NationMaster - Encyclopedia: Pythagoras

Pythagoras of Samos (Greek: Πυθαγόρας; born between 580 and 572 BC, died between 500 and 490 BC) was an Ionian Greek mathematician^[1] and founder of the religious movement called Pythagoreanism. He is often revered as a great mathematician, mystic and scientist; however some have questioned the scope of his contributions to mathematics and natural philosophy.^[2] Herodotus referred to him as "the most able philosopher among the Greeks".^[3] 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.^[4] The Pre-Socratic philosophers were active before Socrates or contemporaneously, but expounding knowledge developed earlier. ... Image File history File links No higher resolution available. ... Michelangelos design for Capitoline Hill, now home to the Capitoline Museums. ... For other uses, see Rome (disambiguation). ... Centuries: 7th century BC - 6th century BC - 5th century BC Decades: 620s BC - 610s BC - 600s BC - 590s BC - 580s BC - 570s BC - 560s BC - 550s BC - 540s BC - 530s BC Events and Trends 589 BC - Apries succeeds Psammetichus II as king of Egypt 588 BC _ Nebuchadnezzar II of... Centuries: 7th century BC - 6th century BC - 5th century BC Decades: 620s BC 610s BC 600s BC 590s BC 580s BC - 570s BC - 560s BC 550s BC 540s BC 530s BC 520s BC Events and Trends 579 BC - Servius Tullius succeeds the assassinated Lucius Tarquinius Priscus as king of Rome. ... Centuries: 7th century BC - 6th century BC - 5th century BC Decades: 550s BC - 540s BC - 530s BC - 520s BC - 510s BC - 500s BC - 490s BC - 480s BC - 470s BC - 460s BC - 450s BC Events and Trends 509 BC - Foundation of the Roman Republic 508 BC - Office of pontifex maximus created... Centuries: 6th century BC - 5th century BC - 4th century BC Decades: 540s BC 530s BC 520s BC 510s BC 500s BC - 490s BC - 480s BC 470s BC 460s BC 450s BC 440s BC Years: 495 BC 494 BC 493 BC 492 BC 491 BC - 490 BC - 489 BC 488 BC... Bust of Pythagoras Pythagoreanism is a term used for the esoteric and metaphysical beliefs held by Pythagoras and his followers, the Pythagoreans, who were much influenced by mathematics and probably a main inspirational source for Plato and platonism. ... Plato (Left) and Aristotle (right), by Raphael (Stanza della Segnatura, Rome) Metaphysics is the branch of philosophy concerned with explaining the ultimate nature of reality, being, and the world. ... For other uses, see Music (disambiguation). ... For other meanings of mathematics or uses of math and maths, see Mathematics (disambiguation) and Math (disambiguation). ... For other uses, see Ethics (disambiguation). ... For other uses, see Politics (disambiguation). ... Musica universalis or music of the spheres is a medieval philosophical concept that regards the proportions in the movements of the celestial bodies - the Sun, Moon and planets - as a form of musica (the medieval Latin name for music). ... Not to be confused with Golden mean (philosophy), the felicitous middle between two extremes, Golden numbers, an indicator of years in astronomy and calendar studies, or the Golden Rule. ... Pythagorean tuning is a system of musical tuning in which the frequency relationships of all intervals are based on the ratio 3:2. ... In mathematics, the Pythagorean theorem (AmE) or Pythagoras theorem (BrE) is a relation in Euclidean geometry among the three sides of a right triangle. ... For the French electronics and defence contractor, see Thales Group Thales (in Greek: Θαλης) of Miletus (circa 635 BC - 543 BC), also known as Thales the Milesian, was a pre-Socratic Greek philosopher and one of the Seven Sages of Greece. ... This article is about the Pre-Socratic philosopher. ... Pherecydes of Syros (in Greek: Φερεχύδης) was a Greek thinker from the island of Siros, Magna Graecia of the 6th century BC. Pherecydes authored the Heptamychia, one of the first attested prose works in Greek literature, which formed an important bridge between mythic and pre-Socratic though. ... Philolaus (circa 480 BC â€“ circa 405 BC) was a Greek mathematician and philosopher. ... Alcmaeon of Croton (mid-fifth century B.C.) was an Ancient Greek philosopher and medical theorist. ... Parmenides of Elea (Greek: , early 5th century BC) was an ancient Greek philosopher born in Elea, a Hellenic city on the southern coast of Italy. ... For other uses, see Plato (disambiguation). ... For other uses, see Euclid (disambiguation). ... Empedocles (Greek: , ca. ... Hippasus of Metapontum, born circa 500 B.C. in Magna Graecia, was a Greek philosopher. ... Kepler redirects here. ... Centuries: 7th century BC - 6th century BC - 5th century BC Decades: 620s BC - 610s BC - 600s BC - 590s BC - 580s BC - 570s BC - 560s BC - 550s BC - 540s BC - 530s BC Events and Trends 589 BC - Apries succeeds Psammetichus II as king of Egypt 588 BC _ Nebuchadnezzar II of... Centuries: 7th century BC - 6th century BC - 5th century BC Decades: 620s BC 610s BC 600s BC 590s BC 580s BC - 570s BC - 560s BC 550s BC 540s BC 530s BC 520s BC Events and Trends 579 BC - Servius Tullius succeeds the assassinated Lucius Tarquinius Priscus as king of Rome. ... Centuries: 7th century BC - 6th century BC - 5th century BC Decades: 550s BC - 540s BC - 530s BC - 520s BC - 510s BC - 500s BC - 490s BC - 480s BC - 470s BC - 460s BC - 450s BC Events and Trends 509 BC - Foundation of the Roman Republic 508 BC - Office of pontifex maximus created... Centuries: 6th century BC - 5th century BC - 4th century BC Decades: 540s BC 530s BC 520s BC 510s BC 500s BC - 490s BC - 480s BC 470s BC 460s BC 450s BC 440s BC Years: 495 BC 494 BC 493 BC 492 BC 491 BC - 490 BC - 489 BC 488 BC... The Ionians were one of the three main ancient Greek ethno-linguistic groups, linked by their use of the Ionic dialect of the Greek language. ... Ancient Greece is the term used to describe the Greek_speaking world in ancient times. ... Leonhard Euler, considered one of the greatest mathematicians of all time A mathematician is a person whose primary area of study and research is the field of mathematics. ... Bust of Pythagoras Pythagoreanism is a term used for the esoteric and metaphysical beliefs held by Pythagoras and his followers, the Pythagoreans, who were much influenced by mathematics and probably a main inspirational source for Plato and platonism. ... Herodotus of Halicarnassus (Greek: HÄ“rÃ³dotos HalikarnÄsseÃºs) was a Greek historian who lived in the 5th century BC (ca. ... For other uses, see Pythia (disambiguation). ... For other uses, see Apollo (disambiguation). ... For the medieval Sicilian translator, scholar, and courtier, see Henry Aristippus. ... Two historical persons go by the name Iamblichus (Greek: Ιάμβλιχος) A Greek novelist; see Iamblichus (novelist) A neoplatonist philosopher; see Iamblichus (philosopher) This is a disambiguation page — a navigational aid which lists other pages that might otherwise share the same title. ...

