LESSON 10

The Infinite

  Chapter 10: 1066a 35-1067a 37

             989. The infinite is either what cannot be spanned because it is not naturally fitted to be spanned (just as the spoken word is invisible); or what is imperfectly spanned; or what is spanned with difficulty; or what is not actually spanned, although it admits of being spanned or of having a terminus.

             990. Further, a thing may be infinite either by addition or by subtraction or by both.

             991. That the infinite should be a separate entity and be perceptible is impossible. For if it is neither a continuous quantity nor a plurality, and is a substance and not an accident, it will be indivisible; for what is divisible is either a continuous quantity or a plurality. But if it is indivisible, it is not infinite, except in the sense in which the spoken word is invisible. But people do not use the term in this sense, nor is this the sense of the infinite which we are investigating, but the infinite in the sense of what cannot be spanned.

             992. Further, how can the infinite exist of itself if number and continuous quantity, of which the infinite is an attribute, do not exist of themselves?

             993. Again, if the infinite is an accident, it cannot, inasmuch as it is infinite, be an element of existing things, just as the invisible is not an element of speech although the spoken word is invisible. It is also evident that the infinite cannot be actual; for any part of it which might be taken would be infinite, since infinity and the infinite are the same if the infinite is a substance and is not predicated of a subject. Hence it is either indivisible, or if it is divisible, the parts into which it is divided are infinite in number. But it is impossible that the same thing should be many infinites; for, just as a part of air is air, so too a part of the infinite must be infinite if the infinite is a substance and principle. Therefore it cannot be divided into parts, and so is indivisible. But this cannot apply to the actually infinite, for it must be a quantity. Hence it is an accidental attribute. But if this is so, then, as we have said, it cannot be it that is a principle, but that of which it is an accident, for example, air or the even. This investigation, then, is universal.

             994. That the infinite does not exist in sensible things is made clear as follows: if it is the nature of a body to be bounded by surfaces, then no body, whether it is perceptible or intelligible, can be infinite.

             995. Nor can there be any separate and infinite number; for a number or that which has a number is numerable.

             996. This is evident from the following argument drawn from nature: the infinite can be neither composite nor simple. It cannot be a composite body if the elements are limited in number; for the contraries must be equal, and no one of them must be infinite; for if the active power of one of two elemental bodies is inferior to that of the other, the finite body will be destroyed by the infinite body. And that each should be infinite is impossible, because a body is what is extended in all directions, and the infinite is what is extended without limit; so if the infinite is a body, it must be infinite in all directions.

             997. Nor can the infinite be a single simple body: neither, as some say, something apart from the elements, from which they generate these (for there is no such body apart from the elements, because everything can be dissolved into that of which it is composed; but there does not appear to be anything apart from the simple bodies), nor fire, nor any of the other elements. For unless some of them are infinite, the whole, even though it is finite, could not be or become any one of them, as Heraclitus says that all things at one time become fire. The same reasoning also applies to "the one," which the philosophers of nature posited as an entity over and above the elements (997). For everything is changed from a contrary, for example, from hot to cold.

             998. Again, a sensible body is somewhere, and the place of the whole and that of a part (of the earth, for example) is the same.

             999. Hence, if the infinite is composed of like parts, it will be immovable or will always be undergoing motion. But this is impossible. For why should it be moved upwards rather than downwards or in some other direction? For example, if it were a clod of earth, where would it move to or where remain at rest? For the place of the body naturally fitted to this will be infinite. Will it then occupy the whole place? And how will it do this? And what then will be its place of rest and of motion? For if it rests everywhere, it will not be in motion. And if it is moved everywhere, it will not be at rest.

             1000. And if the whole is composed throughout of unlike parts, their places will also be unlike. And, first, the body of the whole will be one only by contact and, second, the parts will be either finite or infinite in species. But they cannot be finite, for some would then be infinite and some not (if the whole is infinite), for example, fire or water. But such an infinite element would necessitate the destruction of contrary elements (996). But if the parts are infinite and simple, their places will be infinite, and the elements will be infinite in number. And since this is impossible, their places will be finite and the whole finite.

