The Basic Works of Aristotle (Modern Library Classics) (48 page)

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,
(10)
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.
(15)

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.
(20)
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,
(25)
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,
(30)
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 persists, 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.
[206b]

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.
(5)
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.
(10)
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,
(15)
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
ab 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,
(20)
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,
(25)
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.
(30)
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.
[207a]
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.
(5)
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,
(10)
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.
(15)
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.
(20)
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.
(25)
(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.
(30)

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,
(35)
while it is the form which contains.
[207b]
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,
(5)
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.
(10)
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.
(15)
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.
(20)

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),
(25)
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.)

Our account does not rob the mathematicians of their science, by disproving the actual existence of the infinite in the direction of increase, in the sense of the untraversable. In point of fact they do not need the infinite and do not use it.
(30)
They postulate only that the finite straight line may be produced as far as they wish. It is possible to have divided in the same ratio as the largest quantity another magnitude of any size you like. Hence, for the purposes of proof, it will make no difference to them to have such an infinite instead, while its existence will be in the sphere of real magnitudes.

In the four-fold scheme of causes,
(35)
it is plain that the infinite is a cause in the sense of matter, and that its essence is privation, the subject as such being what is continuous and sensible.
[208a]
All the other thinkers, too, evidently treat the infinite as matter—that is why it is inconsistent in them to make it what contains, and not what is contained.

8
      It remains to dispose of the arguments
⁸
which are supposed to support the view that the infinite exists not only potentially but as a separate thing.
(5)
Some have no cogency; others can be met by fresh objections that are valid.

(1) In order that coming to be should not fail, it is not necessary that there should be a sensible body which is actually infinite.
(10)
The passing away of one thing may be the coming to be of another, the All being limited.

(2)
There is a difference between touching and being limited. The former is relative to something and is the touching of something (for everything that touches touches something), and further is an attribute of some one of the things which are limited. On the other hand, what is limited is not limited in relation to anything. Again, contact is not necessarily possible between any two things taken at random.

(3) To rely on mere thinking is absurd, for then the excess or defect is not in the thing but in the thought.
(15)
One might think that one of us is bigger than he is and magnify him
ad infinitum.
But it does not follow that he is bigger than the size we are, just because some one thinks he is, but only because he
is
the size he is. The thought is an accident.

This concludes my account of the way in which the infinite exists, and of the way in which it does not exist, and of what it is.

¹
viii. 5.

²
Plato in the
Timaeus
(52
E
, 57
E
, 58
A
) makes motion depend on inequality.

³
i. e. we can substitute ‘mover’ and ‘moved’ for ‘agent’ and ‘patient’ in the formulation of the hypothesis.

⁴
Cf.
^a
18–20.

⁵
Aristotle’s general meaning is fairly plain. He is describing
two
constructions: in the one
odd
gnomons are placed round the
one,
in the other
even
gnomons are placed round the
two.

⁶
Aristotle does not regard them as elements.

⁷
The reference is probably to Anaximander.

⁸
Cf. 203
^b
15–30.

1
     The physicist must have a knowledge of Place, too, as well as of the infinite—namely, whether there is such a thing or not, and the manner of its existence and what it is—both because all suppose that things which exist are
somewhere
(the non-existent is nowhere—where is the goat-stag or the sphinx?),
(30)
and because ‘motion’ in its most general and primary sense is change of place, which we call ‘locomotion’.

The question, what is place? presents many difficulties. An examination of all the relevant facts seems to lead to divergent conclusions. Moreover, we have inherited nothing from previous thinkers,
(35)
whether in the way of a statement of difficulties or of a solution.

The existence of place is held to be obvious from the fact of mutual replacement.
[208b]
Where water now is, there in turn, when the water has gone out as from a vessel, air is present. When therefore another body occupies this same place, the place is thought to be different from all the bodies which come to be in it and replace one another.
(5)
What now contains air formerly contained water, so that clearly the place
or space into which and out of which they passed was something different from both.

Further, the typical locomotions of the elementary natural bodies—namely, fire, earth, and the like—show not only that place is something,
(10)
but also that it exerts a certain influence. Each is carried to its own place, if it is not hindered, the one up, the other down. Now these are regions or kinds of place—up and down and the rest of the six directions. Nor do such distinctions (up and down and right and left,
(15)
&c.) hold only in relation to us. To
us
they are not always the same but change with the direction in which we are turned: that is why the same thing may be both right
and
left, up
and
down, before
and
behind. But in
nature
each is distinct, taken apart by itself. It is not every chance direction which is ‘up’,
(20)
but where fire and what is light are carried; similarly, too, ‘down’ is not any chance direction but where what has weight and what is made of earth are carried—the implication being that these places do not differ merely in relative position, but also as possessing distinct potencies. This is made plain also by the objects studied by mathematics. Though they have no real place, they nevertheless, in respect of their position relatively to us, have a right and left as attributes ascribed to them only in consequence of their relative position, not having by nature these various characteristics. Again,
(25)
the theory that the void exists involves the existence of place: for one would define void as place bereft of body.

These considerations then would lead us to suppose that place is something distinct from bodies, and that every sensible body is in place. Hesiod too might be held to have given a correct account of it when he made chaos first.
(30)
At least he says:

First of all things came chaos to being, then broad-breasted earth, implying that things need to have space first, because he thought, with most people, that everything is somewhere and in place. If this is its nature, the potency of place must be a marvellous thing,
(35)
and take precedence of all other things. For that without which nothing else can exist, while it can exist without the others, must needs be first; for place does not pass out of existence when the things in it are annihilated.
[209a]

True, but even if we suppose its existence settled, the question of its
nature
presents difficulty—whether it is some sort of ‘bulk’ of body or some entity other than that, for we must first determine its genus.

(1) Now it has three dimensions,
(5)
length, breadth, depth, the
dimensions by which all body also is bounded. But the place cannot
be
body; for if it were there would be two bodies in the same place.

(2) Further, if body has a place and space, clearly so too have surface and the other limits of body; for the same statement will apply to them: where the bounding planes of the water were, there in turn will be those of the air. But when we come to a point we cannot make a distinction between it and its place.
(10)
Hence if the place of a point is not different from the point, no more will that of any of the others be different, and place will not be something different from each of them.

(3) What in the world then are we to suppose place to be? If it has the sort of nature described, it cannot be an element or composed of elements, whether these be corporeal or incorporeal: for while it has size,
(15)
it has not body. But the elements of sensible bodies are bodies, while nothing that has size results from a combination of intelligible elements.

(4) Also we may ask: of what in things is space the cause? None of the four modes of causation can be ascribed to it. It is neither cause in the sense of the matter of existents (for nothing is composed of it),
(20)
nor as the form and definition of things, nor as end, nor does it move existents.

(5) Further, too, if it is itself an existent,
where
will it be? Zeno’s difficulty demands an explanation: for if everything that exists has a place,
(25)
place too will have a place, and so on
ad infinitum.

(6) Again, just as every body is in place, so, too, every place has a body in it. What then shall we say about
growing
things? It follows from these premisses that their place must grow with them, if their place is neither less nor greater than they are.

By asking these questions, then, we must raise the whole problem about place—not only as to what it is, but even whether there is such a thing.
(30)