1. Introduction
Pythagoras was
an Ionian Greek born on the island of Samos in the sixth century BCE. He
was supposed to have visited Thales in Miletus, who advised him to travel
to Egypt to learn more about mathematics and astronomy. About 535 BCE,
Pythagoras did go to Egypt, but was taken prisoner to Babylon by the
Persian king Cambyses II. Pythagoras returned to Samos eventually, but
then went to the Greek colony in Croton, in southern Italy. There he
founded a religious community and philosophical school, whose inner
circle of followers were called "mathematikoi." Many puzzling
restrictions are said to have been in force in the community, such as
not eating beans and not wearing a ring (see Iamblichus, Protr.
21). These are known as the Akousmata, and in many instances
appear to be superstitions. 3.1. The Archas of Mathematics as the Archas of All Things Aristotle says that Pythagoreanism, as known in his day, is characterized by the view that the principles (archas) of mathematics are the principles of all things. Aristotle writes, Contemporaneously with these philosophers and before them, the so-called Pythagoreans, who were the first to take up mathematics, not only advanced this study, but also having been brought up in it they thought its principles (archas) were the principles of all things. Since of these principles numbers are by nature the first, and in numbers they seemed to see many resemblances to the things that exist and come into being—more than in fire and earth and water (such and such a modification of numbers being justice, another being soul and reason, another being opportunity—and similarly almost all other things being numerically expressible); since, again, they saw that the modifications and the ratios of the musical scales were expressible in numbers;—since, then, all other things seemed in their whole nature to be modeled on numbers, and numbers seemed to be the first things in the whole of nature, they supposed the elements of numbers to be the elements of all things, and the whole heaven to be a musical scale and a number. (Metaphysics, 985b 23-986a 3; see also 1090a20-29) The
Pythagoreans hold that the principles of mathematics are the principles
of all things; in other words, the basic structure of Being is mathematical.
Everything somehow can be explained in mathematical terms. It is the
archas of mathematics that make things what they are, that give
to everything its distinctiveness. It seems that the Pythagoreans
view the principles of mathematics as providing structure or form to
Being. Even though they are immaterial, the principles of mathematics,
nevertheless, are more "real," as it were, than that to which they give
structure or form: that which is an archê is more real
in the sense that without it other things would not be what they are;
their greater reality consists in their logical and ontological priority.
It should be noted that the term archê is Aristolelian,
so that whether Pythagoras or his followers ever described their philosophy
using the term archê is not known.
In the above quotation, Aristotle adds that the Pythagoreans hold that in fact it is the elements (stoicheia) of numbers that are the elements of all things; this makes their teaching more difficult to understand. It seems they reasoned that, since numbers were by nature "first" with respect to mathematics—in other words, since mathematics is impossible without numbers—everything must somehow be numbers. Aristotle explains how impressed the Pythagoreans were to discover that musical scales could be expressed as ratios (using numbers); from this discovery they may have sought to ascribe numbers to all things. Indeed, they conclude that the whole heaven to be a musical scale and a number. This part of Pythagoreanism has perplexed scholars, however, since it is not obvious how numbers resemble things that come into being, including abstract ideas; a possible partial explanation is offered in the next section (3.2).