             1001. And in general there cannot be an infinite body and a place for bodies if every sensible body has either heaviness or lightness; for it will tend either to the center or upwards. But the infinite--either the whole or a half of it--is incapable of any of these motions. For how can you divide it? Or how can one part tend upwards and another downwards, or one part tend to the extreme and another to the center?

             1002. Further, every sensible body is in a place, and there are six kinds of place, but these cannot pertain to an infinite body.

             1003. And in general if a place cannot be infinite, neither can a body be infinite; for to be in a place is to be somewhere, and this means to be either down or up or in some one of the other places, and each of these is a limit.

             1004. And the infinite is not the same in the case of continuous quantity, of motion, and of time, as though it were a single reality; but the secondary member is said to be infinite inasmuch as the primary one is; for example, motion is said to be infinite in reference to the continuous quantity in which it is moved or altered or increased, and time is said to be such in reference to motion.

COMMENTARY

             2314. Having given his views about motion, here the Philosopher deals with the infinite, which is an attribute of motion and of any quantity in general. In regard to this he does three things. First (989:C 2314), he distinguishes the various senses in which the term infinite is used. Second (991:C 2322), he shows that the actually infinite does not exist ("That the infinite"). Third (1004:C 2354), he explains how the infinite is found in different things ("And the infinite").

             In regard to the first he does two things. First, he explains the different senses in which the term infinite is used; and second (990:C 2319), the various senses in which things are said to be potentially infinite ("Further, a thing").

             In regard to the first (989) part it should be borne in mind that every finite thing may be spanned by division. Hence the infinite, properly speaking, is what cannot be spanned by measurement; and therefore the term infinite is used in the same number of senses as the term untraversable.

             2315. Now each of these is used in four ways. First, the infinite or untraversable means what cannot be spanned by measurement because it does not belong to the class of things which are naturally fitted to be spanned; for example, we say that the point or the unit or something which is not a quantity and is not measurable is infinite or untraversable; and in this sense the spoken word is said to be invisible because it does not belong to the class of things which are visible.

             2316. Second, the infinite or untraversable means what has not yet been spanned although it has begun to be spanned. This is his meaning in saying "what is imperfectly spanned."

             2317. Third, the infinite or untraversable means what is spanned with difficulty. Thus we may say that the depth of the sea or the height of the sky is infinite, or that any long distance is immeasurable or untraversable or infinite, because it surpasses our powers of measurement although in itself it is capable of being spanned.

             2318. Fourth, the infinite or untraversable means what belongs to the class of things which are naturally fitted to be spanned, or to have some limit set to them, but are not actually spanned; for example, if a line is limitless. This sense of the infinite is the true and proper one.

             2319. Further, a thing (990).

             Second, he explains the various senses in which things are said to be potentially infinite. He says that in one sense a thing is said to be infinite by addition, as a number; for it is always possible to add a unit to any number, and in this respect number is capable of infinite increase.

             2320. In another sense a thing is said to be infinite by subtraction or division inasmuch as a continuous quantity is said to be infinitely divisible.

             2321. In a third sense it is possible for a thing to be infinite from both points of view; for example, time is said to be infinite both as regards division, because it is continuous, and as regards addition, because it is a number. It is in a similar way that the infinite is found in motion.

             2322. That the infinite (991).

             Then he shows that the actually infinite does not exist; and in regard to this it should be noted that the Platonists held that the infinite is separate from sensible things and is a principle of them, whereas the philosophers of nature held that the infinite exists in sensible things, not in the sense that it is a substance, but rather in the sense that it is an accident of some sensible body. He therefore shows, first (991:C 2322), that the infinite is not a separate entity; and second (994:C 2327), that the actually infinite does not exist in sensible things ("That the infinite does not").

