(Anaxagoras, Empedokles, Atomists) those called Pythagoreans applying
themselves to the sciences, first developed them; and being brought up
in them they thought that the first principles of these (_i.e._ numbers)
were the first principles of all things. And since of these (sciences)
numbers are by nature the first, in numbers rather than in fire and
earth and water they thought they saw many likenesses to things that are
and that are coming to be, as, for instance, justice is such a property
of numbers, and soul and mind are such a property, and another is
opportunity, and of other things one may say the same of each one.
†And further, discerning in numbers the conditions and reasons of
harmonies also†; since, moreover, other things seemed to be like
numbers in their entire nature, and numbers were the first of every
nature, they assumed that the elements of numbers were the elements of
all things, and that the whole heavens were harmony and number. And
whatever characteristics in numbers and harmonies they could show were
in agreement with the properties of the heavens and its parts and with
its whole arrangement, these they collected and adapted; and if there
chanced to be any gap anywhere, they eagerly sought that the whole system
might be connected with these (stray phenomena). To give an example of my
meaning: inasmuch as ten seemed to be the perfect number and to embrace
the whole nature of numbers, they asserted that the number of bodies
moving through the heavens were ten, and when only nine were visible, for
the reason just stated they postulated the counter-earth as the tenth.
We have given a more definite account of these thinkers in other parts
of our writings. But we have referred to them here with this purpose in
view, that we might ascertain from them what they asserted as the first
principles and in what manner they came upon the causes that have been
enumerated. They certainly seem to consider number as the first principle
and as it were the matter in things and in their conditions and states;
and the odd and the even are elements of number, and of these the one is
infinite and the other finite, and unity is the product of both of them,
for it is both odd and even, and number arises from unity, and the whole
heaven, as has been said, is numbers.
A different party in this same school say that the first principles
are ten, named according to the following table:—finite and infinite,
even and odd, one and many, right and left, male and female, rest and
motion, straight and crooked, light and darkness, good and bad, square
and oblong. After this manner Alkmaeon of Kroton seems to have conceived
them, and either he received this doctrine from them or they from him;
for Alkmaeon arrived at maturity when Pythagoras was an old man, and
his teachings resembled theirs. For he says that most human affairs
are twofold, not meaning opposites reached by definition, as did the
former party, but opposites by chance—as, for example, white-black,
sweet-bitter, good-bad, small-great. This philosopher let fall his
opinions indefinitely about the rest, but the Pythagoreans declared the
number of the opposites and what they were. From both one may learn
this much, that opposites are the first principles of things; but from
the latter he may learn the number of these, and what they are. But how
it is possible to bring them into relation with the causes of which we
have spoken if they have not clearly worked out; but they seem to range
their elements under the category of matter, for they say that being is
compounded and formed from them, and that they inhere in it.
987 a 9-27. Down to the Italian philosophers and with the exception of
them the rest have spoken more reasonably about these principles, except
that, as we said, they do indeed use two principles, and the one of
these, whence is motion, some regard as one and others as twofold. The
Pythagoreans, however, while they in similar manner assume two first
principles, add this which is peculiar to themselves: that they do not
think that the finite and the infinite and the one are certain other
things by nature, such as fire or earth or any other such thing, but the
infinite itself and unity itself are the essence of the things of which
they are predicated, and so they make number the essence of all things.
So they taught after this manner about them, and began to discourse and
to define what being is, but they made it altogether too simple a matter.
