We have already explained what the
substance of sensible things is, dealing in our treatise on
physics^{1} with the material substrate, and subsequently
with substance as actuality.^{2}
Now since we are inquiring whether there is or is not some immutable
and eternal substance besides sensible substances, and if there is,
what it is, we must first examine the statements of other thinkers, so
that if they have been mistaken in any respect, we may not be liable
to the same mistakes; and if there is any view which is common to them
and us, we may not feel any private self-irritation on this score. For
we must be content if we state some points better than they have done,
and others no worse.

There are two views on this subject.
Some say that mathematical objects, i.e. numbers and lines, are
substances; and others again that the Ideas are substances.Now since some^{3}
recognize these as two classes—
[20]
the Ideas and the mathematical
numbers—and others^{4} regard both as having one
nature, and yet others^{5} hold that only the
mathematical substances are substances, we must first consider the
mathematical objects, without imputing to them any other
characteristic—e.g. by asking whether they are really Ideas
or not, or whether they are principles and substances of existing
things or not—and merely inquire whether as mathematical
objects they exist or not, and if they do, in what sense; then after
this we must separately consider the Ideas themselves, simply and in
so far as the accepted procedure requires; for most of the arguments
have been made familiar already by the criticisms of other
thinkers.And
further, the greater part of our discussion must bear directly upon
this second question—viz. when we are considering whether
the substances and first principles of existing things are numbers and
Ideas; for after we have dealt with the Ideas there remains this third
question.

Now if the objects of mathematics exist, they must be either in sensible things, as some hold; or separate from them (there are some also who hold this view); or if they are neither the one nor the other, either they do not exist at all, or they exist in some other way. Thus the point which we shall have to discuss is concerned not with their existence, but with the mode of their existence.

That the objects of
mathematics cannot be in sensible things, and that moreover the theory
that they are is a fabrication, has been observed already in our
discussion of difficulties^{6}
[1076b]
[1]
—the reasons being (a) that two solids
cannot occupy the same space, and (b) that on this same theory all
other potentialities and characteristics would exist in sensible
things, and none of them would exist separately. This, then, has been
already stated;but in
addition to this it is clearly impossible on this theory for any body
to be divided. For it must be divided in a plane, and the plane in a
line, and the line at a point; and therefore if the point is
indivisible, so is the line, and so on.For what difference does it make whether
entities of this kind are sensible objects, or while not being the
objects themselves, are yet present in them? the consequence will be
the same, for either they must be divided when the sensible objects
are divided, or else not even the sensible objects can be
divided.

Nor again can entities of this kind exist separately.For if besides sensible solids there are to be other solids which are separate from them and prior to sensible solids, clearly besides sensible planes there must be other separate planes, and so too with points and lines; for the same argument applies. And if these exist, again besides the planes, lines and points of the mathematical solid, there must be others which are separate;for the incomposite is prior to the composite, and if prior to sensible bodies there are other non-sensible bodies, [20] then by the same argument the planes which exist independently must be prior to those which are present in the immovable solids. Therefore there will be planes and lines distinct from those which coexist with the separately-existent solids; for the latter coexist with the mathematical solids, but the former are prior to the mathematical solids.Again, in these planes there will be lines, and by the same argument there must be other lines prior to these; and prior to the points which are in the prior lines there must be other points, although there will be no other points prior to these.Now the accumulation becomes absurd; because whereas we get only one class of solids besides sensible solids, we get three classes of planes besides sensible planes—those which exist separately from sensible planes, those which exist in the mathematical solids, and those which exist separately from those in the mathematical solids—four classes of lines, and five of points;with which of these, then, will the mathematical sciences deal? Not, surely, with the planes, lines and points in the immovable solid; for knowledge is always concerned with that which is prior. And the same argument applies to numbers; for there will be other units besides each class of points, and besides each class of existing things, first the sensible and then the intelligible; so that there will be an infinite number of kinds of mathematical numbers.

Again, there are the problems which we
enumerated in our discussion of difficulties^{7}: how can they be solved?
[1077a]
[1]
For
the objects of astronomy will similarly be distinct from sensible
things, and so will those of geometry; but how can a heaven and its
parts (or anything else which has motion) exist apart from the
sensible heaven? And similarly the objects of optics and of harmonics
will be distinct, for there will be sound and sight apart from the
sensible and particular objects.Hence clearly the other senses and objects of
sense will exist separately; for why should one class of objects do so
rather than another? And if this is so, animals too will exist
separately, inasmuch as the senses will.

Again, there are certain general mathematical theorems which are not restricted to these substances.Here, then, we shall have yet another kind of substance intermediate between and distinct from the Ideas and the intermediates, which is neither number nor points nor spatial magnitude nor time. And if this is impossible, clearly it is also impossible that the aforesaid substances should exist separately from sensible objects.

In general, consequences result which are contrary both to the truth and to received opinion if we thus posit the objects of mathematics as definite separately-existent entities. For if they exist in this way, they must be prior to sensible spatial magnitudes, whereas in truth they must be posterior to them; for the incomplete spatial magnitude is in point of generation prior, but in point of substantiality posterior, [20] as the inanimate is to the animate.

Again, in virtue of what can we possibly regard mathematical magnitudes as one? Things in this world of ours may be reasonably supposed to be one in virtue of soul or part of the soul, or some other influence; apart from this they are a plurality and are disintegrated. But inasmuch as the former are divisible and quantitative, what is the cause of their unity and cohesion?

Again, the ways in which
the objects of mathematics are generated prove our point;for they are generated first
in the dimension of length, then in that of breadth, and finally in
that of depth, whereupon the process is complete. Thus if that which
is posterior in generation^{8} is prior in substantiality, body will be prior
to plane and line, and in this sense it will also be more truly
complete and whole, because it can become animate; whereas how could a
line or plane be animate? The supposition is beyond our powers of
apprehension.

