[
1087a]
[29]
With regard to this kind of
substance,
1 then, let the foregoing account
suffice. All thinkers make the first principles contraries; as in the
realm of natural objects, so too in respect of the unchangeable
substances.Now if
nothing can be prior to the first principle of all things, that first
principle cannot be first principle if it is an attribute of something
else. This would be as absurd as to say that "white" is the first
principle, not qua anything else but qua white, and yet that it is predicable of a
subject, and is white because it is an attribute of something else;
because the latter will be prior to it.Moreover, all things are generated from
contraries as from a substrate, and therefore contraries must most
certainly have a substrate.
[
1087b]
[1]
Therefore all
contraries are predicated of a subject, and none of them exists
separately. But there is no contrary to substance; not only is this
apparent, but it is borne out by reasoned consideration.
2 Thus none of the
contraries is strictly a first principle; the first principle is
something different.
But the Platonists treat one of the
contraries as matter, some opposing "the unequal" to Unity (on the
ground that the former is of the nature of plurality) and others
plurality.For
according to some,
3 numbers
are generated from the unequal dyad of the Great and Small; and
according to another,
4 from plurality; but in both cases they are
generated by the essence of unity. For he who speaks of "the unequal"
and Unity as elements, and describes the unequal as a dyad composed of
Great and Small, speaks of the unequal, i.e. the Great and Small, as
being one; and does not draw the distinction that they are one in
formula but not in number.
5
Again, they state the first principles, which they call elements,
badly; some say that the Great and the Small, together with Unity
(making 3
6 in all), are the elements of numbers; the two former as
matter, and Unity as form. Others speak of the Many and Few, because
the Great and the Small are in their nature more suited to be the
principles of magnitude; and others use the more general term which
covers these—"the exceeding" and "the exceeded."But none of these variations
makes any appreciable difference with respect to some of the
consequences of the theory;
[20]
they only affect the abstract difficulties, which these thinkers
escape because the proofs which they themselves employ are
abstract.There is,
however, this exception: if "the exceeding" and "the exceeded" are the
first principles, and not the Great and the Small, on the same
principle number should be derived from the elements before 2 is
derived; for as "the exceeding and the exceeded" is more universal
than the Great and Small, so number is more universal than 2. But in
point of fact they assert the one and not the other.
Others oppose "the different" or "other" to Unity;
and others contrast Plurality and Unity.Now if, as they maintain, existing things are
derived from contraries, and if there is either no contrary to unity,
or if there is to be any contrary it is plurality; and if the unequal
is contrary to the equal, and the different to the same, and the other
to the thing itself then those who oppose unity to plurality have the
best claim to credibility—but even their theory is
inadequate, because then unity will be few. For plurality is opposed
to paucity, and many to few.
That "unity" denotes a
measure
7 is obvious. And in every case
there is something else which underlies it; e.g., in the scale there
is the quarter-tone; in spatial magnitude the inch or foot or some
similar thing; and in rhythms the foot or syllable. Similarly in the
case of gravity there is some definite weight. Unity is predicated of
all things in the same way;
[
1088a]
[1]
of qualities as a
quality, and of quantities as a quantity.(The measure is indivisible, in the former
case in kind, and in the latter to our senses.) This shows that unity
is not any independent substance. And this is reasonable; because
unity denotes a measure of some plurality, and number denotes a
measured plurality and a plurality of measures. (Hence too it stands
to reason that unity is not a number; for the measure is not measures,
but the measure and unity are starting-points.)The measure must always be something
which applies to all alike; e.g., if the things are horses, the
measure is a horse; if they are men, the measure is a man; and if they
are man, horse and god, the measure will presumably be an animate
being, and the number of them animate beings.If the things are "man," "white" and
"walking," there will scarcely be a number of them, because they all
belong to a subject which is one and the same in number; however,
their number will be a number of genera, or some other such
appellation.