For they made their definitions superficially, and to whatever first
the definition might apply, this they thought to be the essence of the
matter; as if one should say that twofold and two were the same, because
the twofold subsists in the two. But undoubtedly the two and the twofold
are not the same; otherwise the one will be many—a consequence which
even they would not draw. So much then may be learned from the earlier
philosophers and from their successors.
i. 6; 987 b 10. And Plato only changed the name, for the Pythagoreans
say that things exist by imitation of numbers, but Plato, by sharing the
nature of numbers.
i. 6; 987 b 22. But that the one is the real essence of things, and not
something else with unity as an attribute, he affirms, agreeing with the
Pythagoreans; and in harmony with them he affirms that numbers are the
principles of being for other things. But it is peculiar to him that
instead of a single infinite he posits a double infinite, an infinite
of greatness and of littleness; and it is also peculiar to him that he
separates numbers from things that are seen, while they say that numbers
are the things themselves, and do not interpose mathematical objects
between them. This separation of the one and numbers from things, in
contrast with the position of the Pythagoreans, and the introduction of
ideas, are the consequence of his investigation by concepts.
i. 8; 989 b 32-990 a 32. Those, however, who carry on their investigation
with reference to all things, and divide things into what are perceived
and what are not perceived by sense, evidently examine both classes, so
one must delay a little longer over what they say. They speak correctly
and incorrectly in reference to the questions now before us. Now those
who are called Pythagoreans use principles and elements yet stranger than
those of the physicists, in that they do not take them from the sphere of
sense, for mathematical objects are without motion, except in the case of
astronomy. Still, they discourse about everything in nature and study it;
they construct the heaven, they observe what happens in its parts †and
their states and motions†; they apply to these their first principles and
causes, as though they agreed entirely with the other physicists that
being is only what is perceptible and what that which is called heaven
includes. But their causes and first principles, they say, are such as
to lead up to the higher parts of reality, and are in harmony with this
rather than with the doctrines of nature. In what manner motion will take
place when finite and infinite, odd and even, are the only underlying
realities, they do not say; nor how it is possible for genesis and
destruction to take place without motion and change, or for the heavenly
bodies to revolve. Farther, if one grant to them that greatness arises
from these principles, or if this could be proved, nevertheless, how will
it be that some bodies are light and some heavy? For their postulates
and statements apply no more to mathematical objects than to things of
sense; accordingly they have said nothing at all about fire or earth
or any such objects, because I think they have no distinctive doctrine
about things of sense. Farther, how is it necessary to assume that number
and states of number are the causes of what is in the heavens and what
is taking place there from the beginning and now, and that there is no
other number than that out of which the world is composed? For when
opinion and opportune time are at a certain point in the heavens, and
a little farther up or down are injustice and judgment or a mixture of
them, and they bring forward as proof that each one of these is number,
and the result then is that at this place there is already a multitude
of compounded quantities because those states of number have each their
place—is this number in heaven the same which it is necessary to assume
that each of these things is, or is it something different? Plato says it
is different; still, he thinks that both these things and the causes of
them are numbers; but the one class are ideal causes, and the others are
sense causes.
ii. 1; 996 a 4. And the most difficult and perplexing question of all
is whether unity and being are not, as Plato and the Pythagoreans say,
something different from things but their very essence, or whether the
underlying substance is something different, friendship, as Empedokles
says, or as another says, fire, or water, or air.
ii. 4; 1001 a 9. Plato and the Pythagoreans assert that neither being nor
yet unity is something different from things, but that it is the very
nature of them, as though essence itself consisted in unity and existence.
1036 b 17. So it turns out that many things of which the forms appear
different have one form, as the Pythagoreans discovered; and one can say
that there is one form for everything, and the others are not forms; and
thus all things will be one.
ix. 2; 1053 b 11. Whether the one itself is a sort of essence, as first
the Pythagoreans and later Plato affirmed....
xi. 7; 1072 b 31. And they are wrong who assume, as do the Pythagoreans
and Speusippos, that the most beautiful and the best is not in the first
principle, because the first principles of plants and animals are indeed
causes; for that which is beautiful and perfect is in what comes from
these first principles.
xii. 4; 1078 b 21. The Pythagoreans (before Demokritos) only defined
a few things, the concepts of which they reduced to numbers, as for
instance opportunity or justice or marriage....