Further, body is a kind of substance, since it already in some sense possesses completeness; but in what sense are lines substances? Neither as being a kind of form or shape, as perhaps the soul is, nor as being matter, like the body; for it does not appear that anything can be composed either of lines or of planes or of points,whereas if they were a kind of material substance it would be apparent that things can be so composed. [1077b] [1] Let it be granted that they are prior in formula; yet not everything which is prior in formula is also prior in substantiality. Things are prior in substantiality which when separated have a superior power of existence; things are prior in formula from whose formulae the formulae of other things are compounded. And these characteristics are not indissociable.For if attributes, such as "moving" or "white," do not exist apart from their substances, "white" will be prior in formula to "white man," but not in substantiality; for it cannot exist in separation, but always exists conjointly with the concrete whole—by which I mean "white man."Thus it is obvious that neither is the result of abstraction prior, nor the result of adding a determinant posterior—for the expression "white man" is the result of adding a determinant to "white."

Thus we have sufficiently
shown (a) that the objects of mathematics are not more substantial
than corporeal objects; (b) that they are not prior in point of
existence to sensible things, but only in formula; and (c) that they
cannot in any way exist in separation.And since we have seen^{9} that they cannot exist in
sensible things, it is clear that either they do not exist at all, or
they exist only in a certain way, and therefore not absolutely; for
"exist" has several senses.

The general propositions in mathematics are not concerned with objects which exist separately apart from magnitudes and numbers; they are concerned with magnitudes and numbers, [20] but not with them as possessing magnitude or being divisible. It is clearly possible that in the same way propositions and logical proofs may apply to sensible magnitudes; not qua sensible, but qua having certain characteristics.For just as there can be many propositions about things merely qua movable, without any reference to the essential nature of each one or to their attributes, and it does not necessarily follow from this either that there is something movable which exists in separation from sensible things or that there is a distinct movable nature in sensible things; so too there will be propositions and sciences which apply to movable things, not qua movable but qua corporeal only; and again qua planes only and qua lines only, and qua divisible, and qua indivisible but having position, and qua indivisible only.Therefore since it is true to say in a general sense not only that things which are separable but that things which are inseparable exist, e.g., that movable things exist, it is also true to say in a general sense that mathematical objects exist, and in such a form as mathematicians describe them.And just as it is true to say generally of the other sciences that they deal with a particular subject—not with that which is accidental to it (e.g. not with "white" if "the healthy" is white, and the subject of the science is "the healthy"), but with that which is the subject of the particular science; [1078a] [1] with the healthy if it treats of things qua healthy, and with man if qua man—so this is also true of geometry. If the things of which it treats are accidentally sensible although it does not treat of them qua sensible, it does not follow that the mathematical sciences treat of sensible things—nor, on the other hand, that they treat of other things which exist independently apart from these.

Many attributes are essential properties of things as possessing a
particular characteristic; e.g., there are attributes peculiar to an
animal qua female or qua
male, although there is no such thing as female or male in separation
from animals. Hence there are also attributes which are peculiar to
things merely qua lines or planes.And in proportion as the things
which we are considering are prior in formula and simpler, they admit
of greater exactness; for simplicity implies exactness. Hence we find
greater exactness where there is no magnitude, and the greatest
exactness where there is no motion; or if motion is involved, where it
is primary, because this is the simplest kind; and the simplest kind
of primary motion is uniform motion.^{10}

The same principle applies to
both harmonics and optics, for neither of these sciences studies
objects qua sight or qua
sound, but qua lines and numbers^{11}; yet the latter are affections peculiar to
the former. The same is also true of mechanics.

Thus
if we regard objects independently of their attributes and investigate
any aspect of them as so regarded, we shall not be guilty of any error
on this account, any more than when we draw a diagram on the ground
and say that a line is a foot long when it is not;
[20]
because the error is not in the
premisses.^{12} The best way to
conduct an investigation in every case is to take that which does not
exist in separation and consider it separately; which is just what the
arithmetician or the geometrician does.For man, qua man, is
one indivisible thing; and the arithmetician assumes man to be one
indivisible thing, and then considers whether there is any attribute
of man qua indivisible. And the geometrician
considers man neither qua man nor qua indivisible, but qua something solid. For clearly the attributes which would have
belonged to "man" even if man were somehow not indivisible can belong
to man irrespectively of his humanity or indivisibility.Hence for this reason the
geometricians are right in what they maintain, and treat of what
really exists; i.e., the objects of geometry really exist. For things
can exist in two ways, either in complete reality or as matter.^{13}

And since
goodness is distinct from beauty (for it is always in actions that
goodness is present, whereas beauty is also in immovable things),
they^{14} are in error who
assert that the mathematical sciences tell us nothing about beauty or
goodness;for they
describe and manifest these qualities in the highest degree, since it
does not follow, because they manifest the effects and principles of
beauty and goodness without naming them, that they do not treat of
these qualities. The main species of beauty are orderly arrangement,
proportion, and definiteness;
[1078b]
[1]
and these are
especially manifested by the mathematical sciences.And inasmuch as it is evident that
these (I mean, e.g., orderly arrangement and definiteness) are causes
of many things, obviously they must also to some extent treat of the
cause in this sense, i.e. the cause in the sense of the Beautiful. But
we shall deal with this subject more explicitly elsewhere.^{15}

As regards the objects of
mathematics, then, the foregoing account may be taken as sufficient to
show that they exist, and in what sense they exist, and in what sense
they are prior and in what they are not. But as regards the Ideas we
must first consider the actual theory in relation to the Idea, without
connecting it in any way with the nature of numbers, but approaching
it in the form in which it was originally propounded by the first
exponents^{16} of the Ideas.

The
theory of Forms occurred to those who enunciated it because they were
convinced as to the true nature of reality by the doctrine of
Heraclitus, that all sensible things are always in a state of flux; so
that if there is to be any knowledge or thought about anything, there
must be certain other entities, besides sensible ones, which persist.
For there can be no knowledge of that which is in flux.Now
Socrates devoted his attention to
the moral virtues, and was the first to seek a general definition of
these
[20]
(for of the
Physicists Democritus gained only a superficial grasp of the
subject^{17}
and defined, after a fashion, "the hot" and "the cold"; while the
Pythagoreans^{18} at an earlier
date had arrived at definitions of some few things—whose
formulae they connected with numbers—e.g., what
"opportunity" is, or "justice" or "marriage"); and he naturally
inquired into the essence of things;for he was trying to reason logically, and the
starting-point of all logical reasoning is the essence. At that time
there was as yet no such proficiency in Dialectic that men could study
contraries independently of the essence, and consider whether both
contraries come under the same science.There are two innovations^{19}
which, may fairly be ascribed to
Socrates: inductive reasoning and general definition.
Both of these are associated with the starting-point of scientific
knowledge.

