Pythagoras

Pythagoras of Samos (US /pɪˈθæɡərəs/; UK /paɪˈθæɡərəs/; Greek: Πυθαγόρας ὁ Σάμιος Pythagóras ho Sámios "Pythagoras the Samian", or simply Πυθαγόρας; Πυθαγόρης in Ionian Greek; c. 570 – c. 495 BC) was an Ionian Greek philosopher, mathematician, and putative founder of the Pythagoreanism movement. He is often revered as a great mathematician and scientist and is best known for the Pythagorean theorem which bears his name.

Legend and obfuscation cloud his work, so it is uncertain whether he truly contributed much to mathematics or natural philosophy. Many of the accomplishments credited to Pythagoras may actually have been accomplishments of his colleagues or successors. Some accounts mention that the philosophy associated with Pythagoras was related to mathematics and that numbers were important. It was said that he was the first man to call himself a philosopher, or lover of wisdom, and Pythagorean ideas exercised a marked influence on Plato, and through him, all of Western philosophy.

Much of what we know about Pythagoras's life comes from Neoplatonist writers. These writers believed in the Greek gods, so many myths exist around Pythagoras. Myths about Pythagoras include: Apollo was his father, Pythagoras gleamed with a supernatural brightness, Pythagoras had a golden thigh, Abaris once flew to him on a golden arrow, and Pythagoras was seen in different places at the same time. According to Sir William Smith, with the exception of a few remarks by Xenophanes, Heraclitus, Herodotus, Plato, Aristotle, and Isocrates, we are mainly dependent on Diogenes Laërtius, Porphyry, and Iamblichus for biographical details. Burkert (1972, p. 109) states that Aristoxenus and Dicaearchus are the most important accounts.

Aristotle wrote a work On the Pythagoreans, which is no longer extant. Some of it may be available in the Protrepticus. Aristotle's disciples Dicaearchus, Aristoxenus, and Heraclides Ponticus also wrote on the same subject. These writers were among the best sources from whom Porphyry and Iamblichus drew, while still adding some legendary accounts and their own inventions to the mix. Hence, historians are often reduced to considering the statements based on their inherent probability, but even then, if all the credible stories concerning Pythagoras were supposed true, his range of activity would be impossibly vast.

Bust of Pythagoras, Vatican

Herodotus, Isocrates, and other early writers agree that Pythagoras was the son of Mnesarchus and born on the Greek island of Samos, situated in the eastern Aegean. His father is said to have been a gem-engraver or a wealthy merchant, originally from Tyre or Syria. Pythagoras' name led him to be associated with Pythian Apollo; Aristippus of Cyrene explained his name by saying, "He spoke (ἀγορεύω, agoreúo) the truth no less than did the Pythian (Πυθία, Pythía)", and Iamblichus tells the story that the Pythia prophesied that his pregnant mother would give birth to a man supremely beautiful, wise, and beneficial to humankind. A late source gives his mother's name as Pythais. As to the date of his birth, Aristoxenus stated that Pythagoras left Samos in the reign of Polycrates, at the age of 40, which would give a date of birth around 570 BC.

Around 530 BC he arrived in the Greek colony of Croton (today's Crotone, in Calabria) in what was then Magna Graecia. There he founded his own philosophical school the members of which he engaged to a disciplined and simple way of life (the Pythagorean way of life) as well as to mutual loyalty. He furthermore aquired some political influence, on Greeks and non-Greeks of the region. Following a conflict with the neighbouring colony of Sybaris, internal discord drove most of the Pythagoreans out of Croton. Pythagoras left the city before the outbreak of civil unrest and moved to Metapontum (today's Metaponto, in Basilicata), where he stayed for the rest of his life. After his death, his house was transformed into a sanctuary of Demeter, out of veneration for the philosopher, by the local population.

In ancient sources there was much disagreement and inconsistency about the late life of Pythagoras. Some said that he perished in the temple with his disciples, others that he fled first to Tarentum, and that, being driven from there, he escaped to Metapontum, and there according to Diogenes Laërtius, starved himself to death. His tomb was shown at Metapontum in the time of Cicero. According to Walter Burkert,

Family

According to some accounts, Pythagoras married Theano, a lady of Croton. Theano was also a philosopher, and said to have first been Pythagoras' pupil. According to Mary Ritter Beard, Theano told Hippodamus of Thurium (possibly Hippodamus of Miletus, who as per Aristotle planned the city of Thurium), that her treatise On Virtue contained the doctrine of the golden mean. According to Thesleff, Stobaeus and Heeren, in On Piety, Theano wrote that:

I have learned that many of the Greeks believe Pythagoras said all things are generated from number. The very assertion poses a difficulty: How can things which do not exist even be conceived to generate? But he did not say that all things come to be from number; rather, in accordance with number - on the grounds that order in the primary sense is in number and it is by participation in order that a first and a second and the rest sequentially are assigned to things which are counted.

