Pythagoreanism

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Pythagoreanism originated in the 6th century BC, based on and around the teachings and beliefs held by Pythagoras and his followers, the Pythagoreans. Pythagoras established the first Pythagorean community in the ancient Greek colony of Kroton, in modern Calabria (Italy) circa 530 BC. Early Pythagorean communities spread throughout Magna Graecia.

Already during Pythagoras' life it is likely that the distinction between the akousmatikoi ("those who listen"), who is conventionally regarded as more concerned with religious, and ritual elements, and associated with the oral tradition, and the mathematikoi ("those who learn") existed. The ancient biographers of Pythagoras,

Pythagoras was already well known in ancient times for his supposed mathematical achievement of the Pythagorean theorem.^[3] Pythagoras had been credited with discovering that in a right-angled triangle the square of the hypotenuse is equal to the sum of the squares of the other two sides. In ancient times Pythagoras was also noted for his discovery that music had mathematical foundations. Antique sources that credit Pythagoras as the philosopher who first discovered music intervals also credit him as the inventor of the monochord, a straight rod on which a string and a movable bridge could be used to demonstrate the relationship of musical intervals.^[4]

Much of the surviving sources on Pythagoras originated with Aristotle and the philosophers of the Peripatetic school, which founded historiographical academic traditions such as biography, doxography and the history of science. The surviving 5th century BC sources on Pythagoras and early Pythagoreanism are void of supernatural elements, while surviving 4th century BC sources on Pythagoras' teachings introduced legend and fable. Philosophers who discussed Pythagoreanism, such as Anaximander, Andron of Ephesus, Heraclides and Neanthes had access to historical written sources as well as the oral tradition about Pythagoreanism, which by the 4th century BC was in decline. Neopythagorean philosophers, who authored many of the surviving sources on Pythagoreanism, continued the tradition of legend and fantasy.^[5]

The earliest surviving ancient source on Pythagoras and his followers is a satire by Xenophanes, on the Pythagorean beliefs on the transmigration of souls.^[6] Xenophanes wrote of Pythagoras that:

Once they say that he was passing by when a puppy was being whipped,

And he took pity and said:

"Stop! Do not beat it! For it is the soul of a friend

That I recognised when I heard it giving tongue."^[6]

In a surviving fragment from Heraclitus, Pythagoras and his followers are described as follows:

Pythagoras, the son of Mnesarchus, practised inquiry beyond all other men and selecting of these writings made for himself a wisdom or made a wisdom of his own: a polymathy, an imposture.^[7]

Two other surviving fragments of ancient sources on Pythagoras are by Ion of Chios and Empedocles. Both were born in the 490s, after Pythagoras' death. By that time, he was known as a sage and his fame had spread throughout Greece.^[8] According to Ion, Pythagoras was:

... distinguished for his manly virtue and modesty, even in death has a life which is pleasing to his soul, if Pythagoras the wise truly achieved knowledge and understanding beyond that of all men.^[8]

Empedocles described Pythagoras as "a man of surpassing knowledge, master especially of all kinds of wise works, who had acquired the upmost wealth of understanding".^[9] In the 4th century BC the Sophist Alcidamas wrote that Pythagoras was widely honored by Italians.^[10]

Today scholars typically distinguish two periods of Pythagoreanism: early-Pythagoreanism, from the 6th until the 5th century BC, and late-Pythagoreanism, from the 4th until the 3rd century BC.