xii. 6; 1080 b 16. The Pythagoreans say that there is but one number,
the mathematical, but things of sense are not separated from this,
for they are composed of it; indeed, they construct the whole heaven
out of numbers, but not out of unit numbers, for they assume that the
unities have quantity; but how the first unity was so constituted as
to have quantity, they seem at a loss to say. b 31. All, as many as
regard the one as the element and first principle of things, except
the Pythagoreans, assert that numbers are based on the unit; but the
Pythagoreans assert, as has been remarked, that numbers have quantity.
xii. 8; 1083 b 9. The Pythagorean standpoint has on the one hand
fewer difficulties than those that have been discussed, but it has
new difficulties of its own. The fact that they do not regard number
as separate, removes many of the contradictions; but it is impossible
that bodies should consist of numbers, and that this number should be
mathematical. Nor is it true that indivisible elements have quantity;
but, granted that they have this quality of indivisibility, the units
have no quantity; for how can quantity be composed of indivisible
elements? but arithmetical number consists of units. But these say that
things are number; at least, they adapt their speculations to such bodies
as consist of elements which are numbers.
xiii. 3; 1090 a 20. On the other hand the Pythagoreans, because they see
many qualities of numbers in bodies perceived by sense, regard objects as
numbers, not as separate numbers, but as derived from numbers. And why?
Because the qualities of numbers exist in harmony both in the heaven and
in many other things. But for those who hold that number is mathematical
only, it is impossible on the basis of their hypothesis to say any such
thing; and it has already been remarked that there can be no science of
these numbers. But we say, as above, that there is a science of numbers.
Evidently the mathematical does not exist apart by itself, for in that
case its qualities could not exist in bodies. In such a matter the
Pythagoreans are restrained by nothing; when, however, they construct out
of numbers physical bodies—out of numbers that have neither weight nor
lightness, bodies that have weight and lightness—they seem to be speaking
about another heaven and other bodies than those perceived by sense.
_Eth._ i. 4; 1096 b 5. And the Pythagoreans seem to speak more
persuasively about it, putting the unity in the co-ordination of good
things.
ii. 5; 1106 b 29. The evil partakes of the nature of the infinite, the
good of the finite, as the Pythagoreans conjectured.
v. 8; 1132 b 21. Reciprocity seems to some to be absolutely just, as the
Pythagoreans say; for these defined the just as that which is reciprocal
to another.
_Mor._ i. 1; 1118 a 11. First Pythagoras attempted to speak concerning
virtue, but he did not speak correctly; for bringing virtues into
correspondence with numbers, he did not make any distinct.
PYTHAGORAS AND THE PYTHAGOREANS: PASSAGES IN THE DOXOGRAPHISTS.
Aet. _Plac._ i. 3; _Dox._ 280. And again from another starting-point,
Pythagoras, son of Mnesarchos, a Samian, who was the first to call this
matter by the name of philosophy, assumed as first principles the numbers
and the symmetries existing in them, which he calls harmonies, and the
elements compounded of both, that are called geometrical. And again he
includes the monad and the undefined dyad among the first principles;
and for him one of the first principles tends toward the creative and
form-giving cause, which is intelligence, that is god, and the other
tends toward the passive and material cause, which is the visible
universe. And he says that the starting-point of number is the decad;
for all Greeks and all barbarians count as far as ten, and when they get
as far as this they return to the monad. And again, he says, the power
of the ten is in the four and the tetrad. And the reason is this: if any
one †returning† from the monad adds the numbers in a series as far as the
four, he will fill out the number ten (_i.e._ 1 + 2 + 3 + 4 = 10); but if
he goes beyond the number of the tetrad, he will exceed the ten. Just as
if one should add one and two and should add to these three and four, he
will fill out the number ten; so that according to the monad number is
in the ten, but potentially in the four. Wherefore the Pythagoreans were
wont to speak as though the greatest oath were the tetrad: ‘By him that
transmitted to our soul the tetraktys, which has the spring and root of
ever-flowing nature.’ And our soul, he says, is composed of the tetrad;
for it is intelligence, understanding, opinion, sense, from which things
come every art and science, and we ourselves become reasoning beings.
