Pythagoras

Pythagoras of Samos (Greek: Πυθαγόρας ὁ Σάμιος , or just Πυθαγόρας ; Πυθαγόρης in Ionian Greek; Samos, c. 570 – Metapontus, c. 495 BC ) was a Greek philosopher and mathematician called the Ionian movement credited with founding the Pitagorism movement. Most information about Pythagoras was written down centuries after his death, so there is little reliable information about him. He was born on the island of Samos and traveled through Egypt and Greece. In 520 BC he returned to Samos. Around 530 BC he moved to Crotona in Magna Graecia.

Born on the Greek island of Samos, his mother would have been called Pythals and his father Mnesarco, supposedly a merchant from the city of Tyre, who in addition to Pythagoras would have had two or three other children. Pythagoras spent his childhood on Samos although he traveled extensively with his father; he was trained by the best teachers, some of them philosophers. He played the lyre, learned arithmetic, geometry, astronomy and poetry.

Biography

	There is not a single detail in the life of Pythagoras that is uncontroversial. But it is possible, from a more or less critical selection of data, to construct a plausible account.
—Walter Burkert, 1972

Early life

Herodotus, Isocrates and other early writers all agree that Pythagoras was the son of Mnesarchus and that he was born on the Greek island of Samos in the eastern Aegean Sea. His father is said to have been a gem engraver or a wealthy merchant, but his ancestry is controversial and unclear. Pythagoras' name led him to be associated with Apollo Python; Aristippus of Cyrene explained his name by saying, "He spoke ( ἀγορεύω , agoreúō ) the truth no less than the Pythia [ sic ] ( Πῡθῐ́ᾱ , Pūthíā )". Only a late source is given the name of Pythagoras' mother as Pythaïs .Iamblichus tells the story that the Pythia prophesied to her while she was pregnant that she would give birth to a man supremely beautiful, wise, and beneficial to mankind. As for the date of his birth, Aristoxenes stated that Pythagoras left Samos in the reign of Polycrates, at age 40, which would give a birth date around 570 BC.

During Pythagoras' formative years, Samos was a thriving cultural center known for its feats of advanced architectural engineering, including the construction of the Eupalinos Tunnel and its festive culture. It was an important trading center in the Aegean, where traders brought goods from the Near East. According to Christiane L. Joost-Gaugier, these traders almost certainly brought with them ideas and traditions from the Near East. The early life of Pythagoras also coincided with the flowering of early Ionian natural philosophy. He was a contemporary of the philosophers Anaximander, Anaximenes, and the historian Hecataeus, all of whom lived in Miletus, across the Sea from Samos.

Alleged trips

Pythagoras is traditionally believed to have received most of his education in the Near East. Modern studies have shown that the culture of archaic Greece was heavily influenced by the culture of the Near East. Like many other important thinkers in Greece, Pythagoras would have studied in Egypt, where he would have traveled in about 535 BC - a few years after the occupation of Samos by the tyrant Polycrates - there, he got to know the temples and learned about the local priests. By the time of Isocrates, in the 4th century BC, the supposed studies of Pythagoras in Egypt were already taken for granted. The writer Antiphon, who may have lived during the Hellenic Age, stated in his lost work On Men of Exceptional Merit, used as a source by Porphyry, that Pythagoras learned to speak Egyptian from Pharaoh Amosis II himself, that he studied with the Egyptian priests at Diopolis (Thebes), and that he was the only foreigner to receive the privilege of participating in their cult. The Middle Platonic biographer Plutarch (c. 46 – c. 120 CE) writes in his treatise On Isis and Osiris that, during his visit to Egypt, Pythagoras received instructions from the Egyptian priest Enuphis of Heliopolis (meanwhile, Solon received lectures from a Sonchis of Sais). According to the Christian theologian Clement of Alexandria (c. 150 – c. 215 AD), "Pythagoras was a disciple of Soches, an Egyptian archprophet, as Plato was of Secnuphis of Heliopolis."Some ancient writers claimed that Pythagoras learned geometry and the doctrine of metempsychosis from the Egyptians.

Other ancient writers, however, claimed that Pythagoras had learned these teachings from the magicians of Persia or even from Zoroaster himself. Diogenes Laertius claims that Pythagoras later visited Crete, where he went to the Cave of Ida with Epimenides. It is said that the Phoenicians taught Pythagoras arithmetic and that the Chaldeans taught him astronomy. As early as the third century BC, it was said that Pythagoras had already studied under the Jews. Contrary to all these accounts, the novelist Antonio Diogenes, writing in the 2nd century BC, reports that Pythagoras discovered all his doctrines by interpreting dreams. The 3rd century AD sophist Philostratus claims that in addition to the Egyptians, Pythagoras also studied with Hindu sages in India.Iamblichus expands this list even further, stating that Pythagoras would also have studied with the Celts and the Iberians.

