Very little is known about the 6th-century BCE sage Pythagoras and his early followers; but there is a great wealth of (often discordant) material about Pythagorean philosophy from later periods. Indeed, by any definition that seems meaningful to me, it is this later material which constitutes the trend or movement known as “Pythagoreanism”, while the earliest periods are virtually beyond our ken (a shadowy Proto-Pythagoreanism, if you like, and the screen onto which later thinkers project their ideals.).
One especially interesting summary of Pythagoreanism—and one that is more idiosyncratically Pythagorean than the anonymous Life of Pythagoras, which effectively represents a Greco-Roman mainstream worldview—is preserved in a chapter about Pythagoras contained in the massive history of philosophy compiled by Diogenes Laërtius (Lives and Opinions of Eminent Philosophers 8.24–33). He took it from a now lost work about The Successions of the Philosophers by Alexander Polyhistor, who in turn was summarizing a text called the Pythagoric Memoirs (Pythagorika hypomnemata). I translate this chapter from the Greek edition of H.S. Long.
To supply a brief summary, the Memoirs describe the whole world arising from Unity (Monad) as first principle, and from its interaction with Duality (Dyad), which it has produced; from them come mathematical objects which in turn produce the perceptible and living cosmos. Then follows an explanation of cyclical change within the cosmos based on a dualism resembling that of the Pythagorean Table of Opposites. This leads into an account of mortal and immortal life, ensouled and unensouled, and particularly on procreation and the senses. The human soul, its connection to the body and its fate after death are described; the entire air is full of discarnate souls, called daemons and heroes. Finally, some important rules of ethical behavior and ritual propriety are enumerated; compare the more literary Pythagorean Golden Verses on the same subject. Section headings added by me.
Alexander, in his Successions of the Philosophers, says that he also found the following points in the Pythagoric Memoirs.
<Cosmogony and Cosmology (8.24–26)>
The first principle of all things is the Monad; from the Monad comes the Indefinite Dyad, subsisting as matter in relation to the Monad, which is the cause. And from the Monad and the Indefinite Dyad come the numbers; from the numbers, points; from these, lines; from these, two-dimensional shapes; from the two-dimensional, three-dimensional shapes; and from these, the perceptible bodies, whose elements are four: fire, water, earth and air. These change and are turned (into each other) completely, and from them arises the cosmos.
This cosmos is ensouled, thinking (noeron), spherical, and contains the Earth in its midst; and it too is spherical, and inhabited on all sides. For there are antipodeans, and what is down for us is up for them. And within the cosmos, light and dark have an equal share, as do hot and cold, and dry and moist.
And by the predominance of heat, summer is generated, and by that of cold, winter. But the most beautiful parts of the year are when they are equally balanced, and spring is growing and healthful, while autumn is wasting and sickly. But the growing of the day is dawn, but the wasting is the evening; hence, it is more unhealthy.
<Life, Mortal and Immortal (8.26–28)>
The air around the Earth is sluggish and sickly and all things in it are mortal; but the higher air is eternally moved and pure and healthy, and all things in it are immortal, and, on account of this, divine. Sun and Moon and the other stars are gods, since theat predominates in them, and it is the cause of life. And the Moon is illuminated by the Sun.
And humans have a kinship with the gods, because the human being has a share in heat; hence, the gods (gr. ho theos) exercise providential care over us. And fate is the cause oforder, both universally and in particulars.
The ray of the Sun pervades the could and the dense aether—as they call the air the cold aether, and the sea and moisture the dense aether—and this ray descends even into the depths and originates all life. And everything that has a share in heat lives, even though they do not all have souls; and this is why plants are living beings. But the soul is a portion (apospasma) of aether, both the hot and the cold; and the soul differs from life by having a share of cold aether as well. And it is immortal, because that from which it is taken (apespastai) is immortal.
Animals are born from one another through sperm, whereas it is impossible for (spontaneous) generation from the earth to occur. Sperm is a drop of the brain that contains a hot breath (atmos) within itself. When it is brought to the womb, it sends forth ichor, moisture and blood of the brain, of which flesh, sinews, bones, hairs and the whole body consist. And from the breath, it sends forth soul and perception. It first takes solid shape in fourty days, and according to the ratios of harmony, the mature child is born within seven, nine or at most ten months. It has within it all the patterns (logoi) of life, which are formed in a series according to the ratios (logoi) of harmony, and each come to the fore at the designated times (of a person’s lifespan).
<Sense Perception (8.29)>
Perception in general, and sight in specific, is a certain very hot vapor (atmos). And on this account, it is said that seeing is through air and through water, because the hot is resistant to the cold. Because if the vapor in the eyes were cold, it would dissipate in the air, which would be its like. But as it is, (Pythagoras) calls the eyes the gates of the Sun; and he has the same doctrine about hearing and the other senses of perception.
<Soul in the Human Body (8.30–31)>
The soul of the human being is divided threefold, into mind (nous), reason (phrenes) and passion (thymos). Now, mind and spirit are also in the other animals, but reason is only in the human being. Now the rule of the soul is from the heart to the brain, and the part in the heart is passion, but reason and mind are the parts in the brain. The senses of perception are droplets from these parts. And the rational part (phronimon) is immortal, but the other two are mortal.
And the soul is sustained from blood; and the thoughts (logoi) of the soul are air-streams (anemoi). It is invisible, and so are its thoughts, because the aether is also invisible.
And the veins, arteries and sinews are bonds of the soul; but when it is strong and at settled down into itself, thoughts and deeds become its bonds.
<Souls Disincarnate (8.31–32)>
When (the soul) is cast out upon the Earth, it roams around in the air (in a shape) similar to its body.
Hermes is the steward (tamias) of souls, and that is why he is called Pompaios (‘escorter’), Pylaios (‘of the gate’) and Chthonios (‘of the earth, of the underworld’), because he sends the souls away from their bodies, both from the Earth and from the Sea, and the pure are taken to the highest place, while the impure cannot approach these or each other, but are bound in unbreakable chains by the Erinyes.
And the whole air is full of souls, and these are called daemons and heroes; and it is from these that dreams, signs and diseases are sent to people, and not only to people, but also sheep and other cattle.¹ And purifications (katharmoi), apotropaic rites (apotropiasmoi), all divination and omens and the like take place in reference to these (daemons and heroes).
He says that the most weighty thing among people is inducing the soul towards good or towards evil. And humans are fortunate when a good soul is attached to them; [but if it is evil,] they can never be calm or maintain the same course.
The just is (like) an oath (horkion), and for this reason, Zeus is called Horkios. Virtue is harmony, and so is health and everything good, and the gods (gr. ho theos); which is why the universe consists by harmony. And friendship/love (philia) is harmonic equality.
One must honor (nomizein) gods and heroes, but not equally; rather, one must always honor the gods, in silence (euphēmia), in white clothes and in purity, but the heroes only after midday.
Purity (hagneia) is through purifications (katharmoi), baths (loutra) and besprinklings (perirrhantēria); by keeping pure of deaths and births and all pollution; by abstaining of carcasses and flesh as foods, of the fish mullet and black-tail, of eggs and animals born from eggs, of beans and everything else that those who perform the mysteries (teletai) in temples (hiera).
1: This explains both ominous diseases of cattle which can be taken away through ritual, and the (assumed) fact that the innards of animals, and especially the livers of sheep, can be used for divination.
[Work in progress]