Footnote 1This chapter consists of extracts from Aristot. Physics 3.4, 5, 7. The infinite is either (a) that which cannot be traversed because it is not its nature to be traversed (just as sound is by nature invisible); or (b) that which admits of an endless traverse; or (c) scarcely admits of traverse; or (d) which, though it would naturally admit of traverse or limit, does not do so. Further, it may be infinite in respect of addition or of subtraction or of both.
That the infinite should be a separate independent entity, Footnote 2The Pythagorean and Platonic view. and yet imperceptible, is impossible.
For if it is neither magnitude nor plurality, but infinity itself is the essence of it, and not merely an accident, it must be indivisible; because that which is divisible is either magnitude or plurality. And if it is indivisible it cannot be infinite, except in the same way as sound is invisible. But this is not what people mean by infinite; and it is not the infinite in this sense that we are investigating, but the infinite in the sense of the untraversable.
Again, how can the infinite exist independently unless number and magnitude, of which infinity is an attribute, also exist independently? Footnote 3Aristotle has argued that they do not in Aristot. Met. 1.9.16-25. And further, if the infinite is accidental, it cannot, qua infinite, be an element of things; just as the invisible is not an element of speech, although sound is invisible. It is clear also that the infinite cannot exist actually.
Otherwise any part of it which we might take would be infinite; for infinity and the infinite are the same, if the infinite is substance and is not predicated of a subject. Therefore it is either indivisible, or if it is partible, the parts into which it is divisible are infinite. But the same thing cannot be many infinites; for just as a part of air is air, so a part of the infinite will be infinite, if the infinite is a substance and principle.
Therefore it is impartible and indivisible. But this is impossible of the actually infinite, because it must be some quantity. Therefore infinity is an accidental attribute. But if so, as we have said, it cannot be it that is a principle, but that of which it is an accident: air Footnote 4According to Anaximenes; cf. Theophrastus, Phys. Opin. Fr. 2 (Ritter and Preller 26). or the even.
Footnote 5According to the Pythagoreans. Cf. Aristot. Met. 1.5.5. n
The foregoing inquiry is general; but what follows will show that the infinite does not exist in sensible things.
If the definition of a body is that which is bounded by surfaces,
then no body, whether sensible or intelligible, can be infinite nor can there be any separate and infinite number, since number or that which involves number is numerable. This is clearly shown by the following concrete argument. The infinite can neither be composite nor simple. For (a) it cannot be a composite body if the elements are limited in number Footnote 6This is proved in Aristot. Physics 1.6. ;
for the contraries must be equal, and no one of them must be infinite; for if the potency of one of the two corporeal elements is in any way inferior, the finite element will be destroyed by the infinite. And every element cannot be infinite, because body is that which has extension in all directions, and the infinite is that which is extended without limit; so that if the infinite is corporeal it will be infinite in all directions. Footnote 7sc. and so no other body can exist beside it.
Nor (b) can the infinite be any simple body; neither, as some Footnote 8Anaximander. It seems, however, that by ἄπειρον he meant indeterminate
or undifferentiated,
although he no doubt regarded this principle as infinite
as well. Cf. notes on Aristot. Met. 1.7.3, Aristot. Met. 12.2.3. hold, something which is apart from the elements and from which they suppose the elements to be generated (for there is no such body apart from the elements; everything can be resolved into that of which it consists, but we do not see things resolved into anything apart from the simple bodies), nor fire nor any other element.
Apart from the question of how any of them could be infinite, the All, even if it is finite, cannot be or become any one of the elements, as Heraclitus says Footnote 9Cf. Hereclitus Fr. 20-22 (Bywater). all things at certain times become fire. The same argument applies as to the One which the physicists posit besides the elements; for all change proceeds from the contrary, e.g. from hot to cold. Footnote 10The argument seems to be: Since all change is from contrary to contrary, and it is impossible that either (a) one of the elements should be contrary to the rest, or (b) one material principle should be contrary to all four elements, it follows that no one element, and similarly that no one material principle apart from the elements, can be the ultimate material principle of the universe.
Again, a sensible body is in some region, and the region of the whole and of the part (e.g. of the earth) is the same. Footnote 11i.e., the region of the universe which is proper to a given element is proper also to any part of that element. The proper region of earth is the center, of fire the circumference of the universe. Cf. Aristot. De Caelo 1.2. Therefore if the infinite body is homogeneous, it will be immovable or will always be in motion Footnote 12Ross is evidently right in taking this to refer to the rest or motion of the parts. An infinite body cannot move as a whole, because there is no space outside it. ; but this is impossible, for why should there be rest or motion below rather than above or in any other region? E.g., if there were a clod, in what region would it move or be at rest?
The region proper to the body which is homogeneous with the clod is infinite. Then will the clod occupy the whole of that region? How can it? Then what of its rest or motion? It will either rest everywhere—in which case it cannot move—or move everywhere; in which case it cannot rest. Footnote 13If earth is an infinite body, its region must be infinite. But the infinite has no center (cf. sect. 13). Therefore a clod, which cannot occupy the whole region proper to earth, will have no region proper to itself to which it can move or in which it can rest. And if the whole is not alike throughout, the regions proper to its parts are unlike also; and (a) the body of the whole is not one, except in virtue of contact; (b) the parts will be either finite or infinite in kind.
Finite they cannot be, for then those of one kind would be infinite Footnote 14sc. in quantity. If the universe is infinite in quantity, and the elements are limited in kind, some of the elements (or at least one) must be infinite in quantity. But this is impossible, just as it is impossible that all the elements should be infinite in quantity. Cf. sect. 7 above and those of another would not (if the whole is infinite); e.g., fire or water would be infinite. But such a condition would involve the destruction of the contraries. But if the parts are infinite Footnote 15sc. in kind or number. and simple, the regions proper to them are infinite and the elements will be infinite. And since this is impossible, Footnote 16Cf. sect. 6 n. the regions are finite Footnote 17Cf. sect. 14 n. and the whole must be finite.
In general, there cannot be an infinite body and a place for bodies if every body which is sensible has either weight or lightness; for it will have to move either towards the center or upwards, and the infinite—either the whole or the half—cannot do either; for how can you divide it? How can the infinite be part up and part down, or part extreme and part center?
Further, every sensible body is in some place, and of place there are six kinds, Footnote 18i.e., above and below, before and behind, right and left (Aristot. Phys. 205b 31). but these cannot exist in an infinite body. In general, if an infinite place is impossible, so is an infinite body; because that which is in a place is somewhere, and this means either up or down or one of the other kinds of place, and each of these is a limit.
The infinite is not the same in the sense that it is one nature whether it applies to magnitude or to motion or to time; the posterior is derived from the prior sense, e.g. motion is called infinite in virtue of the magnitude involved when a thing is moved or changed or increased, and time is so called on account of motion. Footnote 19Cf. Aristot. Met. 5.13.5.