Book
ΧΙV
1
Regarding this kind of
substance, what we have said must be taken as sufficient. All
philosophers make the first principles contraries: as in natural things,
so also in the case of unchangeable substances.
But since there cannot be anything prior to the first principle of all
things, the principle cannot be the principle and yet be an attribute of
something else. To suggest this is like saying that the white is a first
principle, not qua anything else but qua white, but yet that it is
predicable of a subject, i.e. that its being white presupposes its being
something else; this is absurd, for then that subject will be prior. But
all things which are generated from their contraries involve an
underlying subject; a subject, then, must be present in the case of
contraries, if anywhere.
All contraries, then, are always predicable of a subject, and none can
exist apart, but just as appearances suggest that there is nothing
contrary to substance, argument confirms this. No contrary, then, is the
first principle of all things in the full sense; the first principle is
something different.
But these thinkers make one of the contraries matter,
some making the unequal which they take to be the essence of
plurality-matter for the One, and others making plurality matter for the
One. (The former generate numbers out of the dyad of the unequal, i.e.
of the great and small, and the other thinker we have referred to
generates them out of plurality, while according to both it is generated
by the essence of the One.) For even the philosopher who says the
unequal and the One are the elements, and the unequal is a dyad composed
of the great and small, treats the unequal, or the great and the small,
as being one, and does not draw the distinction that they are one in
definition, but not in number. But they do not describe rightly even the
principles which they call elements, for some name the great and the
small with the One and treat
these three as elements of numbers, two being matter, one the form;
while others name the many and few, because the great and the small are
more appropriate in their nature to magnitude than to number; and others
name rather the universal character common to these-'that which exceeds
and that which is exceeded'. None of these varieties of opinion makes
any difference to speak of, in view of some of the consequences; they
affect only the abstract objections, which these thinkers take care to
avoid because the demonstrations they themselves offer are
abstract,-with this exception, that if the exceeding and the exceeded
are the principles, and not the great and the small, consistency
requires that number should come from the elements before does; for
number is more universal than as the exceeding and the exceeded are more
universal than the great and the small. But as it is, they say one of
these things but do not say the other. Others oppose the different and
the other to the One, and others oppose plurality to the One.
But if, as they claim, things consist of contraries, and to the One
either there is nothing contrary, or if there is to be anything it is
plurality, and the unequal is contrary to the equal, and the different
to the same, and the other to the thing itself, those who oppose the One
to plurality have most claim to plausibility, but even their view is
inadequate, for the One would on their view be a few; for plurality is
opposed to fewness, and the many to the few.
'The one' evidently means a measure. And in every
case there is some underlying thing with a distinct nature of its own,
e.g. in the scale a quarter-tone, in spatial magnitude a finger or a
foot or something of the sort, in rhythms a beat or a syllable; and
similarly in gravity it is a definite weight; and in the same way in all
cases, in qualities a quality, in quantities a quantity (and the measure
is indivisible, in the former case in kind, and in the latter to the
sense); which implies that the one is not in itself the substance of
anything. And this is reasonable; for 'the one' means the measure of
some plurality, and 'number' means a measured plurality and a plurality
of measures. (Thus it is natural that one is not a number; for the
measure is not measures, but both the measure and the one are
starting-points.) The measure must always be some identical thing
predicable of all the things it measures, e.g. if the things are horses,
the measure is 'horse', and if they are men, 'man'. If they are a man, a
horse, and a god, the measure is perhaps 'living being', and the number
of them will be a number of living beings. If the things are 'man' and
'pale' and 'walking', these will scarcely have a number, because all
belong to a subject which is one and the same in number, yet the number
of these will be a number of 'kinds' or of some such term.
