About the time the Ionic philosophy attained its highest development in
Asia Minor, another phase of philosophical thought appeared in the
Greek colonies of Italy. As we turn to the Pythagorean philosophy, the
first philosophy of the West, we are struck with the importance which
the ethico-religious aspect assumes from the outset; philosophy now is
not so much an inquiry into the causes of things as a
rule of life, a way of salvation. It is remarkable, too, that this
notion of philosophy never wholly died out in the subsequent
development of Greek thought. Plato, Aristotle, and the Stoics
constantly referred philosophy to life as well as to knowledge.
The Pythagorean system of speculation is sometimes contrasted with the
Ionian as being an embodiment of the Doric spirit, which was artistic,
conservative, ethical, while the Greeks of the Ionian colonies were
characterized by worldly sense, versatility, curiosity, and commercial
enterprise. Both philosophies, however, are wholly Greek.
Life of Pythagoras. Samos was the home and probably the
birthplace of Pythagoras. It is certain that he journeyed to Italy
about the year 530 B.C., and that he founded in Crotona a
philosophico-religious society. The story of his journey through
Egypt, Persia, India, and Gaul is part of the Neo-Pythagorean legend,
though there is good reason for believing that the account of his
death at Metapontum is true.
Sources. Primary sources. The Neo-Pythagoreans mention an
extensive Pythagorean literature as dating from the days of the
founder. Modern scholarship has, however, shown that (1) the reputed
writings of Pythagoras are certainly spurious; (2) the fragments
of Philolaus (peri phusi˘s) are for the most part
genuine: it was probably from these that Aristotle derived his
knowledge of the Pythagorean doctrines; Philolaus lived towards the end
of the fifth century; (3) the fragments of Archytas of Tarentum
are spurious, with the exception of a few, which do not add to our
knowledge of the Pythagorean doctrines, as they bear too evident marks
of Platonic influence.
Secondary sources. There is no school the history of which is
so overgrown with legend as the Pythagorean. Indeed, Pythagoras and his
disciples are seldom mentioned by writers anterior to Plato and
Aristotle, and even the latter does not mention Pythagoras more than
once or twice; he speaks rather of the Pythagoreans. Thus, the nearer
we approach the time of Pythagoras the more scanty do our data become,
while the farther the tradition is removed from Pythagoras the fuller
they grow. Obviously, therefore, the Neo-Pythagoreans of the first
century B.C. are not to be relied on when they speak of Pythagoras and
 Cf. Burnet, op. cit., pp. 301 ff.
The Pythagorean School was a society formed for an
ethico-religious purpose. It was governed by a set of rules (ho
tropos tou biou). The members recognized one another by means of
secret signs; simplicity of personal attire and certain restrictions in
matter of diet were required. Celibacy and the strict observance of
secrecy in matters of doctrine were also insisted upon. The political
tendency of the school was towards the aristocratic party in Magna
Graecia, a tendency which led to the persecution and final dispersion of
All that can with certainty be traced to Pythagoras is the doctrine of
metempsychosis, the institution of certain ethical rules, and the germ
idea of the mathematico-theological speculation, which was afterwards
carried to a high degree of development. Consequently, by Pythagorean
doctrines we must understand the doctrines of the disciples of
Pythagoras, though these referred nearly all their doctrines to the
founder. (Indeed, they carried this practice so far that they
constantly introduced a question by quoting the autos epha, the
ipse dixit of the Master.)
The Number Theory. The most distinctive of the Pythagorean
doctrines is the principle that number is the essence and basis
(archŕ) of all things. To this conclusion the Pythagoreans were
led "by contemplating with minds trained to mathematical concepts" the
order of nature and the regularity of natural changes.
 Arist., Met., I, 5, 986 a, 23.
To the question, Did the Pythagoreans regard numbers as the physical
substance of things, or merely as symbols or prototypes? the answer
seems to be that they meant number to stand to things in the double
relation of prototype and substance. And if the assertion, "All is
number," sounds strange to us, we must consider how profound was the
impression produced on the minds of these early students of nature by
the first perception of the unalterable universal order of natural
changes. Then we shall cease to wonder at the readiness with which
number -- the formula of the order and regularity of those changes --
was hypostatized into the substance and basis of all things that
Philolaus (frag. 3) distinguishes three natural kinds of number: odd,
even, and the odd-even. Aristotle  says that the Pythagoreans
considered odd and even to be the elements (stoicheia)  of
number. "Of these," he continues, "the one is definite and the other is
unlimited, and the unit is the product of both, for it is both odd and
even, and number arises from the unit, and the whole heaven is
number."  From the dualism which is thus inherent in the unit, and
consequently in number, comes the doctrine of opposites, finite and
infinite, odd and even, left and right, male and female, and so forth.
