is less than the speed of Achilles, - and so on ad infinitum. The worldview meaning of both aporias (according to Zeno) is as follows: if space is infinitely divisible, then the movement can neither begin nor end.

But the meaning of the third and fourth aporias is that motion is impossible even if space is discontinuous. And this means that movement cannot be thought without contradiction, which means that Parmenides is right.

And in the middle of the 5th century. broke out into a catastrophe: in Croton, many Pythagoreans were killed and burned in the house where they gathered; the rout was repeated in other places as well. The survivors were forced to flee, carrying with them the teachings and mysteries of their union. These mysteries gave the union the opportunity to exist even when it lost its former political and philosophical significance. By the end of the 5th century. there is a revival of the political influence of the Pythagoreans in Magna Graecia: Archytas of Tarentum attains great political importance in Tarentum as a military leader and statesman. From the IV century. pythagoreanism falls into decay, and his teachings are absorbed by Platonism.

According to legend, Pythagoras himself did not leave a written presentation of his doctrine, and Philolaus is considered the first writer to give an exposition of the Pythagorean doctrine. The teachings of the early Pythagoreans are known to us from the testimonies of Plato and Aristotle, as well as from the few fragments of Philolaus that are recognized as authentic. Under such conditions, it is difficult to reliably separate the original essence of the Pythagorean teaching from the later layers.

The Pythagorean Union as a Religious Community

Fyodor Bronnikov. Hymn of the Pythagoreans to the sun

There is reason to see in Pythagoras the founder of the mystical union, who taught his followers new cleansing rites. These rites were associated with the doctrine of the transmigration of souls, which can be attributed to Pythagoras on the basis of the testimony of Herodotus and Xenophanes; it is also found in Parmenides, Empedocles and Pindar, who were under the influence of Pythagoreanism.

Did Pythagoreanism give liberation from this "cycle of birth" even to the soul of a philosopher? Gold tablets of the 4th century, found in the graves near Turia - an area that once served as a haven for the Pythagoreans - indicate the possibility of such liberation.

A number of bizarre prescriptions and prohibitions of the Pythagoreans undoubtedly date back to ancient times. Of these prohibitions, the prohibition on eating beans became known most of all, because of which, according to one legend, Pythagoras himself died. The reason for this prohibition was unknown already in antiquity.

The Pythagoreans were also known in antiquity for their vegetarianism associated with the doctrine of the transmigration of souls.

According to tradition, the followers of Pythagoras were divided into acusmatics ("listeners") and mathematicians ("students"). Akusmatists dealt with the religious and ritual aspects of teaching, mathematics - with the study of four Pythagorean "math": arithmetic, geometry, harmonics and spherics. Acumatics did not consider mathematicians "real Pythagoreans", but said that they originated from Hippias, who changed the original Pythagorean tradition, revealed secrets to the uninitiated and began teaching for a fee.

Philosophy of the Pythagoreans

Pythagoras was the first thinker who, according to legend, called himself a philosopher, that is, "a lover of wisdom." He was also the first to call the universe space, that is, "beautiful order." The subject of his teaching was the world as a harmonious whole, subject to the laws of harmony and number.

The basis of the subsequent philosophical teaching of the Pythagoreans was formed by the categorical pair of two opposites - the limit and the limitless. “Infinite” cannot be the single beginning of things; otherwise nothing definite, no "limit" would be conceivable. On the other hand, the "limit" also presupposes something that is determined by it. Hence the conclusion that “nature, existing in space, is harmoniously combined from the limitless and defining; this is how the whole cosmos and everything in it is arranged ”(words of Philolaus). How do these opposite principles agree? This is a mystery that is fully accessible only to divine reason; but it is clear that they must agree, that there must be harmony connecting them, otherwise the world would disintegrate.

