**Part 1 **

WE have stated what is the substance of sensible things,
dealing in the treatise on physics with matter, and later with
the substance which has actual existence. Now since our inquiry
is whether there is or is not besides the sensible substances any
which is immovable and eternal, and, if there is, what it is, we
must first consider what is said by others, so that, if there is
anything which they say wrongly, we may not be liable to the same
objections, while, if there is any opinion common to them and us,
we shall have no private grievance against ourselves on that
account; for one must be content to state some points better than
one's predecessors, and others no worse.

Two opinions are held on this subject; it is said that the
objects of mathematics—i.e. numbers and lines and the like—are
substances, and again that the Ideas are substances. And (1)
since some recognize these as two different classes—the Ideas and
the mathematical numbers, and (2) some recognize both as having
one nature, while (3) some others say that the mathematical
substances are the only substances, we must consider first the
objects of mathematics, not qualifying them by any other
characteristic—not asking, for instance, whether they are in fact
Ideas or not, or whether they are the principles and substances
of existing things or not, but only whether as objects of
mathematics they exist or not, and if they exist, how they exist.
Then after this we must separately consider the Ideas themselves
in a general way, and only as far as the accepted mode of
treatment demands; for most of the points have been repeatedly
made even by the discussions outside our school, and, further,
the greater part of our account must finish by throwing light on
that inquiry, viz. when we examine whether the substances and the
principles of existing things are numbers and Ideas; for after
the discussion of the Ideas this remans as a third inquiry.

If the objects of mathematics exist, they must exist either in
sensible objects, as some say, or separate from sensible objects
(and this also is said by some); or if they exist in neither of
these ways, either they do not exist, or they exist only in some
special sense. So that the subject of our discussion will be not
whether they exist but how they exist.

**Part 2 **

That it is impossible for mathematical objects to exist in
sensible things, and at the same time that the doctrine in
question is an artificial one, has been said already in our
discussion of difficulties we have pointed out that it is
impossible for two solids to be in the same place, and also that
according to the same argument the other powers and
characteristics also should exist in sensible things and none of
them separately. This we have said already. But, further, it is
obvious that on this theory it is impossible for any body
whatever to be divided; for it would have to be divided at a
plane, and the plane at a line, and the line at a point, so that
if the point cannot be divided, neither can the line, and if the
line cannot, neither can the plane nor the solid. What
difference, then, does it make whether sensible things are such
indivisible entities, or, without being so themselves, have
indivisible entities in them? The result will be the same; if the
sensible entities are divided the others will be divided too, or
else not even the sensible entities can be divided.

But, again, it is not possible that such entities should exist
separately. For if besides the sensible solids there are to be
other solids which are separate from them and prior to the
sensible solids, it is plain that besides the planes also there
must be other and separate planes and points and lines; for
consistency requires this. But if these exist, again besides the
planes and lines and points of the mathematical solid there must
be others which are separate. (For incomposites are prior to
compounds; and if there are, prior to the sensible bodies, bodies
which are not sensible, by the same argument the planes which
exist by themselves must be prior to those which are in the
motionless solids. Therefore these will be planes and lines other
than those that exist along with the mathematical solids to which
these thinkers assign separate existence; for the latter exist
along with the mathematical solids, while the others are prior to
the mathematical solids.) Again, therefore, there will be,
belonging to these planes, lines, and prior to them there will
have to be, by the same argument, other lines and points; and
prior to these points in the prior lines there will have to be
other points, though there will be no others prior to these. Now
(1) the accumulation becomes absurd; for we find ourselves with
one set of solids apart from the sensible solids; three sets of
planes apart from the sensible planes—those which exist apart
from the sensible planes, and those in the mathematical solids,
and those which exist apart from those in the mathematical
solids; four sets of lines, and five sets of points. With which
of these, then, will the mathematical sciences deal? Certainly
not with the planes and lines and points in the motionless solid;
for science always deals with what is prior. And (the same
account will apply also to numbers; for there will be a different
set of units apart from each set of points, and also apart from
each set of realities, from the objects of sense and again from
those of thought; so that there will be various classes of
mathematical numbers.

Again, how is it possible to solve the questions which we have
already enumerated in our discussion of difficulties? For the
objects of astronomy will exist apart from sensible things just
as the objects of geometry will; but how is it possible that a
heaven and its parts—or anything else which has movement—should
exist apart? Similarly also the objects of optics and of
harmonics will exist apart; for there will be both voice and
sight besides the sensible or individual voices and sights.
Therefore it is plain that the other senses as well, and the
other objects of sense, will exist apart; for why should one set
of them do so and another not? And if this is so, there will also
be animals existing apart, since there will be senses.

Again, there are certain mathematical theorems that are
universal, extending beyond these substances. Here then we shall
have another intermediate substance separate both from the Ideas
and from the intermediates,—a substance which is neither number
nor points nor spatial magnitude nor time. And if this is
impossible, plainly it is also impossible that the former
entities should exist separate from sensible things.

And, in general, conclusion contrary alike to the truth and to
the usual views follow, if one is to suppose the objects of
mathematics to exist thus as separate entities. For because they
exist thus they must be prior to sensible spatial magnitudes, but
in truth they must be posterior; for the incomplete spatial
magnitude is in the order of generation prior, but in the order
of substance posterior, as the lifeless is to the living.

Again, by virtue of what, and when, will mathematical
magnitudes be one? For things in our perceptible world are one in
virtue of soul, or of a part of soul, or of something else that
is reasonable enough; when these are not present, the thing is a
plurality, and splits up into parts. But in the case of the
subjects of mathematics, which are divisible and are quantities,
what is the cause of their being one and holding together?

Again, the modes of generation of the objects of mathematics
show that we are right. For the dimension first generated is
length, then comes breadth, lastly depth, and the process is
complete. If, then, that which is posterior in the order of
generation is prior in the order of substantiality, the solid
will be prior to the plane and the line. And in this way also it
is both more complete and more whole, because it can become
animate. How, on the other hand, could a line or a plane be
animate? The supposition passes the power of our senses.