Those
8 who regard the unequal as a
unity, and the dyad as an indeterminate compound of great and small,
hold theories which are very far from being probable or possible. For
these terms represent affections and attributes, rather than
substrates, of numbers and magnitudes—"many" and "few"
applying to number, and "great" and "small" to
magnitude—
[20]
just as odd and even, smooth and rough, straight and crooked, are
attributes.Further,
in addition to this error, "great" and "small" and all other such
terms must be relative. And the relative is of all the categories in
the least degree a definite entity or substance; it is posterior to
quality and quantity. The relative is an affection of quantity, as we
have said, and not its matter; since there is something else distinct
which is the matter both of the relative in general and of its parts
and kinds.There is
nothing great or small, many or few, or in general relative, which is
many or few, great or small, or relative to something else without
having a distinct nature of its own. That the relative is in the
lowest degree a substance and a real thing is shown by the fact that
of it alone
9 there is
neither generation nor destruction nor change in the sense that in
respect of quantity there is increase and decrease, in respect of
quality, alteration, in respect of place, locomotion, and in respect
of substance, absolute generation and destruction.There is no real change in respect of
the relative; for without any change in itself, one term will be now
greater, now smaller or equal, as the other term undergoes
quantitative change.
[
1088b]
[1]
Moreover, the matter of every
thing, and therefore of substance, must be that which is potentially
of that nature; but the relative is neither potentially substance nor
actually.
It is absurd, then, or rather
impossible, to represent non-substance as an element of substance and
prior to it; for all the other categories are posterior to substance.
And further, the elements are not predicated of those things of which
they are elements; yet "many" and "few" are predicated, both
separately and together, of number; and "long" and "short" are
predicated of the line, and the Plane is both broad and
narrow.If, then,
there is a plurality of which one term, viz. "few," is always
predicable, e.g. 2 (for if 2 is many, 1 will be few
10), then there will be an absolute
"many"; e.g., 10 will be many (if there is nothing more than 10
11), or 10,000. How, then, in this
light, can number be derived from Few and Many? Either both ought to
be predicated of it, or neither; but according to this view only one
or the other is predicated.
But we must inquire in general
whether eternal things can be composed of elements. If so, they will
have matter; for everything which consists of elements is
composite.Assuming,
then, that that which consists of anything, whether it has always
existed or it came into being, must come into being <if at
all> out of that of which it consists; and that everything
comes to be that which it comes to be out of that which is it
potentially (for it could not have come to be out of that which was
not potentially such, nor could it have consisted of it); and that the
potential can either be actualized or not;
[20]
then however everlasting number or
anything else which has matter may be, it would be possible for it not
to exist, just as that which is any number of years old is as capable
of not existing as that which is one day old. And if this is so, that
which has existed for so long a time that there is no limit to it may
also not exist.Therefore
things which contain matter cannot be eternal, that is, if that which
is capable of not existing is not eternal, as we have had occasion to
say elsewhere.
12 Now if what we have just been
saying—that no substance is eternal unless it is
actuality—is true universally, and the elements are the
matter of substance, an eternal substance can have no elements of
which, as inherent in it, it consists.
There are some
who, while making the element which acts conjointly with unity the
indeterminate dyad, object to "the unequal," quite reasonably, on the
score of the difficulties which it involves. But they are rid only of
those difficulties
13 which necessarily attend the theory of those who
make the unequal, i.e. the relative, an element; all the difficulties
which are independent of this view must apply to their theories also,
whether it is Ideal or mathematical number that they construct out of
these elements.
There are many causes for their
resorting to these explanations,
[
1089a]
[1]
the chief being that
they visualized the problem in an archaic form. They supposed that all
existing things would be one, absolute Being, unless they encountered
and refuted Parmenides' dictum:
It will
ne'er be proved that things which are not, are,
14i.e., that
they must show that that which is not, is; for only so—of
that which is, and of something else—could existing things
be composed, if they are more than one.
