Pythagoreanism was the system of esoteric and metaphysical beliefs held by Pythagoras and his followers, the Pythagorean cult, who were considerably influenced by mathematics, music and astronomy. Pythagoreanism originated in the 5th century BCE and greatly influenced Platonism. Later revivals of Pythagorean doctrines led to what is now called Neopythagoreanism.
According to tradition, Pythagoreanism developed at some point into two separate schools of thought:
- the mathēmatikoi (μαθηματικοί, Greek for "learners") and
- the akousmatikoi (ἀκουσματικοί, Greek for "listeners").
The mathēmatikoi were supposed to have extended and developed the more mathematical and scientific work begun by Pythagoras. The mathēmatikoi allowed that the akousmatikoi were Pythagorean, but felt that their own group was more representative of Pythagoras.
The akousmatikoi focused on the more religious and ritualistic aspects of his teachings: they claimed that the mathēmatikoi were not genuinely Pythagorean, but followers of the "renegade" Pythagorean Hippasus.
Pythagorean thought was dominated by mathematics, and it was profoundly mystical. In the area of cosmology there is less agreement about what Pythagoras himself actually taught, but most scholars believe that the Pythagorean idea of the transmigration of the soul is too central to have been added by a later follower of Pythagoras. The Pythagorean conception of substance, on the other hand, is of unknown origin, partly because various accounts of his teachings are conflicting. The Pythagorean account actually begins with Anaximander's teaching that the ultimate substance of things is "the boundless," or what Anaximander called the "apeiron." The Pythagorean account holds that it is only through the notion of the "limit" that the "boundless" takes form.
Pythagoras wrote nothing down, and relying on the writings of Parmenides, Empedocles, Philolaus and Plato (people either considered Pythagoreans, or whose works are thought deeply indebted to Pythagoreanism) results in a very diverse picture in which it is difficult to ascertain what the common unifying Pythagorean themes were. Relying on Philolaus, whom most scholars agree is highly representative of the Pythagorean school, one has a very intricate picture. Aristotle explains how the Pythagoreans (by which he meant the circle around Philolaus) developed Anaximander's ideas about the apeiron and the peiron, the unlimited and limited, by writing that:
... for they [the Pythagoreans] plainly say that when the one had been constructed, whether out of planes or of surface or of seed or of elements which they cannot express, immediately the nearest part of the unlimited began to be drawn in and limited by the limit.
Continuing with the Pythagoreans:
The Pythagoreans, too, held that void exists, and that it enters the heaven from the unlimited breath – it, so to speak, breathes in void. The void distinguishes the natures of things, since it is the thing that separates and distinguishes the successive terms in a series. This happens in the first case of numbers; for the void distinguishes their nature.
When the apeiron is inhaled by the peiron it causes separation, which also apparently means that it "separates and distinguishes the successive terms in a series." Instead of an undifferentiated whole we have a living whole of inter-connected parts separated by "void" between them. This inhalation of the apeiron is also what makes the world mathematical, not just possible to describe using maths, but truly mathematical since it shows numbers and reality to be upheld by the same principle. Both the continuum of numbers (that is yet a series of successive terms, separated by void) and the field of reality, the cosmos — both are a play of emptiness and form, apeiron and peiron. What really sets this apart from Anaximander's original ideas is that this play of apeiron and peiron must take place according to harmonia (harmony), about which Stobaeus commentated:
About nature and harmony this is the position. The being of the objects, being eternal, and nature itself admit of divine, not human, knowledge – except that it was not possible for any of the things that exist and are known by us to have come into being, without there existing the being of those things from which the universe was composed, the limited and the unlimited. And since these principles existed being neither alike nor of the same kind, it would have been impossible for them to be ordered into a universe if harmony had not supervened – in whatever manner this came into being. Things that were alike and of the same kind had no need of harmony, but those that were unlike and not of the same kind and of unequal order – it was necessary for such things to have been locked together by harmony, if they are to be held together in an ordered universe.
