and another (treating it as the
Equal) as contrary to the Unequal, number must be derived as from
contraries.Hence
there is something else which persists from which, together with one
contrary, number is or has been derived.The objection is directed against the Platonist
treatment of the principles as contraries (cf. Aristot. Met. 14.4.12), and may be illustrated by
Aristot. Met. 12.1.5-2.2. Plurality, as
the contrary of unity, is privation, not matter; the Platonists
should have derived numbers from unity and some other principle
which is truly material.Further, why on earth is it that whereas all other things which are
derived from contraries or have contraries perish, even if the
contrary is exhausted in producing them,Because it may be regarded as still potentially
present. number does not perish? Of this no explanation
is given; yet whether it is inherent or not, a contrary is
destructive; e.g., Strife destroys the mixture.According to Empedocles Fr.
17 (Diels). It should not, however, do this;
because the mixture is not its contrary. Nor is it in any
way defined in which sense numbers are the causes of substances and of
Being; whether as bounds,The
theories criticized from this point onwards to Aristot. Met. 14.6.11 are primarily Pythagorean.
See Introduction. e.g. as points are the bounds of
spatial magnitudes,e.g. the
line by 2 points, the triangle (the simplest plane figure) by 3,
the tetrahedron (the simplest solid figure) by 4. and as
EurytusDisciple of
Philolaus; he "flourished" in the early fourth century
B.C. determined which number belongs to which
thing—e.g. this number to man, and this to
horse—by using pebbles to copy the shape of natural objects,
like those who arrange numbers in the form of geometrical figures, the
triangle and the square.cf.
Burnet, E.G.P. sect. 47. Or is it because harmony is a ratio of
numbers, and so too is man and everything else? But in what sense are
attributes—white, and sweet, and hot—numbers?This is an objection to the
view that numbers are causes as bounds. And clearly
numbers are not the essence of things, nor are they causes of the
form; for the ratioOr
"formula." is the essence, and numberIn the sense of a number of
material particles. is matter.E.g. the essence of flesh or bone is number
only in the sense that it is three parts of fire and two of
earth.Cf. Empedocles Fr. 96 (Diels).
And the number,whatever it
is, is always a number of something; of particles of fire or earth, or
of units. But the essence is the proportion of one quantity to another
in the mixture; i.e. no longer a number, but a ratio of the mixture of
numbers, either of corporeal particles or of any other kind. Thus
number is not an efficient cause—neither number in general,
nor that which consists of abstract units—nor is it the
matter, nor the formula or form of things. Nor again is it a final
cause. The question might also be raised as to what
the good is which things derive from numbers because their mixture can
be expressed by a number, either one which is easily calculable,i.e., a simple ratio. or
an odd number.It is hard to see
exactly what this means. If the terms of a ratio are rational,
one of them must be odd. Alexander says a ratio like 1 : 3 is
meant. Oddness was associated with goodness (cf. Aristot. Met. 1.5.6). For in point of fact
honey-water is no more wholesome if it is mixed in the proportion
"three times three"Apparently
the Pythagoreans meant by this "three parts of water to three of
honey." Aristotle goes on to criticize this way of expressing
ratios.; it would be more beneficial mixed in no
particular proportion, provided that it be diluted, than mixed in an
arithmetical proportion, but strong.Again, the ratios of mixtures are expressed by
the relation of numbers, and not simply by numbers; e.g., it is 3 : 2,
not 3 X 2Cf. previous
note.; for in products of multiplication the units must belong
to the same genus. Thus the product of 1 x 2 x 3 must be measurable by
1, and the product of 4 X 5 x 7 by 4. Therefore all products which
contain the same factor must be measurable by that factor. Hence the
number of fire cannot be 2 X 5 X 3 X 7 if the number of water is 2 x
3.sc. because if so, a
particle of fire would simply equal 35 particles of water.