Early-Pythagorean sects were closed societies and new Pythagoreans were chosen based on merit and discipline. Ancient sources record that early-Pythagoreans underwent a five-year initiation period of listening to the teachings (akousmata) in silence. Initiates could through a test become members of the inner circle. However, Pythagoreans could also leave the community if they wished.[13] Iamblichus listed 235 Pythagoreans by name, among them 17 women whom he described as the "most famous" women practitioners of Pythagoreanism. It was customary that family members became Pythagoreans, as Pythagoreanism developed into a philosophic tradition that entailed rules for everyday life and Pythagoreans were bound by secrets. The home of Pythagoras was known as the site of mysteries.^[14]

Pythagoras had been born on the island of

Following Pythagoras' death, disputes about his teachings led to the development of two philosophical traditions within Pythagoreanism in Italy: akousmatikoi and mathēmatikoi. The mathēmatikoi recognised the akousmatikoi as fellow Pythagoreans, but because the mathēmatikoi allegedly followed the teachings of Hippasus, the akousmatikoi philosophers did not recognise them. Despite this, both groups were regarded by their contemporaries as practitioners of Pythagoreanism.^[21]

The akousmatikoi were superseded in the 4th century BC as a significant

The akousmatikoi believed that humans had to act in appropriate ways. The Akousmata (translated as "oral saying") was the collection of all the sayings of Pythagoras as divine dogma. The tradition of the akousmatikoi resisted any reinterpretation or philosophical evolution of Pythagoras' teachings. Individuals who strictly followed most akousmata were regarded as wise. The akousmatikoi philosophers refused to recognise that the continuous development of mathematical and scientific research conducted by the mathēmatikoi was in line with Pythagoras's intention. Until the demise of Pythagoreanism in the 4th century BC, the akousmatikoi continued to engage in a pious life by practicing silence, dressing simply and avoiding meat, for the purpose of attaining a privileged afterlife. The akousmatikoi engaged deeply in questions of Pythagoras' moral teachings, concerning matters such as harmony, justice,^[24] ritual purity and moral behavior.McKirahan 2011, p. 90

The mathēmatikoi acknowledged the religious underpinning of Pythagoreanism and engaged in mathēma (translated as "learning" or "studying") as part of their practice. While their scientific pursuits were largely mathematical, they also promoted other fields of scientific study in which Pythagoras had engaged during his lifetime. A sectarianism developed between the dogmatic akousmatikoi and the mathēmatikoi, who in their intellectual activism became regarded as increasingly progressive. This tension persisted until the 4th century BC, when the philosopher Archytas engaged in advanced mathematics as part of his devotion to Pythagoras' teachings.^[24]

Today, Pythagoras is mostly remembered for his mathematical ideas, and by association with the work early Pythagoreans did in advancing mathematical concepts and theories on harmonic

Early-Pythagorean philosophers such as Philolaus and Archytas held the conviction that mathematics could help in addressing important philosophical problems.^[34] In Pythagoreanism numbers became related to intangible concepts. The one was related to the intellect and being, the two to thought, the number four was related to justice because 2 * 2 = 4 and equally even. A dominant symbolism was awarded to the number three, Pythagoreans believed that the whole world and all things in it are summed up in this number, because end, middle and beginning give the number of the whole. The triad had for Pythagoreans an ethical dimension, as the goodness of each person was believed to be threefold: prudence, drive and good fortune.^[35]

Pythagoreans thought numbers existed "outside of [human] minds" and separate from the world.^[36] They had many mystical and magical interpretations of the roles of numbers in governing existence.^[36]

The Pythagoreans engaged with

The fact that mathematics could explain the human sentimental world had a profound impact on the Pythagorean philosophy. Pythagoreanism became the quest for establishing the fundamental essences of reality. Pythagorean philosophers advanced the unshakable belief that the essence of all things are numbers and that the universe was sustained by harmony.[38] According to ancient sources music was central to the lives of those practicing Pythagoreanism. They used medicines for the purification (katharsis) of the body and, according to Aristoxenus, music for the purification of the soul. Pythagoreans used different types of music to arouse or calm their souls,^[41] and certain stirring songs could have notes that existed in the same ratio as the "distances of the heavenly bodies from the centre of" Earth.^[36]