But whereas
Socrates regarded neither
universals nor definitions as existing in separation, the Idealists
gave them a separate existence, and to these universals and
definitions of existing things they gave the name of Ideas.^{20}
Hence on their view it followed by virtually the same argument that
there are Ideas of all terms which are predicated universally^{21}; and the result was very
nearly the same as if a man who wishes to count a number of things
were to suppose that he could not do so when they are few, and yet
were to try to count them when he has added to them. For it is hardly
an exaggeration to say that there are more Forms than there are
particular sensible things
[1079a]
[1]
(in seeking for
whose causes these thinkers were led on from particulars to Ideas);
because corresponding to each thing there is a synonymous entity,
apart from the substances (and in the case of non-substantial things
there is a One over the Many) both in our everyday world and in the
realm of eternal entities.

Again, not one of the ways in which it is attempted to prove that the Forms exist demonstrates their point; from some of them no necessary conclusion follows, and from others it follows that there are Form of things of which they hold that 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 "One-over-Many" argument, of negations too; and according to the argument that "we have some conception of what has perished" there will be Forms of perishable things, because we have a mental picture of these things. Further, of the most exact arguments some establish Ideas of relations, of which the Idealists deny that there is a separate genus, and others state the "Third Man."And in general the arguments for the Forms do away with things which are more important to the exponents of the Forms than the existence of the Ideas; for they imply that it is not the Dyad that is primary, but Number; and that the relative is prior to number, and therefore to the absolute; and all the other conclusions in respect of which certain persons by following up the views held about the Forms have gone against the principles of the theory.

Again, according to the assumption by which they hold that the Ideas exist, [20] there will be Forms not only of substances but of many other things (since the concept is one not only in the case of substances but in the case of non-substantial things as well; and there can be sciences not only of substances but also of other things; and there are a thousand other similar consequences);but it follows necessarily from the views generally held about them that if the Forms are participated in, there can only be Ideas of substances, because they are not participated in accidentally; things can only participate in a Form in so far as it is not predicated of a subject.I mean, e.g., that if a thing participates in absolute doubleness, it participates also in something eternal, but only accidentally; because it is an accident of "doubleness" to be eternal. Thus the Ideas will be substance. But the same terms denote substance in the sensible as in the Ideal world; otherwise what meaning will there be in saying that something exists besides the particulars, i.e. the unity comprising their multiplicity?If the form of the Ideas and of the things which participate in them is the same, they will have something in common (for why should duality mean one and the same thing in the case of perishable 2's and the 2's which are many but eternal, [1079b] [1] and not in the case of absolute duality and a particular 2?). But if the form is not the same, they will simply be homonyms; just as though one were to call both Callias and a piece of wood "man," without remarking any property common to them.

^{22}And if we profess
that in all other respects the common definitions apply to the Forms,
e.g. that "plane figure" and the other parts of the definition apply
to the Ideal circle, only that we must also state of what the Form is
a Form, we must beware lest this is a quite meaningless
statement.^{23} For to
what element of the definition must the addition be made? to "center,"
or "plane" or all of them? For all the elements in the essence of an
Idea are Ideas; e.g. "animal" and "two-footed."^{24} Further, it is obvious that "being an Idea," just like
"plane," must be a definite characteristic which belongs as genus to
all its species.^{25}

^{26}Above all we
might examine the question what on earth the Ideas contribute to
sensible things, whether eternal or subject to generation and decay;
for they are not the cause of any motion or change in them.Moreover they are no help
towards the knowledge of other things (for they are not the substance
of particulars, otherwise they would be **in** particulars)
or to their existence (since they are not present in the things which
participate in them. If they were, they might perhaps seem to be
causes, in the sense in which the admixture of white causes a thing to
be white.
[20]
But this theory, which was stated
first by Anaxagoras and later by Eudoxus in his discussion of
difficulties, and by others also, is very readily refuted; for it is
easy to adduce plenty of impossibilities against such a view). Again,
other things are not in any accepted sense derived from the
Forms.To say that
the Forms are patterns, and that other things participate in them, is
to use empty phrases and poetical metaphors; for what is it that
fashions things on the model of the Ideas? Besides, anything may both
be and come to be without being imitated from something else; thus a
man may become like Socrates
whether Socrates exists or
not,and even if
Socrates were eternal,
clearly the case would be the same. Also there will be several
"patterns" (and therefore Forms) of the same thing; e.g., "animal" and
"two-footed" will be patterns of "man," and so too will the Idea of
man.Further, the
Forms will be patterns not only of sensible things but of Ideas; e.g.
the genus will be the pattern of its species; hence the same thing
will be pattern and copy. Further, it would seem impossible for the
substance and that of which it is the substance to exist in
separation;
[1080a]
[1]
then how can the Ideas, if they are the
substances of things, exist in separation from them?

In the__Phaedo__^{27} this
statement is made: that the Forms are causes both of being and of
generation. Yet assuming that the Forms exist, still there is no
generation unless there is something to impart motion; and many other
things are generated (e.g. house and ring) of which the Idealists say
that there are no Forms.Thus it is clearly possible that those things of which they say that
there are Ideas may also exist and be generated through the same kind
of causes as those of the things which we have just mentioned, and not
because of the Forms. Indeed, as regards the Ideas, we can collect
against them plenty of evidence similar to that which we have now
considered; not only by the foregoing methods, but by means of more
abstract and exact reasoning.

