Fundamental Philosophy, Vol. 2 (of 2)

by Jaime Luciano Balmes

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 smaller a, 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 it a; 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 it a, and we can never have D = ∞ unless a = 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.