CONCEPTION OF AN INFINITE NUMBER.
69. Can we conceive an infinite number? On one side, it seems not; because we doubt its possibility, and if we possessed this idea we should have no doubt of its existence. On the other side, it seems that we can conceive an infinite number; for we know immediately when a number is not infinite, and we could not know this if we had not the idea of infinite number.
Our observations on infinite series would seem to prove that the idea of infinite number is an illusion; for we find those numbers which we believed infinite, not to be so.
I think this question may be solved on the same principles as those of the last chapter. I see no difficulty in admitting the idea of an infinite number, nor how any contradiction can proceed from it.
70. Number is a collection of units; it is a general idea, because to conceive the number, we do not need to know of what class, or how many the units may be. The idea of number in general abstracts absolutely all such determinations. It is evident that, whatever number we imagine, we can always conceive another still greater, and if we assign a limit to a number, we can always remove it indefinitely, so that the limit of one is not the limit of the other. To the idea of number, we unite the idea of a limit and of the negation of another limit. Therefore, if we unite to theidea of number in general, the idea of the negation of limit in general, we shall obtain the idea of an infinite number.
71. What does this idea represent? It represents nothing determinate: it is an entirely abstract conception, formed of two other abstract conceptions, those of number and the negation of limit. No determinate object corresponds to it; it is a work of our understanding referred to objects in general, without a determination of any sort. We may now solve the difficulties previously intimated.
72. Why is a series of terms presented to us as infinite, which, when we examine it closely, we find wants some of the marks of infinity? Because, in the first instance, we apply the negation of limit under a condition which we take no notice of in the second instance.
Set us the seriesa,b,c,d,e, ..........
It is evident that we may continue it infinitely, and conceive the negation of all limit of this continuation: in this sense, the number of terms is infinite; for the idea of the negation of limit is really applied to the series. When we ask if the number of terms is absolutely infinite, we abstract the condition under which we had united the negation of limit. That, therefore, which is infinite in one instance is not so in another. Still there is not any contradiction because the yes and the no refer to different suppositions.
73. Let us take a line and measure it by feet. Producing this line we multiply the number of feet; and we may conceive the negation of all limit of this multiplication. The number of feet will then be infinite. If instead of a foot we take an inch as the unit of measure, we shall have a number twelve times as great. This number would also be infinite, and thus we should have two infinite numbers, one of them greater than the other. Is there any contradiction in this? Certainly not: there is only a different combination of ideas. In the first case, the idea of the negation of limit was subordinated to the condition of the division of the line into feet: whereas, in the second case, we introduce a different condition; the division of the line into inches.
74. But, it may be said, these numbers, considered in themselves, abstracted from their relation to feet or inches, are equal or they are not equal; consequently they are infinite or not infinite. The objection vanishes as soon as we correct the error which supports it. When we abstract all relation to determinate divisions, we consider number in general; on this supposition there are not two cases, but only one; there cannot then be a relation of greater or less. We have only the conception of number in general combined with the idea of the negation of limit in general; therefore the result must be an infinite number in the abstract.
The difficulty consists in a contradiction which escapes our sight at first. We abstract particular conditions in order to know if the numbers are in themselves infinite or not; and at the same time we do not abstract them, because it is only in reference to them that the objection has any meaning, since it supposes the division into various kinds of units. When, therefore, we speak of particular numbers, and at the same time pretend to consider them in themselves, we fall into a contradiction, because we take the numbers both with and without particular conditions at the same time.
75. From all that has been said, we may conclude that the conception of infinite number, abstracted from the nature and relations of the things numbered, involves no contradiction, since it contains only the two ideas of number, as a collection of beings, and of the absolute negationof limit; but we cannot affirm from this alone, that an infinite number can be realized. Infinite number cannot become actual without an infinite collection of beings; and these beings, when realized, cannot be abstract beings, which contain nothing else but being; they must have characteristic qualities, and must be subject to the conditions imposed by these qualities. As we absolutely abstract these conditions in the general conception, it is not possible to discover, from the conception alone, the contradiction which they may imply. Hence, although there is no contradiction contained in the conception, there may still be in the reality. In the same manner, certain mechanical theories are perfectly conceivable, but they cannot be reduced to practice on account of the opposition of the matter to which they should be applied. Finite beings are the matter on which indeterminate and metaphysical conceptions are to be realized; the possibility of the conceptions does not absolutely prove the possibility of the beings. The reality may draw with it certain determinations involving a contradiction which was latent in the general conception, and is made manifest by the reality.