Now that we have dealt with
the problems concerning the Ideas, we had better re-investigate the
problems connected with numbers that follow from the theory that
numbers are separate substances and primary causes of existing things.
Now if number is a kind of entity, and has nothing else as its
substance, but only number itself, as some maintain; then either (a)
there must be some one part of number which is primary, and some other
part next in succession, and so on, each part being specifically
different^{28}— and this applies directly to units,
and any given unit is inaddible to any other given unit;
[20]
or (b) they^{29} are all
directly successive, and any units can be added to any other units, as
is held of mathematical number; for in mathematical number no one unit
differs in any way from another.Or (c) some units must be addible and others
not. E.g., 2 is first after 1, and then 3, and so on with the other
numbers; and the units in each number are addible, e.g. the units in
the first^{30}2 are addible to one another, and those in the
first 3 to one another, and so on in the case of the other numbers;
but the units in the Ideal 2 are inaddible to those in the Ideal
3;and similarly in
the case of the other successive numbers. Hence whereas mathematical
number is counted thus: after 1, 2 (which consists of another 1 added
to the former) and 3 (which consists of another 1 added to these two)
and the other numbers in the same way, Ideal number is counted like
this: after 1, a distinct 2 not including the original 1; and a 3 not
including the 2, and the rest of the numbers similarly.Or (d) one kind of number must
be such as we first described, and another or such as the
mathematicians maintain, and that which we have last described must be
a third kind.

Again, these numbers must
exist either in separation from things,
[1080b]
[1]
or not
in separation, but in sensible things (not, however, in the way which
we first considered,^{31} but in the
sense that sensible things are composed of numbers which are present
in them^{32}）—either
some of them and not others, or all of them.^{33} These are of necessity the only ways
in which the numbers can exist. Now of those who say that unity is the
beginning and substance and element of all things, and that number is
derived from it and something else, almost everyone has described
number in one of these ways (except that no one has maintained that
all units are inaddible^{34});and
this is natural enough, because there can be no other way apart from
those which we have mentioned. Some hold that both kinds of number
exist, that which involves priority and posteriority being identical
with the Ideas, and mathematical number being distinct from Ideas and
sensible things, and both kinds being separable from sensible
things^{35}; others hold that
mathematical number alone exists,^{36} being the primary reality and separate from
sensible things.

The Pythagoreans also believe in one
kind of number—the mathematical; only they maintain that it
is not separate, but that sensible substances are composed of it. For
they construct the whole universe of numbers, but not of numbers
consisting of abstract units;
[20]
they suppose the units to be extended—but as for how the
first extended unit was formed they appear to be at a loss.^{37}

Another thinker holds that
primary or Ideal number alone exists; and some^{38} identify this with mathematical
number.

The same applies in the
case of lines, planes and solids.Some^{39} distinguish mathematical objects from those which
"come after the Ideas"^{40}; and of those who treat the subject
in a different manner some^{41} speak of the mathematical
objects and in a mathematical way—viz. those who do not
regard the Ideas as numbers, nor indeed hold that the Ideas
exist—and others^{42} speak of the mathematical
objects, but not in a mathematical way; for they deny that every
spatial magnitude is divisible into extended magnitudes, or that any
two given units make 2.But all who hold that Unity is an element and principle of existing
things regard numbers as consisting of abstract units, except the
Pythagoreans; and they regard number as having spatial magnitude, as
has been previously stated.^{43}

It is clear from
the foregoing account (1.) in how many ways it is possible to speak of
numbers, and that all the ways have been described. They are all
impossible, but doubtless some^{44} are more so than others.

First, then, we must inquire whether the limits are addible or
inaddible;
[1081a]
[1]
and if inaddible, in which of the two
ways which we have distinguished.^{45} For it is possible either (a) that any one unit is
inaddible to any other, or (b) that the units in the Ideal 2 are
inaddible to those in the Ideal 3, and thus that the units in each
Ideal number are inaddible to those in the other Ideal
numbers.

Now if all units are addible and do not
differ in kind, we get one type of number only, the mathematical, and
the Ideas cannot be the numbers thus produced;for how can we regard the Idea of Man
or Animal, or any other Form, as a number? There is one Idea of each
kind of thing: e.g. one of Humanity and another one of Animality; but
the numbers which are similar and do not differ in kind are infinitely
many, so that this is no more the Idea of Man than any other 3 is. But
if the Ideas are not numbers, they cannot exist at all;for from what principles can
the Ideas be derived? Number is derived from Unity and the
indeterminate dyad, and the principles and elements are said to be the
principles and elements of number, and the Ideas cannot be placed
either as prior or as posterior to numbers.^{46}

But if the units are inaddible in the
sense that any one unit is inaddible to any other, the number so
composed can be neither mathematical number (since mathematical number
consists of units which do not differ,
[20]
and the facts demonstrated of it fit in with this
character) nor Ideal number. For on this view 2 will not be the first
number generated from Unity and the indeterminate dyad, and then the
other numbers in succession, as they^{47} say 2, 3, because the
units in the primary 2 are generated at the same time,^{48} whether,
as the originator of the theory held, from unequals^{49}(coming into being when these were
equalized), or otherwise— since if we regard the one unit as prior to
the other,^{50} it will be prior
also to the 2 which is composed of them; because whenever one thing is
prior and another posterior, their compound will be prior to the
latter and posterior to the former.^{51}

Further, since the Ideal 1 is first, and then comes a particular 1 which is first of the other 1's but second after the Ideal 1, and then a third 1 which is next

after the second
but third after the first 1, it follows that the units will be prior
to the numbers after which they are called; e.g., there will be a
third unit in 2 before 3 exists, and a fourth and fifth in 3 before
these numbers exist.^{52}

It is true that nobody has represented
the units of numbers as inaddible in this way; but according to the
principles held by these thinkers even this view is quite reasonable,
[1081b]
[1]
although in actual fact it is
untenable.For
assuming that there is a first unit or first 1,^{53} it is reasonable that
the units should be prior and posterior; and similarly in the case of
2's, if there is a first 2. For it is reasonable and indeed necessary
that after the first there should be a second; and if a second, a
third; and so on with the rest in sequence.But the two statements, that there is after 1
a first and a second unit, and that there is a first 2, are
incompatible. These thinkers, however, recognize a first unit and
first 1, but not a second and third; and they recognize a first 2, but
not a second and third.