In 525 BC, the Persian king Cambyses I attacked Egypt and a legend tells that Pythagoras would have been captured and sent to Babylon, where he would have received spiritual teachings of oriental influence, in anecdotal attempts to associate him with a possible contact with the so-called Persian, Babylonian or even Hindu magicians.

By 522 BC both Polycrates and Cambyses had died, so Pythagoras returned to Samos where he founded a school of philosophy called the Semicircle .

Around 518 BC, to avoid political conflicts, he traveled to southern Italy, to the city of Crotona, where he founded a spiritual school; there he would have married.

Family and friends

Diogenes Laërtius claims that Pythagoras "did not indulge in the pleasures of love" and that he warned others to have sex only "when you are willing to become weaker than you are". According to Porphyry, Pythagoras married Theano, a Cretan lady and daughter of Pitonax of Crete, and had several children with her. Porphyry writes that Pythagoras had two sons named Telauge and Arignot and a daughter named Myia who "took precedence among the maidens of Crotona, and, when espoused, among married women". Iamblichus does not mention any of these sons , and instead only mentions a son named Mnesarchus in honor of his grandfather.This son would have been raised by Pythagoras' appointed successor, Aristaeus, and ended up taking over the school when Aristaeus was too old to continue running it.

The fighter Milo of Croton was said to be a close associate of Pythagoras and was credited with saving the philosopher's life when a roof was about to collapse. This association may have been the result of confusion with a different man named Pythagoras, who was an athletics coach. Diogenes Laertius records the name of Milo's wife as Myia. Iamblichus mentions Theano as the wife of Brontino of Crotona. Diogenes Laertius states that this same Theano was a student of Pythagoras and that Pythagoras' wife, Theano, was her daughter. Diogenes Laertius also records that the works supposedly written by Theano still existed during her own lifetime , and cites several opinions attributed to her.These writings are now known to be pseudepigraphs.

Death

Pythagoras' emphasis on dedication and asceticism is credited with aiding Crotona's decisive victory over the neighboring colony of Sybaris in 510 BC. After the victory, some prominent citizens of Crotona proposed a democratic constitution, which the Pythagoreans rejected. The partisans of democracy, led by Cylon and Nynon, the former of whom is said to have been irritated by their exclusion from the brotherhood of Pythagoras, roused the populace against them. Followers of Cylon and Nyno attacked the Pythagoreans during one of their meetings, either at Milan's house or at some other meeting place. Accounts of the attack are often contradictory and many have likely confused it with later anti-Pythagorean rebellions. The building was apparently set on fireand many of the assembled members died; only the youngest and most active members managed to escape.

Sources disagree on whether Pythagoras was present when the attack took place and, if he was, whether or not he managed to escape. In some accounts, Pythagoras was not at the meeting when the Pythagoreans were attacked because he was in Delos, tending to Pherecydes on his deathbed. According to another account by Dicéarchus, Pythagoras was at the meeting and managed to escape, taking a small group of followers to the nearby town of Locris, where they sought refuge but were denied. They reached the city of Metaponto, where they took shelter in the temple of the Muses and died of hunger after forty days without food.Another tale recorded by Porphyry states that while Pythagoras' enemies were burning the house, his devoted students lay on the ground to make a way for him to escape, walking over their bodies through the flames like a bridge. Pythagoras managed to escape, but was so disheartened by the death of his beloved students that he would have committed suicide. A different legend related by Diogenes Laertius and Iamblichus states that Pythagoras almost managed to escape, but that he came to a field of broad beans and refused to go through it, as doing so would violate his teachings, he stopped then and was killed. This story appears to have originated with the writer Neantes, who spoke about the later Pythagoreans, not Pythagoras himself.

The pythagorean school

The Pythagoreans were interested in the study of the properties of numbers. For them, the number, synonymous with harmony, is constituted by the sum of even and odd numbers (the even and odd numbers expressing the relationships that are in a permanent process of mutation), being considered as the essence of things, creating opposing notions (limited and unlimited) and the basis of the theory of the harmony of the spheres. The Pythagorean school was connected with esoteric conceptions and Pythagorean morals emphasized the concept of harmony , ascetic practices and advocated metempsychosis.

According to the Pythagoreans, the cosmos is governed by mathematical relationships. The observation of the stars suggested to them that an order dominates the universe. Evidence of this would be in the day and night, in the changing of the seasons and in the circular and perfect movement of the stars. That is why the world could be called cosmos, a term that contains the ideas of order, correspondence and beauty. In this worldview they also concluded that the Earth is spherical, a star among the stars that move around a Central Fire. Some Pythagoreans even spoke of the Earth's rotation on its axis, but the greatest discovery of Pythagoras or his disciples (since there are obscurities around Pythagoreanism, due to the esoteric and secret character of the school) took place in the domain of geometry. and refers to the relationships between the sides of the right triangle.

