In this case, the subtrahend would be infinite under a certain aspect; but not in the order of diminution, because it wants the quantity which is taken from it.
87. There is another argument against the absolute infinity of extension, which seems to have more weight than any of those which precede, and I cannot see why it has never occurred to those who argue against this possibility. It is this,—we suppose an infinite extension to exist. God can annihilate it, and then create another equally infinite. The sum of both is greater than either alone; therefore neither of them alone is infinite. This annihilation we may suppose as often as we wish; hence we may have a series of infinite extensions. The terms of this series cannot exist at the same time, since one actual infinite extension excludes all others. Therefore, as the sum of the extensions is greater than any number of particular extensions, the absolute infinite extension must be found, not in the particular extensions, but in the sum, and hence an actual infinite extension is intrinsically impossible.
To solve this difficulty we must distinguish between extension and the thing extended: the whole question turns on the intrinsic possibility of the infinity of extension, considered in itself, abstracting absolutely the subject in which it is found. The difficulty places before our sight a series of successive infinite extensions; but in reality this succession is in the beings which are extended, and the number of which goes on increasing; but not in the extension itself. The pure idea of infinite extension in the one case, is not increased by the new extensions which are produced; the extension appears, disappears, reappears, and again disappears, but is not increased. The succession shows the intrinsic possibility of its appearance and its disappearance, its essential contingency, because it is not repugnant for it to cease to exist when it exists, or to pass again from non-existence to existence. If we examine our ideas, we shall find that we cannot increase the infinite extension which we conceive, by any imaginable supposition; and that whatever we may do, is reduced to a succession of productions and annihilations. The idea of infinite extension seems to be a primitive part of our mind; the infinity which we imagine in space, is only the attempt which our mind makes to express its idea in reality. Created with sensible intuition, we have received the power of expanding this intuition on an infinite scale,—to do this we require the idea of an infinite extension.
[CHAPTER XIII.]
EXISTENCE OF INFINITE EXTENSION.
88. The question of the possibility of an infinite extension is very different from that of its existence. The first we answer in the affirmative, the second in the negative.
Descartes maintained that the extension of the world is indefinite; but this is a term which, although it has a very rational meaning when it refers to the compass of our understanding, has no meaning when applied to things. There is no objection to saying that the extension of the world is indefinite, if it only means that we cannot assign its limits; but in the reality, the limits exist or do not exist, indifferently of our power of assigning them; there is no medium between yes and no; therefore there is no medium between the existence and the non-existence of these limits. If they exist, the extension of the world is finite; if they do not exist, it is infinite;—in either case, the word indefinite expresses nothing.
The argument of Descartes proves nothing, or it proves the true infinity of the world. For, if we must remove its limits indefinitely because we always conceive indefinitely an extension beyond every other extension, as, on the other hand, we know that this series of conceptions has no limit, we may at once transfer the unlimitedness to the object which corresponds to those conceptions, and affirm that the extension of the world is absolutely infinite. Unfortunately, the argument of Descartes is without any basis; for it consists in a transition from the ideal, or, rather, imaginary order, to the real order, which is contrary to good logic.[41]