15 However, (i) in the first
place, if "being" has several meanings (for sometimes it means
substance, sometimes quality, sometimes quantity, and so on with the
other categories), what sort of unity will all the things that are
constitute, if not-being is not to be? Will it be the substances that
are one, or the affections (and similarly with the other categories),
or all the categories together? in which case the "this" and the
"such" and the "so great," and all the other categories which denote
some sense of Being, will be one.But it is absurd, or rather impossible, that
the introduction of one thing should account for the fact that "what
is" sometimes means "so-and-so," sometimes "such-and-such," sometimes
"of such-and-such a size," sometimes "in such-and-such a place."
(2) Of what sort of not-being and Being do
real things consist? Not-being, too, has several senses, inasmuch as
Being has; and "not-man" means "not so-and-so," whereas "not straight"
means "not such-and-such," and "not five feet long" means "not of
such-and-such a size." What sort of Being and not-being, then, make
existing things a plurality?
[20]
This thinker
means by the not-being which together with Being makes existing things
a plurality, falsity and everything of this nature
16; and for this reason also it was said
17 that we must assume something which is false, just
as geometricians assume that a line is a foot long when it is
not.But this cannot
be so; for (a) the geometricians do not assume anything that is false
(since the proposition is not part of the logical inference
18), and (b) existing things are not generated from
or resolved into not-being in this sense. But not only has "not-being"
in its various cases as many meanings as there are categories, but
moreover the false and the potential are called "not-being"; and it is
from the latter that generation takes place—man comes to be
from that which is not man but is potentially man, and white from that
which is not white but is potentially white; no matter whether one
thing is generated or many.
Clearly the point at issue is
how "being" in the sense of the substances is many; for the things
that are generated are numbers and lines and bodies. It is absurd to
inquire how Being as substance is many, and not how qualities or
quantities are many.Surely the indeterminate dyad or the Great and Small is no reason
why there should be two whites or many colors or flavors or shapes;
[
1089b]
[1]
for then these too would be numbers and
units. But if the Platonists had pursued this inquiry, they would have
perceived the cause of plurality in substances as well; for the
cause
19 is the same, or analogous.
This deviation of theirs was the reason why in seeking the opposite
of Being and unity, from which in combination with Being and unity
existing things are derived, they posited the relative (i.e. the
unequal), which is neither the contrary nor the negation of Being and
unity, but is a single characteristic of existing things, just like
substance or quality. They should have investigated this question
also; how it is that relations are many, and not one.As it is, they inquire how it
is that there are many units besides the primary unity, but not how
there are many unequal things besides the Unequal. Yet they employ in
their arguments and speak of Great and Small, Many and Few (of which
numbers are composed), Long and Short (of which the line is composed),
Broad and Narrow (of which the plane is composed), Deep and Shallow
(of which solids are composed); and they mention still further kinds
of relation.
20 Now what is the cause of
plurality in these relations?
We must, then, as I say,
presuppose in the case of each thing that which is it potentially. The
author
21 of
this theory further explained what it is that is potentially a
particular thing or substance, but is not per se
existent—that it is the relative (he might as well have said
"quality"); which is neither potentially unity or Being, nor a
negation of unity or Being,
[20]
but just a particular kind of Being. And it was still more necessary, as we have
said,
22
that, if he was inquiring how it is that things are many, he should
not confine his inquiry to things in the same category, and ask how it
is that substances or qualities are many, but that he should ask how
it is that things in general are many; for some things are substances,
some affections, and some relations.Now in the case of the other categories there
is an additional difficulty in discovering how they are many. For it
may be said that since they are not separable, it is because the
substrate becomes or is many that qualities and quantities are many;
yet there must be some matter for each class of entities, only it
cannot be separable from substances.In the case of particular substances, however,
it is explicable how the particular thing can be many, if we do not
regard a thing both as a particular substance and as a certain
characteristic.
23 The real difficulty which arises from
these considerations is how substances are actually many and not
one.
Again, even if a particular
thing and a quantity are not the same, it is not explained how and why
existing things are many, but only how quantities are many;for all number denotes
quantity, and the unit, if it does not mean a measure, means that
which is quantitatively indivisible. If, then, quantity and substance
are different, it is not explained whence or how substance is many;
[
1090a]
[1]
but if they are the same, he who holds
this has to face many logical contradictions.