Pythagoras was expelled from Crotona and took up residence at Metapontum, where he died, probably in 496 BC or 497 BC. According to Pythagoreanism, the essence, which is the fundamental principle that forms all things, is the number . The Pythagoreans did not distinguish between form, law, and substance, considering number to be the link between these elements. For this school there were four elements: earth, water, air and fire.

Thus, Pythagoras and the Pythagoreans investigated mathematical relationships and discovered various foundations of physics and mathematics.

The symbol used by the school was the pentagram, which, as Pythagoras discovered, has some interesting properties. A pentagram is obtained by tracing the diagonals of a regular pentagon; by the intersections of the segments of this diagonal, a new regular pentagon is obtained, which is exactly proportional to the original by the golden ratio.

Pythagoras discovered in what proportions a string must be divided to obtain musical notes at the beginning, without a defined pitch, being one taken as a fundamental (let us think of a long string attached to two ends that, when plucked, will give us the lowest sound) - and from it, the fifth and third will be generated through harmonic reverberation. Harmonic sounds. By attaching half of the string, then the third part and then the fifth part, we will obtain the fifth and third intervals in relation to the fundamental. The so-called harmonic series. As we subdivide the string we get louder sounds and the intervals will be different. And so on. He further discovered that simple fractions of notes, played along with the original note, produce pleasant sounds. The more complicated fractions, played with the original note,

The name is mainly linked to the important theorem that states: In every right triangle, the sum of the squares of the legs is equal to the square of the hypotenuse.

During the 4th century BC, there was a revival of religious life in the Greek world. According to some historians, one of the factors that contributed to this phenomenon was the political line adopted by the tyrants: to guarantee the role of popular leaders and to weaken the old aristocracy, the tyrants encouraged the expansion of popular or foreign cults.

Among these cults, one had great diffusion: Orphism (of Orpheus), originating in Thrace, and which was an essentially esoteric religion. The followers of this doctrine believed in the immortality of the soul, that is, while the body degenerated, the soul migrated to another body, several times, in order to effect purification. Dionysus would guide this cycle of reincarnations, being able to help man to free himself from it.

Pythagoras followed a different doctrine. He would have arrived at the conception that all things are numbers and the process of liberating the soul would be the result of a basically intellectual effort. Purification would result from an intellectual work, which discovers the numerical structure of things and thus makes the soul a harmonious unity. The numbers would not be, in this case, symbols, but the values ​​of magnitudes, that is, the world would not be composed of the numbers 0, 1, 2, etc., but of the values ​​they express. So, therefore, a thing would externally manifest the numerical structure, this thing being what it is because of this value.

Metempsychosis

Although the exact details of Pythagoras' teachings are uncertain, it is possible to reconstruct a general outline of his main ideas. Aristotle writes at length about the teachings of the Pythagoreans, but without directly mentioning Pythagoras. One of Pythagoras' main doctrines appears to have been metempsychosis , the belief that all souls are immortal and that, after death, a soul is transferred into a new body. This teaching is referenced by Xenophanes, Ion of Chio and Herodotus. However, nothing is known about the nature or mechanism by which Pythagoras believed that metempsychosis occurred.

Empedocles alludes in one of his poems that Pythagoras may have claimed to possess the ability to remember his previous incarnations. Diogenes Laertius reports an account by Heraclides of Pontus that Pythagoras told people that he had lived four previous lives that he could remember in detail. The first of these lives was as Ethalides, son of Hermes, who granted him the ability to remember all of his past incarnations. He then incarnated as Euphorbus, a minor hero of the Trojan War mentioned briefly in the Iliad . He then became the philosopher Hermotimus, who recognized the shield of Euphorbus in the temple of Apollo. His final incarnation was as Pyrrhus, a fisherman from Delos.One of her past lives, as reported by Dicearco, was as a beautiful courtesan. By the way, in his book The Life of Apollonius of Tyana , Philostratus also cites that Pythagoras knew who he had been.

Mysticism

Another belief attributed to Pythagoras was that of the "harmony of the spheres", which held that the planets and stars move according to mathematical equations, which correspond to musical notes and therefore produce an inaudible symphony. According to Porphyry, Pythagoras taught that the seven Muses were actually the seven planets singing together. In his philosophical dialogue Protrepticus , Aristotle says for his literary double:

When Pythagoras was asked [why human beings exist], he said, "to observe the heavens", and used to claim that he himself was an observer of nature and that was why he passed into life.

