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Pythagoras

, Mathematician/Philosopher
Pythagoras
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  • Born: 570 B.C.
  • Birthplace: Samos, Greece
  • Died: c. 500 B.C.
  • Best Known As: The guy they named the Pythagorean Theorem after

Pythagoras was the thinker who discovered the Pythagorean Theorem in geometry (although none of his actual writings are extant). The theorem states that the sum of the squares of the two legs of a right triangle is equal to the square of the hypotenuse. Not much is known about Pythagoras, other than that he was a mathematician and philosopher who founded a community in southern Italy sometime in the 6th century B.C. His followers were extremely secretive and loyal, and held a mystical view of numbers and their relation to nature.

 
 
Scientist: Pythagoras

Greek mathematician and philosopher (c. 580 bc–c. 500 bc)

All that is known of the life of Pythagoras with any certainty is that he left his birthplace, Samos, in about 520 bc to settle in Croton (now Crotone) in southern Italy and, as a result of political trouble, made a final move to Metapontum in about 500.

In Croton Pythagoras established his academy and became a cult leader. His community was governed by a large number of rules, some dietary, such as those commanding abstinence from meat and from beans, and others of obscure origin, such as the commands not to let a swallow nest under the roof or not to sit on a quart measure.

The movement was united by the belief that “all is number.” While the exact meaning of this may be none too clear, that it led to one of the great periods of mathematics is beyond doubt. Not only were the properties of numbers explored in a totally new way and important theorems discovered, of which the familiar theorem of Pythagoras is the best example, but there also emerged what is arguably the first really deep mathematical truth – the discovery of irrational numbers with the realization of the incommensurability of the square root of 2.

 
Music Encyclopedia: Pythagoras

Greek philosopher and religious teacher. He emigrated from Samos to Croton in southern Italy c 531bc. Doctrines of his school include the harmony of the spheres and a belief in the importance of numbers as a guide to the interpretation of the world. The discovery of the numerical ratios corresponding to the principal intervals of the musical scale is attributed to him. He became an almost legendary figure, and from the 5th century onwards his followers constituted one of the principal schools of Greek musical theory.


 
Biography: Pythagoras

The Greek philosopher, scientist, and religious teacher Pythagoras (ca. 575-ca. 495 B.C.) evolved a school of thought that accepted the transmigration of souls and established number as the principle in the universe.

Born on the island of Samos, Pythagoras was the son of Mnesarchus. He fled to southern Italy to escape the tyranny of Polycrates, who came to power about 538, and he is said to have traveled to Egypt and Babylon. He and his followers became politically powerful in Croton in southern Italy, where Pythagoras had established a school for his newly formed sect. It is probable that the Pythagoreans took positions in the local government in order to lead men to the pure life which their teachings set forth. Eventually, however, a rival faction launched an attack on the Pythagoreans at a gathering of the sect, and the group was almost completely annihilated. Pythagoras either had been banished from Croton or had left voluntarily shortly before this attack. He died in Metapontum early in the 5th century.

Religious Teachings

Pythagoras and his followers were important for their contributions to both religion and science. His religious teachings were based on the doctrine of metempsychosis, which held that the soul was immortal and was destined to a cycle of rebirths until it could liberate itself from the cycle through the purity of its life. A number of precepts were drawn up as inviolable rules by which initiates must live.

Pythagoreanism differed from the other philosophical systems of its time in being not merely an intellectual search for truth but a whole way of life which would lead to salvation. In this respect it had more in common with the mystery religions than with philosophy. Several taboos and mystical beliefs were taught which sprang from a variety of primitive sources such as folk taboo, ritual, and sympathetic magic and were examples of the traditional beliefs that the Greeks continued to hold while developing highly imaginative and rational scientific systems.

An important underlying tenet of Pythagoreanism was the kinship of all life. A universal life spirit was thought to be present in animal and vegetable life, although there is no evidence to show that Pythagoras believed that the soul could be born in the form of a plant. It could be born, however, in the body of an animal, and Pythagoras claimed to have heard the voice of a dead friend in the howl of a dog being beaten.