Again, the solid is a sort of substance; for it already has in
a sense completeness. But how can lines be substances? Neither as
a form or shape, as the soul perhaps is, nor as matter, like the
solid; for we have no experience of anything that can be put
together out of lines or planes or points, while if these had
been a sort of material substance, we should have observed things
which could be put together out of them.

Grant, then, that they are prior in definition. Still not all
things that are prior in definition are also prior in
substantiality. For those things are prior in substantiality
which when separated from other things surpass them in the power
of independent existence, but things are prior in definition to
those whose definitions are compounded out of their definitions;
and these two properties are not coextensive. For if attributes
do not exist apart from the substances (e.g. a 'mobile' or a
pale'), pale is prior to the pale man in definition, but not in
substantiality. For it cannot exist separately, but is always
along with the concrete thing; and by the concrete thing I mean
the pale man. Therefore it is plain that neither is the result of
abstraction prior nor that which is produced by adding
determinants posterior; for it is by adding a determinant to pale
that we speak of the pale man.

It has, then, been sufficiently pointed out that the objects
of mathematics are not substances in a higher degree than bodies
are, and that they are not prior to sensibles in being, but only
in definition, and that they cannot exist somewhere apart. But
since it was not possible for them to exist in sensibles either,
it is plain that they either do not exist at all or exist in a
special sense and therefore do not 'exist' without qualification.
For 'exist' has many senses.

**Part 3 **

For just as the universal propositions of mathematics deal not
with objects which exist separately, apart from extended
magnitudes and from numbers, but with magnitudes and numbers, not
however *qua* such as to have magnitude or to be divisible, clearly
it is possible that there should also be both propositions and
demonstrations about sensible magnitudes, not however *qua*
sensible but *qua* possessed of certain definite qualities. For as
there are many propositions about things merely considered as in
motion, apart from what each such thing is and from their
accidents, and as it is not therefore necessary that there should
be either a mobile separate from sensibles, or a distinct mobile
entity in the sensibles, so too in the case of mobiles there will
be propositions and sciences, which treat them however not *qua*
mobile but only *qua* bodies, or again only *qua* planes, or only *qua*
lines, or *qua* divisibles, or *qua* indivisibles having position, or
only *qua* indivisibles. Thus since it is true to say without
qualification that not only things which are separable but also
things which are inseparable exist (for instance, that mobiles
exist), it is true also to say without qualification that the
objects of mathematics exist, and with the character ascribed to
them by mathematicians. And as it is true to say of the other
sciences too, without qualification, that they deal with such and
such a subject—not with what is accidental to it (e.g. not with
the pale, if the healthy thing is pale, and the science has the
healthy as its subject), but with that which is the subject of
each science—with the healthy if it treats its object *qua*
healthy, with man if *qua* man:—so too is it with geometry; if its
subjects happen to be sensible, though it does not treat them *qua*
sensible, the mathematical sciences will not for that reason be
sciences of sensibles—nor, on the other hand, of other things
separate from sensibles. Many properties attach to things in
virtue of their own nature as possessed of each such character; e.g.
there are attributes peculiar to the animal *qua* female or *qua*
male (yet there is no 'female' nor 'male' separate from animals);
so that there are also attributes which belong to things merely
as lengths or as planes. And in proportion as we are dealing with
things which are prior in definition and simpler, our knowledge
has more accuracy, i.e. simplicity. Therefore a science which
abstracts from spatial magnitude is more precise than one which
takes it into account; and a science is most precise if it
abstracts from movement, but if it takes account of movement, it
is most precise if it deals with the primary movement, for this
is the simplest; and of this again uniform movement is the
simplest form.

The same account may be given of harmonics and optics; for
neither considers its objects *qua* sight or *qua* voice, but *qua*
lines and numbers; but the latter are attributes proper to the
former. And mechanics too proceeds in the same way. Therefore if
we suppose attributes separated from their fellow attributes and
make any inquiry concerning them as such, we shall not for this
reason be in error, any more than when one draws a line on the
ground and calls it a foot long when it is not; for the error is
not included in the premisses.

Each question will be best investigated in this way—by setting
up by an act of separation what is not separate, as the
arithmetician and the geometer do. For a man *qua* man is one
indivisible thing; and the arithmetician supposed one indivisible
thing, and then considered whether any attribute belongs to a man
*qua* indivisible. But the geometer treats him neither *qua* man nor
*qua* indivisible, but as a solid. For evidently the properties
which would have belonged to him even if perchance he had not
been indivisible, can belong to him even apart from these
attributes. Thus, then, geometers speak correctly; they talk
about existing things, and their subjects do exist; for being has
two forms—it exists not only in complete reality but also
materially.

Now since the good and the beautiful are different (for the
former always implies conduct as its subject, while the beautiful
is found also in motionless things), those who assert that the
mathematical sciences say nothing of the beautiful or the good
are in error. For these sciences say and prove a great deal about
them; if they do not expressly mention them, but prove attributes
which are their results or their definitions, it is not true to
say that they tell us nothing about them. The chief forms of
beauty are order and symmetry and definiteness, which the
mathematical sciences demonstrate in a special degree. And since
these (e.g. order and definiteness) are obviously causes of many
things, evidently these sciences must treat this sort of
causative principle also (i.e. the beautiful) as in some sense a
cause. But we shall speak more plainly elsewhere about these
matters.