For Pythagoreans, harmony signified the "unification of a multifarious composition and the agreement of unlike spirits". In Pythagoreanism, numeric harmony was applied in mathematical, medical, psychological, aesthetic, metaphysical and cosmological problems. For Pythagorean philosophers, the basic property of numbers was expressed in the harmonious interplay of opposite pairs. Harmony assured the balance of opposite forces.^[42] Pythagoras had in his teachings named numbers and the symmetries of them as the first principle and called these numeric symmetries harmony.^[43] This numeric harmony could be discovered in rules throughout nature. Numbers governed the properties and conditions of all beings and were regarded the causes of being in everything else. Pythagorean philosophers believed that numbers were the elements of all beings and the universe as a whole was composed of harmony and numbers.^[35]

Unity and harmony is extended to all the opposites, which descent from the so-called Pythagorean "Table of ten Opposites", mentioned by Aristotle. Supreme opposites are the following ten couples: limit-unlimited, odd-even, one-many, right-left, male-female , rest-motion, straight-curved, light-darkness, good-evil, and square-oblong.^[44]

The philosopher

It remains to speak of the earth, of its position, of the question whether it is at rest or in motion, and of its shape. As to its position, there is some difference of opinion. Most people–all, in fact, who regard the whole heaven as finite–say it lies at the center. But the Italian philosophers known as Pythagoreans take the contrary view. At the centre, they say, is fire, and the earth is one of the stars, creating night and day by its circular motion about the center. They further construct another earth in opposition to ours to which they give the name counterearth.[49]

It is not known whether Philolaus believed Earth to be round or flat,

Justice

Pythagoreans equated

universal justice was later referenced by Plato. For Pythagorean philosophers the soul was the source of justice and through the harmony of the soul, divinity could be achieved. Injustice inverted the natural order. According to the 4th century BC philosopher Heraclides Ponticus, Pythagoras taught that "happiness consists in knowledge of the perfection of the numbers of the soul.^[51] A surviving fragment from the 3rd century BC by the late-Pythagorean philosopher Aesara

reasoned that:

I think human nature provides a common standard of law and justice for both the family and the city. Whoever follows the paths within and searches will discover; for within is law and justice, which is the proper arrangement of the soul.[53]

Body and soul

See also: Pythagoras § Metempsychosis

Pythagoreans believed that body and soul functioned together, and a healthy body required a healthy psyche.^[54] Early Pythagoreans conceived of the soul as the seat of sensation and emotion. They regarded the soul as distinct from the intellect.^[55] However, only fragments of the early Pythagorean texts have survived, and it is not certain whether they believed the soul was immortal. The surviving texts of the Pythagorean philosopher Philolaus indicate that while early Pythagoreans did not believe that the soul contained all psychological faculties, the soul was life and a harmony of physical elements. As such the soul passed away when certain arrangements of these elements ceased to exist.^[56]

However, the teaching most securely identified with Pythagoras is

Orphics, although its version contains substantial differences. Unlike the Orphics, who considered metempsychosis a cycle of grief that could be escaped by attaining liberation from it, Pythagoras seems to postulate an eternal, endless reincarnation where subsequent lives would not be conditioned by any action done in the previous.^[58]

Vegetarianism

Some medieval authors refer to a "Pythagorean diet", entailing the abstention from eating meat, beans or fish.^[60] Pythagoreans believed that a vegetarian diet fostered a healthy body and enhanced the search for Arete. The purpose of vegetarianism in Pythagoreanism was not self-denial; instead, it was regarded as conducive to the best in a human being. Pythagoreans advanced a grounded theory on the treatment of animals. They believed that any being that experienced pain or suffering should not have pain inflicted on it unnecessarily. Because it was not necessary to inflict pain on animals for humans to enjoy a healthy diet, they believed that animals should not be killed for the purpose of eating them. The Pythagoreans advanced the argument that unless an animal posed a threat to a human, it was not justifiable to kill an animal and that doing so would diminish the moral status of a human. By failing to show justice to the animal, humans diminish themselves.^[54]

