physicists suppose to exist alongside the elements: for everything
changes from contrary to contrary, e.g. from hot to cold).
The preceding consideration of the various cases serves to show us
whether it is or is not possible that there should be an infinite
sensible body. The following arguments give a general demonstration
that it is not possible.
It is the nature of every kind of sensible body to be somewhere, and
there is a place appropriate to each, the same for the part and for
the whole, e.g. for the whole earth and for a single clod, and for
fire and for a spark.
Suppose (a) that the infinite sensible body is homogeneous. Then
each part will be either immovable or always being carried along.
Yet neither is possible. For why downwards rather than upwards or in
any other direction? I mean, e.g, if you take a clod, where will it be
moved or where will it be at rest? For ex hypothesi the place of the
body akin to it is infinite. Will it occupy the whole place, then? And
how? What then will be the nature of its rest and of its movement,
or where will they be? It will either be at home everywhere-then it
will not be moved; or it will be moved everywhere-then it will not
come to rest.
But if (b) the All has dissimilar parts, the proper places of the
parts will be dissimilar also, and the body of the All will have no
unity except that of contact. Then, further, the parts will be
either finite or infinite in variety of kind. (i) Finite they cannot
be, for if the All is to be infinite, some of them would have to be
infinite, while the others were not, e.g. fire or water will be
infinite. But, as we have seen before, such an element would destroy
what is contrary to it. (This indeed is the reason why none of the
physicists made fire or earth the one infinite body, but either
water or air or what is intermediate between them, because the abode
of each of the two was plainly determinate, while the others have an
ambiguous place between up and down.)
But (ii) if the parts are infinite in number and simple, their
proper places too will be infinite in number, and the same will be
true of the elements themselves. If that is impossible, and the places
are finite, the whole too must be finite; for the place and the body
cannot but fit each other. Neither is the whole place larger than what
can be filled by the body (and then the body would no longer be
infinite), nor is the body larger than the place; for either there
would be an empty space or a body whose nature it is to be nowhere.
Anaxagoras gives an absurd account of why the infinite is at rest.
He says that the infinite itself is the cause of its being fixed. This
because it is in itself, since nothing else contains it-on the
assumption that wherever anything is, it is there by its own nature.
But this is not true: a thing could be somewhere by compulsion, and
not where it is its nature to be.
Even if it is true as true can be that the whole is not moved (for
what is fixed by itself and is in itself must be immovable), yet we
must explain why it is not its nature to be moved. It is not enough
just to make this statement and then decamp. Anything else might be in
a state of rest, but there is no reason why it should not be its
nature to be moved. The earth is not carried along, and would not be
carried along if it were infinite, provided it is held together by the
centre. But it would not be because there was no other region in which
it could be carried along that it would remain at the centre, but
because this is its nature. Yet in this case also we may say that it
fixes itself. If then in the case of the earth, supposed to be
infinite, it is at rest, not because it is infinite, but because it
has weight and what is heavy rests at the centre and the earth is at
the centre, similarly the infinite also would rest in itself, not
because it is infinite and fixes itself, but owing to some other
cause.
Another difficulty emerges at the same time. Any part of the
infinite body ought to remain at rest. Just as the infinite remains at
may take will remain in itself. The appropriate places of the whole
and of the part are alike, e.g. of the whole earth and of a clod the
appropriate place is the lower region; of fire as a whole and of a
spark, the upper region. If, therefore, to be in itself is the place
of the infinite, that also will be appropriate to the part.
Therefore it will remain in itself.
In general, the view that there is an infinite body is plainly
incompatible with the doctrine that there is necessarily a proper
place for each kind of body, if every sensible body has either
weight or lightness, and if a body has a natural locomotion towards
the centre if it is heavy, and upwards if it is light. This would need
to be true of the infinite also. But neither character can belong to
it: it cannot be either as a whole, nor can it be half the one and
half the other. For how should you divide it? or how can the
infinite have the one part up and the other down, or an extremity
and a centre?