The number of lives which the soul had to live before being liberated from the cycle is uncertain. Its liberation came through an ascetic life of high moral and ethical standards and strict adherence to the teachings and practices of the sect. Pythagoras himself claimed to remember four different lives. Followers of the sect were enjoined to secrecy, although the discussions of Pythagoras's teachings in other writers proved that the injunction was not faithfully observed.

Mathematical Teachings

The Pythagoreans posited the dualism between Limited and Unlimited. It was probably Pythagoras himself who declared that number was the principle in the universe, limiting and giving shape to matter. His study of musical intervals, leading to the discovery that the chief intervals can be expressed in numerical ratios between the first four integers, also led to the theory that the number 10, the sum of the first four integers, embraced the whole nature of number.

So great was the Pythagoreans' veneration for the "Tetractys of the Decad" (the sum of 1 + 2 + 3 + 4) that they swore their oaths by it rather than by the gods, as was conventional. Pythagoras may have discovered the theorem which still bears his name (in right triangles, the square on the hypotenuse equals the sum of the squares on the other sides), although this proposition has been discovered on a tablet dating from the time of the Babylonian king Hammurabi. Regardless of their sources, the Pythagoreans did important work in systematizing and extending the body of mathematical knowledge.

As a more general scheme, the Pythagoreans posited the two contraries, Limited and Unlimited, as ultimate principles. Numerical oddness and evenness are equated with Limited and Unlimited, as are one and plurality, right and left, male and female, motionlessness and movement, straight and crooked, light and darkness, good and bad, and square and oblong. It is not clear whether an ultimate One, or Monad, was posited as the cause of the two categories.

Cosmological Views

As a result of their religious beliefs and their careful study of mathematics, the Pythagoreans developed a cosmology which differed in some important respects from the world views of their contemporaries, the most important of which was their view of the earth as a sphere which circled the center of the universe. The center of this system was fire, which was invisible to man because his side of the earth was turned from it. The sun reflected that fire; there was a counterearth closer to the center, and the other five planets were farther away and followed longer courses around the center. It is not known how much of this theory was attributable to Pythagoras himself. Later writers ascribe much of it to Philolaos (active 400 B.C.), although it circulated as a view of the school as a whole.

The systematization of mathematical knowledge carried out by Pythagoras and his followers would have sufficed to make him an important figure in the history of Western thought. However, his religious sect and the asceticism which he taught, embracing as it did a vast number of ancient beliefs, make him one of the great teachers of religion in the ancient Greek world.

Further Reading

Pythagoras left no written works. A first-rate technical book, J. A. Philip, Pythagoras and Early Pythagoreanism (1966), separates the valid from the spurious among the legends that surround Pythagoras and his views. An excellent and thorough treatment of the evidence for his life and teachings is in W. K. C. Guthrie, A History of Greek Philosophy (3 vols., 1962-1969). A good account of Pythagoras and his followers is in Kathleen Freeman, The Pre-Socratic Philosophers (1946; 3d ed. 1953), and G. S. Kirk and J. E. Raven, The Presocratic Philosophers (1962). Briefer treatments of the Pythagoreans and the intellectual currents of their time are in the standard histories of Greek literature, such as Albin Lesky, A History of Greek Literature (trans. 1966), or in accounts of Greek philosophy, such as John Burnet, Greek Philosophy (1914).

 
Britannica Concise Encyclopedia: Pythagoras

(born c. 580 BC, Samos, Ionia — died c. 500, Metapontum, Lucania) Greek philosopher and mathematician. He established a community of followers in Croton who adhered to a way of life he prescribed. His school of philosophy reduced all meaning to numerical relationships and proposed that all existing objects are fundamentally composed of form and not material substance. The principles of Pythagoreanism, including belief in the immortality and reincarnation of the soul and in the liberating power of abstinence and asceticism, influenced the thought of Plato and Aristotle and contributed to the development of mathematics and Western rational philosophy. The proportions of musical intervals and scales were first studied by Pythagoras, and he was the first influential Western practitioner of vegetarianism. None of his writings survive, and it is difficult to distinguish the ideas he originated from those of his disciples. His memory is kept alive partly by the Pythagorean theorem, probably developed by his school after he died.