Those who treat the unequal as one thing, and the
dyad as an indefinite compound of great and small, say what is very far
from being probable or possible. For (a) these are modifications and
accidents, rather than substrata, of numbers and magnitudes-the many and
few of number, and the great and small of magnitude-like even and odd,
smooth and rough, straight and curved. Again, (b) apart from this
mistake, the great and the small, and so on, must be relative to
something; but what is relative is least of all things a kind of entity
or substance, and is posterior to quality and quantity; and the relative
is an accident of quantity, as was said, not its matter, since something
with a distinct nature of its own must serve as matter both to the
relative in general and to its parts and kinds.
For there is nothing either great or small, many or few, or, in
general, relative to something else, which without having a nature of
its own is many or few, great or small, or relative to something else. A
sign that the relative is least of all a substance and a real thing is
the fact that it alone has no proper generation or destruction or
movement, as in respect of quantity there is increase and diminution, in
respect of quality alteration, in respect of place locomotion, in respect
of substance simple generation and destruction. In respect of relation
there is no proper change; for, without changing, a thing will be now
greater and now less or equal, if that with which it is compared has
changed in quantity.
And (c) the matter of each thing, and therefore of substance, must be
that which is potentially of the nature in question; but the relative is
neither potentially nor actually substance. It is strange, then, or
rather impossible, to make not-substance an element in, and prior to,
substance; for all the categories are posterior to substance.
Again, (d) elements are not predicated of the things of which they are
elements, but many and few are predicated both apart and together of
number, and long and short of the line, and both broad and narrow apply
to the plane. If there is a plurality, then, of which the one term, viz.
few, is always predicated, e.g. 2 (which cannot be many, for if it were
many, 1 would be few), there must be also one which is absolutely many,
e.g. 10 is many (if there is no number which is greater than 10), or
10,000. How then, in view of this, can number consist of few and many?
Either both ought to be predicated of it, or neither; but in fact only
the one or the other is predicated.
2
We must inquire
generally, whether eternal things can consist of elements. If they do,
they will have matter; for everything that consists of elements is
composite. Since, then, even if a thing exists for ever, out of that of
which it consists it would necessarily also, if it had come into being,
have come into being, and since everything comes to be what it comes to
be out of that which is it potentially (for it could not have come to be
out of that which had not this capacity, nor could it consist of such
elements), and since the potential can be either actual or not,-this
being so, however everlasting number or anything else that has matter
is, it must be capable of not existing, just as that which is any number
of years old is as capable of not existing as that which is a day old;
if this is capable of not existing, so is that which has lasted for a
time so long that it has no limit. They cannot, then, be eternal, since
that which is capable of not existing is not eternal, as we had occasion
to
show in another context.
If that which we are now saying is true universally-that no substance is
eternal unless it is actuality-and if the elements are matter that
underlies substance, no eternal substance can have elements present in
it, of which it consists.
There are some who describe the element which acts
with the One as an indefinite dyad, and object to 'the unequal',
reasonably enough, because of the ensuing difficulties; but they have
got rid only of those objections which inevitably arise from the
treatment of the unequal, i.e. the relative, as an element; those which
arise apart from this opinion must confront even these thinkers, whether
it is ideal number, or mathematical, that they construct out of those
elements.
There are many causes which led them off into these
explanations, and especially the fact that they framed the difficulty in
an obsolete form. For they thought that all things that are would be one
(viz. Being itself), if one did not join issue with and refute the
saying of Parmenides:
'For never will this he proved, that things that are not are.'
They thought it
necessary to prove that that which is not is; for only thus-of that
which is and something else-could the things that are be composed, if
they are many.
But, first, if 'being' has many senses (for it means
sometimes substance, sometimes that it is of a certain quality,
sometimes that it is of a certain quantity, and at other times the other
categories), what sort of 'one', then, are all the things that are, if
non-being is to be supposed not to be? Is it the substances that are
one, or the affections and similarly the other categories as well, or
all together-so that the 'this' and the 'such' and the 'so much' and the
other categories that indicate each some one class of being will all be
one? But it is strange, or rather impossible, that the coming into play
of a single thing should bring it about that part of that which is is a
'this', part a 'such', part a 'so much', part a 'here'.