Pythagoras was said to have practiced divination and prophecy. On visits to various places in Greece - Delos, Sparta, Phlius, Crete, etc. - attributed to him, he usually appears in his religious or priestly guise, or as a lawgiver.

Communal lifestyle

Both Plato and Isocrates claim that, above all, Pythagoras was known as the founder of a new way of life. The organization Pythagoras founded at Croton was called a "school", but in many ways it looked like a monastery. The adepts were bound by a vow to Pythagoras and to one another, for the purpose of pursuing religious and ascetic observances, and of studying his religious and philosophical theories. The sect members shared all their possessions in common and were devoted to one another, excluding foreigners. Ancient sources record that the Pythagoreans had meals together, in the manner of the Spartans. A Pythagorean maxim was " koinà tà phílōn " ("All things in common among friends").Iamblichus and Porphyry provide detailed accounts of the school's organization, although the primary interest of both writers is not historical accuracy, but presenting Pythagoras as a divine figure, sent by the gods to benefit mankind. Iamblichus, in particular, presents the "Pythagorean Way of Life" as a pagan alternative to the Christian monastic communities of his own time.

Two groups existed in early Pythagoreanism: the mathematikoi ("learners") and the akousmatikoi ("listeners"). Akousmatikoi are traditionally identified by scholars as "old believers" in mysticism, numerology, and religious teachings; while the mathematikoi are traditionally identified as a modernist and more intellectual, more rationalist and scientific faction. Gregory warns that there was probably no sharp distinction between them and that many Pythagoreans probably believed the two approaches were compatible. The study of mathematics and music may have been related to the cult of Apollo.The Pythagoreans believed that music was a purification for the soul, just as medicine was a purification for the body. An anecdote from Pythagoras relates that when he found some drunken youths trying to break into the home of a virtuous woman, he sang a solemn tune with long spondees, and the boys' "furious will" was suppressed. The Pythagoreans also particularly emphasized the importance of physical exercise; therapeutic dancing, daily morning walks along scenic routes, and athletics were major components of the Pythagorean lifestyle. Moments of contemplation at the beginning and end of each day were also advised.

Prohibitions and regulations

Pythagoras' teachings were known as "symbols" ( symbola ) and members took a vow of silence that they would not reveal these symbols to non-members. Those who did not obey the community's laws were expelled and the remaining members erected tombstones for them as if they had died. A number of "oral sayings" ( akoúsmata ) attributed to Pythagoras have survived, dealing with how members of the Pythagorean community should perform sacrifices, how they should honor the gods, how they should "move" here, and how they should be buried. Many of these sayings emphasize the importance of ritual purity and avoiding contamination.For example, a saying that Leonid Zhmud concludes can probably be traced genuinely to Pythagoras himself, forbidding his followers to wear woolen clothing. Other existing oral sayings forbid the Pythagoreans from breaking bread, poking at fires with swords or picking up crumbs and teaching that a person should always put the right sandal before the left. The exact meanings of these sayings, however, are often obscure. Iamblichus preserves Aristotle's descriptions of the original ritualistic intentions behind some of these sayings, but these apparently later fell out of fashion because Porphyry provides markedly different ethico-philosophical interpretations for them:

Pythagorean proverb	Original Ritual Purpose According to Aristotle/Iamblichus	Philosophical interpretation of Porphyry
"Do not take roads traveled by the public."	"Fear of being contaminated by the unclean"	"By this he forbade following the opinions of the masses, but following those of the educated few."
"and don't use images of the gods in rings"	"Fear of contaminating them, using them."	"One should not have the teaching and knowledge of the gods quickly at hand and visible [to all], nor communicate them to the masses."
"and pour libations to the gods by holding on to the handle of a drinking cup [the 'ear']"	"Efforts to Keep the Divine and the Human Strictly Separate"	"thus he enigmatically suggests that the gods are to be honored and praised with music; for it passes through the ears."

Allegedly, new initiates were not allowed to meet with Pythagoras until they had completed a five-year initiation period, during which they were to remain silent. Sources indicate that Pythagoras himself was extraordinarily progressive in his attitudes towards women and the women of the Pythagorean school appear to have played an active role in his operations. Iamblichus provides a list of 235 famous Pythagoreans, seventeen of whom are women. In later times, many prominent philosophers contributed to the development of Neo-Pythagoreanism.

Pythagoreanism also involved a series of food prohibitions. It is more or less in agreement that Pythagoras issued a ban against the consumption of fava beans and the meat of non-sacrifice animals such as fish and poultry. Both assumptions, however, have since been contradicted or associated with symbolic meaning. Pythagorean dietary restrictions may have been motivated by belief in the doctrine of metempsychosis. Some ancient writers present Pythagoras as enforcing a strictly vegetarian diet.Eudoxus of Cnidus, a student of Archytas, writes: "Pythagoras distinguished himself by this purity, and therefore avoided killing and those who killed, so that he not only abstained from food of animal origin, but also kept his distance from cooks and hunters". Other authorities contradict this claim. According to Aristoxenes, Pythagoras allowed the use of all types of animal feed, except the meat of oxen used for plowing and sheep. According to Heraclides Ponticus, Pythagoras ate the meat of sacrifices and established a meat-dependent diet for athletes.