A musical scale presupposes an unlimited continuum of pitches, which must be limited in some way in order for a scale to arise. The crucial point is that not just any set of limiters will do. One may not simply choose pitches at random along the continuum and produce a scale that will be musically pleasing. The diatonic scale, also known as "Pythagorean," is such that the ratio of the highest to the lowest pitch is 2:1, which produces the interval of an octave. That octave is in turn divided into a fifth and a fourth, which have the ratios of 3:2 and 4:3 respectively and which, when added, make an octave. If we go up a fifth from the lowest note in the octave and then up a fourth from there, we will reach the upper note of the octave. Finally the fifth can be divided into three whole tones, each corresponding to the ratio of 9:8 and a remainder with a ratio of 256:243 and the fourth into two whole tones with the same remainder. This is a good example of a concrete applied use of Philolaus’ reasoning. In Philolaus' terms the fitting together of limiters and unlimiteds involves their combination in accordance with ratios of numbers (harmony). Similarly the cosmos and the individual things in the cosmos do not arise by a chance combination of limiters and unlimiteds; the limiters and unlimiteds must be fitted together in a "pleasing" (harmonic) way in accordance with number for an order to arise.
This teaching was recorded by Philolaus' pupil Archytas in a lost work entitled On Harmonics or On Mathematics, and this is the influence that can be traced in Plato. Plato's pupil Aristotle made a distinction in his Metaphysics between Pythagoreans and "so-called" Pythagoreans. He also recorded the Table of Opposites, and commented that it might be due to Alcmaeon of the medical school at Croton, who defined health as a harmony of the elements in the body.
After attacks on the Pythagorean meeting-places at Croton, the movement dispersed, but regrouped in Tarentum, also in Southern Italy. A collection of Pythagorean writings on ethics collected by Taylor show a creative response to the troubles.
The legacy of Pythagoras, Socrates and Plato was claimed by the wisdom tradition of the Hellenized Jews of Alexandria, on the ground that their teachings derived from those of Moses. Through Philo of Alexandria this tradition passed into the Medieval culture, with the idea that groups of things of the same number are related or in sympathy. This idea evidently influenced Hegel in his concept of internal relations.
The ancient Pythagorean pentagram was drawn with two points up and represented the doctrine of Pentemychos. Pentemychos means "five recesses" or "five chambers," also known as the pentagonas — the five-angle, and was the title of a work written by Pythagoras' teacher and friend Pherecydes of Syros.
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 souls to achieve a higher rank among the gods. Much of their mysticism concerning the soul seems inseparable from the Orphic tradition. The Orphics included various purifactory rites and practices as well as incubatory rites of descent into the underworld. Apart from being linked with this, 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 Pherecydes was Pythagoras's most "intimate" teacher. Pherecydes 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 (eugieia Eudaimonia).
"Wheel of Birth" and scientific contemplation
The Pythagoreans believed that a release from the "wheel of birth" was possible. They followed the Orphic traditions and practices to purify the soul but at the same time they suggested a deeper idea of what such a purification might be. Aristoxenus said that music was used to purify the soul just like medicine was used to purge the body. But in addition to this, Pythagoreans distinguished three kinds of lives: Theoretic, Practical and Apolaustic. Pythagoras is said to have used the example of Olympic games to distinguish between these three kind of lives. Pythagoras suggests that the lowest class of people who come to the games are the people who come to buy or sell. The next higher class comprises people who come to participate in the games. And the highest class contains people who simply come to look on. Thus Pythagoras suggests that the highest purification of a life is in pure contemplation. It is the philosopher who contemplates about science and mathematics who is released from the "cycle of birth." The pure mathematician's life is, according to Pythagoras, the life at the highest plane of existence.
Thus the root of mathematics and scientific pursuits in Pythagoreanism is also based on a spiritual desire to free oneself from the cycle of birth and death. It is this contemplation about the world that forms the greatest virtue in Pythagorean philosophy.
The Pythagoreans were well known in antiquity for their vegetarianism, which they practised for religious, ethical and ascetic reasons, in particular the idea of metempsychosis – the transmigration of souls into the bodies of other animals. "Pythagorean diet" was a common name for the abstention from eating meat and fish, until the coining of "vegetarian" in the 19th century.