There were Pythagoreans who limited themselves to this general position; others compiled a table of 10 opposites - categories under which all things were summed up. Aristotle gives this table in his "Metaphysics" (I, 5):

limit - limitless
odd - even
one - many
right - left
male - female
rest is movement
straight - curved
light is darkness
good evil
square - elongated rectangle

World harmony, which is the law of the universe, is unity in many and many in unity - έν καί πολλά ... How to think about this truth? The immediate answer to this is number: it unites many, it is the beginning of every measure. Experiments on a monochord show that number is the principle of sound harmony, which is determined by mathematical laws. Isn't sound harmony a special case of universal harmony, as it were, its musical expression? Astronomical observations show us that celestial phenomena, with which all the major changes in earthly life are associated, occur with mathematical correctness, repeating themselves in precisely defined cycles.

“The so-called Pythagoreans, having taken up the mathematical sciences, were the first to push them forward; nurtured by these sciences, they recognized the mathematical principles as the beginning of all that exists. Of these principles, of course, numbers are the first. They saw in numbers many analogies or similarities with things ... so that one property of numbers appeared to them as justice, another as a soul or mind, another as an auspicious occasion, etc. Then they suggested in numbers the properties and relations of musical harmony, and since all other things were by their nature similar to numbers, and numbers were the first of all nature, they recognized that the elements of number are the elements of everything, and that the whole sky is harmony and number ”(Aristotle, Met., I, 5).

Thus, the Pythagorean numbers have not a simple quantitative meaning: if for us a number is a certain sum of units, then for the Pythagoreans it is, rather, the force that sums these units into a certain whole and imparts certain properties to it. One is the cause of unity, two is the cause of bifurcation, division, four is the root and source of the whole number (1 + 2 + 3 + 4 \u003d 10). Apparently, the fundamental opposition of even and odd was seen at the basis of the doctrine of number: even numbers are multiples of two, and therefore “even” is the beginning of divisibility, bifurcation, discord; "Odd" signifies opposite properties. Hence, it is clear that numbers can also have moral forces: 4 and 7, for example, as averages proportional between 1 and 10, are numbers, or principles, of proportionality, and the next, and harmony, health, rationality.

Pythagorean cosmology and astronomy

In the cosmology of the Pythagoreans, we meet with the same two basic principles of limit and infinity. The world is a limited sphere running in infinity. “The initial unity, arising from an unknown source,” says Aristotle, “draws into itself the nearest parts of infinity, limiting them by the force of the limit. Breathing into itself the parts of the infinite, the unity forms in itself a definite empty space or definite gaps that split the original unity into separate parts - extended units ( ὡς όντος χωρισμοϋ τινος τών ἐφεξής ) ". This view is undoubtedly original, since already Parmenides and Zeno are polemicizing against him. Inhaling the boundless emptiness, the central unity gives birth to a series of celestial spheres and sets them in motion. According to Philolaus, "the world is one and began to form from the center."

In the center of the world there is fire, separated by a series of empty intervals and intermediate spheres from the outermost sphere that encompasses the universe and consists of the same fire. The central fire, the hearth of the universe, is Hestia, the mother of the gods, the mother of the universe and the connection of the world; the upper part of the world between the firmament and peripheral fire is called Olympus; under it is the cosmos of planets, sun and moon. Around the center “10 divine bodies are dancing in circles: the sky of fixed stars, five planets, behind them the Sun, under the Sun - the Moon, under the Moon - the Earth, and under it - the counter-earth ( ἀντίχθων ) "- a special tenth planet, which the Pythagoreans took for round-the-clock counting, and maybe for explaining solar eclipses. The sphere of fixed stars revolves most slowly; more rapidly and with a constantly increasing speed as it approaches the center - the spheres of Saturn, Jupiter, Mars, Venus and Mercury.

The planets revolve around the central fire, always facing it with the same side, which is why the inhabitants of the earth, for example, do not see the central fire. Our hemisphere perceives the light and warmth of the central fire through the solar disk, which only reflects its rays, not being an independent source of heat and light.

The Pythagorean doctrine of the harmony of spheres is peculiar: transparent spheres, to which the planets are attached, are separated by intervals that relate to each other as harmonic intervals; celestial bodies sound in their motion, and if we do not distinguish between their consonance, it is only because it is heard incessantly.