Main findings

In addition to being great mystics, the Pythagoreans were great mathematicians. They discovered interesting and curious properties about numbers.

Figured numbers

The Pythagoreans studied and demonstrated various properties of figurative numbers. Among these the most important was the triangular number 10, called by the Pythagoreans tetraktys , tetrad in Portuguese. This number was seen as a mystical number as it contained the four elements fire, water, air and earth: 10=1 + 2 + 3 + 4, and served as a representation for the completeness of the whole.αα αα α αα α α α

The tetrad, which the Pythagoreans drew with an α above, two below, then three and finally four at the base, was one of the main symbols of their advanced knowledge of theoretical realities.

Perfect numbers

The sum of the divisors of a given number, with the exception of itself, is the number itself. Examples:

1. The divisors of 6 are: and . Then, .
2. The divisors of 28 are: and . Then, .

Pythagorean theorem

An unsolved problem in Pythagoras' time was to determine the relationships between the sides of a right triangle. Pythagoras proved that the sum of the squares of the legs is equal to the square of the hypotenuse.

The first irrational number to be discovered was the square root of the number 2, which came exactly from the application of the Pythagorean theorem in a triangle of legs worth 1:

The Greeks did not know the square root symbol and simply said: "the number that multiplied by itself is 2".

From the discovery of the root of 2 many other irrational numbers were discovered.

rector of the first university

The word Mathematics (Mathematike, in Greek) came up with Pythagoras, who was the first to conceive of it as a system of thought, based on deductive proofs.

There are, however, indications that the so-called Pythagorean Theorem (c²= a²+b²) was already known to the Babylonians in 1600 BC with an empirical scope. These used sexagesimal notation systems for measuring time (1h=60min) and measuring angles (60º, 120º, 180º, 240º, 360º).

Pythagoras traveled for 30 years through Egypt, Babylon, Syria, Phoenicia and perhaps India and Persia, where he accumulated eclectic knowledge: astronomy, mathematics, science, philosophy, mysticism and religion. He was a contemporary of Thales of Miletus, Buddha, Confucius and Lao-Tzu.

When he returned to Samos, he fell out with the tyrant Polycrates and emigrated to Crotona in southern Italy. There he founded the Pythagorean School, which is given the glory of being the "first University in the world".

The Pythagorean School and its activities have since been shrouded in a veil of legends. It was a partially secret entity with hundreds of students who made up a religious and intellectual brotherhood. Among the concepts they defended, the following stand out:

• practice of purification rituals and belief in the doctrine of metempsychosis, that is, in the transmigration of the soul after death, from one body to another. Therefore, they advocated the reincarnation and immortality of the soul;
• loyalty among members and community distribution of material goods;
• austerity, asceticism and obedience to the hierarchy of the School;
• prohibition of drinking wine and eating meat (so the information that the disciples had 100 oxen killed when demonstrating the so-called Pythagorean Theorem is false);
• purification of the mind by the study of Geometry, Arithmetic, Music and Astronomy;
• arithmetic classification of numbers into even, odd, prime and factorable;
• "the creation of a model of definitions, axioms, theorems and proofs, according to which the intricate structure of Geometry is obtained from a small number of explicitly made statements and the action of rigorous deductive reasoning" (George Simmons);
• a great stir arose among the disciples of Pythagoras about the irrationality of the 'root of 2'. Using algebraic notation, the Pythagoreans did not accept any numerical solution for x² = 2, as they only admitted rational numbers. Given the mystical connotation attributed to the numbers, it is said that when the unfortunate Hippasus of Metapontum proposed a solution to the impasse, the other disciples expelled him from the School and drowned him in the sea;
• in Astronomy, innovative ideas, although not always true: the Earth is spherical, the planets move at different speeds in the various orbits around the Earth. Through the careful observation of the stars, the idea was crystallized that there is an order that dominates the Universe;
• the Pythagoreans are probably due to the construction of the cube, tetrahedron, octahedron, dodecahedron and the well-known "golden section";
• in Music, a remarkable discovery that musical intervals are so placed that they admit of expressions through arithmetical proportions. Pythagoras - as well as other pre-Socratic Greek philosophers - also described the power of sound and its effects on the human psyche. This music therapy experience was possibly later used by Aristotle as a theoretical basis for his definition of music, which, according to him, was a "medicinal art".