Pythagoreans believed that human beings were animals, but with an advanced intellect and therefore humans had to purify themselves through training. Through purification humans could join the psychic force that pervaded the cosmos. Pythagoreans reasoned that the logic of this argument could not be avoided by killing an animal painlessly. The Pythagoreans also thought that animals were sentient and minimally rational.

blood sacrifice by offering a substitute sacrifice after his victory in a horse race in Olympia.^[47]

Late-Pythagorean philosophers were absorbed into the Platonic school of philosophy and in the 4th century BC the head of the

Porphyry would rely on the teachings of the Pythagoreans when arguing that abstinence from eating meat for the purpose of spiritual purification should be practiced only by philosophers, whose aim was to reach a divine state.^[62]

Female philosophers

The biographical tradition on Pythagoras holds that his mother, wife and daughters were part of his inner circle.[63] Women were given equal opportunity to study as Pythagoreans and learned practical domestic skills in addition to philosophy.^[64]

Many of the surviving texts of women Pythagorean philosophers are part of a collection, known as pseudoepigrapha Pythagorica, which was compiled by Neopythagoreans in the 1st or 2nd century. Some surviving fragments of this collection are by early-Pythagorean women philosophers, while the bulk of surviving writings are from late-Pythagorean women philosophers who wrote in the 4th and 3rd century BC.^[11] Female Pythagoreans are some of the first female philosophers from which texts have survived.

Phintys of Sparta.^[13]

Scholars believe that

Spartan and is believed to have been the daughter of a Spartan admiral killed in the battle of Arginusae in 406 BC.^[13] Phyntis authored the treatise Moderation of Women, in which she assigned the virtue of moderation to women, but asserted that "courage and justice and wisdom are common to both" men and women. Phyntis defended the right of women to philosophise.^[65]

Influence on Plato and Aristotle

Pythagoras' teachings and Pythagoreanism influenced

Platonic solids has been discussed extensively. Plato's dialogues have become an important surviving source of Pythagorean philosophic arguments.^[67] Plato referenced Philolaus in Phaedo and wrote a Platonic adaptation of Philolaus' metaphysical system of limiters and unlimiteds. Plato also quoted from one of the surviving Archytas fragments in the Republic. However, Plato's views that the primary role of mathematics was to turn the soul towards the world of forms, as expressed in Timaeus, is regarded as Platonic philosophy, rather than Pythagorean.^[34]

ontological value.^[66] Aristotle's discussion of Pythagorean philosophy is difficult to interpret, because he had little patience for Pythagorean philosophic arguments, and Pythagoreanism does not fit with his philosophic doctrine.^[68] In On the Heavens, Aristotle refuted the Pythagorean doctrine on the harmony of the spheres.^[69] Nevertheless, he wrote a treatise on the Pythagoreans of which only fragments survive, in which he treats Pythagoras as a wonder-working religious teacher.^[70]

Neopythagoreanism

Main article: Neopythagoreanism

The Neopythagoreans were a school and a religious community. The revival of Pythagoreanism has been attributed to

Nicomachus of Gerasa emerged as leading teachers of Neopythagoreanism.^[71][72] The most significant Neopythagorean teacher was Apollonius of Tyana in the 1st century AD, who was regarded as a sage and lived as an ascetic. The last Neopythagorean philosopher was Numenius of Apamea in the 2nd century. Neopythagoreanism remained an elite movement which in the 3rd century merged into Neoplatonism.^[71]

Neopythagoreans combined Pythagorean teachings with

dualism. Neopythagoreans refined the idea of God and located him beyond the finite so that God could not come into contact with anything corporeal. Neopythagoreans insisted on a spiritual worship of God and that life had to be purified through abstinence.^[71]

Neopythagoreans manifested a strong interest in numerology and the superstitious aspects of Pythagoreanism. They combined this with the teachings of Plato's philosophic successors. Neopythagorean philosophers engaged in the common ancient practice of ascribing their doctrines to the designated founder of their philosophy and by crediting their doctrines to Pythagoras himself, they hoped to gain authority for their views.[67]