He is best known for the Pythagorean theorem, which bears his name. Known as "the father of numbers", Pythagoras made influential contributions to philosophy and religious teaching in the late 6th century BC. 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. We do know 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." In mathematics, the Pythagorean theorem (AmE) or Pythagoras theorem (BrE) is a relation in Euclidean geometry among the three sides of a right triangle. ... For other uses, see Philosophy (disambiguation). ... (2nd millennium BC - 1st millennium BC - 1st millennium) The 6th century BC started on January 1, 600 BC and ended on December 31, 501 BC. // Monument 1, an Olmec colossal head at La Venta The 5th and 6th centuries BC were a time of empires, but more importantly, a time... Pre-Socratic philosophers are often very hard to pin down, and it is sometimes very difficult to determine the actual line of argument they used in supporting their particular views. ... For other meanings of mathematics or uses of math and maths, see Mathematics (disambiguation) and Math (disambiguation). ... Iamblichus, also known as Iamblichus Chalcidensis, (ca. ...

He was the first man to call himself a philosopher, or lover of wisdom,^[5] and Pythagorean ideas exercised a marked influence on Plato. Unfortunately, very little is known about Pythagoras because none of his writings have survived. Many of the accomplishments credited to Pythagoras may actually have been accomplishments of his colleagues and successors. For other uses, see Plato (disambiguation). ...

