The Pythagorean school founded by Pythagoras (580-495 BC) in Magna Graecia constitutes a brotherhood that is both scientific and religious: Pythagoreanism is indeed based on an initiation and offers its followers a way of ethics and food life, as well as scientific research on the cosmos. Although the term of philosophical school is disputed and that one generally prefers to speak of sect for Pythagoreanism, this religious, political and philosophical association lasted nine or ten generations, and enjoyed a very great notoriety as well in the Greek and Roman antiquity. Its members adopted the term studies, in Greek μαθήματα / mathemata, to designate the multiple branches of knowledge that constituted their particular science: they explored the science of numbers, the bases of acoustics and musical theory, the elements of geometry, star movement and cosmology, while adhering to the doctrine of the Orphic religion on the transmigration of souls.

Pythagoreanism and the legend that has formed around it are not without obscurities and controversial subjects. By distinguishing between “those called Pythagoreans” and Pythagoras himself, Aristotle avoids pronouncing on the exact links between their thought and that of the philosopher; the later tradition, ignoring this distinction, undoubtedly favored the production of a large number of pseudepigraphic texts attributed to Pythagoras or to the ancient Pythagoreans. It was not until the 3rd century AD that the first presentations relating to the Pythagorean way of life appeared. After the death of Pythagoras, the school was run by his wife, the mathematician Théano.

Raphael's fresco The School of Athens: Pythagoras(In Raphael’s fresco The School of Athens, Pythagoras is shown writing in a book as a young man presents him with a tablet showing a diagrammatic representation of a lyre above a drawing of the sacred tetractys.)

Teaching rules

According to Plato’s testimony in the Republic, Pythagoras would have been an influential and beloved master, founder of a lifestyle capable of guaranteeing a happy destiny of the soul in the hereafter. We can glimpse his teaching through the Pythagorean maxims cited by Aristotle and largely transmitted by Jamblique; they were designated by the terms akousmata (άκούσματα), heard things, and symbola (σύμβολα), passwords or things to be interpreted. According to an indication of Jamblique which would go back to Aristotle, the Pythagorean teaching could thus be divided into two parts: one part for the “acousmaticiens”, (άκουσματικοί), the not yet initiated, and one for the initiated, the “ mathematicians ”. But this distinction could also be the sign of the schism experienced by the Pythagorean communities in crisis, the “acousmaticians” remaining attached to the Orphic teachings, and holding as fundamental the prescriptions and the prohibitions of the brotherhood, while the “mathematicians”, placing the doctrine of number at the center of Pythagoreanism, were oriented towards science, as can be seen for the very lively Pythagoreanism of Taranto and the colonies of Thebes and Phliontus. Teaching was oral; was it secret? Isocrates in his Praise of Busiris reports that the Pythagoreans were famous for their silence, without knowing whether he alludes to their self-control or to a ban on speaking made to neophytes. What is certain is that to all the questions that were put to them, the initiates answered, referring to their Master: “It is so because he said it (in Greek αὺτὸς ἒφα) “. At least part of the doctrine was to remain secret, for example the division of rational animals into three groups, while the philosophical ideas and mathematical demonstrations could be disclosed which were indeed published by Philolaos or Archytas. As for the transmission of knowledge between disciples, it is inseparable from respect for the moral rules of fraternal friendship as a whole: rule of silence, respect for the initiation grade of disciples. The Pythagorean school is thus both a religious and a scientific brotherhood.


Aristotle in the Protrepticus cites Pythagoras as a founder, model of the ideal of contemplative life, βίος θεωρητικός, and ancestor of Plato’s philosophy. Better yet, as Simone Weil recognized, Pythagorean doctrine permeates all aspects of Greek civilization, “almost all poetry, almost all philosophy, music, architecture, sculpture, all science proceeds from it, arithmetic, geometry, astronomy, mechanics, biology. Plato’s political thought (in its authentic form, that is, as it is set out in the Politics) follows from this. It embraced almost all secular life”: this shows the importance of Pythagorean thought for understanding Greek antiquity.


  • The limiting and the unlimited: For the Pythagoreans, the entire universe is constituted from the mixture of two principles, which limits, which determines, which stops, τὸ πέρας / τὸ περαίνον, and what is unlimited, τὸ ἂπειρον. It is the principle which limits which always dominates.