Later influence

On early Christianity

Sacrobosco

Dante and in the Renaissance a new translation of it was produced by Marsilio Ficino.^[74]

Early Christian theologians, such as

Saint Jerome. In the 2nd century many of the Sentences of Sextus were cited by Plutarch as Pythagorean aphorisms. The Sentences of Sextus were translated into Syriac, Latin and Arabic, then the written language of both Muslims and Jews, but only in the Latin world did they become a guide to daily life that was widely circulated.^[75]

On numerology

1st century treatises of Philo and Nicomachus popularised the mystical and cosmological symbolism Pythagoreans attributed to numbers. This interest in Pythagorean views on the importance of numbers was sustained by mathematicians such as Theon of Smyrna, Anatolius and Iamblichus. These mathematicians relied on Plato's Timaeus as their source for Pythagorean philosophy.^[78]

In the

Alexander Neckham referenced classical writers that had discussed Pythagoreanism, including Cicero, Ovid and Pliny, leading them to believe that mathematics was the key to understanding astronomy and nature. Another important text on Pythagorean numerology was Boethius's De arithmetica, which was widely reproduced in the West. Boethius had relied on Nicomachus's writings as a source of Pythagoreanism.^[79]

The 11th-century Byzantine professor of philosophy

Michael Psellus popularised Pythagorean numerology in his treatise on theology, arguing that Plato was the inheritor of the Pythagorean secret. Psellus also attributed arithmetical inventions by Diophantus to Pythagoras. Psellus thought to reconstruct Iamblichus' 10 book encyclopedia on Pythagoreanism from surviving fragments, leading to the popularisation of Iamblichus' description of Pythagorean physics, ethics and theology at the Byzantine court. Psellus was reputably in the possession of the Hermetica, a set of texts thought to be genuinely antique and which would be prolifically reproduced in the late Middle Ages. Manuel Bryennios introduced Pythagorean numerology to Byzantine music with his treatise Harmonics. He argued that the octave was essential in attaining perfect harmony.^[80]

In the Jewish communities the development of the

Philo of Alexandria, developed a Jewish Pythagoreanism. In the 3rd century Hermippus popularised the belief that Pythagoras had been the basis for establishing key dates in Judaism. In the 4th century this assertion was further developed by Aristobulus. The Jewish Pythagorean numerology developed by Philo held that God as the unique One was the creator of all numbers, of which seven was the most divine and ten the most perfect. The medieval edition of the Kabbalah focused largely on a cosmological scheme of creation, in reference to early Pythagorean philosophers Philolaus and Empedocles and helped to disseminate Jewish Pythagorean numerology.^[81]

On mathematics

taxation, measurement, the estimation of agricultural values and business applications for the buying and selling of goods. There was little interest for the Pythagorean numerology that developed in the Latin world. The primary arithmetical system used by Islamic mathematicians was based on Hindu arithmetic, which rejected the notion that the relations between numbers and geometrical forms were symbolic.^[83]

Besides the enthusiasm that developed in the Latin and Byzantine worlds in the Middle Ages for Pythagorean numerology, the Pythagorean tradition of perfect numbers inspired profound scholarship in

square numbers always arise through the addition of consecutive odd numbers starting with unity. Fibonacci put forward a method of generating sets of three square numbers that satisfied the relationship first attributed to Pythagoras by Vitruvius, that a² + b² = c². This equation is now known as the Pythagorean triple.^[84]

In the Middle Ages

In the

Saint Benedict, as authoritative Christian doctrine. In the Latin medieval western world, the Golden Verses became a widely reproduced text.^[75]

archivolts over the right door of the west portal at Chartres Cathedral.^[85]

Although the concept of the

Seven liberal arts.^[86]

In the early 6th century the Roman philosopher

French cathedrals of the 11th, 12th and 13th century.^[73]

Arabic translations of the Golden Verses were produced in the 11th and 12th centuries.