[edit] Life

Pythagoras was born on 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 Phoenician 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 head 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. It may have been in Egypt where he learned some geometric principles which eventually inspired his formulation of the theorem that is now called by his name. This possible inspiration is presented as an example problem in the Berlin Papyrus. Samos (Greek: Î£Î¬Î¼Î¿Ï‚) is a Greek island in the Eastern Aegean sea, located between the island of Chios to the North and the archipelagic complex of the Dodecanese to the South and in particular the island of Patmos and off the coast of Turkey, on what was formerly known as Ionia. ... Anatolia (Greek: ανατολη anatole, rising of the sun or East; compare Orient and Levant, by popular etymology Turkish Anadolu to ana mother and dolu filled), also called by the Latin name of Asia Minor, is a region of Southwest Asia which corresponds today to the Asian portion of Turkey. ... Phoenicia (or Phenicia ,[1] from Biblical Phenice [1]) was an ancient civilization centered in the north of ancient Canaan, with its heartland along the coast of modern day Lebanon and Syria. ... The Triumphal Arch Tyre (Arabic , Phoenician , Hebrew Tzor, Tiberian Hebrew , Akkadian , Greek TÃ½ros) is a city in the South Governorate of Lebanon. ... Crotone is a city in Calabria, southern Italy, on the Gulf of Taranto. ... For other uses, see Calabria (disambiguation). ... This page is about the religious concept of Tyranny. ... For the bishop, see Polycrates of Ephesus. ... Two historical persons go by the name Iamblichus (Greek: Ιάμβλιχος) A Greek novelist; see Iamblichus (novelist) A neoplatonist philosopher; see Iamblichus (philosopher) This is a disambiguation page — a navigational aid which lists other pages that might otherwise share the same title. ... For the Defense and Security Company, see Thales Group. ... For other uses, see Memphis. ... Look up theorem in Wiktionary, the free dictionary. ... The Berlin papyrus is an ancient Egyptian papyrus document that was created circa 1800 BCE. This papyrus was found at the Saqqara ancient Egyptian burial ground in the early 19th Century. ...

Upon his migration from Samos to Croton, Calabria, Italy, Pythagoras established a secret religious society very similar to (and possibly influenced by) the earlier Orphic cult. Crotone is a city in Calabria, southern Italy, on the Gulf of Taranto. ... For other uses, see Calabria (disambiguation). ... Orphism or (more rarely) Orphicism seems to have been a mystery religion in the ancient Greek world. ...

Pythagoras undertook a reform of the cultural life of Croton, urging the citizens to follow virtue and form an elite circle of followers around himself called Pythagoreans. Very strict rules of conduct governed this cultural center. 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 mainly vegetarian diet (meat that could be sacrificed was allowed to be eaten). Other students who lived in neighboring areas were also permitted to attend Pythagoras's school. Known as Akousmatikoi, these students were permitted to eat meat and own personal belongings. Richard Blackmore, in his book The Lay Monastery (1714), saw in the religious observances of the Pythagoreans, "the first instance recorded in history of a monastic life." A variety of vegetarian food ingredients Vegetarianism is the practice of a diet that excludes all animal flesh, including poultry, game, fish, shellfish or crustacea, and slaughter by-products. ...

According to Iamblichus, the Pythagoreans followed a structured life of religious teaching, common meals, exercise, reading and philosophical study. Music featured as an essential organizing 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. Image File history File links Download high resolution version (864x1152, 181 KB)Bust of Pythagoras at the Vatican Museum, in Rome File history Legend: (cur) = this is the current file, (del) = delete this old version, (rev) = revert to this old version. ... Image File history File links Download high resolution version (864x1152, 181 KB)Bust of Pythagoras at the Vatican Museum, in Rome File history Legend: (cur) = this is the current file, (del) = delete this old version, (rev) = revert to this old version. ... Bust of Richard Bently by Roubiliac A bust is a sculpture depicting a persons chest, shoulders, and head, usually supported by a stand. ... Iamblichus (ca. ... For other uses, see Apollo (disambiguation). ... â€œLyresâ€ redirects here. ...

Flavius Josephus, in his polemical Against Apion, in defence of Judaism against Greek philosophy, mentions that according to Hermippus of Smyrna, Pythagoras was familiar with Jewish beliefs, incorporating some of them in his own philosophy. Josephus, also known as Flavius Josephus (c. ... Against Apion was a work written by Flavius Josephus as a defense of Judaism as a classical religion and philosophy, stressing its antiquity against the relatively more recent traditions of the Greeks. ... This article or section does not cite its references or sources. ... Greek philosophy focused on the role of reason and inquiry. ...