It is also
evident that if all units are inaddible, there cannot be an Ideal 2
and 3, and similarly with the other numbers;for whether the units are indistinguishable or
each is different in kind from every other, numbers must be produced
by addition; e.g. 2 by adding 1 to another 1, and 3 by adding another
1 to the 2, and 4 similarly.^{54}
This being so, numbers cannot be generated as these thinkers try to
generate them, from Unity and the dyad; because 2 becomes a part of
3,^{55} and 3 of 4,
[20]
and the same applies to the following
numbers.But
according to them 4 was generated from the first 2 and the
indeterminate dyad, thus consisting of two 2's apart from the Ideal
2.^{56}
Otherwise 4 will consist of the Ideal 2 and another 2 added to it, and
the Ideal 2 will consist of the Ideal 1 and another 1; and if this is
so the other element cannot be the indeterminate dyad, because it
produces one unit and not a definite 2.^{57}

Again, how can there be other 3's and 2's besides the Ideal numbers
3 and 2, and in what way can they be composed of prior and posterior
units? All these theories are absurd and fictitious, and there can be
no primary 2 and Ideal 3. Yet there must be, if we are to regard Unity
and the indeterminate dyad as elements.^{58} But if the consequences are
impossible, the principles cannot be of this nature.

If, then, any one unit differs in kind from any
other, these and other similar consequences necessarily follow. If, on
the other hand, while the units in different numbers are different,
those which are in the same number are alone indistinguishable from
one another, even so the consequences which follow are no less
difficult.
[1082a]
[1]
For example, in the Ideal number 10 there are
ten units, and 10 is composed both of these and of two 5's. Now since
the Ideal 10 is not a chance number,^{59} and is not composed of chance 5's, any more than of
chance units, the units in this number 10 must be different;for if they are not different,
the 5's of which the 10 is composed will not be different; but since
these are different, the units must be different too. Now if the units
are different, will there or will there not be other 5's in this 10,
and not only the two? If there are not, the thing is absurd^{60}; whereas if there are, what
sort of 10 will be composed of them? for there is no other 10 in 10
besides the 10 itself:

Again, it must also be true
that 4 is not composed of chance 2's. For according to them the
indeterminate dyad, receiving the determinate dyad, made two dyads;
for it was capable of duplicating that which it received.^{61}

Again, how is it possible that 2 can be a definite entity existing besides the two units, and 3 besides the three units? Either by participation of the one in the other, as "white man" exists besides "white" and "man," because it partakes of these concepts; or when the one is a differentia of the other, as "man" exists besides "animal" and "two-footed."

[20] Again, some things are one by contact, others by mixture, and others by position; but none of these alternatives can possibly apply to the units of which 2 and 3 consist. Just as two men do not constitute any one thing distinct from both of them, so it must be with the units.The fact that the units are indivisible will make no difference; because points are indivisible also, but nevertheless a pair of points is not anything distinct from the two single points.

Moreover we must
not fail to realize this: that on this theory it follows that 2's are
prior and posterior, and the other numbers similarly.Let it be granted that the 2's
in 4 are contemporaneous; yet they are prior to those in 8, and just
as the <determinate> 2 produced the 2's in 4, so^{62} they produced
the 4's in 8. Hence if the original 2 is an Idea, these 2's will also
be Ideas of a sort.And
the same argument applies to the units, because the units in the
original 2 produce the four units in 4; and so all the units become
Ideas, and an Idea will be composed of Ideas. Hence clearly those
things also of which these things are Ideas will be composite;
[1082b]
[1]
e.g., one might say that animals are
composed of animals, if there are Ideas of animals.

In general, to regard units as different in any way whatsoever is absurd and fictitious (by "fictitious" I mean "dragged in to support a hypothesis"). For we can see that one unit differs from another neither in quantity nor in quality; and a number must be either equal or unequal—this applies to all numbers, but especially to numbers consisting of abstract units.Thus if a number is neither more nor less, it is equal; and things which are equal and entirely without difference we assume, in the sphere of number, to be identical. Otherwise even the 2's in the Ideal 10 will be different, although they are equal; for if anyone maintains that they are not different, what reason will he be able to allege?

Again, if every unit plus
another unit makes 2, a unit from the Ideal 2 plus one from the Ideal
3 will make 2—a 2 composed of different units^{63}; will this be prior or posterior to 3?
It rather seems that it must be prior, because one of the units is
contemporaneous with 3, and the other with 2.^{64} We
assume that in general 1 and 1, whether the things are equal or
unequal, make 2; e.g. good and bad, or man and horse; but the
supporters of this theory say that not even two units make
2.

If the number of the Ideal 3 is
not greater than that of the Ideal 2,
[20]
it is strange; and if it is greater, then clearly
there is a number in it equal to the 2, so that this number is not
different from the Ideal 2.But this is impossible, if there is a first and second number.^{65} Nor will the Ideas be numbers. For on this
particular point they are right who claim that the units must be
different if there are to be Ideas, as has been already stated.^{66} For the
form is unique; but if the units are undifferentiated, the 2's and 3's
will be undifferentiated.Hence they have to say that when we count like this, l, 2, we do not
add to the already existing number; for if we do, (a) number will not
be generated from the indeterminate dyad, and (b) a number cannot be
an Idea; because one Idea will pre-exist in another, and all the Forms
will be parts of one Form.^{67} Thus in relation to their hypothesis they are
right, but absolutely they are wrong, for their view is very
destructive, inasmuch as they will say that this point presents a
difficulty: whether, when we count and say "1, 2, 3," we count by
addition or by enumerating distinct portions.^{68} But we do both;
and therefore it is ridiculous to refer this point to so great a
difference in essence.
[1083a]
[1]

First of all it
would be well to define the differentia of a number; and of a unit, if
it has a differentia. Now units must differ either in quantity or in
quality; and clearly neither of these alternatives can be true. "But
units may differ, as number does, in quantity." But if units also
differed in quantity, number would differ from number, although equal
in number of units.Again,
are the first units greater or smaller, and do the later units
increase in size, or the opposite? All these suggestions are absurd.
Nor can units differ in quality; for no modification can ever be
applicable to them, because these thinkers hold that even in numbers
quality is a later attribute than quantity.^{69} Further, the units cannot
derive quality either from unity or from the dyad; because unity has
no quality, and the dyad produces quantity, because its nature causes
things to be many. If, then, the units differ in some other way, they
should most certainly state this at the outset, and explain, if
possible, with regard to the differentia of the unit, why it must
exist; or failing this, what differentia they mean.