**Part 4 **

So much then for the objects of mathematics; we have said that
they exist and in what sense they exist, and in what sense they
are prior and in what sense not prior. Now, regarding the Ideas,
we must first examine the ideal theory itself, not connecting it
in any way with the nature of numbers, but treating it in the
form in which it was originally understood by those who first
maintained the existence of the Ideas. The supporters of the
ideal theory were led to it because on the question about the
truth of things they accepted the Heraclitean sayings which
describe all sensible things as ever passing away, so that if
knowledge or thought is to have an object, there must be some
other and permanent entities, apart from those which are
sensible; for there could be no knowledge of things which were in
a state of flux. But when Socrates was occupying himself with the
excellences of character, and in connexion with them became the
first to raise the problem of universal definition (for of the
physicists Democritus only touched on the subject to a small
extent, and defined, after a fashion, the hot and the cold; while
the Pythagoreans had before this treated of a few things, whose
definitions—e.g. those of opportunity, justice, or marriage—they
connected with numbers; but it was natural that Socrates should
be seeking the essence, for he was seeking to syllogize, and
'what a thing is' is the starting-point of syllogisms; for there
was as yet none of the dialectical power which enables people
even without knowledge of the essence to speculate about
contraries and inquire whether the same science deals with
contraries; for two things may be fairly ascribed to Socrates—inductive
arguments and universal definition, both of which are concerned
with the starting-point of science):—but Socrates did not make
the universals or the definitions exist apart: they, however,
gave them separate existence, and this was the kind of thing they
called Ideas. Therefore it followed for them, almost by the same
argument, that there must be Ideas of all things that are spoken
of universally, and it was almost as if a man wished to count
certain things, and while they were few thought he would not be
able to count them, but made more of them and then counted them;
for the Forms are, one may say, more numerous than the particular
sensible things, yet it was in seeking the causes of these that
they proceeded from them to the Forms. For to each thing there
answers an entity which has the same name and exists apart from
the substances, and so also in the case of all other groups there
is a one over many, whether these be of this world or eternal.

Again, of the ways in which it is proved that the Forms exist,
none is convincing; for from some no inference necessarily
follows, and from some arise Forms even of things of which they
think there are no Forms. For according to the arguments from the
sciences there will be Forms of all things of which there are
sciences, and according to the argument of the 'one over many'
there will be Forms even of negations, and according to the
argument that thought has an object when the individual object
has perished, there will be Forms of perishable things; for we
have an image of these. Again, of the most accurate arguments,
some lead to Ideas of relations, of which they say there is no
independent class, and others introduce the 'third man'.

And in general the arguments for the Forms destroy things for
whose existence the believers in Forms are more zealous than for
the existence of the Ideas; for it follows that not the dyad but
number is first, and that prior to number is the relative, and
that this is prior to the absolute—besides all the other points
on which certain people, by following out the opinions held about
the Forms, came into conflict with the principles of the theory.

Again, according to the assumption on the belief in the Ideas
rests, there will be Forms not only of substances but also of
many other things; for the concept is single not only in the case
of substances, but also in that of non-substances, and there are
sciences of other things than substance; and a thousand other
such difficulties confront them. But according to the necessities
of the case and the opinions about the Forms, if they can be
shared in there must be Ideas of substances only. For they are
not shared in incidentally, but each Form must be shared in as
something not predicated of a subject. (By 'being shared in
incidentally' I mean that if a thing shares in 'double itself',
it shares also in 'eternal', but incidentally; for 'the double'
happens to be eternal.) Therefore the Forms will be substance.
But the same names indicate substance in this and in the ideal
world (or what will be the meaning of saying that there is
something apart from the particulars—the one over many?). And if
the Ideas and the things that share in them have the same form,
there will be something common: for why should '2' be one and the
same in the perishable 2's, or in the 2's which are many but
eternal, and not the same in the '2 itself' as in the individual
2? But if they have not the same form, they will have only the
name in common, and it is as if one were to call both Callias and
a piece of wood a 'man', without observing any community between
them.

But if we are to suppose that in other respects the common
definitions apply to the Forms, e.g. that 'plane figure' and the
other parts of the definition apply to the circle itself, but
'what really is' has to be added, we must inquire whether this is
not absolutely meaningless. For to what is this to be added? To
'centre' or to 'plane' or to all the parts of the definition? For
all the elements in the essence are Ideas, e.g. 'animal' and 'two-footed'.
Further, there must be some Ideal answering to 'plane' above,
some nature which will be present in all the Forms as their genus.

**Part 5 **

Above all one might discuss the question what in the world the
Forms contribute to sensible things, either to those that are
eternal or to those that come into being and cease to be; for
they cause neither movement nor any change in them. But again
they help in no wise either towards the knowledge of other things
(for they are not even the substance of these, else they would
have been in them), or towards their being, if they are not in
the individuals which share in them; though if they were, they
might be thought to be causes, as white causes whiteness in a
white object by entering into its composition. But this argument,
which was used first by Anaxagoras, and later by Eudoxus in his
discussion of difficulties and by certain others, is very easily
upset; for it is easy to collect many and insuperable objections
to such a view.

But, further, all other things cannot come from the Forms in
any of the usual senses of 'from'. And to say that they are
patterns and the other things share in them is to use empty words
and poetical metaphors. For what is it that works, looking to the
Ideas? And any thing can both be and come into being without
being copied from something else, so that, whether Socrates
exists or not, a man like Socrates might come to be. And
evidently this might be so even if Socrates were eternal. And
there will be several patterns of the same thing, and therefore
several Forms; e.g. 'animal' and 'two-footed', and also 'man-himself',
will be Forms of man. Again, the Forms are patterns not only of
sensible things, but of Forms themselves also; i.e. the genus is
the pattern of the various forms-of-a-genus; therefore the same
thing will be pattern and copy.

Again, it would seem impossible that substance and that whose
substance it is should exist apart; how, therefore, could the
Ideas, being the substances of things, exist apart?