One might fasten also upon the question with
respect to numbers, whence we should derive the belief that they
exist.For one
24 who posits Ideas, numbers supply a kind of
cause for existing things; that is if each of the numbers is a kind of
Idea, and the Idea is, in some way or other, the cause of existence
for other things; for let us grant them this assumption.But as for him
25 who does not
hold this belief, because he can see the difficulties inherent in the
Ideal theory (and so has not this reason for positing numbers), and
yet posits mathematical number, what grounds have we for believing his
statement that there is a number of this kind, and what good is this
number to other things? He who maintains its existence does not claim
that it is the cause of anything, but regards it as an independent
entity; nor can we observe it to be the cause of anything; for the
theorems of the arithmeticians will all apply equally well to sensible
things, as we have said.
26 Those,
then, who posit the Ideas and identify them with numbers, by their
assumption (in accordance with their method of abstracting each
general term from its several concrete examples) that every general
term is a unity, make some attempt to explain why number exists.
27 Since, however,
their arguments are neither necessarily true nor indeed
possible,
[20]
there is no
justification on this ground for maintaining the existence of
number.The
Pythagoreans, on the other hand, observing that many attributes of
numbers apply to sensible bodies, assumed that real things are
numbers; not that numbers exist separately, but that real things are
composed of numbers.
28 But why? Because the attributes of numbers
are to be found in a musical scale, in the heavens, and in many other
connections.
29 As for those who hold that
mathematical number alone exists,
30 they cannot allege anything of this kind
31 consistently with their hypotheses; what they
did say was that the sciences could not have sensible things as their
objects. But we maintain that they can; as we have said before. And
clearly the objects of mathematics do not exist in separation; for if
they did their attributes would not be present in corporeal
things.Thus in this
respect the Pythagoreans are immune from criticism; but in so far as
they construct natural bodies, which have lightness and weight, out of
numbers which have no weight or lightness, they appear to be treating
of another universe and other bodies, not of sensible ones.
32
But those who treat number as separable assume that it exists and is
separable because the axioms will not apply to sensible objects;
whereas the statements of mathematics are true and appeal to the
soul.
33
[
1090b]
[1]
The same applies to mathematical extended
magnitudes.
It is clear, then, both that the
contrary theory
34 can make
out a case for the contrary view, and that those who hold this theory
must find a solution for the difficulty which was recently raised
35—why
it is that while numbers are in no way present in sensible things,
their attributes are present in sensible things.
There are some
36 who think that, because the point is the limit
and extreme of the line, and the line of the plane, and the plane of
the solid, there must be entities of this kind.We must, then, examine this argument
also, and see whether it is not exceptionally weak. For (1.) extremes
are not substances; rather all such things are merely limits. Even
walking, and motion in general, has some limit; so on the view which
we are criticizing this will be an individual thing, and a kind of
substance. But this is absurd. And moreover (2.) even if they are
substances, they will all be substances of particular sensible things,
since it was to these that the argument applied. Why, then, should
they be separable?
Again, we may, if we are not unduly
acquiescent, further object with regard to all number and mathematical
objects that they contribute nothing to each other, the prior to the
posterior. For if number does not exist, none the less spatial
magnitudes will exist for those who maintain that only the objects of
mathematics exist; and if the latter do not exist, the soul and
sensible bodies will exist.
37 But it does not appear, to judge from the
observed facts, that the natural system lacks cohesion,
[20]
like a poorly constructed drama.
Those
38 who
posit the Ideas escape this difficulty, because they construct spatial
magnitudes out of matter and a number—2 in the case of
lines, and 3, presumably, in that of planes, and 4 in that of solids;
or out of other numbers, for it makes no difference.But are we to regard these
magnitudes as Ideas, or what is their mode of existence? and what
contribution do they make to reality? They contribute nothing; just as
the objects of mathematics contribute nothing. Moreover, no
mathematical theorem applies to them, unless one chooses to interfere
with the principles of mathematics and invent peculiar theories
39 of one's own. But it is not difficult to take any
chance hypotheses and enlarge upon them and draw out a long string of
conclusions.