Secondly, of what sort of non-being and being do the
things that are consist? For 'nonbeing' also has many senses, since
'being' has; and 'not being a man' means not being a certain substance,
'not being straight' not being of a certain quality, 'not being three
cubits long' not being of a certain quantity. What sort of being and
non-being, then, by their union pluralize the things that are? This
thinker means by the non-being the union of which with being pluralizes
the things that are, the false and the character of falsity. This is
also why it used to be said that we must assume something that is false,
as geometers assume the line which is not a foot long to be a foot long.
But this cannot be so. For neither do geometers assume anything false
(for the enunciation is extraneous to the inference), nor is it
non-being in this sense that the things that are are generated from or
resolved into. But since 'non-being' taken in its various cases has as
many senses as there are categories, and besides this the false is said
not to be, and so is the potential, it is from this that generation
proceeds, man from that which is not man but potentially man, and white
from that which is not white but potentially white, and this whether it
is some one thing that is generated or many.
The question evidently is, how being, in the sense of
'the substances', is many; for the things that are generated are numbers
and lines and bodies. Now it is strange to inquire how being in the
sense of the 'what' is many, and not how either qualities or quantities
are many. For surely the indefinite dyad or 'the great and the small' is
not a reason why there should be two kinds of white or many colours or
flavours or shapes; for then these also would be numbers and units. But
if they had attacked these other categories, they would have seen the
cause of the plurality in substances also; for the same thing or
something analogous is the cause. This aberration is the reason also why
in seeking the opposite of being and the one, from which with being and
the one the things that are proceed, they posited the relative term
(i.e. the unequal), which is neither the contrary nor the contradictory
of these, and is one kind of being as 'what' and quality also are.
They should have asked this question also, how
relative terms are many and not one. But as it is, they inquire how
there are many units besides the first 1, but do not go on to inquire
how there are many unequals besides the unequal. Yet they use them and
speak of great and small, many and few (from which proceed numbers),
long and short (from which proceeds the line), broad and narrow (from
which proceeds the plane), deep and shallow (from which proceed solids);
and they speak of yet more kinds of relative term. What is the reason,
then, why there is a plurality of these?
It is necessary, then, as we say, to presuppose for
each thing that which is it potentially; and the holder of these views
further declared what that is which is potentially a 'this' and a
substance but is not in itself being-viz. that it is the relative (as if
he had said 'the qualitative'), which is neither potentially the one or
being, nor the negation of the one nor of being, but one among beings.
And it was much more necessary, as we said, if he was inquiring how
beings are many, not to inquire about those in the same category-how
there are many substances or many qualities-but how beings as a whole
are many; for some are substances, some modifications, some relations.
In the categories other than substance there is yet another problem
involved in the existence of plurality. Since they are not separable
from substances, qualities and quantities are many just because their
substratum becomes and is many; yet there ought to be a matter for each
category; only it cannot be separable from substances. But in the case
of 'thises', it is possible to explain how the 'this' is many things,
unless a thing is to be treated as both a 'this' and a general
character. The difficulty arising from the facts about substances is
rather this, how there are actually many substances and not one.
But further, if the 'this' and the quantitative are
not the same, we are not told how and why the things that are are many,
but how quantities are many. For all 'number' means a quantity, and so
does the 'unit', unless it means a measure or the quantitatively
indivisible. If, then, the quantitative and the 'what' are different, we
are not told whence or how the 'what' is many; but if any one says they
are the same, he has to face many inconsistencies.
One might fix one's attention also on the question,
regarding the numbers, what justifies the belief that they exist. To the
believer in Ideas they provide some sort of cause for existing things,
since each number is an Idea, and the Idea is to other things somehow or
other the cause of their being; for let this supposition be granted
them. But as for him who does not hold this view because he sees the
inherent objections to the Ideas (so that it is not for this reason that
he posits numbers), but who posits mathematical number, why must we
believe his statement that such number exists, and of what use is such
number to other things? Neither does he who says it exists maintain that
it is the cause of anything (he rather says it is a thing existing by
itself), nor is it observed to be the cause of anything; for the
theorems of arithmeticians will all be found true
even of sensible things, as was said before.