In the Phaedo the case is stated in this way—that the Forms
are causes both of being and of becoming. Yet though the Forms
exist, still things do not come into being, unless there is
something to originate movement; and many other things come into
being (e.g. a house or a ring) of which they say there are no
Forms. Clearly therefore even the things of which they say there
are Ideas can both be and come into being owing to such causes as
produce the things just mentioned, and not owing to the Forms.
But regarding the Ideas it is possible, both in this way and by
more abstract and accurate arguments, to collect many objections
like those we have considered.

**Part 6 **

Since we have discussed these points, it is well to consider
again the results regarding numbers which confront those who say
that numbers are separable substances and first causes of things.
If number is an entity and its substance is nothing other than
just number, as some say, it follows that either (1) there is a
first in it and a second, each being different in species,—and
either (a) this is true of the units without exception, and any
unit is inassociable with any unit, or (b) they are all without
exception successive, and any of them are associable with any, as
they say is the case with mathematical number; for in
mathematical number no one unit is in any way different from
another. Or (c) some units must be associable and some not; e.g.
suppose that 2 is first after 1, and then comes 3 and then the
rest of the number series, and the units in each number are
associable, e.g. those in the first 2 are associable with one
another, and those in the first 3 with one another, and so with
the other numbers; but the units in the '2-itself' are
inassociable with those in the '3-itself'; and similarly in the
case of the other successive numbers. And so while mathematical
number is counted thus—after 1, 2 (which consists of another 1
besides the former 1), and 3 which consists of another 1 besides
these two), and the other numbers similarly, ideal number is
counted thus—after 1, a distinct 2 which does not include the
first 1, and a 3 which does not include the 2 and the rest of the
number series similarly. Or (2) one kind of number must be like
the first that was named, one like that which the mathematicians
speak of, and that which we have named last must be a third kind.

Again, these kinds of numbers must either be separable from
things, or not separable but in objects of perception (not
however in the way which we first considered, in the sense that
objects of perception consists of numbers which are present in
them)—either one kind and not another, or all of them.

These are of necessity the only ways in which the numbers can
exist. And of those who say that the 1 is the beginning and
substance and element of all things, and that number is formed
from the 1 and something else, almost every one has described
number in one of these ways; only no one has said all the units
are inassociable. And this has happened reasonably enough; for
there can be no way besides those mentioned. Some say both kinds
of number exist, that which has a before and after being
identical with the Ideas, and mathematical number being different
from the Ideas and from sensible things, and both being separable
from sensible things; and others say mathematical number alone
exists, as the first of realities, separate from sensible things.
And the Pythagoreans, also, believe in one kind of number—the
mathematical; only they say it is not separate but sensible
substances are formed out of it. For they construct the whole
universe out of numbers—only not numbers consisting of abstract
units; they suppose the units to have spatial magnitude. But how
the first 1 was constructed so as to have magnitude, they seem
unable to say.

Another thinker says the first kind of number, that of the
Forms, alone exists, and some say mathematical number is
identical with this.

The case of lines, planes, and solids is similar. For some
think that those which are the objects of mathematics are
different from those which come after the Ideas; and of those who
express themselves otherwise some speak of the objects of
mathematics and in a mathematical way—viz. those who do not make
the Ideas numbers nor say that Ideas exist; and others speak of
the objects of mathematics, but not mathematically; for they say
that neither is every spatial magnitude divisible into
magnitudes, nor do any two units taken at random make 2. All who
say the 1 is an element and principle of things suppose numbers
to consist of abstract units, except the Pythagoreans; but they
suppose the numbers to have magnitude, as has been said before.
It is clear from this statement, then, in how many ways numbers
may be described, and that all the ways have been mentioned; and
all these views are impossible, but some perhaps more than others.

**Part 7 **

First, then, let us inquire if the units are associable or
inassociable, and if inassociable, in which of the two ways we
distinguished. For it is possible that any unity is inassociable
with any, and it is possible that those in the 'itself' are
inassociable with those in the 'itself', and, generally, that
those in each ideal number are inassociable with those in other
ideal numbers. Now (1) all units are associable and without
difference, we get mathematical number—only one kind of number,
and the Ideas cannot be the numbers. For what sort of number will
man-himself or animal-itself or any other Form be? There is one
Idea of each thing e.g. one of man-himself and another one of
animal-itself; but the similar and undifferentiated numbers are
infinitely many, so that any particular 3 is no more man-himself
than any other 3. But if the Ideas are not numbers, neither can
they exist at all. For from what principles will the Ideas come?
It is number that comes from the 1 and the indefinite dyad, and
the principles or elements are said to be principles and elements
of number, and the Ideas cannot be ranked as either prior or
posterior to the numbers.

But (2) if the units are inassociable, and inassociable in the
sense that any is inassociable with any other, number of this
sort cannot be mathematical number; for mathematical number
consists of undifferentiated units, and the truths proved of it
suit this character. Nor can it be ideal number. For 2 will not
proceed immediately from 1 and the indefinite dyad, and be
followed by the successive numbers, as they say '2,3,4' for the
units in the ideal are generated at the same time, whether, as
the first holder of the theory said, from unequals (coming into
being when these were equalized) or in some other way—since, if
one unit is to be prior to the other, it will be prior also to 2
the composed of these; for when there is one thing prior and
another posterior, the resultant of these will be prior to one
and posterior to the other. Again, since the 1-itself is first,
and then there is a particular 1 which is first among the others
and next after the 1-itself, and again a third which is next
after the second and next but one after the first 1,—so the units
must be prior to the numbers after which they are named when we
count them; e.g. there will be a third unit in 2 before 3 exists,
and a fourth and a fifth in 3 before the numbers 4 and 5 exist.—Now
none of these thinkers has said the units are inassociable in
this way, but according to their principles it is reasonable that
they should be so even in this way, though in truth it is
impossible. For it is reasonable both that the units should have
priority and posteriority if there is a first unit or first 1,
and also that the 2's should if there is a first 2; for after the
first it is reasonable and necessary that there should be a
second, and if a second, a third, and so with the others
successively. (And to say both things at the same time, that a
unit is first and another unit is second after the ideal 1, and
that a 2 is first after it, is impossible.) But they make a first
unit or 1, but not also a second and a third, and a first 2, but
not also a second and a third. Clearly, also, it is not possible,
if all the units are inassociable, that there should be a 2-itself
and a 3-itself; and so with the other numbers. For whether the
units are undifferentiated or different each from each, number
must be counted by addition, e.g. 2 by adding another 1 to the
one, 3 by adding another 1 to the two, and similarly. This being
so, numbers cannot be generated as they generate them, from the 2
and the 1; for 2 becomes part of 3 and 3 of 4 and the same
happens in the case of the succeeding numbers, but they say 4
came from the first 2 and the indefinite which makes it two 2's
other than the 2-itself; if not, the 2-itself will be a part of 4
and one other 2 will be added. And similarly 2 will consist of
the 1-itself and another 1; but if this is so, the other element
cannot be an indefinite 2; for it generates one unit, not, as the
indefinite 2 does, a definite 2.