For more information on Pythagoras, visit Britannica.com.

 
Classical Literature Companion: Pȳthagoras

Pȳthagoras, Greek polymath, philosopher, and mystic of the sixth century BC. He wrote no books, but so impressive were his doctrines, his learning, and his way of life that by the end of the fifth century he had become a figure of mystery and legend with a reputation as a great sage and a possessor of miraculous powers (as well as of a golden thigh). The traditions concerning his life are contradictory and confused, but it is believed that he was born at Samos c.580 BC and emigrated (perhaps through hostility to the tyranny of Polycratēs) to Croton in Magna Graecia. There he attracted followers, both men and women, who formed a community and lived according to his rule of life. Even after his death, c.500 BC, Pythagorean societies continued to flourish in Croton and elsewhere in Magna Graecia. The members seem to have been active in the politics of the time and, presenting a united front, were no doubt a powerful force; they became unpopular and eventually (c.450 BC) the societies were broken up and the members killed or exiled.

Part of Pythagoras' teaching was religious and mystical, and it was presumably this aspect which led his contemporary Heracleitus to regard him as a fraud. Another contemporary, Xenophanēs, mocked the most celebrated aspect of his teaching, his doctrine of reincarnation or the transmigration of souls (metempsychōsis), with the story that Pythagoras once claimed to recognize a friend's voice in the howling of a puppy which was being beaten. Pythagoras also declared that he remembered his own previous incarnations, including that as the Trojan Euphorbus, killed in the siege. Pythagoras taught that the soul (a combination of life-principle, self, and mind) is immortal, a fallen divinity imprisoned in the body as in a tomb. Since the soul is rational and responsible for its actions, the choices it makes determine the kind of body into which it is reincarnated, human or animal (perhaps even plant; Empedoclēs, who was greatly influenced by Pythagorean ideas, declares that in one of his incarnations he was a bush). By keeping itself pure from the pollutions of the body the soul may eventually win release from it (see also ORPHEUS and compare Orphic beliefs, with which Pythagoras obviously had much in common). Pythagoras and his followers adhered to a rule of life by which release for the soul might be attained; this was an austere regimen the details of which are not clear but which perhaps entailed silence, self-examination, and abstention from eating flesh and beans (no reason is known for this latter prohibition). The idea of metempsychosis is foreign to Greek tradition and its source uncertain; it may have reached Greece from central Asia or even India. Many precepts of Pythagoras were collected at some time under the name of acusmata (Gk. akousmata, ‘ (oral) instructions’). Some sound like taboo-prohibitions, e.g. ‘Do not poke the fire with a sword’ (which Porphyry interpreted as meaning, ‘Do not vex with sharp words a man swollen with anger’). Others sound like traditional wisdom: ‘What is the wisest of the things in our power? Medicine. What is the fairest thing? Harmony. What is the most powerful? Knowledge. What is the best? Happiness.’

Pythagoras' name is also linked to the study of numbers and proportions as well as astronomy. It is impossible to ascertain what discoveries should be attributed to him personally, but he is credited with the discovery that the relation between the chief musical intervals produced on a vibrating string can be expressed as ratios between the first four whole numbers: octave, 2 : 1; fifth, 3 : 2; fourth, 4 : 3. From this evolved the idea that the explanation of the universe is to be sought in numbers and their relations, of which the objects of sense are representations. According to Aristotle, even abstracts like ‘opinion’ or ‘opportunity’ or ‘injustice’ were numbers in the Pythagorean system, and had their place in the cosmos. Since the first four whole numbers are important in expressing musical harmonies and since their sum can be represented as an equilateral triangular array of ten dots in rows of one, two, three, and four, it was thought that this pattern, the tetraktys, ‘foursome’, of the decad, was of mystical significance, embracing the whole nature of number: the number one could be identified with the point, two with the line, three with the surface, and four with the solid. Pythagoras is also credited with the theorem that still goes under his name, namely that the square on the hypotenuse of a right-angled triangle is equal to the sum of the squares on the other two sides; on discovering it he is said to have made the important sacrifice of a hecatomb. The Pythagoreans believed that the earth is a sphere; later Pythagoreans had an astronomical system in which the heavenly bodies (the sphere of the fixed stars, the five planets, the sun, moon, earth, and counter-earth, the last included to bring the number of bodies up to ten) revolve around a central fire, a system to which an earlier belief in a ‘harmony of the spheres’ was accommodated.

