by Aristotle
[30] Suppose then that infinity belongs accidentally. But, if so, it cannot, as we have said, be described as a principle, but rather that of which it is an accident—the air or the even number.
Thus the view of those who speak after the manner of the Pythagoreans is absurd. With the same breath they treat the infinite as substance, and divide it into parts.
This discussion, however, involves the more general question whether the [204b1] infinite can be present in mathematical objects and things which are intelligible and do not have extension. Our inquiry is limited to our special subject-matter, the objects of sense, and we have to ask whether there is or is not among them a body which is infinite in the direction of increase.
We may begin with a dialectical argument and show as follows that there is no such thing.
[5] If ‘bounded by a surface’ is the definition of body there cannot be an infinite body either intelligible or sensible. Nor can number taken in abstraction be infinite; for number or that which has number is numerable. If then the numerable can be numbered, it would also be possible to go through the infinite.
[10] If, on the other hand, we investigate the question more in accordance with principles appropriate to physics, we are led as follows to the same result.
The infinite can be either compound, or simple.
It will not be compound, if the elements are finite in number. For they must be more than one, and the contraries must always balance, and no one of them can be [15] infinite. If one of the bodies falls in any degree short of the other in potency—suppose fire is finite in amount while air is infinite and a given quantity of fire exceeds in power the same amount of air in any ratio provided it is numerically definite—the infinite body will obviously prevail over and annihilate the finite body. On the other hand, it is impossible that each should be infinite. Body is what has extension in all directions and the infinite is what is boundlessly extended, so that [20] the infinite body would be extended in all directions ad infinitum.
Nor can an infinite body be one and simple, whether it is, as some hold, a thing over and above the elements (from which they generate the elements) or is not thus qualified. There are some people who make this the infinite, and not air or water, in [25] order that the other elements may not be annihilated by the element which is infinite. They have contrariety with each other—air is cold, water moist, fire hot; if one were infinite, the others by now would have ceased to be. As it is, they say, the infinite is different from them and is their source.
It is impossible, however, that there should be such a body; not because it is infinite—on that point a general proof can be given which applies equally to all, air, [30] water, or anything else—but because there is no such sensible body, alongside the so-called elements. Everything can be resolved into the elements of which it is composed. Hence the body in question would have been present in our world here, alongside air and fire and earth and water; but nothing of the kind is observed.
Nor can fire or any other of the elements be infinite. For generally, and apart [205a1] from the question how any of them could be infinite, the universe, even, if it were limited, cannot either be or become one of them, as Heraclitus says that at some time all things become fire. (The same argument applies also to the one which the physicists suppose to exist alongside the elements: for everything changes from [5] contrary to contrary, e.g. from hot to cold.)
In each case, we should consider along these lines whether it is or is not possible that it should be infinite. The following arguments give a general demonstration that it is not possible for there to be an infinite sensible body.
It is the nature of every kind of sensible body to be somewhere, and there is a [10] 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 that the infinite sensible body is homogeneous. Then each 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 the place of the body [15] 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 rest everywhere—then it will not be moved; or it will be moved everywhere—then it will not come to rest.
But if the universe has dissimilar parts, the proper places of the parts will be [20] dissimilar also, and the body of the universe will have no unity except that of contact. Then, further, the parts will be either finite or infinite in variety of kind.
Finite they cannot be; for if the universe 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 such an element would destroy what is contrary to it.
But if the parts are infinite in number and simple, their proper places too will [25] 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 [30] nature it is to be nowhere. 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.
[205b1] 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 [5] its own nature. But this is not true: a thing could be somewhere by compulsion, and not where it is its nature to be.
Thus however true it may 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. For it might be not moving because there is nowhere else for it to move, even though there [10] 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, 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 [15] to be infinite, it is at rest, not for this reason, 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.
It is clear at the same time that 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 [20] 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 [25] the doctrine that there is 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 [30] 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 convention, but also in the whole itself. But in the infinite body [206a1] 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 place is somewhere, and what is somewhere is in a 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 a place means that it is somewhere, and that is either up or down or in some [5] 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 [10] 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.
Now things are said to exist both potentially and in fulfilment. Further, a thing is infinite either by addition or by division. Now, as we have seen, magnitude is not [15] 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 we must not construe potential existence in the way we do when we say that it is possible for this to be a statue—this will be a statue, but something [20] infinite will not be in actuality. Being is spoken of in many ways, 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 [25] 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’ is spoken of in several ways, so that we must not regard the infinite as a ‘this’, such as a man or a horse, but must [30] 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, finite, yet always different.]22
But in spatial magnitudes, what is taken persists, while in the succession of [206b1] 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 just as we see division going on ad infinitum, so we see addition being [5] made in the same proportion 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), we shall not [10] 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 in fulfillment in the sense in which we say ‘it is [15] 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 ab extra. Yet the sum of the parts taken will not exceed every determinate magnitude, just as in the direction of division every [20] determinate magnitude is surpassed and there will always be a smaller part.
But in respect of addition there cannot even potentially be an infinite which exceeds every assignable magnitude, unless it is accidentally infinite in fulfillment, as the physicists hold to be true of the body which is outside the world, whose substance is air or something of the kind. But if there cannot be in this way a [25] sensible body which is infinite in fulfilment, 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 [30] 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 he makes numbers only up to the decad.
The infinite turns out to be the contrary of what it is said to be. It is not what [207a1] 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 infinite,23 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 [5] word. This condition alone is not sufficient: it is necessary also that the same part should never be taken twice. In the circle, the latter condition is not satisfied: it is true only that the next part is always different.
Thus something is infinite if, taking it quantity by quantity, we can always take something outside. 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 [10] whole man or box. What is true of each particular is true of the whole properly speaking—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 which has no end and the end is a limit.
[15] 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’.24 For to connect the infinite with the universe 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 universe in itself—from its having a certain similarity to the whole. It is in fact the matter of [20] the completeness which belongs to size, and what is potentially a whole, though not in fulfilment. 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 something else. 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 [25] 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 [30] 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 [207b1] reasonable too to suppose that in nu
mber there is a limit in the direction of the minimum, and that in the other direction every amount is always surpassed. In magnitude, on the contrary, every magnitude is surpassed in the direction of smallness, while in the other direction there is no infinite magnitude. The reason is [5] 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 derivative terms, and so with each of the other numbers. But in the direction of largeness it is [10] always possible to think of a large 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 definite amount. But this number is not separable, and its infinity does not persist but consists in a process of coming to be, like time and the number of time. [15]
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 actually be. Hence since no sensible magnitude is infinite, it is impossible to exceed every definite magnitude; for if it were possible there [20] 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 the posterior depends on the prior, e.g. 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. [25] Later I shall explain what each of them means, and also why every magnitude is divisible into magnitudes.)