Pythagorean arithmetic

The Pythagoreans considered the properties of numbers, among which the most important were even, odd, even-odd, square and non-square, they studied the compilation of arithmetic progressions and new number series arising from the successive sums of their members. So, the successive addition of the number 2 to itself or to one and to the results obtained then gave in the first case a series of even numbers, and in the second - a series of odd ones. Successive summations of the members of the first row, consisting in adding each of them to the sum of all the members that preceded it, gave a series of heteromek numbers representing the product of two factors that differ from one another by one. The same sums of the terms of the second row gave a series of squares of consecutive natural numbers.

Pythagorean geometry

Of the geometric works of the Pythagoreans, the famous Pythagorean theorem is in the first place. The proof of the theorem was to be the result of the work that required a considerable period of time both by Pythagoras himself and by other mathematicians of his school, which began on an arithmetic basis. A member of a series of odd numbers, which is always the difference between two corresponding members of a series of square numbers, could itself be a square number: 9 \u003d 25 - 16, 25 \u003d 169 - 144, ... The content of the Pythagorean theorem was thus first discovered by rational right-angled triangles with a leg , expressed as an odd number. At the same time, Pythagorov should have revealed the method of forming these triangles, or their formula (n is an odd number expressing the smaller leg; (n 2 - 1) / 2 - the larger leg; (n 2 - 1) / 2 + 1 - hypotenuse ).

The question of a similar property for other right-angled triangles also required the comparison of their sides. In this case, the Pythagoreans for the first time had to meet with incommensurable lines. We have not come down to any indication of either the original general proof or the way in which it was found. According to Proclus, this initial proof was more difficult than Euclid's in the Beginnings and was also based on a comparison of areas.

The Pythagoreans were still engaged in the issue of the so-called "application" ( παραβάλλειν ) areas, that is, building on a given segment of a parallelogram with a given apex angle, having a given area. The nearest development of this issue consisted in the construction on a given segment of a straight rectangle having a given area, under the condition that ( ἔλλειψις ) or missing ( ὑπερβολή ) square.

The Pythagoreans were the first to give a general proof of the theorem on the equality of the interior angles of triangles to two straight lines; they were familiar with the properties and construction of regular 3-, 4-, 5-, and 6-gons.

In stereometry, the Pythagoreans studied regular polyhedra. The Pythagoreans' own research added a dodecahedron to them. Studying the methods of forming the solid angles of polyhedra should have led the Pythagoreans directly to the theorem that “a plane about one point is completely filled with six equilateral triangles, four squares or three regular hexagons, so that it becomes possible to decompose any whole plane into figures of each of these three childbirth ".

Pythagorean harmonica

Raphael Santi. Pythagoras (detail Athenian school). On the black board you can see the image of Pythagorean harmony - a system in which an octave is composed of a quint and a quarter.

All the information that has come down to our time about the emergence in ancient Greece of the mathematical doctrine of musical harmony definitely associates this emergence with the name of Pythagoras. His achievements in this area are summarized in the following passage from Xenocrates, which has come down to us through Porphyry:

“Pythagoras, as Xenocrates says, also discovered that in music intervals are inseparable from number, since they arise from the correlation of quantity with quantity. He investigated, as a result of which there are consonant and dissonant intervals and everything harmonious and inharmonious. "

In the field of harmonics, Pythagoras made important acoustic research, which led to the discovery of the law that all consonant musical intervals are determined by the simplest numerical ratios 2/1, 3/2, 4/3. So, half of the string sounds in an octave, 2/3 in a fifth, 3/4 in a fourth with a whole string. Thus, the structure of harmony is set by four mutually prime numbers 6, 8, 9, 12, where the extreme numbers form an octave among themselves, the numbers taken in one - two fifths, and the edges with neighbors - two quarts.