             In treating the first member of this division he gives three arguments. The first is as follows: if the infinite is a substance which exists of itself and is not an accident of some subject, the infinite must lack continuous quantity and plurality, because continuous quantity and number constitute the subject of the infinite. But if it lacks continuous quantity and plurality, it must be indivisible, because everything divisible is either a continuous quantity or a plurality. But if it is indivisible, it is infinite only in the first sense of the term, as a spoken word is said to be invisible. However, we are not investigating this sense of the term here, nor did they use the term in this sense; but we are considering the fourth sense, i.e., what is untraversable. Therefore, all things considered, if the infinite were an independently existing substance, it would not be truly infinite. This position destroys itself in this way.

             2323. Further, how can (992).

             Then he gives the second argument, which runs thus: infinity is an attribute of number and of continuous quantity. But number and continuous quantity are not things which have separate existence, as has been shown in Book I (122:C 239) and will be shown below (993:C 2324). Therefore much less is the infinite a separate substance.

             2324. Again, if the infinite (993).

             Here he gives the third argument, which runs as follows. Let us suppose that the infinite is either a substance which is separate from sensible things or an accident belonging to some separate subject, for example, to continuous quantity or to number--which are separate according to the Platonists. Now if the infinite is assumed to be an accident, it cannot be the infinite as infinite that is a principle of existing things, but rather the subject of the infinite; just as what is invisible is not said to be a principle of speech, but the spoken word, although the spoken word is invisible in this sense.

             2325. And if the infinite is assumed to be a substance and is not predicated of a subject, it is also evident that it cannot be actually infinite; for it is either divisible or indivisible. But if the infinite itself as infinite is a substance and is divisible, any part of it which might be taken would necessarily be infinite; because infinity and the infinite are the same "if the infinite is a substance," i.e., if infinity expresses the proper intelligible structure of the infinite. Hence, just as a part of water is water and a part of air is air, so too any part of the infinite is infinite if the infinite is a divisible substance. We must say, then, that the infinite is either indivisible or divisible into many infinites. But many infinite things cannot possibly constitute one finite thing; for the infinite is not greater than the infinite, but every whole is greater than any of its parts.

             2326. It follows, then, that the infinite is indivisible. But that any indivisible thing should be actually infinite is impossible, because the infinite must be a quantity. Therefore it remains that it is not a substance but an accident. But if the infinite is an accident, it is not the infinite that is a principle, but the subject of which it is an accident (as was said above), whether it be air, as some of the natural philosophers claimed, or the even, as the Pythagoreans claimed. Thus it follows that the infinite cannot be both a substance and a principle of beings at the same time. Last, he concludes that this investigation is a general one which goes beyond the study of natural things.

             2327. That the infinite does not exist (994).

             Then he proves that the actually infinite does not exist in sensible things. First (994:C 2327), he proves this by probable arguments; and second (996:C 2330), by arguments drawn from nature ("This is evident").

             He accordingly says, first (994), that it is obvious that the actually infinite is not found in sensible things; and he proves two points. First, he says that there is no infinite body in the sensible world, for it is the nature of a body to be bounded by surfaces. But no body with a definite surface is infinite. Therefore no body is infinite, "whether it be perceptible," i.e., a natural body, "or intelligible," i.e., a mathematical one.

             2328. Nor can there be (995).

             Second, he shows in the following way that there is no infinite number in sensible things. Every number and everything which has a number is numerable. But nothing numerable is infinite, because what is numerable can be spanned by numeration. Therefore no number is infinite.

             2329. Now these arguments do not pertain to natural philosophy, because they are not based on the principles of a natural body but on certain principles which are common and probable and not necessary. For anyone who would claim that a body is infinite would not maintain that its surface has limits, for this characteristic belongs to the nature of a finite body. And anyone who would claim that there is an infinite multitude would not hold that it is a number, because number is multitude measured by one, as has been explained in Book X (875:C 2090). But nothing measured is infinite.