The monad, however, is intelligence, for intelligence sees according
to the monad. As for example, men are made up of many parts, and part
by part they are devoid of sense and comprehension and experience, yet
we perceive that man as one alone, whom no being resembles, possesses
these qualities; and we perceive that a horse is one, but part by part
it is without experience. For these are all forms and classes according
to monads. Wherefore, assigning this limit with reference to each one
of these, they speak of a reasoning being and a neighing being. On
this account then the monad is intelligence by which we perceive these
things. And the undefined dyad is science; fittingly, for all proof and
all persuasion is part of science, and farther every syllogism brings
together what is questioned out of some things that are agreed upon, and
easily proves something else; and science is the comprehension of these
things, wherefore it would be the dyad. And opinion as the result of
comprehending them is the triad; fittingly, for opinion has to do with
many things; and the triad is quantity, as ‘The thrice-blessed Danaoi.’
On this account then he includes the triad.... And their sect is called
Italic because Pythagoras taught in Italy, for he removed from Samos, his
fatherland, because of dissatisfaction with the tyranny of Polykrates.
Aet. i. 7; _Dox._ 302. 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 divinity and the
bad, surrounding which is the mass of matter. i. 8; 307. Divine spirits
[δαίμονες] are psychical beings; and heroes are souls separated from
bodies, good heroes are good souls, bad heroes are bad souls. i. 9;
307. The followers of Thales and Pythagoras and the Stoics held that
matter is variable and changeable and transformable and in a state of
flux, the whole through the whole. i. 10; 309. Pythagoras asserted that
the so-called forms and ideas exist in numbers and their harmonies, and
in what are called geometrical objects, apart from bodies. i. 11; 310.
Pythagoras and Aristotle asserted that the first causes are immaterial,
but that other causes involve a union or contact with material substance
[so that the world is material]. i. 14; 312. The followers of Pythagoras
held that the universe is a sphere according to the form of the four
elements; but the highest fire alone is conical. i. 15; 314. The
Pythagoreans call colour the manifestation of matter. i. 16; 314. Bodies
are subject to change of condition, and are divisible to infinity. i. 18;
316. (After quotation from Arist. _Phys._ iv. 4; 212 a 20) And in his
first book on the philosophy of Pythagoras he writes that the heaven is
one, and that time and wind and the void which always defines the places
of each thing, are introduced from the infinite. And among other things
he says that place is the immovable limit of what surrounds the world,
or that in which bodies abide and are moved; and that it is full when it
surrounds body on every side, and empty when it has absolutely nothing in
itself. Accordingly it is necessary for place to exist, and body; and it
is never empty except only from the standpoint of thought, for the nature
of it in perpetuity is destructive of the interrelation of things and of
the combination of bodies; and motions arise according to place of bodies
that surround and oppose each other; and no infiniteness is lacking,
either of quantity or of extent. i. 20; 318. Pythagoras said that time is
the sphere of what surrounds the world. i. 21; 318. Pythagoras, Plato:
Motion is a certain otherness or difference in matter. [This is the
common limit of all motion.] i. 24; 320. Pythagoras and all that assume
that matter is subject to change assert that genesis and destruction
in an absolute sense take place; for from change of the elements and
modification and separation of them there take place juxtaposition and
mixture, and intermingling and melting together.
Aet. _Plac._ ii. 1; 327. Pythagoras first named the circumference of all
things the universe by reason of the order in it. ii. 4; 330. Pythagoras,
Plato, and the Stoics held that the universe is brought into being by
god. And it is perishable so far as its nature is concerned, for it
is perceived by sense, and therefore material; it will not however be
destroyed in accordance with the foreknowledge and plan of god. ii. 6;
334. Pythagoras: The universe is made from five solid figures, which are
called also mathematical; of these he says that earth has arisen from
the cube, fire from the pyramid, air from the octahedron, and water from
the icosahedron, and the sphere of the all from the dodecahedron. ii.