These thinkers, then, are quite wrong
in thus striving to connect the objects of mathematics with the Ideas.
But those who first recognized two kinds of number, the Ideal and the
mathematical as well, neither have explained nor can explain in any
way how mathematical number will exist and of what it will be
composed; for they make it intermediate between Ideal and sensible
number.For if it is
composed of the Great and Small, it will be the same as the former,
i.e. Ideal, number. But of what other Great and Small can it be
composed? for Plato makes spatial magnitudes out of a Great and
Small.
40
[
1091a]
[1]
And if he speaks of some other component, he will
be maintaining too many elements; while if some one thing is the first
principle of each kind of number, unity will be something common to
these several kinds.We
must inquire how it is that unity is these many things, when at the
same time number, according to him, cannot be derived otherwise than
from unity and an indeterminate dyad.
41All these views
are irrational; they conflict both with one another and with sound
logic, and it seems that in them we have a case of Simonides' "long
story
42"; for men
have recourse to the "long story," such as slaves tell, when they have
nothing satisfactory to say.The very elements too, the Great and Small,
seem to protest at being dragged in; for they cannot possibly generate
numbers except rising powers of 2.
43It
is absurd also, or rather it is one of the impossibilities of this
theory, to introduce generation of things which are eternal.There is no reason to doubt
whether the Pythagoreans do or do not introduce it; for they clearly
state that when the One had been constituted—whether out of
planes or superficies or seed or out of something that they cannot
explain—immediately the nearest part of the Infinite began
to be drawn in and limited by the Limit.
44 However, since they are here
explaining the construction of the universe and meaning to speak in
terms of physics, although we may somewhat criticize their physical
theories,
[20]
it is only
fair to exempt them from the present inquiry; for it is the first
principles in unchangeable things that we are investigating, and
therefore we have to consider the generation of this kind of
numbers.
They
45 say that there is no generation of odd
numbers,
46 which clearly implies that there is generation of
even ones; and some hold that the even is constructed first out of
unequals—the Great and Small—when they are
equalized.
47 Therefore the
inequality must apply to them before they are equalized. If they had
always been equalized they would not have been unequal before; for
there is nothing prior to that which has always been.Hence evidently it is not for
the sake of a logical theory that they introduce the generation of
numbers
A difficulty, and a discredit
to those who make light of the difficulty, arises out of the question
how the elements and first principles are related to the the Good and
the Beautiful. The difficulty is this: whether any of the elements is
such as we mean when we
48 speak
of the Good or the Supreme Good, or whether on the contrary these are
later in generation than the elements.It would seem that there is an agreement
between the mythologists and some present-day thinkers,
49
who deny that there is such an element, and say that it was only after
some evolution in the natural order of things that both the Good and
the Beautiful appeared. They do this to avoid a real difficulty which
confronts those who hold, as some do, that unity is a first principle.
[
1091b]
[1]
This difficulty arises not from ascribing
goodness to the first principle as an attribute, but from treating
unity as a principle, and a principle in the sense of an element, and
then deriving number from unity. The early poets agree with this view
in so far as they assert that it was not the original
forces—such as Night, Heaven, Chaos or Ocean—but
Zeus who was king and ruler.It was, however, on the ground of the changing of the rulers of the
world that the poets were led to state these theories; because those
of them who compromise by not describing everything in mythological
language—e.g. Pherecydes
50 and
certain others—make the primary generator the Supreme Good;
and so do the Magi,
51 and some of the later
philosophers such as Empedocles and Anaxagoras: the one making Love an
element,
52 and the other making
Mind a first principle.
53 And of those who hold that
unchangeable substances exist, some
54 identify absolute unity with absolute
goodness; but they considered that the essence of goodness was
primarily unity.
This, then, is the
problem: which of these two views we should hold.Now it is remarkable if that which is
primary and eternal and supremely self-sufficient does not possess
this very quality, viz. self-sufficiency and immunity, in a primary
degree and as something good. Moreover, it is imperishable and
self-sufficient for no other reason than because it is good.