Their children are variously stated to have included a son, Telauges, and three daughters, Damo, Arignote, and Myia (who married to a famous wrestler, Milo of Croton). Milo was said to be an associate of Pythagoras. One story tells of the wrestler saving the philosopher's life when a roof was about to collapse.

Arignote wrote a Bacchica concerning the mysteries of Demeter, and a work called The Rites of Dionysus. Among the Pythagorean Sacred Discourses there is a dictum attributed to Arignote: "The eternal essence of number is the most providential cause of the whole heaven, earth and the region in between. Likewise it is the root of the continued existence of the gods and daimones, as well as that of divine men." Brewer (1894) mentioned that "Pythagoras taught that the sun is a movable sphere in the centre of the universe, and that all the planets revolve round it." Thus, it would appear that Arignote's quote above is not entirely in alignment with his model of the universe, since it is limited to Earth orbit.

Influences

Before 520 BC, on one of his visits to Egypt or Greece, Pythagoras might have met the c. 54 years older Thales of Miletus. Thales was a philosopher, scientist, mathematician, and engineer, also known for a special case of the inscribed angle theorem. It can be marked that Pythagoras' birthplace, the island of Samos, is situated in the Northeast Aegean Sea not far from Miletus.

Scholars disagree about who taught Pythagoras due to an absence of reliable information. Some say his training was almost entirely Greek, others exclusively Egyptian and Oriental. The various names seem to call attention for different aspects of Pythagoras' own teachings. Thus, we find mentioned as his instructors Hermodamas of Samos, or his father Creophylus of Samos (who both stand for a domestic rhapsodic tradition of Samos, competing with Homer's more renowned), Bias of Priene, Thales, Anaximander (a pupil of Thales), and Pherecydes of Syros (all exponents of the Greek philosophical tradition). Of the various claims regarding his Greek teachers, Pherecydes of Syros is mentioned most often.

A scene at the Chartres Cathedral shows a philosopher, on one of the archivolts over the right door of the west portal at Chartres, which has been attributed to depict Pythagoras.

The third aspect of Pythagorean learning to be covered were the fields of piety and cult, equally incorporated into the philosopher's teachings. In his Lives and Opinions of Eminent Philosophers, Diogenes Laërtius (3rd century AD) cites the statement of Aristoxenus (4th century BC) that the Delphic Themistoclea taught Pythagoras his moral doctrines: "Aristoxenus says that Pythagoras got most of his moral doctrines from the Delphic priestess Themistoclea." Porphyry (c. 234–305 AD), who calls her Aristoclea (Aristokleia), wrote: "He (Pythagoras) taught much else, which he claimed to have learned from Aristoclea at Delphi." Ancient authorities furthermore note the similarities between the religious and ascetic peculiarities of Pythagoras with the Orphic or Cretan mysteries, or the Delphic oracle.

Following a similar logic, the Egyptians are said to have taught him geometry, the Phoenicians arithmetic, the Chaldeans astronomy, the Magians the principles of religion and practical maxims for the conduct of life. According to Diogenes Laërtius, Pythagoras not only visited Egypt and learnt the Egyptian language (as reported by Antiphon in his On Men of Outstanding Merit), but also "journeyed among the Chaldaeans and Magi." Later in Crete, he went to the Cave of Ida with Epimenides, and entered Egyptian sanctuaries for the purpose to learn information concerning the secret lore of the different gods. The Middle Platonist biographer Plutarch (c. 46–120 AD) asserted in his book On Isis and Osiris that during his visit to Egypt, Pythagoras received instruction from the Egyptian priest Oenuphis of Heliopolis (meanwhile Solon received lectures from a Sonchis of Sais). Other ancient writers asserted his visit to Egypt. According to the Christian theologian Clement of Alexandria (c. 150–215 AD), Pythagoras was a disciple of Soches, an Egyptian archprophet, as well as Plato of Sechnuphis of Heliopolis. Finally, the Neoplatonist philosopher Iamblichus (ca. 245–325 AD) gives a rather fanciful account of the philosopher's 'years of apprenticeship' in his Life of Pythagoras:

Views

There is little direct evidence as to the kind and amount of knowledge which Pythagoras acquired, or as to his definite philosophical views. Everything of the kind mentioned by Plato and Aristotle is attributed not to Pythagoras, but to the Pythagoreans. Heraclitus stated that he was a man of extensive learning; and Xenophanes claimed that he believed in the transmigration of souls. Xenophanes mentions the story of his interceding on behalf of a dog that was being beaten, professing to recognise in its cries the voice of a departed friend. Pythagoras is supposed to have claimed that he had been Euphorbus, the son of Panthus, in the Trojan war, as well as various other characters, a tradesman, a courtesan, etc. In his book The Life of Apollonius of Tyana, Philostratus wrote that Pythagoras knew not only who he was himself, but also who he had been.

Many mathematical and scientific discoveries were attributed to Pythagoras, including his famous theorem, as well as discoveries in the field of music, astronomy, and medicine. It is mentioned that the people of Croton were supposed to have identified him with the Hyperborean Apollo, and he was said to have practised divination and prophecy. In the visits to various places in Greece – Delos, Sparta, Phlius, Crete, etc. – which are ascribed to him, he usually appears either in his religious or priestly guise, or else as a lawgiver. In his philosophical dialogue Protrepticus Aristotle has his literary double say:

Croton on the southern coast of Magna Graecia (Southern Italy), to which Pythagoras ventured after feeling overburdened in Samos.

After traveling to Egypt, Greece, and possibly India, Pythagoras moved (around 530 BC) to Croton, in Italy (Magna Graecia). Possibly the tyranny of Polycrates in Samos made it difficult for him to achieve his schemes there. His later admirers claimed that Pythagoras was so overburdened with public duties in Samos, because of the high estimation in which he was held by his fellow-citizens, that he moved to Croton. On his arrival in Croton, he quickly attained extensive influence, and many people began to follow him. Later biographers tell fantastical stories of the effects of his eloquent speech in leading the people of Croton to abandon their luxurious and corrupt way of life and devote themselves to the purer system which he came to introduce.

According to Diogenes Laërtius, his followers established a select brotherhood or club (see below school) for the purpose of pursuing the religious and ascetic practices which developed. According to Diogenes Laërtius, what was done and taught among the members was kept a secret. The esoteric teachings may have concerned science and mathematics, or religious doctrines, and may have been connected with the worship of Apollo. Temperance of all kinds seems to have been strictly urged. There is disagreement among the biographers as to whether Pythagoras forbade all animal food, or only certain types. The club was in practice at once "a philosophical school, a religious brotherhood, and a political association".

Conflict seems to have broken out between the towns of Sybaris and Croton. The forces of Croton were headed by the Pythagorean Milo, and it is likely that the members of the brotherhood took a prominent part. After the decisive victory by Croton, a proposal for establishing a more democratic constitution, was unsuccessfully resisted by the Pythagoreans. Their enemies, headed by Cylon and Ninon, the former of whom is said to have been irritated by his exclusion from the brotherhood, roused the populace against them. An attack was made upon them while assembled either in the house of Milo, or in some other meeting-place. The building was set on fire, and many of the assembled members perished; only the younger and more active escaping. Similar commotions ensued in the other cities of Magna Graecia in which Pythagorean clubs had been formed.

As an active and organised brotherhood the Pythagorean order was everywhere suppressed, and did not again revive. Still the Pythagoreans continued to exist as a sect, the members of which kept up among themselves their religious observances and scientific pursuits, while individuals, as in the case of Archytas, acquired now and then great political influence.

Excerpt from Philolaus Pythagoras book, (Sir William Smith, 1870)

No texts by Pythagoras are known to have survived, 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") — emphasising the essentially oral nature of his teaching.

According to Sir William Smith (1870) (see book screenshot for full quote)

The Pythagorean theorem: The sum of the areas of the two squares on the legs (a and b) equals the area of the square on the hypotenuse (c).

