“There is geometry in the humming of the strings. There is music in the Spacing of the spheres.”

– Pythagoras

Pythagoras of Samos^[a] (Ancient Greek: Πυθαγόρας ὁ Σάμιος, romanized: Pythagóras ho Sámios, lit. ‘Pythagoras the Samian‘, or simply Πυθαγόρας; Πυθαγόρης in Ionian Greek; c. 570 – c. 495 BC)^[b] was an ancient Ionian Greek philosopher and the eponymous founder of Pythagoreanism. His political and religious teachings were well known in Magna Graecia and influenced the philosophies of Plato, Aristotle, and, through them, Western philosophy. Knowledge of his life is clouded by legend, but he appears to have been the son of Mnesarchus, a gem-engraver on the island of Samos. Modern scholars disagree regarding Pythagoras’s education and influences, but they do agree that, around 530 BC, he travelled to Croton in southern Italy, where he founded a school in which initiates were sworn to secrecy and lived a communal, ascetic lifestyle. This lifestyle entailed a number of dietary prohibitions, traditionally said to have included vegetarianism, although modern scholars doubt that he ever advocated complete vegetarianism.

The teaching most securely identified with Pythagoras is metempsychosis, or the “transmigration of souls”, which holds that every soul is immortal and, upon death, enters into a new body. He may have also devised the doctrine of musica universalis, which holds that the planets move according to mathematical equations and thus resonate to produce an inaudible symphony of music. Scholars debate whether Pythagoras developed the numerological and musical teachings attributed to him, or if those teachings were developed by his later followers, particularly Philolaus of Croton. Following Croton’s decisive victory over Sybaris in around 510 BC, Pythagoras’s followers came into conflict with supporters of democracy and Pythagorean meeting houses were burned. Pythagoras may have been killed during this persecution, or escaped to Metapontum, where he eventually died.

In antiquity, Pythagoras was credited with many mathematical and scientific discoveries, including the Pythagorean theorem, Pythagorean tuning, the five regular solids, the Theory of Proportions, the sphericity of the Earth, and the identity of the morning and evening stars as the planet Venus. It was said that he was the first man to call himself a philosopher (“lover of wisdom”)^[c] and that he was the first to divide the globe into five climatic zones. Classical historians debate whether Pythagoras made these discoveries, and many of the accomplishments credited to him likely originated earlier or were made by his colleagues or successors. Some accounts mention that the philosophy associated with Pythagoras was related to mathematics and that numbers were important, but it is debated to what extent, if at all, he actually contributed to mathematics or natural philosophy.

Pythagoras influenced Plato, whose dialogues, especially his Timaeus, exhibit Pythagorean teachings. Pythagorean ideas on mathematical perfection also impacted ancient Greek art. His teachings underwent a major revival in the first century BC among Middle Platonists, coinciding with the rise of Neopythagoreanism. Pythagoras continued to be regarded as a great philosopher throughout the Middle Ages and his philosophy had a major impact on scientists such as Nicolaus Copernicus, Johannes Kepler, and Isaac Newton. Pythagorean symbolism was used throughout early modern European esotericism, and his teachings as portrayed in Ovid‘s Metamorphoses influenced the modern vegetarian movement.

Metempsychosis

Although the exact details of Pythagoras’s teachings are uncertain,^[99][100] it is possible to reconstruct a general outline of his main ideas.^[99][101] Aristotle writes at length about the teachings of the Pythagoreans,^[16][102] but without mentioning Pythagoras directly.^[16][102] One of Pythagoras’s main doctrines appears to have been metempsychosis,^{[73][103][104][105][106][107]} the belief that all souls are immortal and that, after death, a soul is transferred into a new body.^[103][106] This teaching is referenced by Xenophanes, Ion of Chios, and Herodotus.^[103][108] Nothing whatsoever, however, is known about the nature or mechanism by which Pythagoras believed metempsychosis to occur.^[109]

Empedocles alludes in one of his poems that Pythagoras may have claimed to possess the ability to recall his former incarnations.^[110] Diogenes Laërtius reports an account from Heraclides Ponticus that Pythagoras told people that he had lived four previous lives that he could remember in detail.^{[111][112][113]} The first of these lives was as Aethalides the son of Hermes, who granted him the ability to remember all his past incarnations.^[114] Next, he was incarnated as Euphorbus, a minor hero from the Trojan War briefly mentioned in the Iliad.^[115] He then became the philosopher Hermotimus,^[116] who recognized the shield of Euphorbus in the temple of Apollo.^[116] His final incarnation was as Pyrrhus, a fisherman from Delos.^[116] One of his past lives, as reported by Dicaearchus, was as a beautiful courtesan.^[104][117]

Mysticism

Another belief attributed to Pythagoras was that of the “harmony of the spheres“,^[118][119] which maintained that the planets and stars move according to mathematical equations, which correspond to musical notes and thus produce an inaudible symphony.^[118][119] According to Porphyry, Pythagoras taught that the seven Muses were actually the seven planets singing together.^[120] In his philosophical dialogue Protrepticus, Aristotle has his literary double say:

When Pythagoras was asked [why humans exist], he said, “to observe the heavens,” and he used to claim that he himself was an observer of nature, and it was for the sake of this that he had passed over into life.^[121]

Pythagoras was said to have practiced divination and prophecy.^[122] 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.^[123]

Numerology

The so-called Pythagoreans, who were the first to take up mathematics, not only advanced this subject, but saturated with it, they fancied that the principles of mathematics were the principles of all things.