Further, every sensible body is in place, and the kinds or
differences of place are up-down, before-behind, right-left; and these
distinctions hold not only in relation to us and by arbitrary
agreement, but also in the whole itself. But in the infinite body they
cannot exist. In general, if it is impossible that there should be
an infinite place, and if every body is in place, there cannot be an
infinite body.
Surely what is in a special place is in place, and what is in
place is in a special place. Just, then, as the infinite cannot be
quantity-that would imply that it has a particular quantity, e,g,
two or three cubits; quantity just means these-so a thing's being in
place means that it is somewhere, and that is either up or down or
in some other of the six differences of position: but each of these is
a limit.
It is plain from these arguments that there is no body which is
actually infinite.
6
But on the other hand to suppose that the infinite does not exist in
any way leads obviously to many impossible consequences: there will be
a beginning and an end of time, a magnitude will not be divisible into
magnitudes, number will not be infinite. If, then, in view of the
above considerations, neither alternative seems possible, an arbiter
must be called in; and clearly there is a sense in which the
infinite exists and another in which it does not.
We must keep in mind that the word 'is' means either what
potentially is or what fully is. Further, a thing is infinite either
by addition or by division.
Now, as we have seen, magnitude is not actually infinite. But by
division it is infinite. (There is no difficulty in refuting the
theory of indivisible lines.) The alternative then remains that the
infinite has a potential existence.
But the
phrase 'potential existence' is ambiguous. When we speak
of the potential existence of a statue we mean that there will be an
actual statue. It is not so with the infinite. There will not be an
actual infinite. The word 'is' has many senses, and we say that the
infinite 'is' in the sense in which we say 'it is day' or 'it is the
games', because one thing after another is always coming into
existence. For of these things too the distinction between potential
and actual existence holds. We say that there are Olympic games,
both in the sense that they may occur and that they are actually
occurring.
The infinite exhibits itself in different ways-in time, in the
generations of man, and in the division of magnitudes. For generally
the infinite has this mode of existence: one thing is always being
taken after another, and each thing that is taken is always finite,
but always different. Again, 'being' has more than one sense, so
that we must not regard the infinite as a 'this', such as a man or a
horse, but must suppose it to exist in the sense in which we speak
of the day or the games as existing things whose being has not come to
them like that of a substance, but consists in a process of coming
to be or passing away; definite if you like at each stage, yet
always different.
But when this takes place in spatial magnitudes, what is taken
perists, while in the succession of time and of men it takes place
by the passing away of these in such a way that the source of supply
never gives out.
In a way the infinite by addition is the same thing as the
infinite by division. In a finite magnitude, the infinite by
addition comes about in a way inverse to that of the other. For in
proportion as we see division going on, in the same proportion we
see addition being made to what is already marked off. For if we
take a determinate part of a finite magnitude and add another part
determined by the same ratio (not taking in the same amount of the
original whole), and so on, we shall not traverse the given magnitude.
But if we increase the ratio of the part, so as always to take in
the same amount, we shall traverse the magnitude, for every finite
magnitude is exhausted by means of any determinate quantity however
small.
The infinite, then, exists in no other way, but in this way it
does exist, potentially and by reduction. It exists fully in the sense
in which we say 'it is day' or 'it is the games'; and potentially as
matter exists, not independently as what is finite does.
By addition then, also, there is potentially an infinite, namely,
what we have described as being in a sense the same as the infinite in
respect of division. For it will always be possible to take
something ah extra. Yet the sum of the parts taken will not exceed
every determinate magnitude, just as in the direction of division
every determinate magnitude is surpassed in smallness and there will
be a smaller part.
But in respect of addition there cannot be an infinite which even
potentially exceeds every assignable magnitude, unless it has the
attribute of being actually infinite, as the physicists hold to be
true of the body which is outside the world, whose essential nature is
air or something of the kind. But if there cannot be in this way a
sensible body which is infinite in the full sense, evidently there can
no more be a body which is potentially infinite in respect of
addition, except as the inverse of the infinite by division, as we
have said. It is for this reason that Plato also made the infinites
two in number, because it is supposed to be possible to exceed all
limits and to proceed ad infinitum in the direction both of increase
and of reduction. Yet though he makes the infinites two, he does not
use them. For in the numbers the infinite in the direction of
reduction is not present, as the monad is the smallest; nor is the
infinite in the direction of increase, for the parts number only up to
the decad.