Pythagoras is the first pure mathematician. However, it is difficult to separate the historical from the legendary, since he must be considered a historically inaccurate figure, since everything we know about him is due to oral tradition. He left nothing written, and the first works on the same are due to Philolaus, almost 100 years after the death of Pythagoras. But it is not easy to deny the Pythagoreans - asserts Carl Boyer - "the primordial role in the establishment of mathematics as a rational discipline". Despite some exaggeration, centuries ago a phrase was coined: "If there were no 'Pythagoras theorem', Geometry would not exist".

When biography of Pythagoras, Iamblichus (c. 300 AD) records that the master kept repeating to his disciples: “all things resemble numbers”.

Later influence in antiquity

The Pythagorean School had a strong influence on the powerful words of Euclid, Archimedes and Plato, in the ancient Christian era, in the Middle Ages, in the Renaissance and even in our days with Neo-Pythagoreanism.

in greek philosophy

Considerable Pythagorean communities existed in Magna Graecia, Fliunte, and Thebes during the early 4th century BC. At the same time, the Pythagorean philosopher Archytas had a great influence on the politics of the city of Tarentum, in Magna Graecia. According to later tradition, Archytas was elected strategos ("general") seven times, although others were banned from serving for more than a year. Archytas was also a renowned mathematician and musician. He was a close friend of Plato and is quoted in Plato's Republic . Aristotle claims that Plato's philosophy was heavily dependent on the teachings of the Pythagoreans. Cicero repeats this statement, noting thatPlatonem ferunt didicisse Pythagorea omnia ("It is said that Plato learned all things Pythagorean"). According to Charles H. Kahn, Plato's middle dialogues, including the Meno , Phaedo , and The Republic , have a strong "Pythagorean coloring", and his later dialogues (particularly Philebus and Timaeus ) are extremely Pythagorean in character.

According to RM Hare, Plato's Republic may be partly based on the "strongly organized community of like-minded thinkers" established by Pythagoras at Croton. Furthermore, Plato may have borrowed from Pythagoras the idea that mathematics and abstract thought are a secure foundation for philosophy, science, and morality. Plato and Pythagoras shared a "mystical approach to the soul and its place in the material world" and it is likely that both were influenced by Orphism. The historian of philosophy Frederick Copleston claims that Plato probably borrowed his tripartite theory of the soul from the Pythagoreans. Bertrand Russell, in A History of Western Philosophy, claims that Pythagoras' influence on Plato and others was so great that he should be considered the most influential philosopher of all time. He concludes that "I know of no other man who has been as influential as he was in the school of thought".

A Table of Opposites was attributed by Aristotle to Pythagoras, which shows that it shares the pre-Socratic Greek thought of investigating dualities in nature.

A revival of Pythagorean teachings took place in the first century BC when Middle Platonist philosophers such as Eudorus and Philo of Alexandria welcomed the emergence of a "new" Pythagoreanism in Alexandria. Around the same time, Neo-Pythagoreanism became prominent. The 1st century AD philosopher. Apollonius of Tyana, sought to imitate Pythagoras and live up to the teachings of Pythagoras. The Neo-Pythagorean philosopher Moderato of Gades, at the end of the first century, expanded the Pythagorean philosophy of numbers and probably understood the soul as a "type of mathematical harmony". The Neo-Pathagorean mathematician and musicologist Nicomachus also expanded upon Pythagorean numerology and music theory.Numenius of Apameia interpreted Plato's teachings in the light of Pythagorean doctrines.

About art and architecture

Greek sculpture sought to represent the permanent reality behind surface appearances. Early archaic sculpture represents life in simple ways and may have been influenced by early Greek natural philosophies. The Greeks generally believed that nature expressed itself in ideal forms and was represented by a type ( εἶδος ), which was calculated mathematically. As dimensions shifted, architects sought to relay permanence through mathematics. Maurice Bowra believes that these ideas influenced the theory of Pythagoras and his students, who believed that "all things are numbers".

During the 6th century BC, the numerical philosophy of the Pythagoreans sparked a revolution in Greek sculpture. Greek sculptors and architects tried to find the mathematical relationship (canon) behind aesthetic perfection. Possibly taking advantage of the ideas of Pythagoras, the sculptor Polycletus writes in his canon that beauty consists in proportion, not in the elements (materials), but in the interrelation of the parts with each other and with the whole. In Greek architectural orders, every element was calculated and constructed by mathematical relationships. Rhys Carpenter states that the 2:1 ratio was "the generative ratio of the Doric order, and in Hellenistic times an ordinary Doric colonnade, trumps a rhythm of notes".