Pythagoreanism influenced not only Empedocles but also Plato, whose science and metaphysics are infused with Pythagorean ideas. Later the doctrines were revived at Rome under the early empire, and became confused with Orphic beliefs with which they had affinities.

 
Philosophy Dictionary: Pythagoras

(b. c. 570 BC) Pythagoras was the son of Mnesarchus of Samos, and emigrated c. 531 BC to Croton in southern Italy. Here he founded a religious society, but was forced into exile and died at Metapontum. Membership of the society entailed self-discipline, silence, and the observance of various taboos, especially against eating flesh and beans. Pythagoras taught the doctrine of metempsychosis, or the cycle of reincarnation, and was supposed able to remember former existences. The soul, which has its own divinity and may have existed as an animal or plant, can, however, gain release by a religious dedication to study, after which it may rejoin the universal world-soul. Pythagoras is usually, but doubtfully, credited with having discovered the basis of acoustics, the numerical ratios underlying the musical scale, thereby initiating the arithmetical interpretation of nature. This tremendous success inspired the view that the whole of the cosmos should be explicable in terms of harmonia or number. The view represents a magnificent break from the Milesian attempt to ground physics on a conception of a prime matter, or undifferentiated basis shared by all things, and to concentrate instead on form, meaning that physical natures receive an intelligible grounding in different geometric structures. The view is vulgarized in the doctrine usually attributed to Pythagoras that all things are numbers. However, the association of abstract qualities with numbers reached remarkable heights, with occult attachments, for instance between justice and the number four, and mystical significances, especially of the number ten. Cosmologically Pythagoras explained the origin of the universe in mathematical terms, as the imposition of limit on the limitless by a kind of injection of a unit. Followers of Pythagoras included Philolaus, the earliest cosmologist known to have understood that the earth is a moving planet. It is also likely that the Pythagoreans discovered the irrationality of the square root of two: see Hippasus of Metapontum. See also harmony of the spheres.

 
Columbia Encyclopedia: Pythagoras
(pĭthăg'ərəs) , c.582–c.507 B.C., pre-Socratic Greek philosopher, founder of the Pythagorean school. He migrated from his native Samos to Crotona and established a secret religious society or order similar to, and possibly influenced by, the earlier Orphic cult. We know little of his life and nothing of his writings. Since his disciples came to worship him as a demigod and to attribute all the doctrines of their order to its founder, it is virtually impossible to distinguish his teachings from those of his followers. The Pythagoreans are best known for two teachings: the transmigration of souls and the theory that numbers constitute the true nature of things. The believers performed purification rites and followed moral, ascetic, and dietary rules to enable their souls to achieve a higher rank in their subsequent lives and thus eventually be liberated from the “wheel of birth.” This belief also led them to regard the sexes as equal, to treat slaves humanely, and to respect animals. The highest purification was “philosophy,” and tradition credits Pythagoras with the first use of the term. Beginning with the discovery that the relationship between musical notes could be expressed in numerical ratios (see Greek music), the Pythagoreans elaborated a theory of numbers, the exact meaning of which is still disputed by scholars. Briefly, they taught that all things were numbers, meaning that the essence of things was number, and that all relationships—even abstract ethical concepts like justice—could be expressed numerically. They held that numbers set a limit to the unlimited—thus foreshadowing the distinction between form and matter that plays a key role in all later philosophy. The Pythagoreans were influential mathematicians and geometricians, and the theorem that bears their name is witness to their influence on the initial part of Euclidian geometry. They made important contributions to medicine and astronomy and were among the first to teach that the earth was a spherical planet, revolving about a fixed point. At the end of the 5th cent. B.C. the Pythagoreans were forced to flee Magna Graecia when people grew enraged at their interference with traditional religious customs; many were killed. A short-lived Neo-Pythagoreanism developed at the beginning of the Christian era; it borrowed some elements from Jewish and Hellenistic thought and greatly emphasized the mystical element in Pythagorean ideas.