3
As for those, then,
who suppose the Ideas to exist and to be numbers, by their assumption in
virtue of the method of setting out each term apart from its
instances-of the unity of each general term they try at least to explain
somehow why number must exist. Since their reasons, however, are neither
conclusive nor in themselves possible, one must not, for these reasons
at least, assert the existence of number. Again, the Pythagoreans,
because they saw many attributes of numbers belonging to sensible
bodies, supposed real things to be numbers-not separable numbers,
however, but numbers of which real things consist. But why? Because the
attributes of numbers are present in a musical scale and in the heavens
and in
many other things. Those, however, who say that mathematical number
alone exists cannot according to their hypotheses say anything of this
sort, but it used to be urged that these sensible things could not be
the subject of the sciences. But we maintain that they are, as we said
before. And it is evident that the objects of mathematics do not exist
apart; for if they existed apart their attributes would not have been
present in bodies. Now the Pythagoreans in this point are open to no
objection; but in that they construct natural bodies out of numbers,
things that have lightness and weight out of things that have not weight
or lightness, they seem to speak of another heaven and other bodies, not
of the sensible. But those who make number separable assume that it both
exists and is separable because the axioms would not be true of sensible
things, while the statements of mathematics are true and 'greet the
soul'; and similarly with the spatial magnitudes of mathematics. It is
evident, then, both that the rival theory will say the contrary of this,
and that the difficulty we raised just now, why if numbers are in no way
present in sensible things their attributes are present in sensible
things, has to be solved by those who hold these views.
There are some who, because the point is the limit
and extreme of the line, the line of the plane, and the plane of the
solid, think there must be real things of this sort. We must therefore
examine this argument too, and see whether it is not remarkably weak.
For (i) extremes are not substances, but rather all these things are
limits. For even walking, and movement in general, has a limit, so that
on their theory this will be a 'this' and a substance. But that is
absurd. Not but what (ii) even if they are substances, they will all be
the substances of the sensible things in this world; for it is to these
that the argument applied. Why then should they be capable of existing
apart?
Again, if we are not too easily satisfied, we may,
regarding all number and the objects of mathematics, press this
difficulty, that they contribute nothing to one another, the prior to
the posterior; for if number did not exist, none the less spatial
magnitudes would exist for those who maintain the existence of the
objects of mathematics only, and if spatial magnitudes did not exist,
soul and sensible bodies would exist. But the observed facts show that
nature is not a series of episodes, like a bad tragedy. As for the
believers in the Ideas, this difficulty misses them; for they construct
spatial magnitudes out of matter and number, lines out of the number
planes doubtless out of solids out of or they use other
numbers, which makes no difference. But will these magnitudes be Ideas,
or what is their manner of existence, and what do they contribute to
things? These contribute nothing, as the objects of mathematics
contribute nothing. But not even is any theorem true of them, unless we
want to change the objects of mathematics and invent doctrines of our
own. But it is not hard to assume any random hypotheses and spin out a
long string of conclusions. These thinkers, then, are wrong in this way,
in wanting to unite the objects of mathematics with the Ideas. And those
who first posited two kinds of number, that of the Forms and that which
is mathematical, neither have said nor can say how mathematical number
is to exist and of what it is to consist. For they place it between
ideal and sensible number. If (i) it consists of the great and small, it
will be the same
as the other-ideal-number (he makes spatial magnitudes out of some other
small and great).
And if (ii) he names some other element, he will be making his elements
rather many. And if the principle of each of the two kinds of number is
a 1, unity will be something common to these, and we must inquire how
the one is these many things, while at the same time number, according
to him, cannot be generated except from one and an indefinite dyad.
