Pythagoreanism

The founder of the union was Pythagoras , the son of Mnesarchus, a native of the Ionian island of Samos (therefore, the genesis of Pythagoreanism is attributed to the Ionian cultural and geographical area ^[2] .) Its heyday falls on the reign of the tyrant Polycrates (c. 530 BC). Pythagoras founded a community in the Italian city of Crotone . He died in Metapont , where he moved as a result of the hostile attitude of the Crotonians to his union.

After the death of Pythagoras, hostility against the Pythagorean union intensified in all the democracies of Great Greece , and in the middle of the fifth century. BC e. a catastrophe erupted: in Croton, many Pythagoreans were killed and burned in the house where they gathered; the defeat was repeated in other places. 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 V century. BC e. the political influence of the Pythagoreans in Greater Greece revived: the most important figure was the Architentus of Tarentus , commander and statesman. From the 4th century BC e. Pythagorean decayed, and his teachings were swallowed by Platonism .

Pythagoras himself, according to legend, did not leave a written account of his doctrine (it was strictly esoteric in nature ^[2] ), and Filolaus is considered the first writer to give an account of the Pythagorean doctrine. Moreover, the Pythagoreans had a tradition of building all the achievements of the school to its founder ^[2] . 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 who are recognized as authentic. Under such conditions, it is difficult to reliably separate the original essence of the Pythagorean teachings from the later layers.

The basis of the teachings of Pythagoras was Orphism ^[3] .

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

A number of instructions and prohibitions of the Pythagoreans date back to antiquity. Of these prohibitions, the ban on eating beans became the most famous, because of which, according to one legend, Pythagoras himself died. The reason for this ban is unknown, historians have made very different assumptions about the causes of such a taboo . For example, the philosopher Elena Shulga explains this by the fact that bean, resembling a human fetus, is associated with primogeniture ^[4] .

The Pythagoreans practiced vegetarianism for religious , ethical and ascetic reasons, in particular in connection with the doctrine of the transmigration of souls. Following the Orphics, the Pythagoreans believed that the soul of every person is bisexual and there are male and female halves in it, which are called Eros and Psyche ^[3] .

According to tradition, the followers of Pythagoras were divided into Akusmatiks ("Listeners") and mathematicians ("students"). Akusmatiks dealt with the religious and ritual aspects of teaching, mathematicians with the studies of the four Pythagorean "maths": arithmetic, geometry, harmonics and the sphere. Akusmatiks did not consider mathematicians to be “real Pythagoreans,” but they said that they stem from Gippas , who changed the original Pythagorean tradition, revealed secrets to the uninitiated, and began teaching for a fee.

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

The principle of justice should be considered as an important (cementing) conceptual position in the formation of the philosophy of this school ^[5] . The pinnacle of the development of philosophy is the contemplative mind; the middle of philosophy is the civil mind and the third is the mind associated with the sacraments. The development of these principles in man completes the Pythagorean learning ^[6] .

The basis of the subsequent philosophical teachings of the Pythagoreans was a categorical pair of two opposites - limit and limitless. "Infinite" cannot be a single beginning of things; otherwise nothing definite, no “limit” would be conceivable. On the other hand, the “limit” also implies something that is determined by it. From this follows the conclusion of Philolaus that “nature, existing in the cosmos, is harmoniously harmonious from the infinite and determining; so is the whole cosmos, and everything that is in it. ”

The Pythagoreans compiled a table of 10 opposites; Aristotle leads her in his “ Metaphysics ” (I, 5):

• limit - unlimited
• odd - even
• one ( single ) - many
• right - left
• male - female
• peace is movement
• straight - crooked
• light is darkness
• good is evil
• square - elongated rectangle (oblong shape)