Again, besides the 3-itself and the 2-itself how can there be
other 3's and 2's? And how do they consist of prior and posterior
units? All this is absurd and fictitious, and there cannot be a
first 2 and then a 3-itself. Yet there must, if the 1 and the
indefinite dyad are to be the elements. But if the results are
impossible, it is also impossible that these are the generating
principles.

If the units, then, are differentiated, each from each, these
results and others similar to these follow of necessity. But (3)
if those in different numbers are differentiated, but those in
the same number are alone undifferentiated from one another, even
so the difficulties that follow are no less. E.g. in the 10-itself
their are ten units, and the 10 is composed both of them and of
two 5's. But since the 10-itself is not any chance number nor
composed of any chance 5's—or, for that matter, units—the units
in this 10 must differ. For if they do not differ, neither will
the 5's of which the 10 consists differ; but since these differ,
the units also will differ. But if they differ, will there be no
other 5's in the 10 but only these two, or will there be others?
If there are not, this is paradoxical; and if there are, what
sort of 10 will consist of them? For there is no other in the 10
but the 10 itself. But it is actually necessary on their view
that the 4 should not consist of any chance 2's; for the
indefinite as they say, received the definite 2 and made two 2's;
for its nature was to double what it received.

Again, as to the 2 being an entity apart from its two units,
and the 3 an entity apart from its three units, how is this
possible? Either by one's sharing in the other, as 'pale man' is
different from 'pale' and 'man' (for it shares in these), or when
one is a differentia of the other, as 'man' is different from
'animal' and 'two-footed'.

Again, some things are one by contact, some by intermixture,
some by position; none of which can belong to the units of which
the 2 or the 3 consists; but as two men are not a unity apart
from both, so must it be with the units. And their being
indivisible will make no difference to them; for points too are
indivisible, but yet a pair of them is nothing apart from the two.

But this consequence also we must not forget, that it follows
that there are prior and posterior 2 and similarly with the other
numbers. For let the 2's in the 4 be simultaneous; yet these are
prior to those in the 8 and as the 2 generated them, they
generated the 4's in the 8-itself. Therefore if the first 2 is an
Idea, these 2's also will be Ideas of some kind. And the same
account applies to the units; for the units in the first 2
generate the four in 4, so that all the units come to be Ideas
and an Idea will be composed of Ideas. Clearly therefore those
things also of which these happen to be the Ideas will be
composite, e.g. one might say that animals are composed of
animals, if there are Ideas of them.

In general, to differentiate the units in any way is an
absurdity and a fiction; and by a fiction I mean a forced
statement made to suit a hypothesis. For neither in quantity nor
in quality do we see unit differing from unit, and number must be
either equal or unequal—all number but especially that which
consists of abstract units—so that if one number is neither
greater nor less than another, it is equal to it; but things that
are equal and in no wise differentiated we take to be the same
when we are speaking of numbers. If not, not even the 2 in the 10-itself
will be undifferentiated, though they are equal; for what reason
will the man who alleges that they are not differentiated be able
to give?

Again, if every unit + another unit makes two, a unit from the
2-itself and one from the 3-itself will make a 2. Now (a) this
will consist of differentiated units; and will it be prior to the
3 or posterior? It rather seems that it must be prior; for one of
the units is simultaneous with the 3 and the other is
simultaneous with the 2. And we, for our part, suppose that in
general 1 and 1, whether the things are equal or unequal, is 2, e.g.
the good and the bad, or a man and a horse; but those who hold
these views say that not even two units are 2.

If the number of the 3-itself is not greater than that of the
2, this is surprising; and if it is greater, clearly there is
also a number in it equal to the 2, so that this is not different
from the 2-itself. But this is not possible, if there is a first
and a second number.

Nor will the Ideas be numbers. For in this particular point
they are right who claim that the units must be different, if
there are to be Ideas; as has been said before. For the Form is
unique; but if the units are not different, the 2's and the 3's
also will not be different. This is also the reason why they must
say that when we count thus—'1,2'—we do not proceed by adding to
the given number; for if we do, neither will the numbers be
generated from the indefinite dyad, nor can a number be an Idea;
for then one Idea will be in another, and all Forms will be parts
of one Form. And so with a view to their hypothesis their
statements are right, but as a whole they are wrong; for their
view is very destructive, since they will admit that this
question itself affords some difficulty—whether, when we count
and say —1,2,3—we count by addition or by separate portions. But
we do both; and so it is absurd to reason back from this problem
to so great a difference of essence.

