CONSIDERATIONS ON THE APPLICATION OF THE IDEA OF THE INFINITE TO CONTINUOUS QUANTITIES, AND TO DISCRETE QUANTITIES, IN SO FAR AS THESE LAST ARE EXPRESSED IN SERIES.
31. One of the characteristic properties of the idea of the infinite is application to different orders. This gives occasion to some important considerations which greatly assist to make this idea clear in our mind.
32. From the point where I am situated I draw a line in the direction of the north; it is evident that I may prolong this line infinitely. This line is greater than any finite line can be; for the finite line must have a determinate value, and therefore, if placed on the infinite line, will reach only to a certain point. This line, therefore, seems to be strictly infinite in all the force of the word, because there is no medium between the finite and the infinite, and we have shown that it is not finite, since it is greater than any finite line; therefore it must be infinite.
This demonstration seems to leave nothing to be desired; yet there is a conclusive argument against the infinity of this line. The infinite has no limits, and this line has a limit, because, starting from the point from which it is drawn in the direction of the north, it does not extend in the direction of the south.
33. This line is greater than any finite line; but we may find another line greater still. If we suppose it produced in the direction of the south, it will be greater by how much it is produced towards the south; and if it be infinitely produced in this direction, its length will be twice that of the first line.
34. By the infinite prolongation of a line in two opposite directions we seem to obtain an absolutely infinite line; for we cannot conceive a lineal value greater than that of a right line infinitely prolonged in opposite directions. But it is not so: by the side of this right line another may be drawn, either finite or infinite, and the sum of the two will form a lineal value greater than that of the first line; therefore that line is not infinite, because it is possible to find another still greater. And as, on the other hand, we may draw infinite lines and prolong them infinitely, it follows that none of them can form an infinite lineal value, becauseit is only a part of the lineal sum resulting from the addition of all the lines.
35. Reflecting on this apparent contradiction in our ideas, we discover that the idea of the infinite is indeterminate, and consequently susceptible of different applications. Thus, in the present instance, it cannot be doubted that the right line, prolonged to infinity, has some infinity, since it is certain that it has no limit in its respective directions.
36. This example would lead us to believe that the idea of the infinite represents nothing absolute to us; because even among those objects which are presented the most clearly to our mind, such as the objects of sensible intuition, we find infinity under one aspect which is contradicted one by another.
37. What we have observed of lineal values is also true of numerical values expressed in series. Mathematics speak of infinite series, but there can be no such series. Let the series bea,b,c,d,e, ....: it is called infinite if its terms continuead infinitum. It cannot be denied that the series is infinite under one aspect; for there is no limit which puts an end to it in one sense; but it is evident that the number of its terms will never be infinite, because there are others greater; such, for instance, is the series continued from left to right, if continued from right to left at the same time, in this manner:
..........e,d,c,b, |a,b,c,d,e, ..........
In this case the number of terms is evidently twice as great as in the first series.
Therefore the series which are called infinite are not infinite, and cannot be so, in the strict sense of the term.
38. But what is still more strange is, that the series is not infinite, even though we suppose it continued in oppositedirections; for by its side we may imagine another, and the sum of the terms of both will be greater than the terms of either; therefore neither will be infinite. As it is evident that whatever be the series, we can always imagine others, it follows that there can be no infinite series in the sense in which mathematicians use the word series to express a continuation of terms, not excluding the possibility of other continuations besides the supposed infinite continuation.
39. The objections against lineal infinity apply equally to surfaces. If we suppose an infinite plane, it is evident that we can describe an infinity of planes distinct from the first plain and intersecting it in a variety of angles; the sum of all these surfaces will be greater than any one of them. Therefore the infinite extension of a plain in all directions does not constitute a truly infinite surface.
40. A solid expanding in all directions seems to be infinite; but if we consider that the mathematical idea of a solid does not involve impenetrability, we shall see that inside of the first solid a second may be placed, which, added to the first, will give a value double that of the first alone. Let S be the empty space which we imagine to be infinite; and let W be a world of equal extension placed in it and filling it; it is evident that S + W are greater than S alone. Therefore, although we suppose S to be infinite, = ∞, W also = ∞; therefore S + W = ∞ + ∞ = 2 ∞. And as this value expresses the size, the first is not infinite because it can be doubled. If we take the impenetrability, the operation may proceedad infinitum.
Therefore the first infinite, far from being infinite, seems to be a quantity susceptible of infinite increase.