             2330. This is evident (996).

             Next, he proves that the actually infinite does not exist within sensible things, by using arguments drawn from nature. He does this, first (996:C 2330), with reference to the active and passive powers of bodies; and second (998:C 2339), with reference to place and the thing in place ("Again, a sensible body").

             Now active and passive powers, place and thing in place are proper to natural bodies as such; and therefore he says that these arguments are drawn from nature. He accordingly says, first (996), that, if a body is perceptible and infinite, it will be either a simple body or a composite body or compound.

             2331. First, he shows that a composite body cannot be infinite, if we assume that simple bodies, which are the elements of composite bodies, are finite in number. He proves this as follows: either all the elements must be infinite in quantity, or one must be infinite and the others finite, otherwise an infinite body could not be composed of elements which are finite in number.

             2332. But that one of the elements should be infinite and the rest finite is impossible; because in the case of a compound contraries must somehow be equalized in order that the compound may be preserved in being, for otherwise that contrary which exceeds the others will destroy them. But if one contrary is infinite and the rest finite, no equality will be established, since there is no proportion between the infinite and the finite. A compound, then, could not exist, for the infinite element would destroy the others.

             2333. And since someone might say that a body which is finite in quantity has greater power, and that equality is achieved in this way (for example, if someone were to say that in a compound air is infinite and fire finite), he therefore adds that, even if we suppose that the active power of one body which is assumed to be infinite falls short of the active power of any one of the others, because these are assumed to be finite, the finite element will be destroyed by the infinite one; for a finite body must have a finite power, and then finite fire will have a finite power. Hence, if from infinite air a portion of air equal to the fire is taken out, its power will be less than that of the whole infinite air, but proportioned to the power of fire. Let us suppose, then, that the power of fire is a hundred times greater than that of air. Hence, if we take away a hundredfold of air from infinite air it will be equal to fire in power; and thus the whole infinite air will have a greater infinite power than fire and will destroy it. It is impossible, then, that one element of a compound should be infinite and the rest finite.

             2334. Similarly, it is impossible that all should be infinite, because a body is what is extended in every dimension. But the infinite is what is infinite in dimension. Hence an infinite body must have an infinite dimension in every direction. But two bodies cannot be in the same place. Therefore two infinite bodies cannot be combined into one.

             2335. Nor can the infinite (997).

             Second, he proves that the infinite cannot be a simple body. There cannot be a simple body apart from the elements, from which all of them are generated, as some claimed air to be, because each thing is dissolved into the elements of which it is composed. But we see that compounds are dissolved only into the four elements; and therefore there cannot be a simple body apart from the four elements.

             2336. Nor can fire or any of the other elements be infinite, because no element could possibly exist except the one which is infinite, since it would fill every place. Again, if there were some finite element it would have to be changed into that infinite element because of the very great power of the latter; just as Heraclitus claimed that at some time all things must be changed into the element fire because of its very great power.

             2337. And the same argument therefore applies to the one simple body which the natural philosophers posited as an entity over and above the elements themselves; for it would have to be opposed to the other elements as a kind of contrary, since according to them there is change from that one body alone into the others. But every change in things is from one contrary to another. Therefore, since one of two contraries destroys the other, it follows that, if that body which is supposed to exist apart from the elements is infinite, it will destroy the others.

             2338. The philosopher omits the celestial body here, because, while it is something apart from the four elements, it is not contrary or repugnant to them in any way, nor are these bodies naturally derived from it. For the philosophers of nature who posited an actually infinite body did not attain any knowledge of this fifth essence or nature. Yet in The Heavens Aristotle proves that even a celestial body, which moves circularly, is not actually infinite.

             2339. Again, a sensible body (998).

             Then he proves that a sensible body is not infinite; and he does this by means of arguments based upon place and a thing in place. He gives three arguments. As a sort of preamble to the first he considers two points necessary for its development. The first is that every sensible body is in a place. He emphasizes sensible in order to distinguish this kind of body from a mathematical one, to which place and contact are attributed only figuratively.