Clearly, then, if the Ideas are numbers, the units cannot all be
addible,
[20]
nor can they
all be inaddible in either sense. Nor again is the theory sound which
certain other thinkers^{70} hold concerning
numbers.These are
they who do not believe in Ideas, either absolutely or as being a kind
of numbers, but believe that the objects of mathematics exist, and
that the numbers are the first of existing things, and that their
principle is Unity itself. For it is absurd that if, as they say,
there is a 1 which is first of the 1's,^{71} there should
not be a 2 first of the 2's, nor a 3 of the 3's; for the same
principle applies to all cases.Now if this is the truth with regard to
number, and we posit only mathematical number as existing, Unity is
not a principle. For the Unity which is of this nature must differ
from the other units; and if so, then there must be some 2 which is
first of the 2's; and similarly with the other numbers in
succession.But if
Unity is a principle, then the truth about numbers must rather be as
Plato used to maintain; there must be a first 2 and first 3, and the
numbers cannot be addible to each other. But then again, if we assume
this, many impossibilities result, as has been already stated.^{72} Moreover, the truth must lie one
way or the other; so that if neither view is sound,
[1083b]
[1]
number cannot have a separate abstract
existence.

From these considerations it is also
clear that the third alternative^{73}—that Ideal number and mathematical
number are the same—is the worst; for two errors have to be
combined to make one theory. (1.) Mathematical number cannot be of
this nature, but the propounder of this view has to spin it out by
making peculiar assumptions; (2.) his theory must admit all the
difficulties which confront those who speak of Ideal number.

The Pythagorean view in one way contains fewer difficulties than the
view described above, but in another way it contains further
difficulties peculiar to itself. By not regarding number as separable,
it disposes of many of the impossibilities; but that bodies should be
composed of numbers, and that these numbers should be mathematical, is
impossible.^{74} For (a) it is not true to speak of indivisible magnitudes^{75}; (b) assuming that this view is
perfectly true, still units at any rate have no magnitude; and how can
a magnitude be composed of indivisible parts? Moreover arithmetical
number consists of abstract units. But the Pythagoreans identify
number with existing things; at least they apply mathematical
propositions to bodies as though they consisted of those numbers.^{76}

Thus if number, [20] if it is a self-subsistent reality, must be regarded in one of the ways described above, and if it cannot be regarded in any of these ways, clearly number has no such nature as is invented for it by those who treat it as separable.

Again, does each unit come from the
Great and the Small, when they are equalized^{77}; or does one come from the Small and another from the
Great? If the latter, each thing is not composed of all the elements,
nor are the units undifferentiated; for one contains the Great, and
the other the Small, which is by nature contrary to the
Great.Again, what
of the units in the Ideal 3? because there is one over. But no doubt
it is for this reason that in an odd number they make the Ideal One
the middle unit.^{78} If on the
other hand each of the units comes from both Great and Small, when
they are equalized, how can the Ideal 2 be a single entity composed of
the Great and Small? How will it differ from one of its units? Again,
the unit is prior to the 2; because when the unit disappears the 2
disappears.Therefore the unit must be the Idea of an Idea, since it is prior to
an Idea, and must have been generated before it. From what, then? for
the indeterminate dyad, as we have seen,^{79}
causes duality.

Again, number must be
either infinite or finite (for they make number separable,
[1084a]
[1]
so that one of these alternatives must be
true).^{80} Now
it is obvious that it cannot be infinite, because infinite number is
neither odd nor even, and numbers are always generated either from odd
or from even number. By one process, when 1 is added to an even
number, we get an odd number; by another, when 1 is multiplied by 2,
we get ascending powers of 2; and by another, when powers of 2 are
multiplied by odd numbers, we get the remaining even
numbers.

Again, if every Idea is an Idea of
something, and the numbers are Ideas, infinite number will also be an
Idea of something, either sensible or otherwise. This, however, is
impossible, both logically^{81} and on their own
assumption,^{82} since they regard the Ideas as they
do.

If, on the other hand, number
is finite, what is its limit? In reply to this we must not only assert
the fact, but give the reason.Now if number only goes up to 10, as some
hold,^{83} in the
first place the Forms will soon run short. For example, if 3 is the
Idea of Man, what number will be the Idea of Horse? Each number up to
10 is an Idea; the Idea of Horse, then, must be one of the numbers in
this series, for they are substances or Ideas.But the fact remains that they will run
short, because the different types of animals will outnumber them. At
the same time it is clear that if in this way the Ideal 3 is the Idea
of Man, so will the other 3's be also (for the 3's in the same
numbers^{84} are
similar),
[20]
so that
there will be an infinite number of men; and if each 3 is an Idea,
each man will be an Idea of Man; or if not, they will still be
men.And if the
smaller number is part of the greater, when it is composed of the
addible units contained in the same number, then if the Ideal 4 is the
Idea of something, e.g. "horse" or "white," then "man" will be part of
"horse," if "man" is 2. It is absurd also that there should be an Idea
of 10 and not of 11, nor of the following numbers.

Again, some things exist and come into being of which there are no
Forms^{85}; why, then, are there not Forms of
these too? It follows that the Forms are not the causes of
things.

