Various Works

  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

  rest in itself because it fixes itself, so too any part of it you

  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

&nbsp
; 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.)