The infinite turns out to be the contrary of what it is said to
be. It is not what has nothing outside it that is infinite, but what
always has something outside it. This is indicated by the fact that
rings also that have no bezel are described as 'endless', because it
is always possible to take a part which is outside a given part. The
description depends on a certain similarity, but it is not true in the
full sense of the word. This condition alone is not sufficient: it
is necessary also that the next part which is taken should never be
the same. In the circle, the latter condition is not satisfied: it
is only the adjacent part from which the new part is different.
Our definition then is as follows:
A quantity is infinite if it is such that we can always take a
part outside what has been already taken. On the other hand, what
has nothing outside it is complete and whole. For thus we define the
whole-that from which nothing is wanting, as a whole man or a whole
box. What is true of each particular is true of the whole as
such-the whole is that of which nothing is outside. On the other
hand that from which something is absent and outside, however small
that may be, is not 'all'. 'Whole' and 'complete' are either quite
identical or closely akin. Nothing is complete (teleion) which has
no end (telos); and the end is a limit.
Hence Parmenides must be thought to have spoken better than
Melissus. The latter says that the whole is infinite, but the former
describes it as limited, 'equally balanced from the middle'. For to
connect the infinite with the all and the whole is not like joining
two pieces of string; for it is from this they get the dignity they
ascribe to the infinite-its containing all things and holding the
all in itself-from its having a certain similarity to the whole. It is
in fact the matter of the completeness which belongs to size, and what
is potentially a whole, though not in the full sense. It is
divisible both in the direction of reduction and of the inverse
addition. It is a whole and limited; not, however, in virtue of its
own nature, but in virtue of what is other than it. It does not
contain, but, in so far as it is infinite, is contained. Consequently,
also, it is unknowable, qua infinite; for the matter has no form.
(Hence it is plain that the infinite stands in the relation of part
rather than of whole. For the matter is part of the whole, as the
bronze is of the bronze statue.) If it contains in the case of
sensible things, in the case of intelligible things the great and
the small ought to contain them. But it is absurd and impossible to
suppose that the unknowable and indeterminate should contain and
determine.
7
It is reasonable that there should not be held to be an infinite
in respect of addition such as to surpass every magnitude, but that
there should be thought to be such an infinite in the direction of
division. For the matter and the infinite are contained inside what
; contains them, while it is the form which contains. It is natural
too to suppose that in number there is a limit in the direction of the
minimum, and that in the other direction every assigned number is
surpassed. In magnitude, on the contrary, every assigned magnitude
is surpassed in the direction of smallness, while in the other
direction there is no infinite magnitude. The reason is that what is
one is indivisible whatever it may be, e.g. a man is one man, not
many. Number on the other hand is a plurality of 'ones' and a
certain quantity of them. Hence number must stop at the indivisible:
for 'two' and 'three' are merely derivative terms, and so with each of
the other numbers. But in the direction of largeness it is always
possible to think of a larger number: for the number of times a
magnitude can be bisected is infinite. Hence this infinite is
potential, never actual: the number of parts that can be taken
always surpasses any assigned number. But this number is not separable
from the process of bisection, and its infinity is not a permanent
actuality but consists in a process of coming to be, like time and the
number of time.
With magnitudes the contrary holds. What is continuous is divided ad
infinitum, but there is no infinite in the direction of increase.
For the size which it can potentially be, it can also actually be.
Hence since no sensible magnitude is infinite, it is impossible to
exceed every assigned magnitude; for if it were possible there would
be something bigger than the heavens.
The infinite is not the same in magnitude and movement and time,
in the sense of a single nature, but its secondary sense depends on
its primary sense, i.e. movement is called infinite in virtue of the
magnitude covered by the movement (or alteration or growth), and
time because of the movement. (I use these terms for the moment. Later
I shall explain what each of them means, and also why every
magnitude is divisible into magnitudes.)