             2340. The second point is that the natural place of a whole and that of a part are the same, i.e., the place in which it naturally rests and to which it is naturally moved. This is clear, for instance, in the case of earth and of any part of it, for the natural place of each is down.

             2341. Hence, if the infinite (999).

             After giving these two points he states his argument, which runs as follows. If a sensible body is assumed to be infinite, either its parts will all be specifically the same, as is the case with bodies having like parts, such as air, earth, blood, and so on, or they will be specifically different.

             2342. But if all of its parts are specifically the same, it will follow that the whole will always be at rest or always in motion. Each one of these is impossible and incompatible with the facts of sensory perception.

             2343. For why should it (ibid.).

             Then he shows that the other alternative has to be accepted; for it has already been assumed that the natural place of a whole and that of a part are the same. And it is evident that every body is at rest when it is in its natural place, and that it naturally moves to its natural place when it is outside of it. If, then, the whole place occupied by a body having an infinite number of like parts is natural to it, this place must be natural to each part, and thus the whole and each of its parts will be at rest. But if it is not natural to it, the whole and each of its parts will then be outside their proper place; and thus the whole and any part of it will always be in motion.

             2344. For it cannot be said that some part of a place is natural to the whole and to its parts, and that some part of a place is not; because, if a body were infinite and every body were in a place, its place would also have to be infinite. But in infinite place there is no dividedness by reason of which one part of it is the natural place of the body and another is not, because there must be some fixed proportion and distance between a place which is natural and one which is not, and this cannot apply to an infinite place. This is what he means when he says that an infinite body or one of its parts will not be moved downwards rather than upwards or in some other direction, because in an infinite place it is impossible to find any fixed proportion between these parts.

             2345. He gives an example of this. If we assume that the earth is infinite, it will be impossible to give any reason why it should be in motion or at rest in one place rather than in another, because the whole infinite place will be equally fitted by nature to the infinite body which occupies this place. Hence, if some part of a place is naturally fitted to a clod of earth, the same will apply to another part; and if one part is not naturally fitted to a place, neither will another be. If, then, an infinite body is in a place, it will fill the whole of that infinite place. Yet how can it be at rest and in motion at the same time? For if it rests everywhere, it will not be in motion; or if it is in motion everywhere, it follows that no part of it will be at rest.

             2346. And if the whole (1000).

             Then the Philosopher examines the other alternative, namely, the supposition that the whole is not composed of like parts. He says that it follows, first, that, if "the body of the whole," i.e., of the universe, is composed of specifically unlike parts, it will be one only by contact, as a pile of stones is one. But things specifically different, such as fire, air and water, cannot be continuous; and this is not to be one in an absolute sense.

             2347. Again, if this whole is composed of parts which are specifically unlike, they will be either infinite in species, i.e., so that the different parts of the whole are infinite in species; or they will be finite in species, i.e., so that the diversity of species found among the parts amount to some fixed number.

             2348. But that the elements cannot be finite in species is clear from what was proposed in the preceding argument; for it would be impossible for an infinite whole to be composed of parts which are finite in number, unless either all parts were infinite in quantity, which is impossible, since an infinite body must be infinite in any of its parts, or at least unless some part or parts were infinite. Therefore, if a whole were infinite and its parts were different species infinite in number, it would follow that some of them would be infinite and some finite in quantity--for example, if one were to assume that water is infinite and fire finite. But this position introduces corruption among contraries, because an infinite contrary would destroy other contraries, as has been shown above (996:C 2332). Therefore they cannot be finite in number.

             2349. But if the parts of the universe were infinite in species, and these must be assumed to be simple, it would follow that places would be infinite and that the elements would be infinite. But both of these are impossible; for since each simple body has a place naturally fitted to it which is specifically different from the place of another body, if there were an infinite number of simple bodies which are different in species, it would also follow that there are an infinite number of places which are different in species. This is obviously false; for the species of places are limited in number, and these are up and down, and so on. It is also impossible that the elements should be infinite in number, because it would then follow that they would remain unknown; and if they were unknown, all things would be unknown. Therefore, if the elements cannot be infinite, places must be finite, and consequently the whole must be finite.

