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Pythagoras of Samos (Greek: Πυθαγόρας; c.582–c.507 BCE) was an Ionian (Greek) mathematician, astronomer, scientist, and philosopher, founder of the mathematical, mystic, religious, and scientific society called the Pythagoreans. His name led him to be associated with Pythian Apollo; Aristippus explained his name by saying: "He spoke (agor-) the truth no less than did the Pythian (Pyth-)," and Iamblichus tells the story that the Pythia prophesied that his pregnant mother would give birth to a man supremely beautiful, wise, and of benefit to humankind.
Known as "the father of numbers", Pythagoras made influential contributions to philosophy and religious teaching in the late sixth century BCE. Because legend and obfuscation cloud his work even more than with the other pre-Socratics, one can say little with confidence about his life and teachings. It is not certain that Pythagoras and his students believed that everything was related to mathematics and that numbers were the ultimate reality and, through mathematics, everything could be predicted and measured in rhythmic patterns or cycles. According to Iamblichus, Pythagoras once said that "number is the ruler of forms and ideas and the cause of gods and demons."
Pythagoras was born on the island of Samos (a Greek island in the Eastern Aegean), off the coast of Asia Minor. He was born to Pythais (his mother, a native of Samos) and Mnesarchus (his father, a merchant from Tyre). As a young man, he left his native city for Croton, Calabria, in Southern Italy, to escape the tyrannical government of Polycrates. According to Iamblichus, Thales, impressed with his abilities, advised Pythagoras to go to Memphis in Egypt and study with the priests there who were renowned for their wisdom. He also was discipled in the temples of Tyre and Byblos in Phoenicia. Various writers claim that he traveled to Babylonia; Iamblichus claims that he studied mathematics and the theory of music there.
Pythagoras undertook a reform of the cultural life of Croton, urging the citizens to follow virtue, and forming an elite circle of followers around himself — the Pythagoreans. Very strict rules of conduct governed this cultural centre. He opened his school to male and female students alike. Those who joined the inner circle of Pythagoras's society called themselves the mathematikoi. They lived at the school, owned no personal possessions, and were required to assume a vegetarian diet. Other students who lived in neighbouring areas were also permitted to attend Pythagoras's school. Known as akousmatikoi, these students were permitted to eat meat and own personal belongings.
According to Iamblichus, the Pythagoreans followed a structured life of religious teaching, common meals, exercise, reading and philosophical study. Music featured as an essential organising factor of this life: the disciples would sing hymns to Apollo together regularly; they used the lyre to cure illness of the soul or body; poetry recitations occurred before and after sleep to aid the memory.
Flavius Josephus relates that, according to Hermippus of Smyrna, Pythagoras was familiar with and an admirer of Jewish customs and wisdom (De Pythagora, Contra Apionem I, 162/165). Hermippus is quoted as saying about Pythagoras: "In practising and repeating these precepts he was imitating and appropriating the doctrines of Jews and Thracians. In fact, it is actually said that great man introduced many points of Jewish law into his philosophy" (trans. H. St. J. Thackeray, The Loeb Classical Library, Cambridge, MA).
Pythagoras is commonly given credit for discovering the Pythagorean theorem, a theorem in trigonometry that states that in a right-angled triangle the area of the square whose side is the hypotenuse (the side opposite the right angle), c, is equal to the sum of the areas of the squares of the other two sides, b and a, that is, a2 + b2 = c2.
The history of the Pythagorean theorem that bears his name is complex. Whether Pythagoras himself proved this theorem is not known, as it was common in the ancient world to credit a famous teacher with the discoveries of his students. The earliest known mention of Pythagoras's name in connection with the theorem occurred five centuries after his death, in the writings of Cicero and Plutarch. Instances of the Pythagorean theorem appear in Babylonian sources very early on, and the scribe of Plimpton 322 must have possessed the general statement. The theorem is certainly stated explicitly in Baudhayana's Sulbasutram (ca. 800 BCE); it is unknown whether Vedic mathematicians learned the statement from the Babylonians, or whether Pythagoras himself did.
Pythagoras made an important achievement in astronomy; he was one of the first people to realise that Venus as the morning star and Venus as the evening star were the same planet. He believed in the mistaken geocentric world-view of his age, but on the other hand he recognised that the Moon's orbit around the Earth was inclined towards the equator of the earth.
According to myth, he died from being shot by a soldier, because he refused to trample a bean-field while fleeing.
Pythagoras's followers were commonly called Pythagoreans. For the most part we remember them as philosophical mathematicians who had an influence on the beginning of axiomatic geometry, which after two hundred years of development was written down by Euclid in The Elements.
