Metaphysics
By Aristotle
Part 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.
Part 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.
Part 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 te 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.
Part 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.
Part 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) as boundaries (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.
Part 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. "
END
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