Towards the end of his life he fled to Metapontum because of a plot against him and his followers by a noble of Croton named Cylon. He died in Metapontum around 90 years old from unknown causes. This article needs to be cleaned up to conform to a higher standard of quality. ...

[edit] Pythagoreans

The organization was in some ways a school, in some ways a brotherhood, and in some ways a monastery. It was based upon Pythagoras’ religious teachings and was very secretive. At first, the school was highly concerned with the morality of society. Members were required to live ethically, love one another, share political beliefs, practice pacifism, and devote themselves to the mathematics of nature. The Pythagoreans were a Hellenic organization of astronomers, musicians, mathematicians, and philosophers who believed that all things are, essentially, numeric. ... Download high resolution version (880x1032, 169 KB) Wikipedia does not have an article with this exact name. ... Download high resolution version (880x1032, 169 KB) Wikipedia does not have an article with this exact name. ... The School of Athens or in Italian is one of the most famous paintings by the Italian Renaissance artist Raphael. ... This article is about the Renaissance artist. ...

Pythagoras's followers were commonly called "Pythagoreans". They are generally accepted 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. For other uses, see Euclid (disambiguation). ... The frontispiece of Sir Henry Billingsleys first English version of Euclids Elements, 1570 Euclids Elements (Greek: ) is a mathematical and geometric treatise consisting of 13 books written by the Greek mathematician Euclid in Alexandria circa 300 BC. It comprises a collection of definitions, postulates (axioms), propositions (theorems...

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 by Manly 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 akousmatikoi were the exoteric disciples who listened to lectures that Pythagoras gave out loud from behind a veil. Porphyry of Tyre (Greek: , c. ... Iamblichus (ca. ... Exoteric knowledge is knowledge that is publicly available, in contrast with esoteric knowledge, which is kept from everyone except the initiated. ...

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 behavior and morality in the form of cryptic, brief sayings that had hidden meanings. The akousmatikoi recognized the mathematikoi as real Pythagoreans, but not vice versa. After the murder of a number of the mathematikoi by the cohorts of Cylon, a resentful disciple, the two groups split from each other entirely, with Pythagoras's wife Theano and their two daughters leading the mathematikoi. Theano was one of the few women in ancient mathematics. ...

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. In 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. In philosophy (especially that of Aristotle), the golden mean is the felicitous middle between two extremes, one of excess and the other of deficiency; for this meaning, see golden mean (philosophy). ...

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 prohibition on beans could be linked to favism, which is relatively widespread around the Mediterranean. For other uses, see Superstition (disambiguation). ... Glucose-6-Phosphate Dehydrogenase (G6PD) deficiency is an X-linked recessive hereditary disease featuring nonimmune hemolytic anemia in response to a number of causes. ...

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 allegorizing them. We have 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. We have evidence for Pythagoreans allegorizing in this way at least as far back as the early fifth century BC. This suggests that the strange sayings were riddles for the initiated. For other uses, see Aristotle (disambiguation). ...

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). The head of Orpheus, from an 1865 painting by Gustave Moreau. ... Pherecydes of Syros (in Greek: Φερεχύδης) was a Greek thinker from the island of Siros, Magna Graecia of the 6th century BC. Pherecydes authored the Heptamychia, one of the first attested prose works in Greek literature, which formed an important bridge between mythic and pre-Socratic though. ...

[edit] Musical theories and investigations

Pythagoras was very interested in music, and so were his followers. The Pythagoreans were musicians as well as mathematicians. Pythagoras wanted to improve the music of his day, which he believed was not harmonious enough and was too hectic.

According to legend, the way Pythagoras discovered that musical notes could be translated into mathematical equations was when one day 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." (See Pythagorean tuning.) For other uses, see Anvil (disambiguation). ... Pythagorean tuning is a system of musical tuning in which the frequency relationships of all intervals are based on the ratio 3:2. ...

The Pythagoreans elaborated on 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 the planets and stars moved according to mathematical equations, which corresponded to musical notes and thus produced a symphony.^[6]

[edit] Influence

Pythagoras is commonly given credit for discovering the Pythagorean theorem, a theorem in geometry that states that in a right-angled triangle the square of the hypotenuse (the side opposite the right angle), c, is equal to the sum of the squares of the other two sides, b and a—that is, a² + b² = c². In mathematics, the Pythagorean theorem (AmE) or Pythagoras theorem (BrE) is a relation in Euclidean geometry among the three sides of a right triangle. ...