9; 338. The followers of Pythagoras hold that there is a void outside
the universe into which the universe breathes forth, and from which it
breathes in. ii. 10; 339. Pythagoras, Plato, Aristotle: The right hand
side of the universe is the eastern part from which comes the beginning
of motion, and the left hand side is the west. They say the universe has
neither height nor depth, in which statement height means distance from
below upwards, and depth from above downwards. For none of the distances
thus described exist for the universe, inasmuch as it is disposed around
the middle of itself, from which it extends toward the all, and with
reference to which it is the same on every side. ii. 12; 340. Thales,
Pythagoras, and their followers: The sphere of the whole heaven is
divided into five circles, which they call zones; the first of these
is called the arctic zone and is ever visible; the second the summer
solstice, the third the equinoctial, the fourth the winter solstice, and
fifth the antarctic zone, which is invisible. And the ecliptic called
the zodiac in the three middle ones is projected to touch the three
middle ones. And the meridian crosses all these from the north to the
opposite quarter at right angles. It is said that Pythagoras was the
first to recognise the slant of the zodiacal circle which Oenopides of
Chios appropriated as his own discovery. ii. 13; 343. Herakleides and
the Pythagoreans asserted that each world [κόσμος] of the stars is air
and aether surrounding earth in the infinite aether. And these doctrines
are brought out in the Orphic writings, for they construct each world
of the stars. ii. 22; 352. The Pythagoreans: The sun is spherical. ii.
23; 353. Plato, Pythagoras, Aristotle: The solstices lie along the slant
of the zodiacal circle, through which the sun goes along the zodiac,
and with the accompaniment of the tropic circles; and all these things
also the globe shows. ii. 24; 354. An eclipse takes place when the moon
comes past. ii. 25; 357. Pythagoras: The moon is a mirror-like body. i.
29; 360. Some of the Pythagoreans (according to the Aristotelian account
and the statement of Philip the Opuntian) said that an eclipse of the
moon takes place, sometimes by the interposition of the earth, sometimes
by the interposition of the counter-earth [ἀντίχθων]. But it seems to
some more recent thinkers that it takes place by a spreading of the
flame little by little as it is gradually kindled, until it gives the
complete full moon, and again, in like manner, it grows less until the
conjunction, when it is completely extinguished. ii. 30; 361. Some of
the Pythagoreans, among them Philolaos, said that the earthy appearance
of the moon is due to its being inhabited by animals and by plants, like
those on our earth, only greater and more beautiful; for the animals on
it are fifteen times as powerful, not having any sort of excrement, and
their day is fifteen times as long as ours. But others said that the
outward appearance in the moon is a reflection on the other side of the
inflamed circle of the sea that is on our earth. ii. 32; 364. Some regard
the greater year ... as the sixty year period, among whom are Oenopides
and Pythagoras.
Aet. _Plac._ iii. 1; _Dox._ 364. Some of the Pythagoreans said that the
milky way is the burning of a star that fell from its own foundation,
setting on fire the region through which it passed in a circle, as
Phaethon was burned. And others say that the course of the sun arose in
this manner at the first. And certain ones say that the appearance of the
sun is like a mirror reflecting its rays toward the heaven, and therefore
it happens at times to reflect its rays on the rainbow in the clouds.
Aet. iii. 2; 366. Some of the followers of Pythagoras say that the comet
is one of the stars that are not always shining, but emit their light
periodically through a certain definite time; but others say that it is
the reflection of our vision into the sun, like reflected images. iii.
14; 378. Pythagoras: The earth, after the analogy of the sphere of the
all, is divided into five zones, arctic, antarctic, summer, winter, and
equinoctial; of these the middle one he defines to be the middle of the
earth, called for this very reason the torrid zone; but the inhabited one
[the one between the arctic and the torrid zones] being well-tempered....