“Harmony is a system of three consonances - fourths, fifths and octaves. The numerical proportions of these three consonances are within the four numbers indicated above, that is, within one, two, three and four. Namely, the consonance of a fourth is in the form of a super-third ratio, a fifth - a one and a half, and an octave - a double. Hence the number four, being the super-third of three, since it is composed of three and its third part, embraces the consonance of a fourth. The number three, being one and a half of two, since it contains two and its half, expresses the consonance of a fifth. The number four, being double in relation to two, and the number two, being double in relation to one, determine the consonance of the octave "(Sextus Empiricus, Against logicians, I, 94–97).

The successors of acoustic research, as well as representatives of the desire that arose in the Pythagorean school for a theoretical substantiation of musical harmony were Las Hermione and Hippas Metaponta, who performed many experiments both on strings of various lengths and stretched by various weights, and on vessels filled with water to different heights.

The Pythagorean harmonic concept was embodied in the idea of \u200b\u200ba pure diatonic scale, tuned in only one consonant intervals - octaves and fifths. Here, a discovery was made concerning the fact that a whole tone - the difference between a fifth and a fourth - does not fit into an octave an integer number of times: an octave is equal to six whole tones with some excess, the so-called pythagorean comma.

An outstanding music theorist of the Pythagorean school was Archytas of Tarentum, who tried to bring a mathematical basis to other harmonic systems used in the ancient Greek music of his time.

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Pythagoras, born about 580-570 BC on the island of Samos, the son of a gem cutter or merchant Mnesarchus, was a man gifted with remarkable physical beauty and great mental power.

In the news that has come down to us, his life is clothed with a mythical and mystical fog. In his youth, Pythagoras diligently studied mathematics, geometry and music; according to Heraclitus, there was no person who worked so much and with such success to research the truth and acquired such a vast knowledge. There is news that he studied philosophy with Ferekides. To expand his knowledge, Pythagoras traveled for a long time: he lived in European Greece, Crete, Egypt; tradition says that the priests of the Egyptian religious center, Heliopolis, initiated him into the mysteries of their wisdom.

Pythagoras. Bust in the Capitoline Museum, Rome. Photo by Galilea

When Pythagoras was about 50 years old, he moved from Samos to the southern Italian city of Croton to engage in practical activities there, for which there was no room on Samos, which fell under the dominion tyrant Polycrates... The citizens of Croton were courageous people who did not succumb to the temptations of luxury and voluptuous effeminacy, who loved to do gymnastics, strong in body, active, striving to glorify themselves with brave deeds. Their way of life was simple, their manners were strict. Pythagoras soon gained between them many listeners, friends, adherents to his teachings, preaching self-control, directed towards the harmonious development of the mental and physical strength of a person, his majestic appearance, imposing manners, the purity of his life, his self-control: he ate only honey, vegetables, fruits, bread. Like the Ionian philosophers (Thales, Anaximander and Anaximenes), Pythagoras was engaged in research about nature, about the structure of the universe, but in his research he followed a different path, studied the quantitative relations between objects, tried to formulate them in numbers. Having settled in a Dorian city, Pythagoras gave his activities a Dorian, practical direction. That system of philosophy, which is called Pythagorean, was developed, in all likelihood, not by himself, but by his students - the Pythagoreans. But her main thoughts belong to him. Already Pythagoras himself found a mysterious meaning in numbers and figures, he said that “ number is the essence of things; essence of an object - its number”, Put harmony as the supreme law of the physical world and moral order. There is a legend that he brought the hecatomb to the gods when he discovered a geometric theorem called by his name: "in a right-angled triangle, the square of the hypotenuse is equal to the sum of the squares of the legs."

Pythagoras and the school of the Pythagoreans made bold, although in many ways fantastic, attempts to explain the structure of the universe. They believed that all celestial bodies, including the earth itself, which has a spherical shape, and another planet, which they called the opposite of the earth, move in circular orbits around the central fire, from which they receive life, light and warmth. The Pythagoreans believed that the orbits of the planets are among themselves in proportions corresponding to the intervals of tones of the seven-stringed cithara and that the harmony of the universe arises from this proportionality of the distances and times of revolution of the planets; the goal of human life, they set the soul to acquire a harmonious mood, through which it becomes worthy to return to the realm of eternal order, to the god of light and harmony.