**Part 8 **

First of all it is well to determine what is the differentia
of a number—and of a unit, if it has a differentia. Units must
differ either in quantity or in quality; and neither of these
seems to be possible. But number *qua* number differs in quantity.
And if the units also did differ in quantity, number would differ
from number, though equal in number of units. Again, are the
first units greater or smaller, and do the later ones increase or
diminish? All these are irrational suppositions. But neither can
they differ in quality. For no attribute can attach to them; for
even to numbers quality is said to belong after quantity. Again,
quality could not come to them either from the 1 or the dyad; for
the former has no quality, and the latter gives quantity; for
this entity is what makes things to be many. If the facts are
really otherwise, they should state this quite at the beginning
and determine if possible, regarding the differentia of the unit,
why it must exist, and, failing this, what differentia they mean.

Evidently then, if the Ideas are numbers, the units cannot all
be associable, nor can they be inassociable in either of the two
ways. But neither is the way in which some others speak about
numbers correct. These are those who do not think there are
Ideas, either without qualification or as identified with certain
numbers, but think the objects of mathematics exist and the
numbers are the first of existing things, and the 1-itself is the
starting-point of them. It is paradoxical that there should be a
1 which is first of 1's, as they say, but not a 2 which is first
of 2's, nor a 3 of 3's; for the same reasoning applies to all.
If, then, the facts with regard to number are so, and one
supposes mathematical number alone to exist, the 1 is not the
starting-point (for this sort of 1 must differ from the other
units; and if this is so, there must also be a 2 which is first
of 2's, and similarly with the other successive numbers). But if
the 1 is the starting-point, the truth about the numbers must
rather be what Plato used to say, and there must be a first 2 and
3 and numbers must not be associable with one another. But if on
the other hand one supposes this, many impossible results, as we
have said, follow. But either this or the other must be the case,
so that if neither is, number cannot exist separately.

It is evident, also, from this that the third version is the
worst,—the view ideal and mathematical number is the same. For
two mistakes must then meet in the one opinion. (1) Mathematical
number cannot be of this sort, but the holder of this view has to
spin it out by making suppositions peculiar to himself. And (2)
he must also admit all the consequences that confront those who
speak of number in the sense of 'Forms'.

The Pythagorean version in one way affords fewer difficulties
than those before named, but in another way has others peculiar
to itself. For not thinking of number as capable of existing
separately removes many of the impossible consequences; but that
bodies should be composed of numbers, and that this should be
mathematical number, is impossible. For it is not true to speak
of indivisible spatial magnitudes; and however much there might
be magnitudes of this sort, units at least have not magnitude;
and how can a magnitude be composed of indivisibles? But
arithmetical number, at least, consists of units, while these
thinkers identify number with real things; at any rate they apply
their propositions to bodies as if they consisted of those
numbers.

If, then, it is necessary, if number is a self-subsistent real
thing, that it should exist in one of these ways which have been
mentioned, and if it cannot exist in any of these, evidently
number has no such nature as those who make it separable set up
for it.

Again, does each unit come from the great and the small,
equalized, or one from the small, another from the great? (a) If
the latter, neither does each thing contain all the elements, nor
are the units without difference; for in one there is the great
and in another the small, which is contrary in its nature to the
great. Again, how is it with the units in the 3-itself? One of
them is an odd unit. But perhaps it is for this reason that they
give 1-itself the middle place in odd numbers. (b) But if each of
the two units consists of both the great and the small,
equalized, how will the 2 which is a single thing, consist of the
great and the small? Or how will it differ from the unit? Again,
the unit is prior to the 2; for when it is destroyed the 2 is
destroyed. It must, then, be the Idea of an Idea since it is
prior to an Idea, and it must have come into being before it.
From what, then? Not from the indefinite dyad, for its function
was to double.

Again, number must be either infinite or finite; for these
thinkers think of number as capable of existing separately, so
that it is not possible that neither of those alternatives should
be true. Clearly it cannot be infinite; for infinite number is
neither odd nor even, but the generation of numbers is always the
generation either of an odd or of an even number; in one way,
when 1 operates on an even number, an odd number is produced; in
another way, when 2 operates, the numbers got from 1 by doubling
are produced; in another way, when the odd numbers operate, the
other even numbers are produced. Again, if every Idea is an Idea
of something, and the numbers are Ideas, infinite number itself
will be an Idea of something, either of some sensible thing or of
something else. Yet this is not possible in view of their thesis
any more than it is reasonable in itself, at least if they
arrange the Ideas as they do.

But if number is finite, how far does it go? With regard to
this not only the fact but the reason should be stated. But if
number goes only up to 10 as some say, firstly the Forms will
soon run short; e.g. if 3 is man-himself, what number will be the
horse-itself? The series of the numbers which are the several
things-themselves goes up to 10. It must, then, be one of the
numbers within these limits; for it is these that are substances
and Ideas. Yet they will run short; for the various forms of
animal will outnumber them. At the same time it is clear that if
in this way the 3 is man-himself, the other 3's are so also (for
those in identical numbers are similar), so that there will be an
infinite number of men; if each 3 is an Idea, each of the numbers
will be man-himself, and if not, they will at least be men. And
if the smaller number is part of the greater (being number of
such a sort that the units in the same number are associable),
then if the 4-itself is an Idea of something, e.g. of 'horse' or
of 'white', man will be a part of horse, if man is It is
paradoxical also that there should be an Idea of 10 but not of 11,
nor of the succeeding numbers. Again, there both are and come to
be certain things of which there are no Forms; why, then, are
there not Forms of them also? We infer that the Forms are not
causes. Again, it is paradoxical—if the number series up to 10 is
more of a real thing and a Form than 10 itself. There is no
generation of the former as one thing, and there is of the latter.
But they try to work on the assumption that the series of numbers
up to 10 is a complete series. At least they generate the
derivatives—e.g. the void, proportion, the odd, and the others of
this kind—within the decade. For some things, e.g. movement and
rest, good and bad, they assign to the originative principles,
and the others to the numbers. This is why they identify the odd
with 1; for if the odd implied 3 how would 5 be odd? Again,
spatial magnitudes and all such things are explained without
going beyond a definite number; e.g. the first, the indivisible,
line, then the 2 &c.; these entities also extend only up to
10.