The oldest known building, designed according to the teachings of Pythagoras, is the Porta Maggiore Basilica, an underground basilica that was built during the reign of Roman Emperor Nero as a secret place of worship for the Pythagoreans. The basilica was built underground because of the Pythagorean emphasis on secrecy and also because of the legend that Pythagoras had isolated himself in a cave on Samos. The basilica's apse is on the east and its atrium on the west out of respect for the rising sun. It has a narrow entrance that leads to a small pool where initiates can purify themselves. The building is also designed according to Pythagoras' numerology, with each table in the sanctuary offering seating for seven people.Three aisles lead to a single altar, symbolizing the three parts of the soul that approach the unity of Apollo. The apse shows a scene of the poet Sappho jumping off the Leucadian cliffs, holding her lyre to her chest, while Apollo stands beneath it, extending his right hand in a protective gesture symbolizing the Pythagorean teachings on the immortality of the soul. The interior of the shrine is almost entirely white because the color white was considered sacred by the Pythagoreans.

Emperor Hadrian's pantheon in Rome was also built on Pythagorean numerology. The temple's circular plan, central axis, hemispherical dome and alignment with the four cardinal directions symbolize the Pythagorean visions of the order of the universe. The single oculus at the top of the dome symbolizes the monad and the sun god Apollo. The twenty-eight ribs extending from the oculus symbolize the moon, because twenty-eight were the same number of months in Pythagoras' lunar calendar. The five rings boxed below the ribs represent the marriage of the sun and moon.

at the beginning of christianity

Pythagoras. Eusebius ( c. 260 - c. 340 CE), bishop of Caesarea, praises Pythagoras in his book Against Hierocles for his mastery of silence, his frugality, his "extraordinary" morals and his wise teachings. In another work, Eusebius compares Pythagoras to Moses. In one of his letters, Church Father Jerome ( c. 347 - 420 AD) praises Pythagoras for his wisdom , and in another letter he credits Pythagoras for his belief in the immortality of the soul, which he suggests Christians inherited from him. . Augustine of Hippo (AD 354 - 430) rejected Pythagoras' teachings on metempsychosis without explicitly naming him, but expressed admiration for him. InOn the Trinity , Augustine praises the fact that Pythagoras was humble enough to call himself a philosopher or "lover of wisdom" rather than "sage". In another passage, Augustine defends Pythagoras' reputation, arguing that Pythagoras certainly never taught the doctrine of metempsychosis.

Influence after antiquity

In the Middle Ages

During the Middle Ages, Pythagoras was revered as the founder of mathematics and music, two of the Seven Liberal Arts. It appears in numerous medieval depictions, in illuminated manuscripts, and in the relief carvings on the portal of Chartres Cathedral. The Timaeus was the only dialogue of Plato to survive in Latin translation in western Europe, which led William of Conches (c. 1080-1160) to declare Plato to be a Pythagorean. In the 1430s, the Camaldolese friar Ambrose Traversari translated the Lives and Doctrines of the Illustrious Philosophers of Diogenes Laertius from Greek into Latin , and in the 1460s the philosopher Marsilio Ficino also translated the Lives of Pythagoras by Porphyry and Iamblichus.thus allowing them to be read and studied by Western academics. In 1494 the Greek Neo-Pythagorean scholar Constantine Láscaris published The Golden Verses of Pythagoras , translated into Latin, with a printed edition of his Grammatica , bringing them to a widespread audience. In 1499 he published the first Renaissance biography of Pythagoras in his work Vitae illustrium philosophorum siculorum et calabrorum , published in Messina.

in modern science

Em seu prefácio ao livro A revolução das esferas celestiais (1543), Nicolau Copérnico cita vários pitagóricos como as influências mais importantes no desenvolvimento de seu modelo heliocêntrico do universo, omitindo deliberadamente a menção de Aristarco de Samos, um astrônomo não pitagórico que havia desenvolvido um modelo totalmente heliocêntrico no século IV a.C., em um esforço para retratar seu modelo como fundamentalmente pitagórico. Johannes Kepler considerava-se um pitagórico. Ele acreditava na doutrina pitagórica da musica universalis e foi sua busca pelas equações matemáticas por trás dessa doutrina que levou à descoberta das leis do movimento planetário. Kepler intitulou seu livro sobre o assunto Harmonices Mundi (Harmônicas do Mundo), em homenagem aos ensinamentos pitagóricos que o inspiraram. Perto da conclusão do livro, Kepler descreve-se adormecendo ao som da música celestial, "aquecido por ter bebido um gole generoso ... do copo de Pitágoras".