The philosophy of Pythagoras soon received a practical direction in Croton. The glory of his wisdom attracted many disciples to him, and he formed of them piphagorean union, whose members were raised to purity of life and to the observance of all moral laws ”by religious initiation rites, moral commandments and the adoption of special customs.

According to the legends that have come down to us about the union of the Pythagoreans, it was a religious and political society, which consisted of two classes. The highest class of the Pythagorean union was the Esoterics, the number of which could not exceed 300; they were initiated into the secret teachings of the union and knew the ultimate goals of its aspirations; the lower class of the union were the Exotericists, uninitiated into the sacraments. Acceptance into the category of the Pythagorean Esotericists was preceded by a severe test of the life and character of the student; during this trial, he had to remain silent, examine his heart, work, obey; I had to accustom myself to renunciation of the bustle of life, to asceticism. All members of the Pythagorean union led a moderate, morally strict lifestyle according to the established rules. They were going to do gymnastic exercises and mental labors; dined together, did not eat meat, did not drink wine, performed special liturgical rites; had symbolic sayings and signs, but with which they recognized each other; they wore linen clothes of a special cut. There is a legend that the common property was introduced in the school of the Pythagoreans, but it seems that this is a fiction of later times. The fabulous embellishments that obscure the news of the life of Pythagoras extend to the union founded by him. Unworthy members were shamefully excluded from the union. The moral commandments of the union and the rules of life for its members were set forth in the "Golden Sayings" of Pythagoras, which were probably symbolic and mysterious. The members of the Pythagorean union were devoted to their teacher with such reverence that the words: "he said it himself" were considered undeniable proof of the truth. Inspired by the love of virtue, the Pythagoreans constituted a brotherhood in which the human personality was completely subordinated to the goals of society.

The foundations of Pythagorean philosophy were number and harmony, the concepts of which coincided for the Pythagoreans with the ideas of law and order. The moral commandments of their union had as their purpose to establish law and harmony in life, therefore they intensively studied mathematics and music, as the best means for delivering a calm, harmonious mood to the soul, which was for them the highest goal of education and development; diligently engaged in gymnastics and medicine to give the body strength and health. These rules of Pythagoras and solemn service to Apollo, the god of purity and harmony, corresponded to the general concepts of the Greek people, whose ideal was "a handsome and kind man", and in particular they corresponded to the dominant direction of the citizens of Croton, who have long been famous as athletes and doctors. The Pythagorean moral and religious teachings contained many details that strangely contradicted the claims of the Pythagorean system to mathematical solidity; but the energetic, deep desire of the Pythagoreans to find a “unifying connection”, “the law of the universe,” to bring human life into harmony with the life of the universe, had beneficial results in practical terms.

Members of the school of the Pythagoreans strictly fulfilled the duties that were prescribed to them by the "golden sayings" of the teacher; they not only preached, but in practice observed piety, reverence and gratitude to parents and benefactors, obedience to the law and authorities, loyalty to friendship and marriage, loyalty to the given word, abstinence in pleasures, moderation in everything, meekness, justice and other virtues. The Pythagoreans tried with all their might to curb their passions, to suppress all impure impulses in themselves, “to preserve harmonious peace in the soul; they were friends of order and law. They behaved peacefully, judiciously, tried to avoid any deeds and words that disturb the public silence; from their manners, from the tone of the conversation, it was evident that they were people who enjoyed an imperturbable spiritual peace. The blissful consciousness of the inviolability of peace of mind was the happiness that the Pythagorean aspired to. At the end of the evening, getting ready to go to bed, the Pythagorean was obliged to play the cithara so that its sounds would give his soul a harmonious mood.