Bibliography

See biographies by P. Gorman (1978) and T. Stanley (1988); D. J. O'Meara, Pythagoras Revived: Mathematics and Philosophy in Late Antiquity (1989).

 
Food & Culture Encyclopedia: Pythagoras

Pythagoras (c. 580–c. 580 B.C.E.) was a Greek mathematician, philosopher, and mystic. He wrote nothing himself, so his ideas survive through the writings of others, including Aristotle. Many people are familiar with him as the mathematician who formulated the Pythagorean theorem in geometry that relates the lengths of the sides in a right triangle. Others know him as a mystic and the first person known to be motivated by moral and philosophical concerns to adopt a vegetarian diet.

The schools and societies Pythagoras founded in the southern Italian area of Magna Graecia flourished for a while, and they developed and spread many of his concepts, which were later adopted and expanded by others. These concepts include bodily humors (evident in modern descriptions of melancholic and phlegmatic personalities), a tripartite soul, reincarnation, and the numerical ratios that determine the concordant intervals of the musical scales. Permeating all of his thoughts was the idea that all things are numbers. Numbers (individuals, groups, and series) were imbued with mystical properties that were carefully guarded and only shared among initiates to the Pythagorean schools founded by him or his disciples.

Pythagoras and his followers practiced one of the first recorded diets known as vegetarianism. He advocated a diet devoid of the flesh of slaughtered animals partially because he felt food influenced the distribution of the bodily humors and thereby the health of the individual and partially because it would prevent the killing of a reincarnated individual and its transmigrated soul. Up until the late nineteenth century non–meat eaters were generally known as "Pythagoreans."

Pythagoras is also alleged to have admonished his disciples to abstain from eating beans. Ancient and medieval writers ingeniously ascribed this pronouncement to the belief that beans contained or transmitted souls. The Greek phrase supporting this gastronomic recommendation, however, could also be construed to imply that his followers should avoid politics. Black and white beans were used as counters in voting in Magna Graecia. The school Pythagoras founded there became actively involved in the populist political views that gained ascendancy in the town of Kroton, where he lived for many years. Later an opposing aristocratic party gained control of the city and banished him and his followers for their political views and activism. Pythagoras died in exile. His supposed warning to "abstain from beans" is therefore thought to have meant "avoid politics." Alternatively he may have realized that eating undercooked broad (fave) beans (Vica faba vulgaris), a common food of the Mediterranean region, produced a severe hemolytic anemia (favism) in some people. Interestingly the same mutant gene that makes people sensitive to favism also increases their resistance to the malarial parasite, possibly accounting for the widespread presence of the mutant gene in regions with endemic malaria.

Bibliography

Bamford, Christopher, ed. Homage to Pythagoras. Hudson, N.Y.: Lindisfarne Press, 1994.

Gorman, Peter. Pythagoras. London: Routledge and Kegan Paul, 1979.

Spencer, Colin. The Heretic's Feast: A History of Vegetarianism. London: Fourth Estate, 1993.

Walters, Kerry S., and Lisa Portmess, eds. Ethical Vegetarianism: From Pythagoras to Peter Singer. Albany: State University of New York Press, 1999.

—Mikal E. Saltveit

 
Essay: Mathematics and mysticism

Pythagoras was among the early Greek philosophers. He is said to have visited Thales and studied with Anaximander. He traveled to Egypt and Mesopotamia, learning the basics of philosophy, science, and mathematics. After returning to Samos and teaching there, he moved to Croton (Crotone, Italy). There he formed a secret society of men and women who shared their knowledge of science and mathematics. Today they are known as Pythagoreans. Credit for many discoveries attributed to Pythagoras probably belongs to other members of his secret society.