Again, it is absurd that
number up to 10 should be more really existent, and a Form, than 10
itself; although the former is not generated as a unity, whereas the
latter is. However, they try to make out that the series up to 10 is a
complete number;at least
they generate the derivatives, e.g. the void, proportion, the odd,
etc., from within the decad. Some, such as motion, rest, good and
evil, they assign to the first principles; the rest to numbers.^{86} Hence they identify the odd with Unity;
because if oddness depended on 3, how could 5 be odd?^{87}

Again, they hold that spatial magnitudes and the
like have a certain limit;
[1084b]
[1]
e.g. the first or
indivisible line, then the 2, and so on; these too extending up to
10.^{88}

Again, if
number is separable, the question might be raised whether Unity is
prior, or 3 or 2.Now if
we regard number as composite, Unity is prior; but if we regard the
universal or form as prior, number is prior, because each unit is a
material part of number, while number is the form of the units. And
there is a sense in which the right angle is prior to the acute
angle—since it is definite and is involved in the definition
of the acute angle—and another sense in which the acute
angle is prior, because it is a part of the other, i.e., the right
angle is divided into acute angles.Thus regarded as matter the acute angle and
element and unit are prior; but with respect to form and substance in
the sense of formula, the right angle, and the whole composed of
matter and form, is prior. For the concrete whole is nearer to the
form or subject of the definition, although in generation it is
posterior.^{89}

In what sense, then, is the One a first principle?
Because, they say, it is indivisible.But the universal and the part or element are
also indivisible. Yes, but they are prior in a different sense; the
one in formula and the other in time. In which sense, then, is the One
a first principle? for, as we have just said, both the right angle
seems to be prior to the acute angle, and the latter prior to the
former; and each of them is one.Accordingly the Platonists make the One a
first principle in both senses. But this is impossible; for in one
sense it is the One qua form or
essence,
[20]
and in the
other the One qua part or matter, that is
primary. There is a sense in which both number and unit are one; they
are so in truth potentially—that is, if a number is not an
aggregate but a unity consisting of units distinct from those of other
numbers, as the Platonists hold— but each of the two^{90} units is not one in
complete reality. The cause of the error which befell the Platonists
was that they were pursuing their inquiry from two points of
view—that of mathematics and that of general
definition—at the same time. Hence as a result of the former
they conceived of the One or first principle as a point, for the unit
is a point without position. (Thus they too, just like certain
others,represented
existing things as composed of that which is smallest.)^{91} We get,
then, that the unit is the material element of numbers, and at the
same time is prior to the number 2; and again we get that it is
posterior to 2 regarded as a whole or unity or form. On the other
hand, through looking for the universal, they were led to speak of the
unity predicated of a given number as a part in the formal sense also.
But these two characteristics cannot belong simultaneously to the same
thing.

And if Unity itself must only be without
position^{92}(for it differs only in that it is a principle) and 2
is divisible whereas the unit is not, the unit will be more nearly
akin to Unity itself; and if this is so, Unity itself will also be
more nearly akin to the unit than to 2. Hence each of the units in 2
will be prior to 2. But this they deny; at least they make out that 2
is generated first.^{93}
[1085a]
[1]
Further, if 2 itself and 3 itself are each one
thing, both together make 2. From what, then, does this 2
come?

Since there is no contact in numbers, but units which have nothing between them—e.g. those in 2 or 3—are successive, the question might be raised whether or not they are successive to Unity itself, and whether of the numbers which succeed it 2 or one of the units in 2 is prior.

We
find similar difficulties in the case of the genera posterior to
number^{94}—the line, plane and solid. Some derive these from
the species of the Great and Small; viz. lines from the Long and
Short, planes from the Broad and Narrow, and solids from the Deep and
Shallow. These are species of the Great and Small.As for the geometrical first principle
which corresponds to the arithmetical One, different Platonists
propound different views.^{95} In these too we can
see innumerable impossibilities, fictions and contradictions of all
reasonable probability. For (a) we get that the geometrical forms are
unconnected with each other, unless their principles also are so
associated that the Broad and Narrow is also Long and Short; and if
this is so, the plane will be a line and the solid a plane.
[20]
Moreover, how can angles and figures, etc.,
be explained? And (b) the same result follows as in the case of
number; for these concepts are modifications of magnitude, but
magnitude is not generated from them, any more than a line is
generated from the Straight and Crooked, or solids from the Smooth and
Rough.

Common to all these Platonic theories is the same problem which presents itself in the case of species of a genus when we posit universals—viz. whether it is the Ideal animal that is present in the particular animal, or some other "animal" distinct from the Ideal animal. This question will cause no difficulty if the universal is not separable; but if, as the Platonists say, Unity and the numbers exist separately, then it is not easy to solve (if we should apply the phrase "not easy" to what is impossible).For when we think of the one in 2, or in number generally, are we thinking of an Idea or of something else?

These
thinkers, then, generate geometrical magnitudes from this sort of
material principle, but others^{96} generate them from the point (they regard the point
not as a unity but as similar to Unity) and another material principle
which is not plurality but is similar to it; yet in the case of these
principles none the less we get the same difficulties.For if the matter is one, line,
plane and solid will be the same; because the product of the same
elements must be one and the same.
[1085b]
[1]
If on the other hand
there is more than one kind of matter—one of the line,
another of the plane, and another of the solid—either the
kinds are associated with each other, or they are not. Thus the same
result will follow in this case also; for either the plane will not
contain a line, or it will be a line.

Further, no
attempt is made to explain how number can be generated from unity and
plurality; but howsoever they account for this, they have to meet the
same difficulties as those who generate number from unity and the
indeterminate dyad. The one school generates number not from a
particular plurality but from that which is universally predicated;
the other from a particular plurality, but the first; for they hold
that the dyad is the first plurality.^{97} Thus there is practically no difference
between the two views; the same difficulties will be involved with
regard to mixture, position, blending, generation and the other
similar modes of combination.^{98}

We might very well ask the further question: if
each unit is one, of what it is composed; for clearly each unit is not
absolute unity. It must be generated from absolute unity and either
plurality or a part of plurality.Now we cannot hold that the unit is a
plurality, because the unit is indivisible; but the view that it is
derived from a part of plurality involves many further difficulties,
because (a) each part must be indivisible; otherwise it will be a
plurality and the unit will be divisible,
[20]
and unity and plurality will not be its elements,
because each unit will not be generated from plurality^{99} and unity.(b) The exponent of this theory merely
introduces another number; because plurality is a number of
indivisible parts.^{100}

Again, we must
inquire from the exponent of this theory whether the number^{101} is infinite or finite.There was, it appears, a finite
plurality from which, in combination with Unity, the finite units were
generated; and absolute plurality is different from finite plurality.
What sort of plurality is it, then, that is, in combination with
unity, an element of number?