Hymn of the Pythagoreans to the sun. Artist F. Bronnikov, 1869

It goes without saying that the union to which belonged the noblest and most influential people of Croton and other Greek cities of southern Italy, could not but have an influence on public life, on state affairs; according to the concepts of the Greeks, the dignity of man consisted in his civil activity. Indeed, we find that not only in Croton, but also in Locri, Metapont, Tarentum, and in other cities, the members of the Pythagorean school gained influence in the administration of state affairs, that in the meetings of the government council they usually prevailed because they acted with one accord. The Pythagorean Union, being a religious and moral society, was at the same time a political club ( heterogeneity); they had a systematic mindset about domestic politics; they formed a complete political party. By the nature of the teachings of Pythagoras, this party was strictly aristocratic; they wanted the aristocracy to rule, but the aristocracy of education, not nobility. In an effort to transform state institutions according to their own concepts, to push the old noble families out of government and to prevent democracy that demanded a political disposition from participating in government, they incurred the enmity of both noble families and democrats. It seems, however, that the resistance from the aristocrats was not very stubborn, partly because the teaching of the Pythagoreans itself had an aristocratic direction, partly because almost all Pythagoreans belonged to aristocratic families; however, Cylon, who became the leader of their opponents, was an aristocrat.

The Democratic Party strongly hated the Pythagoreans for their arrogance. Proud of his education, his new philosophy, which showed them heavenly and earthly affairs in a different light than they were presented by popular belief. Proud of their virtues and their rank of initiates in the sacraments, they despised the crowd that took the "ghost" for the truth, irritated the people by being alienated from it and speaking in a mysterious language incomprehensible to him. We have received sayings attributed to Pythagoras; perhaps they do not belong to him, but they express the spirit of the Pythagorean union: “Do what you think is good, even if it exposes you to the danger of exile; the crowd is not able to correctly judge noble people; despise her praise, despise her censure. Respect your brothers as gods, and consider other people a despicable rabble. Fight the democrats irreconcilably. "

With this mentality of the Pythagoreans, their death as a political party was inevitable. The destruction of the city of Sybaris resulted in a catastrophe that destroyed the Pythagorean union. The houses of their public meetings were burned everywhere, they themselves were killed, or driven out. But the teachings of Pythagoras survived. Partly by its inner dignity, partly by the inclination of people to the mysterious and miraculous, it had adherents in later times. The most famous of the Pythagoreans of the following centuries were Philolaus and Archyt, contemporaries of Socrates, and Lysis, teacher of the great Theban commander Epaminondas.

Pythagoras died about 500; legend says that he lived to be 84 years old. The adherents of his teaching considered him a holy man, a miracle worker. The fantastic thoughts of the Pythagoreans, their symbolic language and strange expressions gave rise to the Attic comedians laugh at them; in general, they carried to the extreme the panache of learning, for which Heraclitus condemned Pythagoras. Their wonderful tales of Pythagoras clothed his life with a mythical mist; all news about his personality and activity is distorted by fabulous exaggerations.

The religious beliefs of the Pythagoreans are nothing more than threads that connect this teaching with the East. These threads begin and end in knots, and these knots are difficult to untangle, if not impossible. Did Pythagoras really penetrate the secrets of the Egyptian priests and from there did he take out his conviction that the body is the grave of the soul, as well as the belief in the immortality of souls, in their judgment and their migration? Was the founder of the great Greek teaching in Babylon and was not influenced by Zend-Avesta did he transfer bloodless sacrifices to Greece? Did he penetrate into India and did he borrow the theory of vision from the brahmanas? The travels of Pythagoras are one of the hobbies of Eastern explorers and the subject of attacks for all those who deny the originality of Greek philosophy. Wanting to deny borrowing, these researchers usually deny travel itself.

But the importance of the teachings of Pythagoras lies not in religious beliefs. Its meaning is a deep philosophical worldview.