The society was both a mystical religion and one of the most productive schools of mathematics in history. A fundamental mystical belief of the Pythagorean society was that "all is number," which is to say that the entire universe can be explained in terms of numbers. The Pythagoreans had other mystical beliefs as well, such as an aversion to beans and a belief in the transmigration of souls (that is, that people's souls after death occupy the bodies of animals and vice versa).

But the principal ideas of the Pythagoreans that have been handed down concern numbers. The Pythagoreans assigned gender to numbers, saying that odd numbers are male and even numbers female. They also worked with figurate numbers, numbers that can be identified by a shape when dots are used. Square numbers of dots can be formed into squares, triangular numbers form dot triangles, and so forth.

Although the name Pythagoras is most closely associated with the Pythagorean theorem (the sum of the squares on the legs of a right triangle equals the square on the hypotenuse), this theorem was known to the Chinese and probably to the Babylonians before the time of Pythagoras.

When Pythagoreans talked about numbers, they meant the natural numbers, 1, 2, 3, and so forth. Fractions are simply ratios of natural numbers, so fractions posed no problem. But the Pythagoreans proved that objects in the real world cannot always be measured completely with natural numbers. For example, if the unit chosen to be one is the side of a square, the diagonal of the square cannot be measured exactly using that unit.

 
Word Tutor: Pythagoras
pronunciation

IN BRIEF: n. - Greek philosopher and mathematician who proved the Pythagorean theorem.

 
Quotes By: Pythagoras

Quotes:

"Friends are as companions on a journey, who ought to aid each other to persevere in the road to a happier life."

"Rest satisfied with doing well, and leave others to talk of you as they will."

"In this theater of man's life, it is reserved only for God and angels to be lookers-on."

"Above the cloud with its shadow is the star with its light. Above all things reverence thyself."

"It is better wither to be silent, or to say things of more value than silence. Sooner throw a pearl at hazard than an idle or useless word; and do not say a little in many words, but a great deal in a few."

"Strength of mind rests in sobriety; for this keeps your reason unclouded by passion."

See more famous quotes by Pythagoras

 
Wikipedia: Pythagoras


Pythagoras
Pre-Socratic philosophy
Kapitolinischer_Pythagoras_adjusted.jpg
Bust of Pythagoras of Samos in the Capitoline Museums, Rome

Name

Pythagoras (Πυθαγόρας)

Birth

c. 580 B.C.-572 B.C.

Death

c. 500 B.C.-490 B.C.

School/tradition

Pythagoreanism

Main interests

Metaphysics, Music, Mathematics, Ethics, Politics

Notable ideas

Musica universalis, Golden ratio, Pythagorean tuning, Pythagorean theorem

Influences

Thales, Anaximander, Pherecydes

Influenced

Philolaus, Alcmaeon, Parmenides, Plato, Euclid, Empedocles, Hippasus, Kepler

Pythagoras of Samos (Greek: Πυθαγόρας; between 580 and 572 BC–between 500 and 490 BC) was an Ionian (Greek) philosopher[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 or natural philosophy. [2] 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. [3]

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."

He was the first man to call himself a philosopher, or lover of wisdom,[4] 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.

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.

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.

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.

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.

Bust of Pythagoras, Vatican
Enlarge
Bust of Pythagoras, Vatican

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.

Influence

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 practicing and repeating these precepts he was imitating and appropriating the doctrines of Jews and Thracians. In fact, it is actually said that that great man introduced many points of Jewish law into his philosophy." (trans. H. St. J. Thackeray, The Loeb Classical Library, Cambridge (Mass.)-London)

Pythagoras is commonly given credit for discovering the Pythagorean theorem, a theorem in trigonometry 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.

While the theorem that now bears his name was well known and previously utilized by the Babylonians, Egyptians 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. [5] 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. There are many ancient references to the facts stated in the Pythagorean theorem; Egyptian and Chinese tablets and writings show that they knew the theorem.

Today, Pythagoras is revered as a prophet by the Ahl al-Tawhid or Druze faith along with his fellow Greek, Plato.

Pythagoreans

Main article: 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.

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 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 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 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, 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.

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.

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.