The Pythagoreans observed a rule of silence called echemythia, the breaking of which was punishable by death. This was because the Pythagoreans believed that a man's words were usually careless and misrepresented him and that when someone was "in doubt as to what he should say, he should always remain silent". Another rule that they had was to help a man "in raising a burden, but do not assist him in laying it down, for it is a great sin to encourage indolence", and they said "departing from your house, turn not back, for the furies will be your attendants"; this axiom reminded them that it was better to learn none of the truth about mathematics, God, and the universe at all than to learn a little without learning all. (The Secret Teachings of All Ages Many P. Hall).
In his biography of Pythagoras (written seven centuries after Pythagoras's time), Porphyry stated that this silence was "of no ordinary kind." The Pythagoreans were divided into an inner circle called the mathematikoi ("mathematicians") and an outer circle called the akousmatikoi ("listeners"). Porphyry wrote "the mathematikoi learned the more detailed and exactly elaborate version of this knowledge, the akousmatikoi (were) those which had heard only the summary headings of his (Pythagoras's) writings, without the more exact exposition." According to Iamblichus, the akosmatikoi were the exoteric disciples who listened to lectures that Pythagoras gave out loud from behind a veil.
The akousmatikoi were not allowed to see Pythagoras and they were not taught the inner secrets of the cult. Instead they were taught laws of behaviour and morality in the form of cryptic, brief sayings which had hidden meanings. The akousmatikoi recognised the mathematikoi as real Pythagoreans, but not vice versa. After the murder of Pythagoras and a number of the mathematikoi by the followers of Cylon of Athens, a resentful disciple, the two groups split from each other entirely, with Pythagoras's wife Theano and their two daughters leading the mathematikoi.
Theano, daughter of the Orphic initiate Brontinus, was a mathematician in her own right. She is credited with having written treatises on mathematics, physics, medicine, and child psychology, although nothing of her writing survives. Her most important work is said to have been a treatise on the principle of the golden mean. At a time when women were usually considered property and relegated to the role of housekeeper or spouse, Pythagoras allowed women to function on equal terms in his society.
The Pythagorean society is associated with prohibitions such as not to step over a crossbar, and not to eat beans. These rules seem like primitive superstition, similar to "walking under a ladder brings bad luck". The abusive epithet mystikos logos ("mystical speech") was hurled at Pythagoras even in ancient times to discredit him.
The key here is that akousmata means "rules", so that the superstitious taboos primarily applied to the akousmatikoi, and many of the rules were probably invented after Pythagoras's death and independent from the mathematikoi (arguably the real preservers of the Pythagorean tradition). The mathematikoi placed greater emphasis on inner understanding than did the akousmatikoi, even to the extent of dispensing with certain rules and ritual practices. For the mathematikoi, being a Pythagorean was a question of innate quality and inner understanding.
There was also another way of dealing with the akousmata — by allegorising them. There are a few examples of this, one being Aristotle's explanations of them: "'step not over a balance', i.e., be not covetous; 'poke not the fire with a sword', i.e., do not vex with sharp words a man swollen with anger, 'eat not heart', i.e., do not vex yourself with grief", etc. There is evidence for Pythagoreans allegorising in this way at least as far back as the early fifth century BCE. This suggests that the strange sayings were riddles for the initiated.
The Pythagoreans are known for their theory of the transmigration of souls, and also for their theory that numbers constitute the true nature of things. They performed purification rites and followed and developed various rules of living which they believed would enable their soul to achieve a higher rank among the gods.
Much of their mysticism concerning the soul seem inseparable from the Orphic tradition. The Orphics advocated various purificatory rites and practices as well as incubatory rites of descent into the underworld. Pythagoras is also closely linked with Pherecydes of Syros, the man ancient commentators tend to credit as the first Greek to teach a transmigration of souls. Ancient commentators agree that Pherekydes was Pythagoras's most intimate teacher. Pherekydes expounded his teaching on the soul in terms of a pentemychos ("five-nooks", or "five hidden cavities") — the most likely origin of the Pythagorean use of the pentagram, used by them as a symbol of recognition among members and as a symbol of inner health (ugieia).
Pythagoras was interested in music and the Pythagoreans were musicians as well as mathematicians. He wanted to improve the music of his day, which he believed was not harmonious enough and was too chaotic.