Pythagorean doctrine of the universe

Like the Ionian sages, the Pythagorean school tried to explain the origin and structure of the universe. Thanks to their diligent studies of mathematics, the Pythagorean philosophers formed concepts about the structure of the world that were closer to the truth than those of other ancient Greek astronomers. Their ideas about the origin of the universe were fantastic. The Pythagoreans spoke of him as follows: in the center of the universe a "central fire" was formed; they called him the monad, the "one," because he is the "first heavenly body." He is the “mother of the gods” (celestial bodies), Hestia, the hearth of the universe, the altar of the universe, her guardian, the abode of Zeus, his throne. By the action of this fire, according to the opinion of the Pythagorean school, other celestial bodies were created; he is the center of the power that keeps the order of the universe. It attracted to itself the nearest parts of the "infinite", that is, the nearest parts of the substance located in the infinite space; gradually expanding, the action of this force of him, introducing the infinite into the limits, gave the structure of the universe.

Ten celestial bodies revolve around the central fire, from west to east; the most distant of them is the sphere of fixed stars, which the Pythagorean school considered one continuous whole. The celestial bodies closest to the central fire are the planets; there are five of them. Further from him are located, according to the Pythagorean cosmogony, the sun, moon, earth and a heavenly body, which is the opposite of the earth, antichthon, "anti-earth". The shell of the universe is the "fire of a circle", which the Pythagoreans needed in order for the circumference of the universe to be in harmony with its center. The central fire of the Pythagoreans, the center of the universe, constitutes the basis of order in it; he is the norm of everything, the connection of everything in her. The earth revolves around the central fire; its shape is spherical; you can live only on the upper half of its circumference. The Pythagoreans believed that she and other bodies move in circular paths. The sun and the moon, balls of glass-like substance, receive light and warmth from the central fire and transmit it to the earth. She revolves closer to him than they, but between him and her the counter-earth revolves, having the same path and the same period of its rotation as she; that is why the central fire is constantly closed by this body from the earth and cannot give light and warmth directly to it. When the earth in its daytime rotation is on the same side of the central fire as the sun, then it is day on the earth, and when the sun and it are on different sides, then it is night on the earth. The path of the earth is tilted relative to the path of the sun; by this correct information the Pythagorean school explained the change of the seasons; moreover, if the path of the sun were not inclined relative to the path of the earth, then at each of its daily cycles the earth would pass directly between the sun and the central fire, and every day would produce a solar eclipse. But with the inclination of her path relative to the paths of the sun and moon, she only occasionally happens on a straight line between the central fire and these bodies, and covering them with her shadow, produces their eclipses.

In Pythagorean philosophy, it was believed that heavenly bodies are like the earth, and like it, they are surrounded by air. There are both plants and animals on the moon; they are much larger and more beautiful than on earth. The time of revolution of celestial bodies near the central fire is determined by the size of the circles they pass. The earth and counter-earths bypass their circular paths per day, and the moon needs 30 days for this, the sun, Venus and Mercury need a whole year, etc., and the starry sky makes its circular revolution in a period whose duration was not precisely determined by the Pythagorean school , but was thousands of years old, and which was called the "great year." The invariable correctness of these movements is conditioned by the action of numbers; therefore, number is the supreme law of the structure of the universe, the power that rules it. And the proportionality of numbers is harmony; therefore, the correct movement of celestial bodies should create a harmony of sounds.

Harmony of the spheres

This was the basis of the teaching of the Pythagorean philosophy about the harmony of the spheres; it said that "the celestial bodies, by their rotation around the center, produce a number of tones, the combination of which makes an octave, harmony"; but the human ear does not hear this harmony, just as the human eye does not see the central fire. The harmony of the spheres was heard only by one of all mortals, Pythagoras. For all the fantasticness of its details, the teaching of the Pythagorean school about the structure of the universe is, in comparison with the concepts of previous philosophers, great astronomical progress. Previously, the diurnal course of changes was explained by the movement of the sun near the earth; the Pythagoreans began to explain it by the movement of the earth itself; from their concept of the nature of its daily circulation, it was easy to move to the concept that it rotates about its axis. It was only necessary to discard the fantastic element, and the truth was obtained: the counter-earth turned out to be the western hemisphere of the globe, the central fire turned out to be located in the center of the globe, the rotation of the earth near the central fire turned into a rotation of the earth about the axis.

