Metaphysics
By Aristotle
Written 350 B.C.E
Translated by W. D. Ross
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. "
THE END
Chicago-North Shore Therapy.com
|
|