Again, if number can exist separately, one might ask which is
prior—1, or 3 or 2? Inasmuch as the number is composite, 1 is
prior, but inasmuch as the universal and the form is prior, the
number is prior; for each of the units is part of the number as
its matter, and the number acts as form. And in a sense the right
angle is prior to the acute, because it is determinate and in
virtue of its definition; but in a sense the acute is prior,
because it is a part and the right angle is divided into acute
angles. As matter, then, the acute angle and the element and the
unit are prior, but in respect of the form and of the substance
as expressed in the definition, the right angle, and the whole
consisting of the matter and the form, are prior; for the
concrete thing is nearer to the form and to what is expressed in
the definition, though in generation it is later. How then is 1
the starting-point? Because it is not divisiable, they say; but
both the universal, and the particular or the element, are
indivisible. But they are starting-points in different ways, one
in definition and the other in time. In which way, then, is 1 the
starting-point? As has been said, the right angle is thought to
be prior to the acute, and the acute to the right, and each is
one. Accordingly they make 1 the starting-point in both ways. But
this is impossible. For the universal is one as form or
substance, while the element is one as a part or as matter. For
each of the two is in a sense one—in truth each of the two units
exists potentially (at least if the number is a unity and not
like a heap, i.e. if different numbers consist of differentiated
units, as they say), but not in complete reality; and the cause
of the error they fell into is that they were conducting their
inquiry at the same time from the standpoint of mathematics and
from that of universal definitions, so that (1) from the former
standpoint they treated unity, their first principle, as a point;
for the unit is a point without position. They put things
together out of the smallest parts, as some others also have done.
Therefore the unit becomes the matter of numbers and at the same
time prior to 2; and again posterior, 2 being treated as a whole,
a unity, and a form. But (2) because they were seeking the
universal they treated the unity which can be predicated of a
number, as in this sense also a part of the number. But these
characteristics cannot belong at the same time to the same thing.

If the 1-itself must be unitary (for it differs in nothing
from other 1's except that it is the starting-point), and the 2
is divisible but the unit is not, the unit must be liker the 1-itself
than the 2 is. But if the unit is liker it, it must be liker to
the unit than to the 2; therefore each of the units in 2 must be
prior to the 2. But they deny this; at least they generate the 2
first. Again, if the 2-itself is a unity and the 3-itself is one
also, both form a 2. From what, then, is this 2 produced?

**Part 9 **

Since there is not contact in numbers, but succession, viz.
between the units between which there is nothing, e.g. between
those in 2 or in 3 one might ask whether these succeed the 1-itself
or not, and whether, of the terms that succeed it, 2 or either of
the units in 2 is prior.

Similar difficulties occur with regard to the classes of
things posterior to number,—the line, the plane, and the solid.
For some construct these out of the species of the 'great and
small'; e.g. lines from the 'long and short', planes from the
'broad and narrow', masses from the 'deep and shallow'; which are
species of the 'great and small'. And the originative principle
of such things which answers to the 1 different thinkers describe
in different ways, And in these also the impossibilities, the
fictions, and the contradictions of all probability are seen to
be innumerable. For (i) geometrical classes are severed from one
another, unless the principles of these are implied in one
another in such a way that the 'broad and narrow' is also 'long
and short' (but if this is so, the plane will be line and the
solid a plane; again, how will angles and figures and such things
be explained?). And (ii) the same happens as in regard to number;
for 'long and short', &c., are attributes of magnitude, but
magnitude does not consist of these, any more than the line
consists of 'straight and curved', or solids of 'smooth and
rough'.

(All these views share a difficulty which occurs with regard
to species-of-a-genus, when one posits the universals, viz.
whether it is animal-itself or something other than animal-itself
that is in the particular animal. True, if the universal is not
separable from sensible things, this will present no difficulty;
but if the 1 and the numbers are separable, as those who express
these views say, it is not easy to solve the difficulty, if one
may apply the words 'not easy' to the impossible. For when we
apprehend the unity in 2, or in general in a number, do we
apprehend a thing-itself or something else?).

Some, then, generate spatial magnitudes from matter of this
sort, others from the point —and the point is thought by them to
be not 1 but something like 1—and from other matter like
plurality, but not identical with it; about which principles none
the less the same difficulties occur. For if the matter is one,
line and plane—and soli will be the same; for from the same
elements will come one and the same thing. But if the matters are
more than one, and there is one for the line and a second for the
plane and another for the solid, they either are implied in one
another or not, so that the same results will follow even so; for
either the plane will not contain a line or it will he a line.

Again, how number can consist of the one and plurality, they
make no attempt to explain; but however they express themselves,
the same objections arise as confront those who construct number
out of the one and the indefinite dyad. For the one view
generates number from the universally predicated plurality, and
not from a particular plurality; and the other generates it from
a particular plurality, but the first; for 2 is said to be a
'first plurality'. Therefore there is practically no difference,
but the same difficulties will follow,—is it intermixture or
position or blending or generation? and so on. Above all one
might press the question 'if each unit is one, what does it come
from?' Certainly each is not the one-itself. It must, then, come
from the one itself and plurality, or a part of plurality. To say
that the unit is a plurality is impossible, for it is
indivisible; and to generate it from a part of plurality involves
many other objections; for (a) each of the parts must be
indivisible (or it will be a plurality and the unit will be
divisible) and the elements will not be the one and plurality;
for the single units do not come from plurality and the one.
Again, (,the holder of this view does nothing but presuppose
another number; for his plurality of indivisibles is a number.
Again, we must inquire, in view of this theory also, whether the
number is infinite or finite. For there was at first, as it
seems, a plurality that was itself finite, from which and from
the one comes the finite number of units. And there is another
plurality that is plurality-itself and infinite plurality; which
sort of plurality, then, is the element which co-operates with
the one? One might inquire similarly about the point, i.e. the
element out of which they make spatial magnitudes. For surely
this is not the one and only point; at any rate, then, let them
say out of what each of the points is formed. Certainly not of
some distance + the point-itself. Nor again can there be
indivisible parts of a distance, as the elements out of which the
units are said to be made are indivisible parts of plurality; for
number consists of indivisibles, but spatial magnitudes do not.

