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 negation of 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.
[CHAPTER X.]
CONCEPTION OF INFINITE EXTENSION.
76. Is infinite extension conceivable? This conception includes two ideas: the idea of extension, and the idea of the negation of limit. The idea of extension is a general conception, referring to the intuition which, whatever may be in itself and in its object, represents extension and the union of the three dimensions, the pure form of which is space. It is evident that we can unite, in one conception, the two ideas of extension in general and the negation of limit; and if this is what is called the idea of infinite extension, it is clear that we have this idea. This conception of infinite extension, abstracts all conditions of the reality; we do not know whether there be, in the nature of extended things, any thing which prevents the absolute infinity of their extension; consequently, we are ignorant whether there is or is not any latent contradiction, which the general conception does not reveal to us.
77. It must be remembered that I am speaking of the idea and not of the sensible representation of extension; for although I hold that it is possible for us to have the conception of an infinite extension, I do not think the same with respect to its sensible representation. The latter may be indefinitely expanded, but it cannot become infinite.
Reason demonstrates this impossibility which consciousness makes known to us. Internal sensible representations are only the repetition of the external, or at least are formed from the elements which these latter furnish. Sight and touch are the two senses which produced the representation of extension, and they both imply a limit. Touch only reaches that which is immediate to it, and sight cannot see with a limit which sends the rays of light to it. Internal sensible representations must always retain this limitation; their object may be expanded, or the limit removed to a greater distance, but to destroy this limit would be to destroy themselves. Therefore, the imagination of an infinite extension is impossible to every sensitive being.
78. I have proposed above (§ 40) an objection against the infinity of extension, in so far as we may represent it as a size without limits.
The objection was, that as the idea of impenetrability is not contained in the conception of a solid, we may imagine an infinite series of infinites placed one inside of another. This difficulty is only conclusive when speaking of the conception of a solid which contains something more than the pure idea of extension. The idea of extension necessarily implies that some parts are outside of others, and it is not possible to conceive extension otherwise. It is certain that a body may be situated in a part of space; taking from this body its impenetrability, we may put another body in the same place, and so on to infinity; but in that case we conceive something besides pure extension, we unite something, although in a general and indeterminate manner, to the idea of things situated in space; otherwise we should not distinguish the space, representing pure extension, from the solids placed in it, nor should we distinguish these solids from one another, if we did not recognize in them some difference, although general and undetermined.