The world harmony, which is the law of the universe, is unity in multitude and multitude in unity - --ν καὶ πολλά . How to think this truth? The immediate answer to this is the number: the multitude is united in it, it is the beginning of any measure. Experiments on the monochord show that number is the principle of sound harmony , which is determined by mathematical laws. Is sound harmony a special case of universal harmony, how would its musical expression? Astronomical observations show us that celestial phenomena, with which all the most important 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; nourished in these sciences, they recognized the mathematical principles for the beginnings of everything existing. Of these, naturally, the first are numbers. In the numbers they saw many analogies or similarities with things ... so that one property of numbers appeared to them as justice, the other as soul or mind, another as an opportunity, etc. Then they found the properties and relations of musical harmony in numbers, and since all other things, by their nature, were similar to numbers, while numbers were the first of all nature, they recognized that the elements of a number are the elements of everything that exists, and that the whole sky is harmony and number (Aristotle, Met., I , five).

Thus, the Pythagorean numbers are not of simple quantitative significance: if for us the number is a certain sum of units, then for the Pythagoreans it is, rather, the force that sums up these units into a certain whole and tells it certain properties. One is the cause of unity, two is the cause of bifurcation, separation, four is the root and source of the whole number (1 + 2 + 3 + 4 = 10). The basis of the doctrine of the number was, apparently, the fundamental opposite of the even and the odd: the even numbers are multiples of two, and therefore “even” is the beginning of divisibility, bifurcation, discord; "Odd" signifies opposing properties. From this it is clear that numbers can also have moral powers: 4 and 7, for example, as the average proportional between 1 and 10, are numbers, or principles, of proportionality, and consequently, of harmony, health, and rationality.

In the cosmology of the Pythagoreans, we encounter the same two basic principles of limit and infinity. The world is a limited sphere, rushing in infinity. “The original unity, having arisen out of nowhere,” says Aristotle, “draws in the immediate parts of infinity, limiting them to the power of the limit. Inhaling the parts of the infinite into itself, the one forms in itself a certain empty place or certain gaps that divide the original unity into separate parts - extended units ( ὡς όντος χωρισμοϋ τινος τών ἐφεξής ). ” This view is undoubtedly initial, since Parmenides and Zeno are already polemicizing against him. Inhaling the infinite void, the central unity gives rise to a series of celestial spheres and sets them in motion. The doctrine that the world breathes in air (or void), as well as some of the teachings on celestial bodies, the Pythagoreans learned from Anaximenes ^[7] . According to Philolaus , "the world is one and began to form from the center."

In the center of the world is a fire, separated by a series of empty intervals and intermediate spheres from the extreme sphere enclosing the universe and consisting of the same fire. The central fire, the center 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 stellar firmament and peripheral fire is called Olympus; beneath it is the cosmos of planets, sun and moon. Around the center “round dance of 10 divine bodies: the sky of motionless stars, five planets, followed by the Sun, under the Sun - the Moon, under the Moon - the Earth, and under it - the anti- earth ( ἀντίχθων )” - a special tenth planet, which the Pythagoreans accepted for round-robin counting , and maybe to explain solar eclipses. The sphere of fixed stars rotates slowest of all; more rapidly and with ever-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 on 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 medium of the solar disk, which only reflects its rays, not being an independent source of heat and light.

A peculiar Pythagorean doctrine of the harmony of the spheres : the transparent spheres to which the planets are attached are separated by gaps that relate to each other as musical intervals ; celestial bodies sound in their motion, and if we do not distinguish their consonances, then only because it is heard unceasingly.

The Pythagoreans considered the properties of numbers, between which the most important were even, odd, even-odd, square and non-square, studied arithmetic progressions and new numerical series derived from successive summations of their members. Thus, the sequential addition of the number 2 to himself or to unity 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 preceding it, gave a series of heteromek numbers representing the product of two factors differing by one by one. The same summation of the members of the second row gave a series of squares of consecutive natural numbers.

