1.6: 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 mathe- matical 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 b82-990a 82. 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
140 the First Philosophers of Greece
one must delay alittle longer over what they say. They speak correctly and incorrectly in reference to the ques- tions now before us. Now those who are called Pytha- goreans 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 fand their states and motionst ; 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; accord- ingly they have said nothing at all about fire or earth or any such objects, because I think they have no dis- tinctive 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,
Pythagoras and the Pythagoreans 14]
and a little farther up or down are injustice and judg- ment 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.
li. 1; 996a 4. And the most difficult and perplexing οι 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 un- derlying substance is something different, friendship, as Empedokles says, or as another says, fire, or water, or air.
ii. 4; 1001a9. 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.
1086b17. Soitturns out that many things of which
the forms appear different have one form, as the Pytha-
goreans 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; 1058b11. Whether the one itself is a sort of
essence, as first the Pythagoreans and later Plato
affirmed...
xi. 7; 1072b81. 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.
142 the First Philosophers of Greece
xii. 4; 1078b21. The Pythagoreans (before Demo-
kritos) only defined a few things, the concepts of which
they reduced to numbers, as for instance opportunity or
justice or marriage...
xii.6; 1080b16. The Pythagoreans say that there
is but one number, the mathematical, but things of
sense are not separated from thjs, for they are com-
posed 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 8 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; 1068689509. 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 ean 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; 1090a 20. On the other hand the Pytha-
goreans, 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.
τ And why ? Because the qualities of numbers exist in
Pythagoras and the Pythagoreans 143
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. Ετί- dently 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 light-
ness—they seem to be speaking about another heaven
and other bodies than those perceived by sense.
Eth.i. 4; 1096b5. And the Pythagoreans seem to
speak more persuasively about it, putting the unity in
the co-ordination of good things.
11.5; 1106b29. The evil partakes of the nature of
the infinite, the good of the finite, as the Pythagoreans ~
conjectured.
vy. 8; 1182b21. 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; 118211. 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. 8; 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
144 the First Philosopiiers of Greece
the symmetries existing in them, which he calls har- monies, 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 treturningt 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+8+44=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.
| Pythagoras and the Pythagoreans 145
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. 902. 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; 807. Divine spirits [δαίμονες] are psychical beings; and heroes are als} separated from bodies, good heroes are good souls, bad heroes are bad souls. i. 9; 807. 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. 1.10; 809. 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. 1.11; 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