Philolaos of Croton, central figure of ancient Pythagoreanism, is the first Pythagorean to have left a written work. In his book, Philolaos accounts for the cosmos in these terms using these two types of fundamental entities:

“Nature in the cosmos has been brought into harmony from what limits and what is limitless – the cosmos taken as a whole and all that it contains.”
– Philolaos, (Diels, Fragments of the Presocratic, 2 B 47.)

One can consider the unlimited as continuations without intrinsic limit, among them appear the air, the water, the earth, but also the vacuum and the time. Limiters establish limits within a continuum: for example, these are structuring principles such as forms. Thus, a tree is a combination of an inherently unlimited continuum, wood, and structuring principles, the shape and structure of the tree. Philoloas concludes that particular objects as well as the cosmos are combinations of limiters and unlimited which obey a harmony, according to mathematical reports. Likewise, the origin of the cosmos is a fire (unlimited) in the center of a sphere (limiting). Since the world needs opposites to exist, Theophrastus asserts that the Pythagoreans believed that divine power is limited and that it cannot and does not want to reduce the good to the best.

  • Polarity

From mathematics to the harmony of the world

The Pythagorean doctrine of number is above all a numerical symbolism, which has been influenced both by the pseudo-science of the Chaldeans and the mathematical symbolism of the Milesian philosophy of nature. This is not a simple arithmetic theory. It would find its origin in the discovery by Pythagoras of natural laws establishing a relation between the length of the string of the lyre and the pitch of the note emitted by it; by extrapolation, he generalized these laws by declaring that everything, in human life as in the cosmos, was subject to number, as the qualitative essence of things, and not as a means of expressing measurable quantities. The number is the principle of everything, and each number is associated with a figure; all created things each have a number for symbol, so Aristotle quotes the following Pythagorean formula: “Justice is a number to the second power”, in Greek: ὴ δικαιοσύνη άριθμὸς ίσάκις ἲσος, formula having for key the notions of proportional mean and mediation in the theological sense, hence the restriction to the study of positive integers:

  • 1 represented the divinity: Heraclitus, very close to the Pythagoreans, also said: “The One, this unique sage, wants and does not want to be named Zeus. “
  • 2: the woman
  • 3; the man
  • 4: mother
  • 10: the Pythagorean brotherhood

This number-figure association is the support of a mathematical abstraction, because the number no longer derives from the results of mathematical applications – financial, agricultural, among others – but is therefore posited as the first principle (the Arche) of knowledge. For the Pythagoreans, it is a question of getting as close as possible to the mysticism of numbers, by establishing laws between arithmetic. It is notable that the arithmetic sets known to the Pythagoreans were by iterative constructions: this in fact follows from the figuration of numbers. Starting from a simple figure, such as a triangle formed by three points, we can enlarge the whole while retaining its shape but increasing its parts, to arrive, for example, at a triangle formed by six points. This non-frozen figuration is an important abstraction for Antiquity, especially as it also concerned certain volumes (pyramids with triangular bases, square, cylinder …). The comparison of the sequences thus constructed leads to the discovery of structural and general relations between particular sets of numbers. These natural laws are the hard core of the Pythagorean conception of mathematics, considered esoteric and sectarian, where whole numbers are believed to represent all of nature. This category of number becomes an end in itself, an immutable principle which aims to explain all things, as Philolaos of Crotone affirms it. Capital discovery with a great future; because after the numerical laws governing sounds, research on the structure of music led to the knowledge of the nature of harmony and rhythm, harmony defined as the relationship which unites the parts to the whole.

“Harmony is the unity of a mixture of many, and the single thought of separate thinkers, ἒστι άρμονία πολυμιγέων ἒνωσις καὶ δίχα φρονεόντων συμφρόνησις”
– Philolaos of Croton (Diels, Fragments of the Presocratic I, 410, fr. 10.)

The mathematical idea of ​​proportion was therefore applied to the cosmos, governed by inflexible laws: the notion of world harmony signified both musical harmony and any well-balanced mathematical structure, subject to strict geometric laws: “In all aspects of Greek life, the subsequent influence of this conception was immeasurable. It affected not only sculpture and architecture, but also poetry and rhetoric, religion and morality,” writes the great Hellenist Werner Jaeger.