Aristotle, Metaphysics 1–5 , cc. 350 BC

Pythagorean theorem

A visual proof of the Pythagorean theorem

Since the fourth century AD, Pythagoras has commonly been given credit for discovering the Pythagorean theorem, a theorem in geometry that states that in a right-angled triangle the area of the square on the hypotenuse (the side opposite the right angle) is equal to the sum of the areas of the squares of the other two sides—that is, ${\displaystyle a^{2}+b^{2}=c^{2}}$.

While the theorem that now bears his name was known and previously utilised by the Babylonians and Indians, he, or his students, are often said to have constructed the first proof. It must, however, be stressed that the way in which the Babylonians handled Pythagorean numbers implies that they knew that the principle was generally applicable, and knew some kind of proof, which has not yet been found in the (still largely unpublished) cuneiform sources. 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. 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.

Musical theories and investigations

Medieval woodcut showing Pythagoras with bells and other instruments in Pythagorean tuning

According to legend, the way Pythagoras discovered that musical notes could be translated into mathematical equations was when he passed blacksmiths at work one day and thought that the sounds emanating from their anvils 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 the sounds were produced by looking at their tools. He discovered that it was because the hammers were "simple ratios of each other, one was half the size of the first, another was 2/3 the size, and so on".

This legend has since proven to be false by virtue of the fact that these ratios are only relevant to string length (such as the string of a monochord), and not to hammer weight. However, it may be that Pythagoras was indeed responsible for discovering the properties of string length.

Pythagoreans elaborated on a theory of numbers, the exact meaning of which is still debated among scholars. Another belief attributed to Pythagoras was that of the "harmony of the spheres". Thus the planets and stars moved according to mathematical equations, which corresponded to musical notes and thus produced a symphony.

Brewer (1894), wrote (page 2614):

Tetractys

Pythagoras was also credited with devising the tetractys, the triangular figure of four rows which add up to the perfect number, ten. As a mystical symbol, it was very important to the worship of the Pythagoreans who would swear oaths by it.

Iamblichus, Vit. Pyth., 29

Brewer (1894), wrote (page 2732):

Heraclides Ponticus reports the story that Pythagoras claimed that he had lived four previous lives that he could remember in detail. One of his past lives, as reported by Aulus Gellius, was as a beautiful courtesan. According to Xenophanes, Pythagoras heard the cry of his dead friend in the bark of a dog.

Brewer (1894), wrote (page 2293):

Lore

Pythagoras became the subject of elaborate legends surrounding his historic persona. Aristotle described Pythagoras as a wonder-worker and somewhat of a supernatural figure, attributing to him such aspects as a golden thigh, which he showed to Abaris, the Hyperborean priest, and exhibited in the Olympic games. According to Muslim tradition, Pythagoras was said to have been initiated by Hermes (Egyptian Thoth).

Brewer (1894), wrote (page 2292):

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

Both Plato and Isocrates affirm that, above all else, Pythagoras was famous for leaving behind him a way of life. According to Timaeus of Locri, he was the first to say, Friends have all things in common and Friendship is equality.

Brewer (1894), wrote (page 2685):

According to Walter Burkert (1972, p. 109)

Pythagorean school

Pythagoreans celebrate sunrise, painting by Fyodor Bronnikov(1827–1902)

According to Iamblichus (ca. 245–325 AD, 1918 translation) in The life of Pythagoras

Carl B. Boyer (1968), mentioned that "the Pythagorean school of thought was politically conservative and with a strict code of conduct." Leonid Zhmud (2006), identified two camps with the early Pythagoreans, the scientific mathematici and the religious acusmatici, who engaged in politics. According to Reidwig and Rendall (2005), who cite Antiphon reports, the school name was Semicircle, a place to discuss common interest topics among Samians. Outside of Samos he adapted a cave where he studied and lived day and night, discoursing with a few of his associates. In Samos he may have instructed the small athlete Eurymenes to eat a certain amount of meat every day.

Both Iamblichus and Porphyry give detailed accounts of the organisation of the school, although the primary interest of both writers is not historical accuracy, but rather to present Pythagoras as a divine figure, sent by the gods to benefit humankind.

Pythagoras set up an organisation which was in some ways a school, in some ways a brotherhood (and here it should be noted that sources indicate that as well as men there were many women among the adherents of Pythagoras), and in some ways a monastery. It was based upon the religious teachings of Pythagoras and was very secretive. The adherents were bound by a vow to Pythagoras and each other, for the purpose of pursuing the religious and ascetic observances, and of studying his religious and philosophical theories. There is mentioning of an oath on the Tetractys.