We might ask a similar question with regard to the point, i.e. the element out of which they create spatial magnitudes.This is surely not the one and only point. At least we may ask from what each of the other points comes; it is not, certainly, from some interval and the Ideal point. Moreover, the parts of the interval cannot be indivisible parts, any more than the parts of the plurality of which the units are composed; because although number is composed of indivisible parts, spatial magnitudes are not.

All these and other similar considerations
make it clear that number and spatial magnitudes cannot exist
separately.
[1086a]
[1]
Further, the fact that the leading
authorities^{102} disagree about numbers indicates that it is the
misrepresentation of the facts themselves that produces this confusion
in their views.Those^{103} who recognize only the objects of mathematics as
existing besides sensible things, abandoned Ideal number and posited
mathematical number because they perceived the difficulty and
artificiality of the Ideal theory. Others,^{104} wishing to
maintain both Forms and numbers, but not seeing how, if one posits
these^{105} as first principles,
mathematical number can exist besides Ideal number, identified Ideal
with mathematical number,—but only in theory, since actually
mathematical number is done away with, because the hypotheses which
they state are peculiar to them and not mathematical.^{106} And he^{107} who first assumed that there are Ideas, and that
the Ideas are numbers, and that the objects of mathematics exist,
naturally separated them. Thus it happens that all are right in some
respect, but not altogether right; even they themselves admit as much
by not agreeing but contradicting each other. The reason of this is
that their assumptions and first principles are wrong;and it is difficult to
propound a correct theory from faulty premisses: as Epicharmus says,
"no sooner is it said than it is seen to be wrong."^{108}

We have
now examined and analyzed the questions concerning numbers to a
sufficient extent; for although one who is already convinced might be
still more convinced by a fuller treatment,
[20]
he who is not convinced would be
brought no nearer to conviction.As for the first principles and causes and
elements, the views expressed by those who discuss only sensible
substance either have been described in the __Physics__^{109} or have no place in
our present inquiry; but the views of those who assert that there are
other substances besides sensible ones call for investigation next
after those which we have just discussed.

Since, then, some thinkers hold that the Ideas and numbers are such substances, and that their elements are the elements and principles of reality, we must inquire what it is that they hold, and in what sense they hold it.

Those^{110} who posit only numbers,
and mathematical numbers at that, may be considered later^{111}; but as
for those who speak of the Ideas, we can observe at the same time
their way of thinking and the difficulties which befall them. For they
not only treat the Ideas as universal substances, but also as
separable and particular.(That this is impossible has been already shown^{112} by a consideration of the
difficulties involved.) The reason why those who hold substances to be
universal combined these two views was that they did not identify
substances with sensible things.
[1086b]
[1]
They considered that
the particulars in the sensible world are in a state of flux, and that
none of them persists, but that the universal exists besides them and
is something distinct from them.This theory, as we have said in an earlier
passage,^{113} was initiated by
Socrates as a result of his definitions, but he did
not separate universals from particulars; and he was right in not
separating them. This is evident from the facts; for without the
universal we cannot acquire knowledge, and the separation of the
universal is the cause of the difficulties which we find in the Ideal
theory.Others,^{114}
regarding it as necessary, if there are to be any substances besides
those which are sensible and transitory, that they should be
separable, and having no other substances, assigned separate existence
to those which are universally predicated; thus it followed that
universals and particulars are practically the same kind of thing.
This in itself would be one difficulty in the view which we have just
described.^{115}

Let us now mention a point
which presents some difficulty both to those who hold the Ideal theory
and to those who do not. It has been stated already, at the beginning
of our treatise, among the problems.^{116} If we do
not suppose substances to be separate, that is in the way in which
particular things are said to be separate, we shall do away with
substance in the sense in which we wish to maintain it; but if we
suppose substances to be separable,
[20]
how are we to regard their elements and
principles?If they
are particular and not universal, there will be as many real things as
there are elements, and the elements will not be knowable. For let us
suppose that the syllables in speech are substances, and that their
letters are the elements of substances. Then there must be only one
BA, and only one of each of the other syllables; that is, if they are
not universal and identical in form, but each is numerically one and
an individual, and not a member of a class bearing a common
name.(Moreover, the
Platonists assume that each Ideal entity is unique.) Now if this is
true of the syllables, it is also true of their letters. Hence there
will not be more than one A, nor more than one of any of the other
letters,^{117} on the same
argument by which in the case of the syllable there cannot be more
than one instance of the same syllable. But if this is so, there will
be no other things besides the letters, but only the
letters.

Nor again will the elements be knowable; for they will not be universal, and knowledge is of the universal. This can be seen by reference to proofs and definitions; for there is no logical conclusion that a given triangle has its angles equal to two right angles unless every triangle has its angles equal to two right angles, or that a given man is an animal unless every man is an animal. [1087a] [1]

On the other hand, if the first principles are universal, either the substances composed of them will be universal too, or there will be a non-substance prior to substance; because the universal is not substance, and the element or first principle is universal; and the element or first principle is prior to that of which it is an element or first principle.All this naturally follows when they compose the Ideas of elements and assert that besides the substances which have the same form there are also Ideas each of which is a separate entity.

But if, as in the case of the phonetic elements, there is no reason why there should not be many A's and B's, and no "A itself" or "B itself" apart from these many, then on this basis there may be any number of similar syllables.

The doctrine
that all knowledge is of the universal, and hence that the principles
of existing things must also be universal and not separate substances,
presents the greatest difficulty of all that we have discussed; there
is, however, a sense in which this statement is true, although there
is another in which it is not true.Knowledge, like the verb "to know," has two
senses, of which one is potential and the other actual. The
potentiality being, as matter, universal and indefinite, has a
universal and indefinite object; but the actuality is definite and has
a definite object, because it is particular and deals with the
particular.It is
only accidentally that sight sees universal color,
[20]
because the particular color which it
sees is color; and the particular A which the grammarian studies is an
A. For if the first principles must be universal, that which is
derived from them must also be universal, as in the case of logical
proofs^{118};
and if this is so there will be nothing which has a separate
existence; i.e. no substance. But it is clear that although in one
sense knowledge is universal, in another it is not.