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, the man in the center with the book, teaching music, in The School of Athens by Raphael
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Pythagoras, the man in the center with the book, teaching music, in The School of Athens by Raphael

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.)

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 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.[6]

Academic Genealogy
Notable teachers Notable students
Anaximander

Pherecydes of Samos
Hermodamas of Samos
Thales

 Ameinias

Bathyllus
Brontinus
Calliphon
Cercops
Echecrates
Empedocles
Eurytus
Hippasus
Leon
Lysis of Tarentum
Milon, whose house was used as a Pythagorean meeting place
Parmeniscus
Petron
Philolaus of Croton
Theano, Pythagoras' Wife/Daughter of Milo
Xenophilus of Chaldice
Zalmoxis, Pythagoras' slave

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 Greek Mythology. 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.

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.

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."

Lore

There is another Pythagoras, who is the subject of elaborate legends surrounding his 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.[7] An extract from Brewer's Dictionary of Phrase and Fable's entry entitled "Golden Thigh":

Pythagoras is said to have had a golden thigh, which he showed to Abaris, the Hyperborean priest, and exhibited in the Olympic games.[8]

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

Pythagoras asserted he could write on the moon. His plan of operation was to write on a looking-glass in blood, and place it opposite the moon, when the inscription would appear photographed or reflected on the moon's disc.[9]

Other accomplishments

One of his 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. He assigned roles for the numbers as follows: one was reason, two was opinion, four was justice, five was marriage because it was the sum of the first odd and the first even numbers (one was disregarded), seven was virgin because it neither factors or produces among the numbers one through ten. 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". 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.

Groups influenced by Pythagoras

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.[10]

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.

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.

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 Islamic alchemist Jabir ibn Hayyan grounded his work in an elaborate numerology greatly influenced by Pythagorean theory.

See also

References

Primary 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 from these four books.

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)

Notes

  1. ^ 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).
  2. ^ Walter Burkert's seminal work Lore and Science in Ancient Pythagoreanism (see sources) sheds considerable doubt on the widely held traditions of late Classical Greece, accepted without scrutiny until the beginning of the 20th century, that Pythagoras made substantial contributions to mathematics and science.
  3. ^ Christoph Riedweg, Pythagoras: His Life, Teaching and Influence, trans. Steven Rendall (Cornell UP, 2005), pp. 5-6, 59, 73.
  4. ^ Cicero, Tusculan Disputations, 5.3.8-9 = Heraclides Ponticus fr. 88 Wehrli, Diogenes Laertius 1.12, 8.8, Iamblichus VP 58. Burkert attempted to discredit this ancient tradition, but it has been defended by C.J. De Vogel, Pythagoras and Early Pythagoreanism (1966), pp. 97-102, and C. Riedweg, Pythagoras: His Life, Teaching, And Influence (2005), p. 92.
  5. ^ From Christoph Riedweg , Pythagoras, His Life, Teaching and Influence, Cornell: Cornell University Press, 2005: "Had Pythagoras and his teachings not been since the early Academy overwritten with Plato’s philosophy, and had this ‘palimpsest’ not in the course of the Roman Empire achieved unchallenged authority among Platonists, it would be scarcely conceivable that scholars from the Middle Ages and modernity down to the present would have found the Presocratic charismatic from Samos so fascinating. In fact, as a rule it was the image of Pythagoras elaborated by Neopythagoreans and Neoplatonists that determined the idea of what was Pythagorean over the centuries."
  6. ^ Christoph Riedweg , Pythagoras, His Life, Teaching and Influence, Cornell: Cornell University Press, 2005 .
  7. ^ Huffman, Carl. Pythagoras (Stanford Encyclopedia of Philosophy) [1]
  8. ^ Brewer, E. Cobham, Brewer's Dictionary of Phrase and Fable [2]
  9. ^ Brewer, E. Cobham, Brewer's Dictionary of Phrase and Fable [3]
  10. ^ R.M. Hare, Plato in C.C.W. Taylor, R.M. Hare and Jonathan Barnes, Greek Philosophers, Socrates, Plato, and Aristotle, Oxford: Oxford University Press, 1999 (1982), 103-189, here 117-9.

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