The Pythagoreans had built a whole theory on the relationship of the zodiac with the migration of souls: “Cancer and Capricorn marked the two gates of heaven. Either to descend into generation, or to ascend to God, souls must therefore necessarily cross one of them. Through the Cancer Gate, souls fall to earth; through the gate of Capricorn, ascension of souls in the ether.” (Jérôme Carcopino, La Basilique pythagoricienne de la Porte Majeure, L’Artisan du Livre, 1927)

According to Philippe d’Oponte, some Pythagoreans refuted the idea that it is through the interposition of the Earth or the Moon that lunar eclipses take place.

  • The sky and the numbers
  • The anti-earth

Ethics and metaphysics

The testimonies of Ion of Chios and Herodotus attest Pythagoras’ links with Orphism and his knowledge of the destinies of the soul after death; the philosopher’s zeal in matters of initiation rituals and his concern for the survival of the soul underline the importance of the Pythagorean doctrine of metempsychosis or palingenesis. It promises its initiates that they will escape the painful cycle of metempsychosis if they know how to lead a virtuous life, practice a certain asceticism and perform numerous rites of purification. The capital idea capable of introducing a perfect coherence between the identity of the soul and the cycle of its successive destinies is that of the judgment of souls by a supreme god: this essential notion in a mystical doctrine is already mentioned by Plato. Such teaching also promoted the development of moral virtues such as self-control, the importance of friendship and mutual aid, even if the members of the community did not know each other, and a well-regulated conduct of daily life through an ethic of asceticism and abstinence; certain behaviors were prescribed such as moving with the right foot, wearing specific clothes, having specific behavior towards his fellow citizens: Pythagoras in this regard appears as “a great master in matters of morality, a true precursor of Socrates and of Christ”.

Food prohibitions and vegetarianism

According to the Pythagorean maxims cited by Aristotle, it was forbidden to eat certain fish such as mullet and bugs, and certain animal organs such as the heart and the womb; but it is not known whether these prohibitions were applied literally or whether they should be interpreted to give them a deeper meaning: for example, the rule which asks “not to eat the heart” (or, according to another translation, ” not to gnaw its heart ”) could have meant from the beginning of the 4th century BC that one should not torment oneself in the misfortune. One of the food bans whose authenticity is best documented concerns the consumption of beans. Aristotle provides several obscure explanations about it; it may be explained by an allergy to a certain amino acid they contain. The question of vegetarianism is more difficult to decide, the testimonies of the Ancients already contradicting each other in the 4th century BC. It is possible that Pythagoras was a strict vegetarian, the philosopher Empedocles, in the generation that followed Pythagoras, having clearly condemned the meat diet which he likened to cannibalism. This rule of vegetarianism is still presented, in the imperial period, as an authentic dogma of the Master by the Neopythagorean Ovid who makes Pythagoras say: “Be careful, mortals, not to defile your bodies with the harmful foods that the gods proscribe. Feed yourself without the need to kill and shed blood.” Were there different degrees of initiation within the Pythagorean communities, which could explain why the practice of vegetarianism varied according to the rank of its followers? The question remains open.


The political activity of the Pythagoreans is – it seems – very intense, especially in the city states of Magna Graecia. The social model of the Pythagorean brotherhood would imply a position in favor of the democratic regime where traditionally an aristocracy holds the power, and in this case, the knowledge. However, this democratic commitment is questionable, since the example of Archytas of Taranto shows that the political balance sought by the Pythagoreans did not necessarily imply democratic rule. In addition, Plato makes a clear distinction between Pythagoras and the legislators: it therefore seems very likely that Pythagoreanism did not advocate a particular political line. But, in the eyes of his adversaries, he could appear to form suspicious political circles. Indeed, the Pythagoreans endowed themselves in the 5th century BC of a way of life which separates them from the community: complex prohibitions and practice of a community existence beginning, it seems, by a rule of five years of silence. They exercise power in Crotone for a while, during Pythagoras’ lifetime, but their fellow citizens end up revolting by burning their houses and massacring members of their sect. The master himself had to seek refuge in Metapontus. This popular riot was at the origin of the disappearance of the Pythagorean school, but the disciples, and soon the Neopythagoreans, continued to uphold the doctrine of their masters for a long time.

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