Of the geometric works of the Pythagoreans in the first place is the famous Pythagorean theorem . The proof of the theorem should have been the result of a considerable period of time, the work of both Pythagoras himself and other mathematicians of his school. A member of a series of odd numbers, always the difference between the two corresponding members of a series of square numbers, could be a square number itself: 9 = 25 - 16, 25 = 169-144, ... The content of the Pythagorean theorem was thus first discovered by rational right-angled triangles with a leg expressed by an odd number. At the same time, Pythagoras should also have discovered the way these triangles were formed, or their formula (n is an odd number that expresses a smaller leg; (n² - 1) / 2 is a larger leg; (n² - 1) / 2 + 1 is a hypotenuse).

The question of a similar property of other right-angled triangles also required the measurement of their sides. At the same time, the Pythagoreans first had to meet with disparate lines. No indication has reached us either of the initial general evidence, nor of the way in which it was found. According to Proclus , this initial proof was more difficult than Euclid, who was in the "Principles", and was also based on a comparison of areas.

The Pythagoreans were engaged in the tasks of “applying” ( παραβάλλεεν ) areas, that is, building a rectangle on this segment (in the general case, a parallelogram with a given angle at the apex) having this area. The nearest development of this issue consisted in constructing a straight rectangle having a given area on a given segment, under the condition that (( λλειψις ) remain or the square ( ὑπερβολή ) ^{[ specify ]} .

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

In stereometry, the subject matter of the Pythagoreans was regular polyhedrons. Pythagoreans' own research added to them the dodecahedron . The study of the methods of forming solid angles of polyhedra should directly lead the Pythagoreans to the theorem that “a plane near one point is filled without rest 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. "

All extant information about the emergence in ancient Greece of the mathematical doctrine of harmony (this science was called " harmonica ") definitely connect this occurrence with the name of Pythagoras. His achievements in this area are summarized in the following passage from Xenocrates , which came to us through Porphyry :

Pythagoras, as Xenocrates says, also discovered that in music the intervals are inseparable from the number, since they arise from the correlation of quantity with quantity. He investigated, as a result of which there are harmonious and dissonant intervals and everything harmonious and inharmonious (Porfiry. Commentary on the Ptolemy's Harmonica ) ^[8]

In the field of harmonics, Pythagoras made important acoustic studies that led to the discovery of the law, according to which the first (that is, the most important, most significant) consonances are determined by the simplest numerical ratios 2/1, 3/2, 4/3. So, half the string sounds in an octave , 2/3 - in a fifth , 3/4 - in a quart with a whole string. “The most perfect harmony” is defined by four mutually prime numbers 6, 8, 9, 12, where the extreme numbers form an octave between themselves, the numbers taken through one or two fifths, and the edges with neighbors two quarts.

Harmony is a system of three consonances - quarts, 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 harmony of the quart is in the form of a super-tertiary relation, the fifth is one and a half and the octave is double. Hence the number four, being super-tertiary of three, since it is composed of three and its third beat, embraces the harmony of the quart. The number three, being one and a half from two, since it contains two and its half, expresses the consonance of the fifth. The number four, being double with respect to two, and the number two, being double with respect to unity, determine the harmony of the octave "(Sextus Empiricus, Against Logicians , I, 94-97).

The successors of acoustic research, as well as representatives of the desire for a theoretical justification of musical harmony that arose in the Pythagorean school, were Hippas and Eubulid , who performed many experiments both on strings of different lengths and pulled by various weights, and on vessels filled with water in different ways.

The Pythagorean concept of harmonics was embodied in the idea of ​​the Pythagorean (or Pythagorean) system , tuned only by consonance - octaves and fifths. Among other things, the Pythagoreans discovered that (1) the whole tone is indivisible by 2 equal semitones, and also that (2) 6 whole tones are more than an octave by a negligible amount of coma (later called the "Pythagorean").

The outstanding musical theorists of the Pythagorean school were Philolaus and Archit , who developed the mathematical foundations of ancient Greek (musical) harmony.

• Neophthagoreanism
• Pre-existence
• Hebrew (philosopher)

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