[20]
Hence it is probably true to say
that the first principle is of this nature. But to say that this principle is unity, or
if not that, that it is an element, and an element of numbers, is
impossible; for this involves a serious difficulty, to avoid which
some thinkers
55 have abandoned the theory (viz.
those who agree that unity is a first principle and element, but of
mathematical number). For on this view all units
become identical with some good, and we get a great abundance of
goods.
56 Further, if the Forms are
numbers, all Forms become identical with some good. Again, let us
assume that there are Ideas of anything that we choose. If there are
Ideas only of goods, the Ideas will not be substances
57; and if there are Ideas of
substances also, all animals and plants, and all things that
participate in the Ideas, will be goods.
58 Not only do these absurdities follow, but it also follows that the
contrary element, whether it is plurality or the unequal, i.e. the
Great and Small, is absolute badness. Hence one thinker
59 avoided
associating the Good with unity, on the ground that since generation
proceeds from contraries, the nature of plurality would then
necessarily be bad.Others
60 hold that inequality is the nature of the
bad. It follows, then, that all things partake of the Bad except
one—absolute unity; and that numbers partake of it in a more
unmitigated form than do spatial magnitudes
61;
[
1092a]
[1]
and that the Bad is
the province for the activity of the Good, and partakes of and tends
towards that which is destructive of the Good; for a contrary is
destructive of its contrary.And if, as we said,
62
the matter of each thing is that which is it
potentially—e.g., the matter of actual fire is that which is
potentially fire—then the Bad will be simply the potentially
Good.
Thus all these objections
follow because (1.) they make every principle an element; (2.) they
make contraries principles; (3.) they make unity a principle; and (4.)
they make numbers the primary substances, and separable, and
Forms.
If, then, it is impossible both not to
include the Good among the first principles, and to include it in this
way, it is clear that the first principles are not being rightly
represented, nor are the primary substances. Nor is a certain
thinker
63
right in his assumption when he likens the principles of the universe
to that of animals and plants, on the ground that the more perfect
forms are always produced from those which are indeterminate and
imperfect, and is led by this to assert that this is true also of the
ultimate principles; so that not even unity itself is a real
thing.
64 He
is wrong; for even in the natural world the principles from which
these things are derived are perfect and complete—for it is
man that begets man; the seed does not come first.
65 It is absurd also to generate space
simultaneously with the mathematical solids (for space is peculiar to
particular things, which is why they are separable in space, whereas
the objects of mathematics have no position)
[20]
and to say that they must be somewhere,
and yet not explain what their spatial position is.
Those
who assert that reality is derived from elements, and that numbers are
the primary realities, ought to have first distinguished the senses in
which one thing is derived from another, and then explained in what
way number is derived from the first principles. Is it by mixture? But
(a) not everything admits of mixture
66; (b) the result of mixture is something
different; and unity will not be separable,
67 nor will it be a distinct entity, as
they intend it to be.Is it
by composition, as we hold of the syllable? But (a) this necessarily
implies position; (b) in thinking of unity and plurality we shall
think of them separately. This, then, is what number will
be—a unit
plus plurality, or unity
plus the Unequal.
And since
a thing is derived from elements either as inherent or as not inherent
in it, in which way is number so derived? Derivation from inherent
elements is only possible for things which admit of generation.
68 Is it derived as from seed?But nothing can be emitted from that
which is indivisible.
69 Is it derived from a contrary which does not persist? But
all things which derive their being in this way derive it also from
something else which does persist. Since, therefore, one thinker
70 regards unity as
contrary to plurality,
[
1092b]
[1]
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.
71Further, 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,
72 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.
73 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,
74 e.g. as points are the bounds of
spatial magnitudes,
75 and as
Eurytus
76 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.
77 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?
78 And clearly
numbers are not the essence of things, nor are they causes of the
form; for the ratio
79 is the essence, and number
80 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.
81
And the number,
[20]
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,
82 or
an odd number.