While the theorem that now bears his name was known and previously utilized by the Babylonians, and Indians, he, or his students, are thought to have constructed the first proof. Because of the secretive nature of his school and the custom of its students to attribute everything to their teacher, there is no evidence that Pythagoras himself worked on or proved this theorem. For that matter, there is no evidence that he worked on any mathematical or meta-mathematical problems. Some attribute it as a carefully constructed myth by followers of Plato over two centuries after the death of Pythagoras, mainly to bolster the case for Platonic meta-physics, which resonate well with the ideas they attributed to Pythagoras. This attribution has stuck, down the centuries up to modern times. ^[7] 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. For other uses, see Plato (disambiguation). ... For other uses, see Cicero (disambiguation). ... Mestrius Plutarchus (Greek: Î Î»Î¿ÏÏ„Î±ÏÏ‡Î¿Ï‚; 46 - 127), better known in English as Plutarch, was a Greek historian, biographer, essayist, and Middle Platonist. ...

Today, Pythagoras is revered as a prophet by the Ahl al-Tawhid or Druze faith along with his fellow Greek, Plato. But Pythagoras also had his critics, such as Heraclitus who said that "much learning does not teach wisdom; otherwise it would have taught Hesiod and Pythagoras, and again Xenophanes and Hecataeus".^[8] The People of Monotheism (Arabic: Ahl al-Tawhid) is one a name the Druze use for themselves. ... Religions Druzism Scriptures Rasail al-hikmah (Epistles of Wisdom), Quran Languages Arabic. ... Heraclitus of Ephesus (Ancient Greek - HerÃ¡kleitos ho EphÃ©sios (Herakleitos the Ephesian)) (about 535 - 475 BC), known as The Obscure (Ancient Greek - ho SkoteinÃ³s), was a pre-Socratic Greek philosopher, a native of Ephesus on the coast of Asia Minor. ... Roman bronze bust, the so-called Pseudo-Seneca, now identified by some as possibly Hesiod Hesiod (Hesiodos, ) was an early Greek poet and rhapsode, who presumably lived around 700 BC. Hesiod and Homer, with whom Hesiod is often paired, have been considered the earliest Greek poets whose work has survived... Xenophanes of Colophon (Greek: ÎžÎµÎ½Î¿Ï†Î¬Î½Î·Ï‚, 570 BC-480 BC) was a Greek philosopher, poet, and social and religious critic. ... Hecataeus (c. ...

[edit] Religion and science

Pythagoras’ religious and scientific views were, in his opinion, inseparably interconnected. However, they are looked at separately in the 21st century. Religiously, Pythagoras was a believer of metempsychosis. He believed in transmigration, or the reincarnation of the soul again and again into the bodies of humans, animals, or vegetables until it became moral. His ideas of reincarnation were influenced by ancient Greek religion. He was one of the first to propose that the thought processes and the soul were located in the brain and not the heart. He himself claimed to have lived four lives that he could remember in detail, and heard the cry of his dead friend in the bark of a dog. Metempsychosis is a philosophical term in the Greek language referring to the belief of transmigration of the soul, especially its reincarnation after death. ...

One of Pythagoras' beliefs was that the essence of being is number. Thus, being relies on stability of all things that create the universe. Things like health relied on a stable proportion of elements; too much or too little of one thing causes an imbalance that makes a being unhealthy. Pythagoras viewed thinking as the calculating with the idea numbers. When combined with the Folk theories, the philosophy evolves into a belief that Knowledge of the essence of being can be found in the form of numbers. If this is taken a step further, one can say that because mathematics is an unseen essence, the essence of being is an unseen characteristic that can be encountered by the study of mathematics.

[edit] Literary works

No texts by Pythagoras survive, although forgeries under his name — a few of which remain extant — did circulate in antiquity. Critical ancient sources like Aristotle and Aristoxenus cast doubt on these writings. Ancient Pythagoreans usually quoted their master's doctrines with the phrase autos ephe ("he himself said") — emphasizing 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." Classical antiquity is a broad term for a long period of cultural history centered on the Mediterranean Sea, which begins roughly with the earliest-recorded Greek poetry of Homer (7th century BC), and continues through the rise of Christianity and the fall of the Western Roman Empire (5th century AD... For other uses, see Aristotle (disambiguation). ... Aristoxenus (Greek: á¼ˆÏÎ¹ÏƒÏ„ÏŒÎ¾ÎµÎ½Î¿Ï‚) of Tarentum (4th century BC) was a Greek peripatetic philosopher, and writer on music and rhythm. ... For other uses, see Ovid (disambiguation) Publius Ovidius Naso (March 20, 43 BC â€“ 17 AD) was a Roman poet known to the English-speaking world as Ovid who wrote on topics of love, abandoned women and mythological transformations. ... // Cover of George Sandyss 1632 edition of Ovids Metamorphosis Englished The Metamorphoses by the Roman poet Ovid is a poem in fifteen books that describes the creation and history of the world in terms according to Greek and Roman points of view. ...

