SOLUTION OF VARIOUS OBJECTIONS AGAINST THE POSSIBILITY OF AN INFINITE EXTENSION.
84. The discussions on the possibility of an infinite extension are of a very ancient date. How could it be otherwise? Must not the glorious spectacle of the universe, and the space which we imagine beyond the boundaries of all worlds, naturally have given rise to questions as to the existence or possibility of a limit to this immensity?
Some philosophers think an infinite extension impossible. Let us see on what they found their opinion.
85. Extension is a property of a finite substance, and that which belongs to a finite thing cannot be infinite; therefore it is impossible to conceive infinity of any kind in a finite being. This argument is not conclusive. It is true that an extended substance is finite, in the sense that it does not possess absolute infinity such as is conceived in the Supreme Being; but it does not follow from this that it cannot be infinite under certain aspects. Neither is itcorrect to say that no finite substance can have an infinite property, because the properties flow from the substance, and the infinite cannot proceed from the finite. In order that this argument may be valid, it is necessary to prove that all the properties of a being emanate from its substance: figures are accidental properties of bodies, and yet many of them have no relation to the substance, and are mere accidents which appear or disappear, not by the internal force of the substance, but by the action of an external cause. We see extension in bodies; but as we know not the essence of corporeal substance, we cannot say how far this property is connected with the substance, whether it is an emanation from it, or only something which has been given to it and may be taken from it without any essential alteration.[39]
Moreover, when we say that the infinite cannot proceed from the finite, we do not deny that an infinite property may proceed from a substance finite in its essence.
When we admit the infinite property, we admit at the same time all that is necessary in the substance in order that this property may have its root in it, so long as we do not deny the character of finite which essentially belongs to every creature. When we deny that creatures are or can be infinite, we speak of essential infinity, of that infinity which implies necessity of being and absolute independence under every aspect; but we do not deny them a relative infinity, such as that of extension.
To undertake to prove that infinite extension is impossible, because every property of a finite substance must be finite, is equivalent to supposing the very thing in dispute; for the precise question is, whether one of these properties, namely, extension, can be infinite. In order to establishthe negative proposition, "No property of a finite substance can be infinite," it is necessary to prove this of extension. Hence the argument which we are imposing implies, in some manner, a begging of the question, when they found it on a general proposition which can only be certain when the present question is solved.
86. Infinite extension ought to be the greatest of all extensions, but there is no such extension. From any given extension God can take away a certain quantity; for example, a yard: in that case the infinite extension would become finite, for it would be less than the first; and as the difference between the two extensions is only a yard, it is clear that not even the first could be infinite; for it is impossible that there should be only the difference of one yard between the finite and the infinite.
This difficulty merits a serious consideration: at first sight it seems so conclusive that no possibility of a satisfactory solution is conceivable.
The proposition that the difference between the finite and the infinite cannot be finite, is not wholly correct. We must first of all take notice that the difference between two quantities, whether finite or infinite, cannot be absolutely infinite, in the sense of diminution. Difference is the excess of one quantity over another, and necessity implies a limit; for as the excess only is considered, the quantity exceeded is not contained in the difference. Calling the difference D, the greater quantity A, and the smallera, I say that D can in no hypothesis be infinite. By the supposition D = A -a; therefore D +a= A; in order that D may equal A it is necessary to add to ita; therefore D cannot be infinite. If we suppose A = ∞, we shall have D = A -a= ∞ -a, or D +a= ∞. Therefore to make D infinite we must add to ita, and we can never have D = ∞ unlessa= 0; but in that case there would be no true difference,since the equation, D = A -a, would be converted into D = A - 0 = A, and the difference would not be real but imaginary.
It follows from this that no difference between two positive quantities can be absolutely infinite; if it is so in some sense, it is not so in the sense of diminution; and the union of these two ideas of difference and infinity results in a contradiction.[40]
The difference between an infinite quantity and a given finite quantity cannot be another given finite quantity, but it must be infinite in some sense. Let us suppose an infinite line and a given finite line, the difference between them cannot be expressed by a given finite lineal value. For supposing the second line to be a finite and a given line, we may place it upon the infinite line in any of its directions, and from any point in it it will reach a certain point of the infinite line. If we suppose a second given finite line, representing the difference between the other two lines, we ought to place it upon the infinite line at the point where the other terminates; and it is evident that it will terminate at another point determined by its length; therefore it will not measure the whole of the difference between the infinite and the finite lines.
We obtain the same result in algebraic expressions. If A be a given finite value, the difference between A and ∞ cannot be another given finite value. For, expressing the difference by D, we shall have ∞ - D ± A D. Therefore, D + A = ∞; consequently, if both were given finite values, an infinite would result from two given finite values, which is absurd.
Hence, a difference may be in some sense infinite, according to the meaning we attach to the term infinity. If from the point where we are situated, we draw a line towards the north and produce it infinitely, and then produce it, also, infinitely towards the south, the difference between either of these lines and the sum of them both, will be infinite only in a certain sense. This is also verified by algebraic expressions. If we have the infinite value equal 2∞, and compare it with ∞, the result is 2∞ - ∞ = ∞.
In general, from any infinite value we may subtract any finite difference in relation to it, so long as the subtrahend is not a given finite value. Let ∞ be the infinite value,—I say that we can find in it any finite value; for, ∞ being an infinite value, A contains all finite values of the same order; therefore it contains the finite value, A; consequently we may form the equation, ∞ - A = B. Whatever be the value of B, the relation of B to ∞ is A; for by only adding A to B we obtain ∞. The equation, ∞ - A = B, gives B + A = ∞, and also ∞ - B = A; and as A is a given value according to the supposition, and A is the given finite difference between ∞ and B, it follows that we may find a finite difference to every infinite value.
We may infer from this that the possibility of assigning a finite difference to an infinite extension, does not prove any thing against its true infinity. The infinite, and because it is infinite, contains all that belongs to the order in which it is infinite. We may take any sure value, and considering it as a difference, and we shall obtain a finite difference. But far from proving the absence of infinity, this confirms its existence; for it shows that all the finite is contained in the infinite.
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.