Pythagoras and fava beans, French, 1512/1514

There were ascetic practices (many of which had, perhaps, a symbolic meaning). Some represent Pythagoras as forbidding all animal food, advocating a plant-based diet, and prohibiting consumption of beans. This may have been due to the doctrine of metempsychosis. Other authorities contradict the statement. According to Aristoxenus, he allowed the use of all kinds of animal food except the flesh of oxen used for ploughing, and rams. There is a similar discrepancy as to the prohibition of fish and beans. But temperance of all kinds seems to have been urged. It is also stated that they had common meals, resembling the Spartan system, at which they met in companies of ten.

Considerable importance seems to have been attached to music and gymnastics in the daily exercises of the disciples. Their whole discipline is represented as encouraging a lofty serenity and self-possession, of which, there were various anecdotes in antiquity. Iamblichus (apparently on the authority of Aristoxenus) gives a long description of the daily routine of the members, which suggests many similarities with Sparta. The members of the sect showed a devoted attachment to each other, to the exclusion of those who did not belong to their ranks. There were even stories of secret symbols, by which members of the sect could recognise each other, even if they had never met before.

Commentary from Sir William Smith, Dictionary of Greek and Roman Biography and Mythology (1870, p. 620).

Influence on Plato

Pythagoras, depicted as a medieval scholar in the Nuremberg Chronicle

Pythagoras, or in a broader sense, the Pythagoreans, allegedly exercised an important influence on the work of Plato. According to R. M. Hare, this influence consists of three points: (1) The platonic Republic might be related to the idea of "a tightly organised community of like-minded thinkers", like the one established by Pythagoras in Croton. (2) 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". (3) Plato and Pythagoras shared a "mystical approach to the soul and its place in the material world". It is probable that both were influenced by Orphism.

Aristotle claimed that the philosophy of Plato closely followed the teachings of the Pythagoreans, and Cicero repeats this claim: Platonem ferunt didicisse Pythagorea omnia ("They say Plato learned all things Pythagorean"). Bertrand Russell, in his A History of Western Philosophy, contended that the influence of Pythagoras on Plato and others was so great that he should be considered the most influential of all Western philosophers.

Politics and science

Pythagoras was the first person known to have taught the earth was spherical, with antipodes and that it revolved around the sun. Pythagoras was also said to have spread the seeds of political liberty to Crotona, Sybaris, Metapontum, Rhegium, Sicily, Tauromenium, Catana, Agrigentum and Himera.

Influence on Greek art

Pythagoras, depicted on a 3rd-century coin

In the arts the Greeks searched some reality behind the appearances of things. The early Archaic sculpture represents life in simple forms, and it seems that it was influenced by the earliest Greek natural philosophies. There was a general Greek belief that nature expresses itself in ideal forms, and it was represented by a type (εἶδος), which was mathematically calculated. This can be observed in the construction of the first temples. The original forms were considered divine, and the forms of the later marble or stone elements indicate that there was an original wooden prototype. When the dimensions changed, the architects searched in mathematics some permanent principles behind the appearances of things. Maurice Bowra believes that these ideas influenced the theory of Pythagoras and his students who asserted that "all things are numbers".

During the 6th century BC, there was an evolution in the arts from the natural philosophies to the metaphysical theory of Pythagoras. The Greek sculptors and architects, tried to find the mathematical relation (canon), which would lead to the aesthetic perfection. The sculptor Polykleitos in his Canon wrote that beauty consists in the proportion not of the elements (materials), but of the parts, that is the interrelation of parts with one another and with the whole. It seems that he was influenced by the theories of Pythagoras. The numbers were extensively used in the Greek architectural orders. In the architectural canons every element was calculated and constructed by mathematical relations. The universe was controlled by the order, and even the sounds were functions of number and ratio. Rhys Carpenter says that he ratio 2:1 was the generative ratio of the Doric order, and in Hellenistic times an ordinary Doric colonnade, beats out a rhythm of notes."

Influence on other groups

Pythagoreanism may had an effect on Freemasonry and Rosicrucianism, both of which were groups dedicated to the study of mathematics/geometry and logical reasoning as opposed to religious dogma. Both Freemasonry and Rosicrucianism have claimed to have evolved out of the Pythagorean Brotherhood. Pythagorean mathematics are discussed in a chapter of Manly P. Hall's The Secret Teachings of All Ages entitled "Pythagorean Mathematics".