All these objections, then, and others of the sort make it
evident that number and spatial magnitudes cannot exist apart
from things. Again, the discord about numbers between the various
versions is a sign that it is the incorrectness of the alleged
facts themselves that brings confusion into the theories. For
those who make the objects of mathematics alone exist apart from
sensible things, seeing the difficulty about the Forms and their
fictitiousness, abandoned ideal number and posited mathematical.
But those who wished to make the Forms at the same time also
numbers, but did not see, if one assumed these principles, how
mathematical number was to exist apart from ideal, made ideal and
mathematical number the same—in words, since in fact mathematical
number has been destroyed; for they state hypotheses peculiar to
themselves and not those of mathematics. And he who first
supposed that the Forms exist and that the Forms are numbers and
that the objects of mathematics exist, naturally separated the
two. Therefore it turns out that all of them are right in some
respect, but on the whole not right. And they themselves confirm
this, for their statements do not agree but conflict. The cause
is that their hypotheses and their principles are false. And it
is hard to make a good case out of bad materials, according to
Epicharmus: 'as soon as 'tis said, 'tis seen to be wrong.'

But regarding numbers the questions we have raised and the
conclusions we have reached are sufficient (for while he who is
already convinced might be further convinced by a longer
discussion, one not yet convinced would not come any nearer to
conviction); regarding the first principles and the first causes
and elements, the views expressed by those who discuss only
sensible substance have been partly stated in our works on
nature, and partly do not belong to the present inquiry; but the
views of those who assert that there are other substances besides
the sensible must be considered next after those we have been
mentioning. Since, then, some say that the Ideas and the numbers
are such substances, and that the elements of these are elements
and principles of real things, we must inquire regarding these
what they say and in what sense they say it.

Those who posit numbers only, and these mathematical, must be
considered later; but as regards those who believe in the Ideas
one might survey at the same time their way of thinking and the
difficulty into which they fall. For they at the same time make
the Ideas universal and again treat them as separable and as
individuals. That this is not possible has been argued before.
The reason why those who described their substances as universal
combined these two characteristics in one thing, is that they did
not make substances identical with sensible things. They thought
that the particulars in the sensible world were a state of flux
and none of them remained, but that the universal was apart from
these and something different. And Socrates gave the impulse to
this theory, as we said in our earlier discussion, by reason of
his definitions, but he did not separate universals from
individuals; and in this he thought rightly, in not separating
them. This is plain from the results; for without the universal
it is not possible to get knowledge, but the separation is the
cause of the objections that arise with regard to the Ideas. His
successors, however, treating it as necessary, if there are to be
any substances besides the sensible and transient substances,
that they must be separable, had no others, but gave separate
existence to these universally predicated substances, so that it
followed that universals and individuals were almost the same
sort of thing. This in itself, then, would be one difficulty in
the view we have mentioned.

**Part 10 **

Let us now mention a point which presents a certain difficulty
both to those who believe in the Ideas and to those who do not,
and which was stated before, at the beginning, among the problems.
If we do not suppose substances to be separate, and in the way in
which individual things are said to be separate, we shall destroy
substance in the sense in which we understand 'substance'; but if
we conceive substances to be separable, how are we to conceive
their elements and their principles?

If they are individual and not universal, (a) real things will
be just of the same number as the elements, and (b) the elements
will not be knowable. For (a) let the syllables in speech be
substances, and their elements elements of substances; then there
must be only one 'ba' and one of each of the syllables, since
they are not universal and the same in form but each is one in
number and a 'this' and not a kind possessed of a common name (and
again they suppose that the 'just what a thing is' is in each
case one). And if the syllables are unique, so too are the parts
of which they consist; there will not, then, be more a's than
one, nor more than one of any of the other elements, on the same
principle on which an identical syllable cannot exist in the
plural number. But if this is so, there will not be other things
existing besides the elements, but only the elements.

(b) Again, the elements will not be even knowable; for they
are not universal, and knowledge is of universals. This is clear
from demonstrations and from definitions; for we do not conclude
that this triangle has its angles equal to two right angles,
unless every triangle has its angles equal to two right angles,
nor that this man is an animal, unless every man is an animal.

But if the principles are universal, either the substances
composed of them are also universal, or non-substance will be
prior to substance; for the universal is not a substance, but the
element or principle is universal, and the element or principle
is prior to the things of which it is the principle or element.

All these difficulties follow naturally, when they make the
Ideas out of elements and at the same time claim that apart from
the substances which have the same form there are Ideas, a single
separate entity. But if, e.g. in the case of the elements of
speech, the a's and the b's may quite well be many and there need
be no a-itself and b-itself besides the many, there may be, so
far as this goes, an infinite number of similar syllables. The
statement that an knowledge is universal, so that the principles
of things must also be universal and not separate substances,
presents indeed, of all the points we have mentioned, the
greatest difficulty, but yet the statement is in a sense true,
although in a sense it is not. For knowledge, like the verb 'to
know', means two things, of which one is potential and one actual.
The potency, being, as matter, universal and indefinite, deals
with the universal and indefinite; but the actuality, being
definite, deals with a definite object, being a 'this', it deals
with a 'this'. But per accidens sight sees universal colour,
because this individual colour which it sees is colour; and this
individual a which the grammarian investigates is an a. For if
the principles must be universal, what is derived from them must
also be universal, as in demonstrations; and if this is so, there
will be nothing capable of separate existence—i.e. no substance.
But evidently in a sense knowledge is universal, and in a sense
it is not.