Pythagorean teaching on the transmigration of souls

The doctrine of numbers, of the combination of opposites, replacing disorder with harmony, served in the Pythagorean school of philosophy as the basis for a system of moral and religious duties. As harmony reigns in the universe, so it should reign in the individual and state life of people: unity here should rule over all heterogeneities, the odd, masculine element, over the even, feminine, calm over movement. Therefore, the first duty of man is to bring under harmony all the opposite drives of the soul, to subordinate instincts and passions to the dominion of reason. According to Pythagorean philosophy, the soul is connected to the body and the punishment for sins is buried in it, as in a dungeon. Therefore, she should not self-authoritatively free herself from him. She loves him as long as she is united with him, because she receives impressions only through the senses of the body. Freed from him, she leads a disembodied life in a better world.

But the soul, according to the teachings of the Pythagorean school, enters this better world of order and harmony only if it has established harmony in itself, if it has made itself worthy of bliss by virtue and purity. An inharmonious and impure soul cannot be accepted into the kingdom of light and eternal harmony ruled by Apollo; she must return to earth for a new journey through the bodies of animals and people. So, the Pythagorean school of philosophy had concepts similar to the eastern ones. She believed that earthly life is a time of purification and preparation and future life; unclean souls lengthen this period of punishment for themselves, they must undergo rebirth. The means to prepare the soul for a return to a better world are, according to the Pythagoreans, the same rules of purification and abstinence as in indian, persian and Egyptian religions. They, like the Eastern priests, had the commandments about what formalities must be performed in different everyday situations, what food you can eat, what food you should refrain from, the necessary aids for a person on the path of earthly life. According to the views of the Pythagorean school, a person should pray to the gods in white linen clothes, and he should also be buried in such clothes. The Pythagoreans had many similar rules.

Giving such commandments, Pythagoras conformed to popular beliefs and customs. The Greek people were no strangers to religious formalism. The Greeks had rituals of purification, and their commoners had many superstitious rules. In general, Pythagoras and his philosophical school did not contradict popular religion as sharply as other philosophers. They only tried to cleanse popular concepts and talked about the unity of divine power. Apollo, the god of pure light, giving the world warmth and life, the god of pure life and eternal harmony, was the only god to whom the Pythagoreans prayed and offered their bloodless sacrifices. They served him by dressing in a clean dress, washing their bodies, and taking care to cleanse their thoughts; to his glory, they sang their songs with the accompaniment of music and made solemn processions.

From the Pythagorean kingdom of Apollo everything that was impure, inharmonious, and disorderly was excluded; a person who was immoral, unjust, wicked on earth will not receive access to this kingdom; he will be reborn in the bodies of various animals and people, until by this process of purification he achieves purity and harmony. To shorten the wandering of the soul through different bodies, Pythagorean philosophy invented sacred, mysterious rituals ("orgies"), which improve the fate of the soul after a person's death, gives it eternal peace in the kingdom of harmony.

The followers of Pythagoras said that he himself was gifted with the ability to recognize in new bodies those souls that he knew before, and that he remembered his entire past existence in different bodies. Once in the Arsenal of Arsenal, looking at one of the shields that were there, Pythagoras wept: he remembered that he wore this shield when he fought against the Achaeans besieging Troy; he was then that Euphorbus whom he killed Menelaus in the battle between the Trojans and the Achaeans for the body of Patroclus. The life in which he was the philosopher Pythagoras was his fifth life on earth. Disembodied souls, according to the teachings of the Pythagorean philosophy, are spirits ("demons") that live either under the earth or in the air and quite often enter into intercourse with people. From them the Pythagorean school received its revelations and prophecies. Once Pythagoras, during his visit to the kingdom of Hades, saw that the souls of Homer and Hesiod were subjected to severe torment there for their offensive inventions about the gods.

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