Ibn Sina vehemently rejected this Pythagorean doctrine.^[89] in Kitab al-Musiqa al-Kabir Al-Farabi rejected the notion of celestial harmony on the grounds that it was "plainly wrong" and that it was not possible for the heavens, orbs and stars to emit sounds through their motions.^[69]

The four books of the Corpus Areopagiticum or

signs of the zodiac.^[73]

In the

Alexander Neckham referred not only to Plato but also to other classical authors that had discussed Pythagoreanism, including Cicero, Ovid and Pliny. William of Conches argued that Plato was an important Pythagorean. In this medieval Pythagorean understanding of Plato, God was a craftsman when he designed the universe.^[79]

On Western science

Harmonices Mundi, full title Ioannis Keppleri Harmonices mundi libri V (The Harmony of the World), by Johannes Kepler

In the preface of

Heliocentric Theory

: Hicetas, Philolaus and Ecphantus.

At first I found in

Heraclides of Pontus and Ecphantus the Pythagorean make the earth move, not in a progressive motion, but like a wheel in a rotation from west to east about its own center."^[91][92][93]

In the 16th century

Zarlino. Zarlino supported the theory that if two weights in the ratio of 2 to 1 were attached to two strings, the pitches generated by the two strings would produce the octave. Vincenzo Galilei proclaimed that he had been a committed Pythagorean, until he "ascertained the truth by means of experiment, the teacher of all things". He devised an experiment which showed that the weights attached to the two strings needed to increase as the square of the string length.^[94] This public challenge to prevailing numerology in musical theory triggered an experimental and physical approach to acoustics in the 17th century. Acoustics emerged as a mathematical field of music theory and later an independent branch of physics. In the experimental investigation of sound phenomena, numbers had no symbolic meaning and were merely used to measure physical phenomena and relationships such as frequency and vibration of a string.^[95]

Many of the most eminent 17th century natural philosophers in Europe, including

Stevin and Galileo, had a keen interest in music and acoustics.^[96] By the late 17th century it was accepted that sound travels like a wave in the air at a finite speed and experiments to establish the speed of sound were carried out by philosophers attached to the French Academy of Sciences, the Accademia del Cimento and the Royal Society.^[97]

At the height of the

pre-established harmony.^[42] Albert Einstein believed that through this pre-established harmony, the productive unison between the spiritual and material world was possible.^[42]

The Pythagorean belief that all bodies are composed of numbers and that all properties and causes could be expressed in numbers, served as the basis for a mathematization of science. This mathematization of the physical reality climaxed in the 20th century. The pioneer of physics Werner Heisenberg argued that "this mode of observing nature, which led in part to a true dominion over natural forces and thus contributes decisively to the development of humanity, in an unforeseen manner vindicated the Pythagorean faith".^[98]

See also

Library resources about
Pythagoreanism

---

• Online books
• Resources in your library
• Resources in other libraries

• Dyad (Greek philosophy)
• Esoteric cosmology
• Hippasus
• Ionian School (philosophy)
• Incommensurable magnitudes
• Ipse dixit
• Mathematical beauty
• Mathematicism
• Pyrrhonism
• Sacred geometry
• Tetractys
• Unit-point atomism

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94. ^ Mancosu 2003, p. 603.
95. ^ Mancosu 2003, pp. 597–98.
96. ^ Mancosu 2003, p. 604.
97. ^ Mancosu 2003, p. 609.
98. ^ ^{a b} Vamvacas 2009, p. 77.

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External links

Wikiquote has quotations related to Pythagoreanism.

• Media related to Pythagoreanism at Wikimedia Commons
• Carl Huffman. "Pythagoreanism/". In Zalta, Edward N. (ed.). Stanford Encyclopedia of Philosophy.