For the Pythagoreans, the number ten was a sacred number, the tetraktys,
because it is the sum of the first four integers, which are the numbers
used to express the ratios in the basic intervals
3.2. The Even and the Odd and All Things as Numbers In Aristotle's day, the Pythagoreans traced all things back to their origin to two principles. He explains, "Evidently, then, these thinkers also consider that number is the principle both as matter for things and as forming both their modifications and their permanent states, and hold that the elements of number are the even and the odd, and that of these the latter is limited, and the former unlimited; and that the One proceeds from both of these (for it is both even and odd), and number from the One; and that the whole heaven, as has been said, is numbers." (Metaphysics, 986a 15-21; see also 987a 15)For the Pythagoreans, the elements of number are the even and the odd, the limited and the unlimited; this is because numbers derive from the One and the One from the even/unlimited and the odd/limited. It is important first to understand the Pythagorean principles of the limited/odd and unlimited/even in order to understand Pythagorean philosophy. Although obscure, perhaps owing to an inadequate philosophical lexicon, the Pythagorean position seems to be that originally (temporal and/or logically) there exists the principle of the unlimited (or even), as a type of prime matter, without order or  formal
identity, co-eternal with which is the principle of the limited (or
odd), which imposes order and formal identity on the unlimited (The
unlimited is passive, whereas the limited is active). According to Aristotle,
the Pythagoreans believed that the unlimited is evil, and the limited
good, so that evil is synonymous with the unordered and good with the
ordered: "Evil belongs to the class of the class of the unlimited,
as the Pythagoreans conjectured, and good to that of the limited"
(Nicomachean Ethics 1106b29). In the doxographic tradition,
it is asserted that the Pythagoreans identified the limited with the
Monad (different from the One) and the unlimited with the dyad; the
Monad is a good deity, whereas the Dyad is an evil deity: "Pythagoras
held that one of the first principles, the monad, is god and the good,
which is the origin of the One, and is itself intelligence; but the
undefined dyad is a deity and the evil, surrounding which is the mass
of matter" (Aet. 1. 7; Dox. 302). If this account
is accepted, then Pythagoreanism operated with an eternal dualism of
two divine principles. From the unlimited/even and the limited/odd arises the One, since it is both even and odd; number then comes from the One, since the One is the basic unit of which all numbers consist (see also Physics 203a 10). According to Pythagorean teaching, number is the principle (or cause) both of something's matter—that of which it is made—and its formal characteristics, both permanent and accidental. Thus it seems that the Pythagoreans conceive numbers not only as physical and extended units, but also intelligible realities or formal principles. In other words, number as composed of units are not only the primary material components of all things, but also, since the One partakes of the limited/odd, give order and formal identity to all material things. In the latter case, the units are the points in geometrical shapes that define shape. So, for example, from two units comes a line; from three the minimum surface (triangle) and from four the minimum solid (tetrahedron); from solids come all material bodies (things that are perceptible). This at least sheds light on Aristotle's statement in the quotation above that for the Pythagoreans "Number is the principle both as matter for things and as forming both their modifications and their permanent states" and his explanation that the Pythagoreans differ from Plato because for the latter numbers do not exist apart from sensible things but "The things themselves are numbers" (Metaphysics 987b 29-30; see also Metaphysics 990a 12, 30; 1080b16-22; 1083b 8; 1090a 20-35). Number is the principle of matter because things consist of extended units, which are the basic elements of numbers. These extended units or points are, however, also the component parts of all geometric shapes, which give order and formal identity to all things. Aristotle explains that for the Pythagoreans numbers are not "monadic" (unextended and incorporeal) but have magnitude, so that they occupy space (Metaphysics 1080b 16); from these basic, extended units then all things are constructed as expressions of the basic geometrical shapes. (This does not explain, however, how immaterial things, such as abstract ideas, can be numbers.) In another passage from Metaphysics, Aristotle says that the Pythagoreans hold that things imitate numbers: "For the Pythagoreans say that things exist by imitation (mimesis) of numbers" (987b 11). This imitation should be understood as synonymous with the formulations above that all things are numbers, and not as contradictory. What Aristotle says about Pythagoreanism is similar to what Diogenes Laertius reports that Alexander wrote about it in a later period (1st c. BCE) in his Successions of Philosophers: "The principle of all things is the monad or unit; arising from this monad, the undefined dyad or two serves as material substratum to the monad, which is cause; from the monad and undefined dyad spring numbers." (Diogenes Laertius, Lives VIII. 25) (According to Diogenes, Alexander had access to the "Pythagorean memoirs.") It appears that the monad in Alexander's description is equivalent to Aristotle's odd/limited, whereas the dyad is the even/unlimited; unlike Aristotle, however, he says that the monad and dyad are not equal principles, but that the monad-odd/limited produces the dyad and acts upon it to produce the number One. From the number One all the other numbers are composed, which are the elements of all things. Alexander also explains the nature of all things as composed of the One, which has its origin in the monad and dyad: "From the monad and the undefined dyad spring numbers; from numbers point; from points, lines; from line plane figures; from plane figures, solid figures; from solid figures, sensible bodies, the elements which are four, fire, water, earth and air; these elements interchange and turn into one another completely, and combine to create a universe animate, intelligent, spherical" (Diogenes Laertius, Lives VIII. 25). All things literally are made of number because they are composed of extended units, many instances of the One. Whether Pythagoreanism in Aristotle's day was the same as it was in Alexander's time is an open question; it appears, however, that Aristotle's description is similar enough to Alexander's in this matter that there was probably no significant doctrinal development between the time of Aristotle and that of Alexander.