[edit] Lore

There is another side to Pythagoras, as he became the subject of elaborate legends surrounding his historic persona. Aristotle described Pythagoras as wonder-worker and somewhat of a supernatural figure, attributing to him such aspects as a golden thigh, which was a sign of divinity. According to Aristotle and others' accounts, some ancients believed that he had the ability to travel through space and time, and to communicate with animals and plants.^[9] An extract from Brewer's Dictionary of Phrase and Fable's entry entitled "Golden Thigh": Brewers Dictionary of Phrase and Fable - sometimes referred to simply as Brewers - is a reference work containing definitions and explanations of many famous phrases, allusions and figures, whether historical or mythical. ...

Another legend, also taken from Brewer's Dictionary, describes his writing on the moon:

[edit] Other accomplishments

One of Pythagoras's major accomplishments was the discovery that music was based on proportional intervals of the numbers one through four. He believed that the number system, and therefore the universe system, was based on the sum of these numbers: ten. Pythagoreans swore by the Tetrachtys of the Decad, or ten, rather than by the gods. Odd numbers were masculine and even were feminine. He discovered the theory of mathematical proportions, constructed from three to five geometrical solids. One of his order, Hippasos, also discovered irrational numbers, but the idea was unthinkable to Pythagoras, and according to one version this member was executed. Pythagoras (or the Pythagoreans) also discovered square numbers. They found that if one took, for example, four small stones and arranged them into a square, each side of the square was not only equivalent to the other, but that when the two sides were multiplied together, they equaled the sum total of stones in the square arrangement, hence the name "Square Root"^[12]. He was one of the first to think that the earth was round, that all planets have an axis, and that all the planets travel around one central point. He originally identified that point as Earth, but later renounced it for the idea that the planets revolve around a central “fire” that he never identified as the sun. He also believed that the moon was another planet that he called a “counter-Earth” – furthering his belief in the Limited-Unlimited. Hippasus of Metapontum, born circa 500 B.C. in Magna Graecia, was a Greek philosopher. ... In mathematics, an irrational number is any real number that is not a rational number, i. ...

[edit] Groups influenced by Pythagoras

[edit] Influence on Plato

Pythagoras or in a broader sense, the Pythagoreans, allegedly exercised an important influence on the work of Plato. According to R. M. Hare, his influence consists of three points: a) the platonic Republic might be related to the idea of "a tightly organized community of like-minded thinkers", like the one established by Pythagoras in Croton. b) there is evidence that Plato possibly 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". c) 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.^[13] R.M. Hare Richard Mervyn Hare (March 21, 1919 â€“ January 29, 2002) was an English moral philosopher, who held the post of Whites Professor of Moral Philosophy at the University of Oxford from 1966 until 1983. ... The Republic (Greek: ) is an influential work of philosophy and political theory by the Greek philosopher Plato, written in approximately 360 BC. It is written in the format of a Socratic dialogue. ... A magnet levitating above a high-temperature superconductor demonstrates the Meissner effect. ... Morality is a complex of principles based on cultural, religious, and philosophical concepts and beliefs, by which an individual determines whether his or her actions are right or wrong. ... For other uses, see Soul (disambiguation). ... For alternative meanings, see nature (disambiguation). ... Orphism or Orphicism is a secret religious movement in the classical Greek world. ...

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. Archytas Archytas (428 BC - 347 BC) was a Greek philosopher, mathematician, astronomer, statesman, strategist and commander-in-chief. ... For other uses, see Euclid (disambiguation). ...

[edit] Roman influence

In the legends of ancient Rome, Numa Pompilius, the second King of Rome, is said to have studied under Pythagoras. This is unlikely, since the commonly accepted dates for the two lives do not overlap. Ancient Rome was a civilization that grew from a small agricultural community founded on the Italian Peninsula circa the 9th century BC to a massive empire straddling the Mediterranean Sea. ... rome hotel According to legend, Numa Pompilius was the second of the Kings of Rome, succeeding Romulus. ...

