LECTURE 7 (203 b 15-204 b 3)

ARGUMENTS WHICH PERSUADE US THAT THE INFINITE EXISTS. THE MEANINGS OF 'INFINITE'. AN INFINITE SEPARATED FROM SENSIBLE THINGS MUST BE DENIED

             336. Having set forth the opinions of the ancients concerning the infinite, he begins here to seek the truth.

             First he raises objections against each part. Secondly, where he says, 'It is plain from these arguments . . .' (206 a 8), he answers these difficulties.

             Concerning the first part he makes two points. First, he sets forth arguments to show that there is an infinite. Secondly, where he says, 'But the problem of the infinite . . .' (203 b 30), he sets forth arguments to show that there is no infinite.

             337. With reference to the first part he sets forth five arguments.

             The first of these is taken from time, which according to the common opinion of the ancients is infinite. For Plato alone generated time, as will be explained in Book VIII.

             He says, therefore, first that there are five arguments to show that there is an infinite.

             The first of these is from time, which is infinite according to those who said that time always was and always will be.

             338. The second argument is taken from the division of magnitude to infinity. For mathematicians in their demonstrations also use the infinite in magnitude. And this would not be so if the infinite were altogether removed from things. Therefore, it is necessary to posit an infinite.

             339. A third argument according to the opinion of many is taken from the perpetuity of generation and corruption. For if the infinite were completely removed, it could not be said that generation and corruption endure to infinity. Hence it would be necessary to say that at some time generation ceases altogether, and this is contrary to the opinion of many. Therefore, it is necessary to posit an infinite.

             340. The fourth argument is taken from the apparent nature [ratio] of the finite. For it seems to many that it is of the very nature [ratio] of the finite that it is always contained by something else. For around us we see that every finite thing is extended as far as some other thing. Therefore, if some designated body is infinite, then the proposition is established [i.e., there is an infinite]. If, however, the designated body is finite, then it must be terminated by another body; and if that body is finite, it must be terminated by still another body. Therefore, this will either proceed to infinity or will end at some infinite body. In either case an infinite is posited. Therefore, it must be that there is no end to bodies if it is always necessary to every finite body to be contained by another.

             341. The fifth argument is taken from the apprehension of the intellect or the imagination. He says that that which most of all raises the question leading men to posit the infinite is that the intellect never fails in being able to add something to any given finite thing. Now the ancient philosophers thought that things corresponded to the apprehension of intellect and sense. Hence they said that everything which is seen is true, as is said in Metaphysics, IV. And because of this they believed that there is also an infinite in things. It seems that number is infinite because the intellect by adding unity to any given number forms another species. And for the same reason mathematical magnitudes, which are formed in the imagination, seem to be infinite. For when we are given any magnitude, we can imagine one which is greater. And for the same reason there seems to be a certain infinite space outside the heavens. For we are able to imagine certain dimensions [reaching] to infinity outside the heavens.

             If, however, there is an infinite space outside the heavens, it seems to be necessary that there is an infinite body and that there are infinite worlds. This is so for two reasons.

             The first reason is as follows. If a whole infinite space is considered, the whole considered in itself is uniform. Therefore, we cannot assign any reason why that space should be void of body in one part more than in another. Therefore, if the corporeal magnitude of this world is found in any part of that space, then it is necessary that in every part of that space be found some corporeal magnitude such as what is of this world. And thus it is necessary that body as well as space be infinite. Or else it is necessary that worlds be infinite, as Democritus held.

             The other reason for showing the same thing is as follows. If there is an infinite space, it is either a void or a plenum. If it is a plenum, then our proposition is established: there is an infinite body. If, however, it is a void, then since a void is nothing but a place which is not occupied by a body, though it is possible for it to be occupied, it is necessary, if there is an infinite space, that there also be an infinite place which can be occupied by a body. And thus it will be necessary for there to be an infinite body, because 'to be' and 'to happen' do not differ in perpetual things. Hence, if an infinite place happens to be occupied by a body, it is necessary to say that it is occupied by an infinite body. Therefore, it seems necessary to say that there is an infinite body.

             342. Next where he says, 'But the problem . . .' (203 b 30), he raises objections to the contrary. Concerning this he makes three points.

             First he shows that the question has its difficulty, lest the arguments set forth above should seem to conclude to the truth. Secondly, where he says, 'We must begin . . .' (204 a 2), he explains how the word 'infinite' is used. Thirdly, where he says, 'Now it is impossible . . .' (204 a 8), he gives arguments to show that there is no infinite.

             343. He says, therefore, first that it is difficult to decide whether there is or is not an infinite. For many impossibilities follow from the position of those who hold that the infinite in no way is, as is clear from what has already been said. And many impossibilities also follow from the position of those who hold that there is an infinite, as will be clear from the arguments which follow.

             Moreover, there is also the question of what sort of thing the infinite is, i.e., whether it is something existing in itself as some kind of substance, or some accident belonging per se to some nature, or neither of these (i.e., neither existing in itself as substance, nor a per se accident). Nevertheless, if it is an accident, is it an infinite continuum or something infinite in respect to multitude? Now the problem of whether there is an infinite sensible magnitude especially pertains to the consideration of the natural philosopher. For sensible magnitude is natural magnitude.