             2350. And in general (1001).

             Here he gives the second argument. He says that, since every sensible body has a place, it is impossible for any sensible body to be infinite, granted the assumption that every sensible body has heaviness and lightness--which would be true according to the opinion of the ancient natural philosophers, who claimed that bodies are actually infinite. Aristotle, however, is of the opinion that there is a sensible body which does not have heaviness or lightness, namely, a celestial body, as he proved in The Heavens. He introduces this circumstantially, as admitted by his opponents, but not in the sense that it is unqualifiedly true. If every sensible body, then, is either heavy or light and some sensible body is infinite, it must be heavy or light; and therefore it must be moved upwards or towards the center; for a light thing is defined as one that rises upwards, and a heavy thing as one that tends towards the center. But this cannot apply to the infinite, either to the whole of it or to a part; for the center of a body is found only when a proportion is established between the boundaries by dividing the whole. But the infinite cannot be divided according to any proportion; and therefore neither up and down nor boundary and center can be found there.

             2351. This argument must be understood to apply even if one assumes that there is a third kind of body which is neither heavy nor light; for such a body is naturally moved around the center, and this could not be the case with an infinite body.

             2352. Further, every sensible body (1002).

             The Philosopher now gives the third argument, which runs thus: every sensible body is in a place. But there are six kinds of place: up and down, right and left, before and behind; and it is impossible to attribute these to an infinite body, since they are the limits of distances. Thus it is impossible that a place should be attributed to an infinite body; and therefore no sensible body is infinite. However, in saying that there are six kinds of place he does not mean that these places are distinguished because of the elements (for their motions are distinguished merely in terms of up and down) but only because, just as up and down are out of the question so far as an infinite body is concerned, so are all the other differences of place.

             2353. And in general if (1003).

             He gives the fourth argument, which is as follows. Every sensible body is in a place; but it is impossible for a place to be infinite; and therefore it is impossible for a body to be infinite. The way in which it is impossible for a place to be infinite he proves thus: whatever has a common term predicated of it must also have predicated of it any of the things which fall under that common term; for example, whatever is an animal must belong to some particular species of animal, and whatever is man must be some particular man. Similarly, whatever occupies an infinite place must be "somewhere," i.e., it must occupy some place. But to occupy some place is to be up or down or to be in some one of the other kinds of place. However, none of these can be infinite because each is the limit of some distance. It is impossible, then, that a place should be infinite, and the same applies to a body.

             2354. And the infinite (1004).

             Then he shows how the potentially infinite is found in different things. He says that it is found in continuous quantity, in motion, and in time, and it is not predicated of them univocally but in a primary and a secondary way. And the secondary member among them is always said to be infinite inasmuch as the primary member is; for example, motion is said to be infinite in reference to the continuous quantity in which something is moved locally or increased or altered; and time is said to be infinite in reference to motion. This must be understood as follows: infinite divisibility is attributed to what is continuous, and this is done first with reference to continuous quantity, from which motion derives its continuity. This is evident in the case of local motion because the parts of local motion are considered in relation to the parts of continuous quantity. The same thing is evident in the case of the motion of increase, because increase is noted in terms of the addition of continuous quantity. However, this is not as evident in the case of alteration, although in a sense it also applies there; because quality, which is the realm of alteration, is divided accidentally upon the division of continuous quantity. Again, the intensification and abatement of a quality is also noted inasmuch as its subject, which has continuous quantity, participates in some quality to a greater or lesser degree. And motion is referred to continuity, and so is a continuous time; for since time in itself is a number, it is continuous only in a subject, just as ten measures of cloth are continuous because the cloth is continuous. The term infinite, then, must be used of these three things in the same order of priority as the term continuous is.