All this is absurd, and conflicts both with itself
and with the probabilities, and we seem to see in it Simonides 'long
rigmarole' for the long rigmarole comes into play, like those of slaves,
when men have nothing sound to say. And the very elements-the great and
the small-seem to cry out against the violence that is done to them; for
they cannot in any way generate numbers other than those got from 1 by
doubling.
It is strange also to attribute generation to things
that are eternal, or rather this is one of the things that are
impossible.
There need be no doubt whether the Pythagoreans attribute generation to
them or not; for they say plainly that when the one had been
constructed, whether out of planes or of surface or of seed or of
elements which they cannot express, immediately the nearest part of the
unlimited began to be constrained and limited by the limit. But since
they are constructing a world and wish to speak the language of natural
science, it is fair to make some examination of their physical theorics,
but to let them off from the present inquiry; for we are investigating
the principles at work in unchangeable things, so that it is numbers of
this kind whose genesis we must study.
4
These thinkers say there is no generation of the odd
number, which evidently implies that there is generation of the even;
and some present the even as produced first from unequals -the great and
the small-when these are equalized. The inequality, then, must belong to
them before they are equalized. If they had always been equalized, they
would not have been unequal before; for there is nothing before that
which is always. Therefore evidently they are not giving their account
of the generation of numbers merely to assist contemplation of their
nature.
A difficulty, and a reproach to any one who finds it
no difficulty, are contained in the question how the elements and the
principles are related to the good and the beautiful; the difficulty is
this, whether any of the elements is such a thing as we mean by the good
itself and the best, or this is not so, but these are later in origin
than the elements. The theologians seem to agree with some thinkers of
the present day, who answer the question in the negative, and say that
both the good and the beautiful appear in the nature of things only when
that nature has made some progress. (This they do to avoid a real
objection which confronts those who say, as some do, that the one is a
first principle. The objection arises not from their ascribing goodness
to the first principle as an attribute, but from their making the one a
principle-and a principle in the sense of an element-and generating
number from the one.) The old poets agree with this inasmuch as they say
that not those who
are first in time, e.g. Night and Heaven or Chaos or Ocean, reign and
rule, but Zeus. These poets, however, are led to speak thus only because
they think of the rulers of the world as changing; for those of them who
combine the two characters in that they do not use mythical language
throughout, e.g. Pherecydes and some others, make the original
generating agent the Best, and so do the Magi, and some of the later
sages also, e.g. both Empedocles and Anaxagoras, of whom one made love
an element, and the other made reason a principle. Of those who maintain
the existence of the unchangeable substances some say the One itself is
the good itself; but they thought its substance lay mainly in its unity.
This, then, is the problem,-which of the two ways of
speaking is right. It would be strange if to that which is primary and
eternal and most self-sufficient this very quality--self-sufficiency and
self-maintenance--belongs primarily in some other way than as a good.
But indeed it can be for no other reason indestructible or
self-sufficient than because its nature is good. Therefore to say that
the first principle is good is probably correct; but that this principle
should be the One or, if not that, at least an element, and an element
of numbers, is impossible. Powerful objections arise, to avoid which
some have given up the theory (viz. those who agree that the One is a
first principle and element, but only of mathematical number). For on
this view all the units become identical with species of good, and there
is a great profusion of goods. Again, if the Forms are numbers, all the
Forms are identical with species of good. But let a man assume Ideas of
anything he pleases. If these are Ideas only of goods, the Ideas will
not be substances; but if the Ideas are also Ideas of substances, all
animals and plants and all individuals that share in Ideas will be good.
These absurdities follow, and it also follows that
the contrary element, whether it is plurality or the unequal, i.e. the
great and small, is the bad-itself. (Hence one thinker avoided attaching
the good to the One, because it would necessarily follow, since
generation is from contraries, that badness is the fundamental nature of
plurality; while others say inequality is the nature of the bad.) It
follows, then, that all things partake of the bad except one -- the One
itself, and that numbers partake of it in a more undiluted form than
spatial magnitudes, and that the bad is the space in which the good is
realized, and that it partakes in and desires that which tends to
destroy it; for contrary tends to destroy contrary. And if, as we were
saying, the matter is that which is potentially each thing, e.g. that of
actual fire is that which is potentially fire, the bad will be just the
potentially good.