— Aristotle, Metaphysics 1–5, c. 350 BC

According to Aristotle, the Pythagoreans used mathematics for solely mystical reasons, devoid of practical application.^[128] They believed that all things were made of numbers.^[129][130] The number one (the monad) represented the origin of all things^[131] and the number two (the dyad) represented matter.^[131] The number three was an “ideal number” because it had a beginning, middle, and end^[132] and was the smallest number of points that could be used to define a plane triangle, which they revered as a symbol of the god Apollo.^[132] The number four signified the four seasons and the four elements.^[133] The number seven was also sacred because it was the number of planets and the number of strings on a lyre,^[133] and because Apollo’s birthday was celebrated on the seventh day of each month.^[133] They believed that odd numbers were masculine,^[134] that even numbers were feminine,^[134] and that the number five represented marriage, because it was the sum of two and three.^[135][136]

Ten was regarded as the “perfect number”^[128] and the Pythagoreans honored it by never gathering in groups larger than ten.^[137] Pythagoras was credited with devising the tetractys, the triangular figure of four rows which add up to the perfect number, ten.^[124][125] The Pythagoreans regarded the tetractys as a symbol of utmost mystical importance.^{[124][125][126]} Iamblichus, in his Life of Pythagoras, states that the tetractys was “so admirable, and so divinised by those who understood [it],” that Pythagoras’s students would swear oaths by it.^{[98][125][126][138]} Andrew Gregory concludes that the tradition linking Pythagoras to the tetractys is probably genuine.^[139]

Modern scholars debate whether these numerological teachings were developed by Pythagoras himself or by the later Pythagorean philosopher Philolaus of Croton.^[140] In his landmark study Lore and Science in Ancient Pythagoreanism, Walter Burkert argues that Pythagoras was a charismatic political and religious teacher,^[141] but that the number philosophy attributed to him was really an innovation by Philolaus.^[142] According to Burkert, Pythagoras never dealt with numbers at all, let alone made any noteworthy contribution to mathematics.^[141] Burkert argues that the only mathematics the Pythagoreans ever actually engaged in was simple, proofless arithmetic,^[143] but that these arithmetic discoveries did contribute significantly to the beginnings of mathematics.^[144]

No authentic writings of Pythagoras have survived,^[5][6][7] and almost nothing is known for certain about his life.^[8][9][10] The earliest sources on Pythagoras’s life are brief, ambiguous, and often satirical.^[7][11][12] The earliest source on Pythagoras’s teachings is a satirical poem probably written after his death by Xenophanes of Colophon, who had been one of his contemporaries.^[13][14] In the poem, Xenophanes describes Pythagoras interceding on behalf of a dog that is being beaten, professing to recognize in its cries the voice of a departed friend.^{[12][13][15][16]} Alcmaeon of Croton, a doctor who lived in Croton at around the same time Pythagoras lived there,^[13] incorporates many Pythagorean teachings into his writings^[17] and alludes to having possibly known Pythagoras personally.^[17] The poet Heraclitus of Ephesus, who was born across a few miles of sea away from Samos and may have lived within Pythagoras’s lifetime,^[18] mocked Pythagoras as a clever charlatan,^[11][18] remarking that “Pythagoras, son of Mnesarchus, practiced inquiry more than any other man, and selecting from these writings he manufactured a wisdom for himself—much learning, artful knavery.”^[11][18]

The Greek poets Ion of Chios (c. 480 – c. 421 BC) and Empedocles of Acragas (c. 493 – c. 432 BC) both express admiration for Pythagoras in their poems.^[19] The first concise description of Pythagoras comes from the historian Herodotus of Halicarnassus (c. 484 – c. 420 BC),^[20] who describes him as “not the most insignificant” of Greek sages^[21] and states that Pythagoras taught his followers how to attain immortality.^[20] The accuracy of the works of Herodotus is controversial.^{[22][23][24][25][26]} The writings attributed to the Pythagorean philosopher Philolaus of Croton, who lived in the late fifth century BC, are the earliest texts to describe the numerological and musical theories that were later ascribed to Pythagoras.^[27] The Athenian rhetorician Isocrates (436–338 BC) was the first to describe Pythagoras as having visited Egypt.^[20] Aristotle wrote a treatise On the Pythagoreans, which is no longer extant.^[28] Some of it may be preserved in the Protrepticus. Aristotle’s disciples Dicaearchus, Aristoxenus, and Heraclides Ponticus also wrote on the same subject.^[29]

Most of the major sources on Pythagoras’s life are from the Roman period,^[30] by which point, according to the German classicist Walter Burkert, “the history of Pythagoreanism was already… the laborious reconstruction of something lost and gone.”^[29] Three lives of Pythagoras have survived from late antiquity,^[10][30] all of which are filled primarily with myths and legends.^[10][30][31] The earliest and most respectable of these is the one from Diogenes Laërtius‘s Lives and Opinions of Eminent Philosophers.^[30][31] The two later lives were written by the Neoplatonist philosophers Porphyry and Iamblichus^[30][31] and were partially intended as polemics against the rise of Christianity.^[31] The later sources are much lengthier than the earlier ones,^[30] and even more fantastic in their descriptions of Pythagoras’s achievements.^[30][31] Porphyry and Iamblichus used material from the lost writings of Aristotle’s disciples^[29] and material taken from these sources is generally considered to be the most reliable.^[29]