From the doctrine of opposites proceeds the notion of harmony, which
plays such an important part in the Pythagorean philosophy, for harmony
is the union of opposites.
 Met., I, 5, 985 b, 24.
 The term was first used in the technical scientific sense by Plato.
 On the Pythagorean concept of the Infinite, cf.Archiv f.
Gesch. der Phil. (April, 1901), Bd. VII, Heft 3.
Application of the Doctrine of Number: 1. To physics.
True to their mathematical concept of the world, the Pythagoreans
analyzed bodies into surfaces, surfaces into lines, and lines into
points. From this, however, we must not conclude that they conceived
the numerical unit of all things as material; they apparently used
numbers and geometrical quantities merely as quantities, abstracting
from their contents, that is, without determining whether the contents
were material or immaterial, a distinction which belongs to a later date.
Every body is an expression of the number four; the surface is three,
because the triangle is the simplest of figures; the line is two,
because of its terminations; and the point is
one. Ten is the perfect number, because it is the sum of the numbers
from one to four.
2. To the theory of music. The application of the number theory
to the arrangement of tones is obvious. The story,  however, of the
discovery of the musical scale by Pythagoras, as told by Iamblichus and
others, is one of many instances in which discoveries made by the
successors of Pythagoras were attributed to Pythagoras himself.
 Cf. Zeller, op. cit., I, 431, n.
3. To cosmology. Not only is each body a number, but the entire
universe is an arrangement of numbers, the basis of which is the
perfect number, ten. For the universe consists of ten bodies, -- the
five planets, the sun, the moon, the heaven of the fixed stars, the
earth, and the counter-earth (antichth˘n). The earth is a
sphere; the counter-earth, which is postulated in order to fill up the
number ten, is also a sphere, and moves parallel to the earth. In the
center of the universe is the central fire, around which the heavenly
bodies, fixed in their spheres, revolve from west to east, while around
all is the peripheral fire. This motion of the heavenly bodies is
regulated as to velocity, and is therefore a harmony. We do not,
however, perceive this harmony of the spheres, either because we are
accustomed to it, or because the sound is too intense to affect our
organs of hearing.
4. To psychology. It would seem that the early Pythagoreans
taught nothing definite regarding the nature of the soul. In the
Phaedo,  Plato introduces into the dialogue a disciple of
Philolaus, who teaches that the soul is a harmony, while Aristotle 
says: "Some of them (the Pythagoreans) say that the soul is identified
with the corpuscles in the air, and others say that it is that which
moves (to kinoun) the corpuscles." The idea, however, that the soul
is a harmony seems to be part of the doctrine of the Pythagoreans. The
transmigration of souls
is, as has been said, traceable to the founder of the school, though
it was probably held as a tradition, being derived from the mysteries
without being scientifically connected with the idea of the soul or
with the number theory.
 Phaedo, 85 E.
 De An., I, 2, 404, a, 26.
5. To theology. The Pythagoreans did not make extensive
application of their number theory to their theological beliefs. They
seem to have conformed, externally at least, to the popular religious
notions, though there are indications of a system of purer religious
concepts which were maintained esoterically.
6. To ethics. The ethical system of the Pythagoreans was
thoroughly religious. The supreme good of man is to become godlike.
This assimilation is to be accomplished by virtue. Now virtue is a
harmony: it essentially consists in a harmonious equilibrium of the
faculties, by which what is lower in man's nature is subordinated to
what is higher. Knowledge, the practice of asceticism, music, and
gymnastics are the means by which this harmony is attained. Finally,
the Pythagoreans used numbers to define ethical notions. Thus, they
said, justice is a number squared, arithmos isakis isos.
Historical Position. The chief importance of the Pythagorean
movement lies in this, that it marks a deepening of the moral
consciousness in Greece. The old-time buoyancy of religious feeling as
seen in the Homeric poems has given way to a calmer and more reflective
mood, in which the sense of guilt and the consequent need of atonement
and purification assert themselves.
As a system of philosophy, the body of Pythagorean doctrine must, like
all the pre-Socratic systems, be regarded as primarily intended to be a
philosophy of nature, and this is how Aristotle describes it.  It is
not concerned with the conditions of knowledge, and although the
society which Pythagoras founded was ethical, the philosophy which is
with that society treats of ethical problems only incidentally and in
a superficial manner.
 Met., I, 8, 989 b, 29.
As an investigation of nature the Pythagorean philosophy must be
pronounced a very decided advance on the speculative attempts of the
Ionians. The Pythagoreans leave the concrete, sense-perceived basis of
existence, and substitute for it the abstract notion of number, thus
preparing the way for a still higher notion -- that of Being.