The Pythagorean code further restricted the diet of its followers, prohibiting the consumption or even touching of any sort of bean. It is probable that this is due to their belief in the soul, and the fact that beans obviously showed the potential for life. Some, for example Cicero, say perhaps the flatulence caused by beans is an emergency response system, as protection from potential favism, perhaps because they resemble the kidneys and genitalia, but most likely for magico-religious reasons, such as the belief that beans and human beings were created from the same material. Most stories of Pythagoras' murder revolve around his aversion to beans. According to legend, enemies of the Pythagoreans set fire to Pythagoras' house, sending the elderly man running toward a bean field, where he halted, declaring that he would rather die than enter the field – whereupon his pursuers slit his throat. Susceptible persons may develop a hemolytic anemia by eating the beans, or even by walking through a field where the plants are in flower. 
Views on women
Women were given equal opportunity to study as Pythagoreans, and learned practical domestic skills in addition to philosophy. Women were held to be different from men, but sometimes in good ways. The priestess, philosopher and mathematician Themistoclea is regarded as Pythagoras' teacher; Theano, Damo and Melissa as female disciples. Pythagoras is also said to have preached that men and women ought not to have sex during the summer, holding that winter was the appropriate time.
Neopythagoreanism was a revival in the 2nd century BC – 2nd century CE period of various ideas traditionally associated with the followers of Pythagoras, the Pythagoreans. Notable Neopythagoreans include Nigidius Figulus, Apollonius of Tyana and Moderatus of Gades. Middle and Neo-Platonists such as Numenius and Plotinus also showed some Neopythagorean influence.
They emphasized the distinction between the soul and the body. God must be worshipped spiritually by prayer and the will to be good. The soul must be freed from its material surroundings by an ascetic habit of life. Bodily pleasures and all sensuous impulses must be abandoned as detrimental to the spiritual purity of the soul. God is the principle of good; Matter the groundwork of Evil. The non-material universe was regarded as the sphere of mind or spirit.
In 1915, a subterranean basilica where 1st century Neo-Pythagoreans held their meetings was discovered near Porta Maggiore on Via Praenestina, Rome. The groundplan shows a basilica with three naves and an apse similar to early Christian basilicas that did not appear until much later, in the 4th century. The vaults are decorated with white stuccoes symbolizing Neopythagorean beliefs but its exact meaning remains a subject of debate.
- The Pythagorean idea that whole numbers and harmonic (euphonic) sounds are intimately connected in music, must have been well known to lute-player and maker Vincenzo Galilei, father of Galileo Galilei. While possibly following Pythagorean modes of thinking, Vincenzo is known to have discovered a new mathematical relationship between string tension and pitch, thus suggesting a generalization of the idea that music and musical instruments can be mathematically quantified and described. This may have paved the way to his son's crucial insight that all physical phenomena may be described quantitatively in mathematical language (as physical "laws"), thus beginning and defining the era of modern physics.
- Pythagoreanism has had a clear and obvious influence on the texts found in the hermetica corpus and thus flows over into hermeticism, gnosticism and alchemy.
- The Pythagorean cosmology also inspired the Arab gnostic Monoimus to combine this system with monism and other things to form his own cosmology.
- The pentagram (five-pointed star) was an important religious symbol used by the Pythagoreans, which is often seen as being related to the elements theorized by Empedocles to comprise all matter.
- The Pythagorean school doubtless had a monumental impact on the development of numerology and number mysticism, an influence that still resonates today. For example, it is from the Pythagoreans that the number 3 acquires its modern reputation as the noblest of all digits.
- The Pythagoreans were advised to "speak the truth in all situations," which Pythagoras said he learned from the Magi of Babylon.
- The Pythagorean theory of harmonic ratios is basis of studies on music theory in the Islamic world, for example the al-Farabi's Kitab al-Musiqa al-kabir.