All these objections, then, follow, partly because
they make every principle an element, partly because they make
contraries principles, partly because they make the One a principle,
partly because they treat the numbers as the first substances, and as
capable of existing apart, and as Forms.
5
If, then, it is
equally impossible not to put the good among the first principles and to
put it among them in this way, evidently the principles are not being
correctly described, nor are the first substances. Nor does any one
conceive the matter correctly if he compares the principles of the
universe to that of animals and plants, on the ground that the more
complete always comes from the indefinite and incomplete-which is what
leads this thinker to say that this is also true of the first principles
of reality, so that the One itself is not even an existing thing. This
is incorrect, for even in this world of animals and plants the
principles from which these come are complete; for it is a man that
produces a man, and the seed is not first.
It is out of place, also, to generate place
simultaneously with the mathematical solids (for place is peculiar to
the individual things, and hence they are separate in place; but
mathematical objects are nowhere), and to say that they must be
somewhere, but not say what kind of thing their place is.
Those who say that existing things come from elements
and that the first of existing things are the numbers, should have first
distinguished the senses in which one thing comes from another, and then
said in which sense number comes from its first principles.
By intermixture? But (1) not everything is capable of
intermixture, and (2) that which is produced by it is different from
its elements, and on this view the one will not remain separate or a
distinct entity; but they want it to be so.
By juxtaposition, like a syllable? But then (1) the
elements must have position; and (2) he who thinks of number will be
able to think of the unity and the plurality apart; number then will be
this - a unit and plurality, or the one and the unequal.
Again, coming from certain things means in one sense
that these are still to be found in the product, and in another that
they are not; which sense does number come from these elements? Only
things that are generated can come from elements which are present in
them. Does number come, then, from its elements as from seed? But
nothing can be excreted from that which is indivisible. Does it come
from its contrary, its contrary not persisting? But all things that come
in
this way come also from something else which does persist. Since, then,
one thinker places the 1 as contrary to plurality, and another places it
as contrary to the unequal, treating the 1 as equal, number must be
being treated as coming from contraries. There is, then, something else
that persists, from which and from one contrary the compound is or has
come to be. Again, why in the world do the other things that come from
contraries, or that have contraries, perish (even when all of the
contrary is used to produce them), while number does not? Nothing is
said about this.
Yet whether present or not present in the compound the contrary destroys
it, e.g. 'strife' destroys the 'mixture' (yet it should not; for it is
not to that that is contrary).
Once more, it has not been determined at all in which
way numbers are the causes of substances and of being-whether (1)
asboundaries (as points are of spatial magnitudes). This is how Eurytus
decided what was the number of what (e.g. one of man and another of
horse), viz. by imitating the figures of living things with pebbles, as
some people bring numbers into the forms of triangle and square. Or (2)
is it because harmony is a ratio of numbers, and so is man and
everything else? But how are the attributes-white and sweet and
hot-numbers? Evidently it is not the numbers that are the essence or the
causes of the form; for the ratio is the essence, while the number the
causes of the form; for the ratio is the essence, while the number is
the matter. E.g. the essence of flesh or bone is number only in this
way, 'three parts of fire and two of earth'. And a number, whatever
number it is, is always a number of certain things, either of parts of
fire or earth or of units; but the essence is that there is so much of
one thing to so much of another in the mixture; and this is no longer a
number but a ratio of mixture of numbers, whether these are corporeal or
of any other kind.
Number, then, whether it be number in general or the
number which consists of abstract units, is neither the cause as agent,
nor the matter, nor the ratio and form of things. Nor, of course, is it
the final cause.