Isaac Newton was a firm believer in the Pythagorean teaching of the mathematical harmony and order of the universe. Although Newton was notorious for rarely giving other people credit for his discoveries, he attributed the discovery of the Law of Universal Gravitation to Pythagoras. Albert Einstein believed that a scientist can also be "a Platonist or a Pythagorean, insofar as he regards the point of view of logical simplicity as an indispensable and effective tool of his research". The English philosopher Alfred North Whitehead argued that "In some ways Plato and Pythagoras are closer to modern physical science than Aristotle. The first two were mathematicians, while Aristotle was the son of a physician."By this measure, Whitehead declared that Einstein and other modern scientists like him are "following the pure Pythagorean tradition."

about vegetarianism

A fictionalized portrayal of Pythagoras appears in Book XV of Ovid 's Metamorphoses, in which he gives a speech imploring his followers to adhere to a strictly vegetarian diet. It was through Arthur Golding's 1567 English translation of Ovid 's Metamorphoses that Pythagoras was best known to English speakers during the early modern period. John Donne 's Progress of the Soul discusses the implications of the doctrines expounded in the speech and Michel de Montaigne quoted the speech at least three times in his treatise "On Cruelty" to express his moral objections against the mistreatment of animals. William Shakespeare references the speech in his playThe Merchant of Venice . John Dryden included a translation of the scene with Pythagoras in his 1700 work Fables, Ancient and Modern, and John Gay's 1726 fable "Pythagoras and the Peasant" reiterates its main themes, linking carnivorism with tyranny. Lord Chesterfield records that his conversion to vegetarianism was motivated by reading Pythagoras' speech in Ovid 's Metamorphoses. Until the word vegetarianism was coined in the 1840s, vegetarians were referred to in English as "Pythagoreans". Percy Bysshe Shelley wrote an ode entitled "To the Pythagorean Diet" and Leo Tolstoy adopted the Pythagorean Diet.

About Western Esotericism

Modern European esotericism drew heavily from the teachings of Pythagoras. The German humanist scholar Johannes Reuchlin (1455-1522) synthesized Pythagoreanism with Christian theology and Jewish Kabbalah, arguing that Kabbalah and Pythagoreanism were both inspired by the Mosaic tradition and that Pythagoras was therefore a Kabbalist. In his dialogue De verbo mirifico (1494), Reuchlin compared the Pythagorean tetractyls to the ineffable divine name YHWH, attributing to each of the four letters of the tetragrammaton a symbolic meaning in keeping with the mystical teachings of Pythagoras.

Heinrich Cornelius Agrippa's popular and influential three-volume treatise De Occulta Philosophia cites Pythagoras as a "religious magician" and indicates that Pythagoras' mystical numerology operates on a supercelestial level. Freemasons deliberately modeled their society on the community founded by Pythagoras at Crotona. Rosicrucianism used Pythagorean symbolism , as did Robert Fludd (1574-1637), who believed that his own musical writings had been inspired by Pythagoras. John Dee was heavily influenced by Pythagorean ideology, particularly the teaching that all things are made of numbers. Adam Weishaupt, the founder of the Illuminati, was a strong admirer of Pythagorasand, in his book Pythagoras (1787), he argued that society should be reformed to resemble more like the community of Pythagoras at Crotona. Wolfgang Amadeus Mozart incorporated Masonic and Pythagorean symbolism into his opera The Magic Flute . Sylvain Maréchal, in his 1799 six-volume biography The Voyages of Pitthagoras , declared that all revolutionaries in all time periods are the "heirs of Pythagoras".

In literature

Dante Alighieri was fascinated by Pythagorean numerology and based his descriptions of hell, purgatory and heaven on Pythagorean numbers. Dante wrote that Pythagoras saw Unity as good and plurality as evil , and in Paradise XV, 56–57, he declares: "five and six, if understood, radiate from unity". The number eleven and its multiples are found throughout the Divine Comedy , each book having thirty-three cantos, with the exception of Inferno , having thirty-four, the first of which serves as a general introduction. Dante describes the ninth and tenth bolgias in the eighth circle of hell as being twenty-two miles and eleven miles respectively, which corresponds to the fraction of227, which was Pythagoras' approximation of pi. Hell, Purgatory and Heaven are all described as circular and Dante compares the wonder of God's majesty to the mathematical puzzle of squaring the circle. The number three also features prominently: the Divine Comedy has three parts and Beatrice is associated with the number nine, which is equal to three times three.

Transcendentalists read the ancient Lives of Pythagoras as guides on how to live a model life. Henry David Thoreau was impacted by Thomas Taylor's translations of Iamblichus ' Life of Pythagoras and Stobeeus' Pythagorean Sayings and his views on nature may have been influenced by the Pythagorean idea of ​​images corresponding to archetypes. The Pythagorean teaching of universal music is a recurring theme throughout Thoreau 's magnum opus Walden .

The Brazilian philosopher Mário Ferreira dos Santos wrote more than one work on Pythagoreanism, from a very particular perspective and called himself "Pythagorean".