             344. Next where he says, 'We must begin . . .' (204 a 2), he explains the ways in which the term 'infinite' is used. He posits two divisions of the infinite.

             The first of these is common to the infinite and to all things spoken of as privations.

             The term 'invisible' is used in three ways, either as that which is not apt to be seen, such as the voice which does not belong to the genus of visible things, or as that which is poorly seen, as what is seen in a dim light or from a distance, or as that which is apt to be seen but is not seen, as what is altogether in the dark.

             Thus in one way the infinite is said of that which is not apt to be passed through (for the infinite is the same as the impassable). And this is so because it belongs to the genus of impassable things, such as indivisible things, e.g., a point and a form. It is also in this way that the voice is said to be invisible.

             In another way the infinite is said of that which of itself can be passed through, even though this passing through cannot be accomplished by us. For example we might say that the depth of the sea is infinite. Or we might refer to something which can be accomplished but only scarcely and with difficulty, as if we were to say that a journey to India is infinite. Each of these pertains to that which is poorly passable.

             In a third way the infinite is said of that which is apt to be passed through as existing in the genus of passable things, but is not passed through to its end, as if there were a line or any other quantity which does not have an end. And this is the infinite properly speaking.

             He sets forth the other proper division of the infinite where he says, 'Further, everything that is infinite . . .' (204 a 7). He says that the infinite is called such either by addition, as in numbers, or by division, as in magnitudes, or in both ways, as in time.

             345. Next where he says, 'Now it is impossible . . .' (204 a 8), he sets forth arguments for denying the infinite. First he sets forth arguments for denying a separated infinite, which the Platonists posit. Secondly, where he says, 'We may begin . . .' (204 b 4), he sets forth arguments for denying an infinite in sensible things.

             Concerning the first part he sets forth three arguments. In the first argument he says that it is impossible for there to be an infinite separated from sensible things in such a way that this infinite would be something existing in itself, as the Platonists held. For if the infinite is held to be something separated, it either has some quantity (i.e., it is either continuous which is magnitude or discrete which is multitude) or it does not. If it is a substance without the accident of magnitude or multitude, then the infinite must be indivisible. For every divisible thing is either a number or a magnitude. If, however, it is something indivisible, it will not be infinite except in the first way, i.e., as a thing is said to be infinite which is not apt to be passed through, just as the voice is said to be invisible. But this lies outside the meaning of the present question, wherein we inquire about the infinite, and outside the intention of those who posit an infinite. For they do not intend to posit the infinite as something indivisible, but as something impassable, i.e., as something apt to be passed through, though not passed through.

             If, on the other hand, it were not only a substance, but would also have the accident of magnitude and multitude to which the infinite pertains (and thus the infinite would be in the substance by reason of that accident), then the infinite as such will not be a principle of the things which are, as the ancients held. Likewise we do not say that the invisible is a principle of speech, even though it is an accident of voice which is a principle of speech.

             346. He sets forth the second argument where he says, 'Further, how can the infinite . . .' (204 a 17). The argument is as follows.

             A passion is less separable and existent in itself than its subject. But the infinite is a passion of magnitude and number. But magnitude and number cannot be separated and existing in themselves, as is proved in metaphysics. Therefore neither can the infinite.

             347. He sets forth the third argument where he says, 'It is plain . . .' (204 a 20). He says that it is clear that it cannot be held that the infinite exists in act and that it exists as a substance and as a principle of things.

             For the infinite will either be divisible into parts or not divisible into parts. If it will be divisible into parts, it is necessary that every part of it be infinite, if the infinite is a substance. For if the infinite is a substance and is not predicated of a subject as an accident, then it will be necessary that 'the infinite' be the same as 'to be infinite', i.e., the same as the essence and the nature [ratio] of the infinite. For that which is white and the nature of white are not the same. But that which is man is that which is the nature of man. Hence, if the infinite is a substance, it must be either indivisible or divided into infinite parts, which is impossible. For it is impossible for one and the same thing to be composed of many infinite things, because it would be necessary for the infinite to be terminated by another infinite.

             It is clear not only from reason but also from an example that, if the infinite is a substance and is divided, then every part of it is infinite.

             For as every part of air is air, so every part of the infinite will be infinite, if the infinite is a substance and a principle. For if it is a principle, the infinite must be a simple substance not composed of diverse parts, just as no part of a man is a man. Since, therefore, it is impossible for any part of the infinite to be infinite, it must be that the infinite is not divisible into parts and is indivisible. But that which is indivisible cannot be infinite in act. For that which is infinite in act is a quantity, and every quantity is divisible. It follows, therefore, that if there is an infinite in act, it is not infinite as a substance, but as under the intelligibility [ratio] of the accident of quantity. And if this is the infinite, it will not be a principle; rather that of which it is an accident [will be a principle]. And this is either some sensible substance, such as air, as the natural philosopher held, or some intelligible substance, such as the even, as the Pythagoreans held.

             Hence it is clear that the Pythagoreans inconsistently said that the infinite is a substance, maintaining at the same time that it is divisible. For it follows that every part of the infinite is infinite, which is impossible, as was said above.

             348. Lastly he says that the question of whether there is an infinite in mathematical quantities and in intelligible things which have no magnitude is more general than what is being considered at present. For at the present we intend to consider whether in sensible things, concerning which we develop natural science, there is a body infinite in size, as the ancient natural philosophers held.