There was an ethical side to Pythagoreanism. (Ethics is the drawing out of implications for intentional behavior from one's view of the basic structure of Being: if Being is a certain way, then it follows that one should live in a certain way.) In brief, the Pythagoreans believe that human beings are composed of a body and a soul, that the soul is immortal and that souls transmigrate upon the death of the body. (Xenophanes, a contemporary of Pythagoras, tells the following story about him: "And now I will turn to another tale and point the way.... Once they say that he (Pythagoras) was passing by when a dog was being beaten and spoke this word: "Stop! Don't beat it! For it is the soul of a friend that I recognized when I heard its voice." [Fr. 7].) Now what the soul is exactly is unclear, except that it is a number. Their beliefs about the soul in conjunction with their view the cosmos is harmonious leads them to the ethical position that the task for human beings is to ensure that they live in conformity with the harmony of the cosmos. This is done by following the ethical principles laid down by the community.
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It
is difficult to differentiate between what Pythagoras taught and what
his followers added to his teaching, because there was a tendency in
the history of Pythagoreanism to attribute all new ideas to the founder,
Pythagoras. In addition, since he was such a revered figure among his
followers, events in the life of Pythagoras tend to be idealized, and
it is probable that many unhistorical stories about him were created.
For example, some of Pythagoras's later biographers portray him as a
god-like figure. Thus, accurate biographical information about Pythagoras
is difficult to isolate from the available data. There is a third obstacle
to a historically satisfactory understanding of Pythagoras and the movement
he spawned. Pythagoras founded not merely a philosophical school but
a religious community; the members were required to keep important Pythagorean
teaching from the uninitiated. The existence of esoteric teaching, therefore,
makes it difficult for the historian of ideas to reconstruct the exact
nature of Pythagoreanism, especially in the earliest period of
its existence: because their teaching was esoteric, Pythagoras
and his followers left behind no publicly-accessible writings that explain
systematically the doctrinal foundation of their religious community.
The focus of this study is on the nature of Pythagoreanism at the time
of Aristotle; the best source for this stage in the evolution of this
philosophy is Aristotle himself, in particular his Metaphysics. There
is much about Pythagoreanism that is obscure. 
of
ancient Greek music (1:2, 3:2, 4:3). Aristotle says that for the Pythagoreans,
this number is perfect and comprises the whole nature of numbers (Metaphysics
906a 8-14); all other numbers are reducible somehow to the first four
integers. Thus the Pythagoreans seek to find the number ten exemplified
in the cosmos, even if it means falsifying empirical data. Aristotle
explains, "They say that the bodies which move
through the heavens are ten, but as the visible bodies are only nine,
to meet this they invent a tenth—the 'counter-earth'" (Metaphysics
906a 10-12). At the time of the ancient Greeks it was thought that there
were nine spheres—containing nine heavenly bodies—but, the
Pythagoreans invented a tenth heavenly body—the "counter earth,"
in order to have cosmic perfection.