[edit] Influence on esoteric groups

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. The mystical and occult qualities of Pythagorean mathematics are discussed in a chapter of Manly P. Hall's The Secret Teachings of All Ages entitled "Pythagorean Mathematics".

Pythagorean theory was tremendously influential on later numerology, which was extremely popular throughout the Middle East in the ancient world. The 8th-century Muslim alchemist Jabir ibn Hayyan grounded his work in an elaborate numerology greatly influenced by Pythagorean theory. Look up numerology in Wiktionary, the free dictionary. ... A map showing countries commonly considered to be part of the Middle East The Middle East is a region comprising the lands around the southern and eastern parts of the Mediterranean Sea, a territory that extends from the eastern Mediterranean Sea to the Persian Gulf. ... The 8th century is the period from 701 - 800 in accordance with the Julian calendar in the Christian Era. ... There is also a collection of Hadith called Sahih Muslim A Muslim (Arabic: Ù…Ø³Ù„Ù…, Persian: Mosalman or Mosalmon Urdu: Ù…Ø³Ù„Ù…Ø§Ù†, Turkish: MÃ¼slÃ¼man, Albanian: Mysliman, Bosnian: Musliman) is an adherent of the religion of Islam. ... For other uses, see Alchemy (disambiguation). ... 15th century European portrait of Geber, Codici Ashburnhamiani 1166, Biblioteca Medicea Laurenziana, Florence Abu Musa Jabir Ibn Hayyan, in Latin Geber, was one of the most notable Islamic alchemists. ...

[edit] See also

Hippasus of Metapontum, born circa 500 B.C. in Magna Graecia, was a Greek philosopher. ... Bust of Pythagoras Pythagoreanism is a term used for the esoteric and metaphysical beliefs held by Pythagoras and his followers, the Pythagoreans, who were much influenced by mathematics and probably a main inspirational source for Plato and platonism. ... In music, when ascending from an initial (low) pitch by a cycle of justly tuned perfect fifths (ratio 3:2), leapfrogging twelve times, one eventually reaches a pitch approximately seven whole octaves above the starting pitch. ... Cut-through illustration of a Pythagorean cup A Pythagorean cup (also known as a Pythagoras cup) is a form of drinking cup which forces its user to imbibe only in moderation. ... The Parthenons facade showing an interpretation of golden rectangles in its proportions. ... For other uses, see Heliopolis. ... Pen and ink drawing of the Lute of Pythagoras Lute of Pythagoras is a geometric form made of pentagons with inscribed pentagrams where the sides of the pentagrams are the sides of the smaller pentagons. ... The Pythagoras tree is a plane fractal constructed from squares. ...

[edit] References

[edit] Primary sources

No primary sources about Pythagoras have survived. This article describes the classical interpretation of Pythagoras, which is based on a small set of texts written between 150 AD and 450 AD. As these texts were written 600 to 1000 years after Pythagoras is said to have lived, their accuracy is uncertain.

It is postulated that the classical Pythagoras did not exist prior to these biographies: many of the discoveries and life details they attributed to Pythagoras may have been those of other Pythagoreans, if not fiction. This would explain the lack of reference to a man Pythagoras until 150 AD, given that he would have been of interest to contemporary philosophers (Aristotle referred to the so-called Pythagoreans). It is suggested that the mathematical significance of the early Pythagoreans (pre 450 BC) has been exaggerated (with the exception of their theory of harmonics), and that the Pythagoreans were an Orphic-like cult with an emphasis on numerology who only later evolved into serious mathematicians as geometry became popular across Greece. The head of Orpheus, from an 1865 painting by Gustave Moreau. ... Numerology is any of many systems, traditions or beliefs in a mystical or esoteric relationship between numbers and physical objects or living things. ...

[edit] Classical secondary sources

Only a few relevant source texts deal with Pythagoras and the Pythagoreans, most are available in different translations. Other texts usually build solely on information in these works.