According to legend, the way Pythagoras discovered that musical notes could be translated into mathematical equations was one day as he passed blacksmiths at work, and thought that the sounds emanating from their anvils being hit were beautiful and harmonious and decided that whatever scientific law caused this to happen must be mathematical and could be applied to music. He went to the blacksmiths to learn how this had happened by looking at their tools. He discovered that it was because the anvils were "simple ratios of each other, one was half the size of the first, another was 2/3 the size, and so on." The Pythagoreans elaborated a theory of numbers, the exact meaning of which is still debated among scholars. Pythagoras believed in something called the harmony of the spheres. He believed that since planets and the stars all moved in the universe according to mathematical equations that these mathematical equations could be translated into musical notes and thus produce a symphony.
No texts by Pythagoras survive, although forgeries under his name – a few of which have survived – did circulate in antiquity. Critical ancient sources such as Aristotle and Aristoxenus cast doubt on these writings. Ancient Pythagoreans usually quoted their master's doctrines with the phrase autos ephe ("he himself said") — emphasising the essentially oral nature of his teaching. Pythagoras appears as a character in the last book of Ovid's Metamorphoses, where Ovid has him expound upon his philosophical viewpoints. Pythagoras has been quoted as saying: "No man is free who cannot command himself".
Influence on Plato
Pythagoras or in a broader sense, the Pythagoreans, exercised an important influence on the work of Plato. Indeed, much of Pythagoras reappears in Plato. For example, at a time when such thoughts were rare, Pythagoras insisted that men and women were equal, that property should be held in common, and that adherents live and eat communally. All of this reappears in the Republic as Plato's recommended lifestyle for the Guardians alongside the Pythagorean doctrines of the heavenly forms and the split between the world of knowledge and the world of matter (of which philosophers must remain aloof).
- In the Meno, Pythagoras' view of how learning is really recollection appears, as the 'slave boy' recalls the geometrical theorem that bears Pythagoras' name.
- In the Gorgias, there is the Pythagorean doctrine that the better one knows something, the more one becomes like it.
- The Timaeus is a Pythagorean description of the universe in terms of (musical) harmonies, and matter which is revealed mystically here as being made up of geometrical shapes, notably triangles.
- In the Phaedo is the Pythagorean view that philosophy is a preparation for death and immortality 
Philosophy is sometimes said to be a series of footnotes to Plato, which in many ways is true. But mysteriously enough, Plato himself seems on closer examination to consist largely of footnotes to Pythagoras.
The contemporary British philosopher R. M. Hare also notes the influence in three places: first, the Platonic Republic reflects the idea of "a tightly organized community of like-minded thinkers", like the one established by Pythagoras in Croton; secondly, there is evidence that Plato took from Pythagoras the idea that mathematics and, generally speaking, abstract thinking is a secure basis for philosophical thinking as well as "for substantial theses in science and morals"; thirdly, Plato and Pythagoras shared a "mystical approach to the soul and its place in the material world". It is probable that both have been influenced by Orphism.
Plato's harmonics were clearly influenced by the work of Archytas, a genuine Pythagorean of the third generation, who made important contributions to geometry, reflected in Book VIII of Euclid's Elements.
Esotericism and numerology
Pythagoras started a secret society called the Pythagorean brotherhood devoted to the study of mathematics. This had a great effect on future esoteric traditions such as Rosicrucianism and Freemasonry, both of which were occult groups dedicated to the study of mathematics, and both of which claimed to have evolved out of the Pythagorean brotherhood.
Pythagorean theory was tremendously influential on later numerology, which was extremely popular throughout the Middle East in the ancient world. Eighth-century Islamic alchemist Jabir ibn Hayyan, inventor of numerous important chemical processes still in use today, grounded his work in an elaborate numerology greatly influenced by Pythagorean theory.
- According to Diogenes Laertius: "Pythagoras was the first person who invented the term philosophy, and called himself a philosopher" (Φιλοσοφίαν δὲ πρῶτος ὠνόμασε Πυθαγόρας καὶ ἑαυτὸν φιλόσοφον: Lives of Philosophers 1.12 (Greek).
- Christoph Riedweg, Pythagoras: His Life, Teaching and Influence, trans. Steven Rendall (Cornell UP, 2005), pp 5–6, 59, 73.
- Christoph Reidwig, Pythagoras, His Life, Teaching and Influence. Cornell: Cornell University Press, 2005.
- Philosophical Tales (2008), by Martin Cohen, p40
- R.M. Hare, "Plato", in C.C.W. Taylor, R.M. Hare, and Jonathan Barnes [edd] Greek Philosophers, Socrates, Plato, and Aristotle. Oxford: Oxford University Press, 1999 (1982).
- Manly P. Hall, The Secret Teachings of All Ages