6
One might also raise
the question what the good is that things get from numbers because their
composition is expressible by a number, either by one which is easily
calculable or by an odd number. For in fact honey-water is no more
wholesome if it is mixed in the proportion of three times three, but it
would do more good if it were in no particular ratio but well diluted
than if it were numerically expressible but strong. Again, the ratios of
mixtures are expressed by the adding of numbers, not by mere numbers;
e.g. it is 'three parts to two', not 'three times two'. For in any
multiplication the genus of
the things multiplied must be the same; therefore the product 1X2X3 must
be measurable by 1, and 4X5X6 by 4 and therefore all products into which
the same factor enters must be measurable by that factor. The number of
fire, then, cannot be 2X5X3X6 and at the same time that of water 2X3.
If all things must share in number, it must follow
that many things are the same, and the same number must belong to one
thing and to another. Is number the cause, then, and does the thing
exist because of its number, or is this not certain? E.g. the motions of
the sun have a number, and again those of the moon,-yes, and the life
and prime of each animal. Why, then, should not some of these numbers be
squares, some cubes, and some equal, others double? There is no reason
why they should not, and indeed they must move within these limits,
since all things were assumed to share in number.
And it was assumed that things that differed might fall under the same
number. Therefore if the same number had belonged to certain
things, these would have been the same as one another, since they
would have had the same form of number; e.g. sun and moon would have
been the same. But why need these numbers be causes? There are seven
vowels, the scale consists of seven strings, the Pleiades are seven, at
seven animals lose their teeth (at least some do, though some do
not),
and the champions who fought against Thebes were seven. Is it then
because the number is the kind of number it is, that the champions were
seven or the Pleiad consists of seven stars? Surely the champions were
seven because there were seven gates or for some other reason, and the
Pleiad we count as seven, as we count the Bear as twelve, while other
peoples count more stars in both. Nay they even say that X, Ps and Z are
concords and that because there are three concords, the double
consonants also are three. They quite neglect the fact that there might
be a thousand such letters; for one symbol might be assigned to GP. But
if they say that each of these three is equal to two of the other
letters, and no other is so, and if the cause is that there are three
parts of the mouth and one letter is in each applied to sigma, it is for
this reason that there are only three, not because the concords are
three; since as a matter of fact the concords are more than three, but
of double consonants there cannot be more.
These people are like the old-fashioned Homeric
scholars, who see small resemblances but neglect great ones. Some say
that there are many such cases, e.g. that the middle strings are
represented by nine and eight, and that the epic verse has seventeen
syllables, which is equal in number to the two strings, and that the
scansion is, in the right half of the line nine syllables, and in the
left eight. And they say that the distance in the letters from alpha to
omega is equal to that from the lowest note of the flute to the highest,
and that the number of this note is equal to that of the whole choir of
heaven. It may be suspected that no one could find difficulty either in
stating such analogies or in finding them in eternal things, since they
can be found even in perishable things.
But the lauded characteristics of numbers, and the
contraries of these, and generally the mathematical relations, as some
describe them, making them causes of nature, seem, when we inspect them
in this way, to vanish; for none of them is a cause in any of the senses
that have been distinguished in reference to the first principles. In a
sense, however, they make it plain that goodness belongs to numbers, and
that the odd, the straight, the square, the potencies of certain
numbers, are in the column of the beautiful. For the seasons and a
particular kind of number go together; and the other agreements that
they collect from the theorems of mathematics all have this meaning.
Hence they are like coincidences.
For they are accidents, but the things that agree are all appropriate to
one another, and one by analogy.
For in each category of being an analogous term is found-as the straight
is in length, so is the level in surface, perhaps the odd in number, and
the white in colour.
Again, it is not the ideal numbers that are the
causes of musical phenomena and the like (for equal ideal numbers differ
from one another in form; for even the units do); so that we need not
assume Ideas for this reason at least.
These, then, are the results of the theory, and yet
more might be brought together. The fact that our opponnts have much
trouble with the generation of numbers and can in no way make a system
of them, seems to indicate that the objects of mathematics are not
separable from sensible things, as some say, and that they are not the
first principles.
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