58. It would be easy to show by other examples, how fruitful some general ideas are, and how they may undergo innumerable combinations, without presenting any thing determinate to the intellect. This is precisely what happens with the idea of the infinite: in vain we ask what there is within us which corresponds to it: the conception of being in general and of the negation of limit present nothing fixed, except certain abstract conditions to which we continually reduce the objects which come under our intuition, or are presented to us with certain characteristic properties which permit us to form a less vague idea of the negation of limit.
[CHAPTER VIII.]
THE DEFINITION OF INFINITY CONFIRMED BY APPLICATION TO EXTENSION.
59. We have explained the idea of infinity in general, by the indeterminate conceptions of being and the negation of limit. In order to assure ourselves that the explanation is well grounded, and that we have pointed out the essential marks of the conception, let us examine whether their application to determinate objects corresponds to what we have established in general.
If the idea of infinity is what we have defined it to be, we may apply it to all objects of sensible intuition or of the pure understanding, and we shall obtain the results which we ought to obtain, including the anomalies already referred to.[38]
60. The anomalies, or, rather, the contradictions which we seem to find in the applications of the idea of the infinite, when any thing is presented to us as infinite which we afterwards discover not to be so, originate in the application of this idea under different conditions. This variety would not be possible if the idea represented any thing determinate; but as it only contains the negation of limit in general joined to being in general, it follows that we subject this negation to particular conditions in each case, and therefore when we pass to other conditions, the general idea cannot give us the same result.
61. A line drawn from the point where we are situated in the direction of the north, and produced infinitely, gives us an infinite and a not-infinite. This contradiction is only apparent; there is really only the difference of result caused by the condition under which the general idea is applied.
When we consider a line infinitely produced towards the north, we do not apply the idea of the infinite to a lineal value in the abstract, but to a right line starting from a point and produced only in one direction. The result is what it should be. The negation of limit is affirmed under a condition; the infinite which results is subject to that condition. It may be said that there is no medium between the infinite and the not-infinite; but it is easy to solve this difficulty, if we observe that yes and no, to be contradictory, must be referred to the same thing, which is not the case when the conditions of the object are changed.
62. If instead of a line produced in one direction only, we had wished to apply the negation of limit to a right line in general, it is evident that we should have been obliged to produce the line in the two opposite directions: which would have given us another infinite under a new condition.