83 For in point of fact
honey-water is no more wholesome if it is mixed in the proportion
"three times three"
84; 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 2
85; 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.
86
[
1093a]
[1]
If all things must share in number, it
must follow that many things are the same; i.e., that the same number
belongs both to this thing and to something else. Is number, then, a
cause; i.e., is it because of number that the object exists? Or is
this not conclusive? E.g., there is a certain number of the sun's
motions, and again of the moon's,
87 and indeed
of the life and maturity of every animate thing. What reason, then, is
there why some of these numbers should not be squares and others
cubes, some equal and others double?There is no reason; all things must fall
within this range of numbers if, as was assumed, all things share in
number, and different things may fall under the same number. Hence if
certain things happened to have the same number, on the Pythagorean
view they would be the same as one another, because they would have
the same form of number; e.g., sun and moon would be the same.
88 But why
are these numbers causes? There are seven vowels,
89
seven strings to the scale,
90 seven Pleiads; most animals (though not
all
91) lose their teeth in the seventh year; and there were
seven heroes who attacked
Thebes. Is it, then, because the number 7 is such as it
is that there were seven heroes, or that the Pleiads consist of seven
stars? Surely there were seven heroes because of the seven gates, or
for some other reason, and the Pleiads are seven because we count them
so; just as we count the Bear as 12, whereas others count more stars
in both.
[20]
Indeed, they assert also that
Ξ, Ψ and
Ζ are concords,
92 and that because there are three concords, there
are three double consonants. They ignore the fact that there might be
thousands of double consonants—because there might be one
symbol for
ΓΡ. But if they say that
each of these letters is double any of the others, whereas no other
is,
93 and that the reason is
that there are three regions
94 of the mouth, and that one
consonant is combined with
ς in each
region, it is for this reason that there are only three double
consonants, and not because there are three concords—because
there are really more than three; but there cannot be more than three
double consonants.
Thus these thinkers are like the
ancient Homeric scholars, who see minor similarities but overlook
important ones.
Some say that there are
many correspondences of this kind; e.g., the middle notes
95 of the octave are respectively 8 and 9,
and the epic hexameter has seventeen syllables, which equals the sum
of these two;
[
1093b]
[1]
and the line scans in the first half
with nine syllables, and in the second with eight.
96 And they point out that the
interval from
α to
ω in the alphabet is equal to that from
the lowest note of a flute to the highest, whose number is equal to
that of the whole system of the universe.
97 We
must realize that no one would find any difficulty either in
discovering or in stating such correspondences as these in the realm
of eternal things, since they occur even among perishable
things.
As for the celebrated characteristics of
number, and their contraries, and in general the mathematical
properties, in the sense that some describe them and make them out to
be causes of the natural world, it would seem that if we examine them
along these lines, they disappear; for not one of them is a cause in
any of the senses which we distinguished with until respect to the
first Principles.
98
There is a sense,
however, in which these thinkers make it clear that goodness is
predicable of numbers, and that the odd, the straight, the
equal-by-equal,
99 and the powers
100 of
certain numbers, belong to the series of the Beautiful.
101 For the seasons are connected with a
certain kind of number
102; and the other examples which they adduce from mathematical
theorems all have the same force.Hence they would seem to be mere coincidences,
for they are accidental; but all the examples are appropriate to each
other, and they are one by analogy. For there is analogy between all
the categories of Being—as "straight" is in
length,
[20]
so is "level"
in breadth, perhaps "odd" in number, and "white" in color.
Again, it is not the Ideal numbers that are the causes of harmonic
relations, etc. (for Ideal numbers, even when they are equal, differ
in kind, since their units also differ in kind)
103; so on this ground at least we need not posit
Forms.
Such, then, are the consequences of the
theory, and even more might be adduced. But the mere fact that the
Platonists find so much trouble with regard to the generation of Ideal
numbers, and can in no way build up a system, would seem to be a proof
that the objects of mathematics are not separable from sensible
things, as some maintain, and that the first principles are not those
which these thinkers assume.