Diogenes Laertius, Vitae philosophorum VIII (Lives of Eminent Philosophers), c. 200 AD, which in turn reference the lost work Successions of Philosophers by Alexander Polyhistor) — Pythagoras, Translation by C.D. Yonge
Porphyry, Vita Pythagorae (Life of Pythagoras), c. 270 AD
Iamblichus, De Vita Pythagorica (On the Pythagorean Life), c. 300 AD
Apuleius also writes about Pythagoras in Apologia, including a story of him being taught by Babylonian disciples of Zoroaster, c. 150 AD
Hierocles of Alexandria, Golden Verses of Pythagoras, Concord Grove Pr., 1983 c.430 AD

Diogenes Laërtius, the biographer of the Greek philosophers, is supposed by some to have received his surname from the town of Laerte in Cilicia, and by others from the Roman family of the Laërtii. ... Lives and Opinions of Eminent Philosophers is a biography of the Greek philosophers by Diogenes LaÃ«rtius. ... A lost work is a document or literary work produced some time in the past of which no surviving copies are known to exist. ... Successions of Philosophers or Philosophers Successions is a lost book written by Alexander Polyhistor, and referenced several times in Diogenes Laertius book Vitae philosophorum (Lives and Opinions of Eminent Philosophers). ... Lucius Cornelius Alexander Polyhistor was a Greek scholar who was enslaved by the Romans during the war of Sulla and taken to Rome as a tutor. ... Porphyry of Tyre (Greek: , c. ... Iamblichus (ca. ... Lucius Apuleius (c. ... For other uses, see Babylon (disambiguation). ... Zoroaster (Greek Î–Ï‰ÏÎ¿Î¬ÏƒÏ„ÏÎ·Ï‚, ZÅroastrÄ“s) or Zarathustra (Avestan: ZaraÎ¸uÅ¡tra), also referred to as Zartosht (Persian: ; Kurdish: ), was an ancient Iranian prophet and religious poet. ... Hierocles of Alexandria, Neoplatonist writer, flourished c. ...

[edit] Modern secondary sources

Burkert, Walter. Lore and Science in Ancient Pythagoreanism. Harvard University Press, June 1, 1972. ISBN 0-674-53918-4
Burnyeat, M. F. "The Truth about Pythagoras". London Review of Books, 22 February 2007.
Guthrie, W. K. A History of Greek Philosophy: Earlier Presocratics and the Pythagoreans, Cambridge University Press, 1979. ISBN 0-521-29420-7
Kingsley, Peter. Ancient Philosophy, Mystery, and Magic: Empedocles and the Pythagorean Tradition. Oxford University Press, 1995.
Hermann, Arnold. To Think Like God: Pythagoras and Parmenides—the Origins of Philosophy. Parmenides Publishing, 2005. ISBN 978-1-930972-00-1
O'Meara, Dominic J. Pythagoras Revived. Oxford University Press, 1989. ISBN 0-19-823913-0 (paperback), ISBN 0-19-824485-1 (hardcover)

Walter Burkert (born Neuendettelsau (Bavaria), February 2, 1931), the most eminent living scholar of Greek myth and cult, is an emeritus professor of classics at the University of Zurich, Switzerland who has also taught in the United Kingdom and the United States. ... is the 152nd day of the year (153rd in leap years) in the Gregorian calendar. ... Year 1972 (MCMLXXII) was a leap year starting on Saturday (link will display full calendar) of the Gregorian calendar. ... Myles Fredric Burnyeat (born 1939) is an English classicist and philosopher. ... There are very few or no other articles that link to this one. ...

[edit] Notes

Diogenes Laërtius, the biographer of the Greek philosophers, is supposed by some to have received his surname from the town of Laerte in Cilicia, and by others from the Roman family of the Laërtii. ... For other uses, see Philosophy (disambiguation). ... Walter Burkert (born Neuendettelsau (Bavaria), February 2, 1931), the most eminent living scholar of Greek myth and cult, is an emeritus professor of classics at the University of Zurich, Switzerland who has also taught in the United Kingdom and the United States. ... For other uses, see Cicero (disambiguation). ... The Tusculanae Quaestiones, also known as the Tusculan Disputations, is a series of books written by Cicero, around 45 BC 1, attempting to popularise philosophy in Ancient Rome. ... Heraclides Ponticus (387 - 312 BCE), also known as Heraklides, was a Greek philosopher who lived and died at Heraclea, now Eregli, Turkey. ... Diogenes Laërtius, the biographer of the Greek philosophers, is supposed by some to have received his surname from the town of Laerte in Cilicia, and by others from the Roman family of the Laërtii. ... Two historical persons go by the name Iamblichus (Greek: Ιάμβλιχος) A Greek novelist; see Iamblichus (novelist) A neoplatonist philosopher; see Iamblichus (philosopher) This is a disambiguation page — a navigational aid which lists other pages that might otherwise share the same title. ...

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