- Pythagorean philosophy had a marked impact on the thoughts of early modern scholars involved within the Scientific Revolution. Of particular interest is the focus applied to the Platonic Solids derived from the Pythagorean theories of geometry and numbers by Plato. Within the work of Leonardo  fascination can be found within manuscripts describing the Platonic Solids, and also within the work of Kepler who supported the Copernican theory of heliocentrism and attempted a theory of the universe based on musical, geometrical harmony.
- Esoteric cosmology
- Incommensurable magnitudes
- Mathematical Beauty
- Orphism (religion)
- Pythagorean tuning
- Sacred geometry
- Unit-point atomism
- On the two schools and these differences, see Charles Kahn, p. 15, Pythagoras and the Pythagoreans, Hackett 2001.
- This is actually a lost book whose contents are preserved in Damascius, de principiis, quoted in Kirk and Raven, The Pre-Socratic Philosophers, Cambridge Univ. Press, 1956, page 55.
- Burnet J. (1892) Early Greek Philosophy A. & C. Black, London, OCLC 4365382, and subsequent editions, 2003 edition published by Kessinger, Whitefish, Montana, ISBN 0-7661-2826-1
- Russell, Bertrand, History of Western Philosophy
- "Vegetarianism". The Oxford Encyclopedia of Food and Drink. OUP. 2004[dead link]
- See for instance the popular treatise by Antonio Cocchi, Del vitto pitagorico per uso della medicina, Firenze 1743, which initiated a debate on the "Pythagorean diet".
- Cicero, On Divination, I xxx 62, quoted in
- Seife, Charles. Zero p 26
- Gabrielle Hatfield, review of Frederick J. Simoons, Plants of Life, Plants of Death, University of Wisconsin Press, 1999. ISBN 0-299-15904-3. In Folklore 111:317-318 (2000).
- Riedweg, Christoph, Pythagoras: his life, teaching, and influence. Ithaca : Cornell University Press, pp. 39, 70. (2005), ISBN 0-8014-4240-0
- Seife, p 38
- Glenn, Cheryl, Rhetoric Retold: Regendering the Tradition from Antiquity Through the Renaissance. Southern Illinois University, 1997. 30–31.
- Seife, Charles. Zero p. 27
- This article incorporates text from a publication now in the public domain: Chisholm, Hugh, ed. (1911). "Neopythagoreanism". Encyclopædia Britannica (11th ed.). Cambridge University Press.
- Ball Platner, Samuel. "Basilicae". penelope.uchicago.edu.
- Cohen, Mark, Readings In Ancient Greek Philosophy: From Thales To Aristotle. Indianapolis, IN: Hackett Publishing Company, 2005. 15–20.
- Zammattio, Carlo, "Leonardo The Scientist." Maidenhead, England: Mcgraw-Hill Book Company, 1980. p.98-99
- Koestler, Arthur, "The Sleepwalkers." London, England: Penguin Books, 1959. p.250-251
- Jacob, Frank Die Pythagoreer: Wissenschaftliche Schule, religiöse Sekte oder politische Geheimgesellschaft?, in: Jacob, Frank (Hg.): Geheimgesellschaften: Kulturhistorische Sozialstudien/ Secret Societies: Comparative Studies in Culture, Society and History, Globalhistorische Komparativstudien Bd.1, Comparative Studies from a Global Perspective Vol. 1, Königshausen&Neumann, Würzburg 2013, S.17-34.
- O'Meara, Dominic J. Pythagoras Revived: Mathematics and Philosophy in Late Antiquity , Clarendon Press, Oxford, 1989. ISBN 0-19-823913-0
- Riedweg, Christoph Pythagoras : his life, teaching, and influence ; translated by Steven Rendall in collaboration with Christoph Riedweg and Andreas Schatzmann, Ithaca : Cornell University Press, (2005), ISBN 0-8014-4240-0
- Pythagoreanism Web Article
- Pythagoreanism Discussion Group
- Pythagoreanism Web Site
- Pythagoreanism Web Site
- Pythagoreanism entry by Carl Huffman in the Stanford Encyclopedia of Philosophy