106. This variety of motion and form can be numbered. At every step we measure motion, applying to it the idea of number; at every instant we count the forms of a portion of matter, as for example, a piece of wax, to which different forms have been given successively: whatever be the number of the beings which we suppose to exist, every one of them will be susceptible of transformations which may be counted. Therefore, in the very nature of things, there is an intrinsic impossibility of the existence of an actual infinite number.
107. I believe that these arguments fully demonstrate the impossibility of an actual infinite number; and if I do not dare to say that I am sure of having given a complete demonstration, it is because the nature of the question presents so many and so great difficulties, it so bewilders and confounds the weak understanding of man, that there is always reason to fear that even those arguments, which seem the clearest and most conclusive, may conceal some fault which vitiates their force, and makes an illusion appear an incontestible truth. Still I cannot but observe that to combat this demonstration, it seems, to me that it would be necessary to deny our primary ideas, the exclusion of being and not-being, and the necessity of succession, of time, to the realization of contradictory things.
108. Perhaps it may be objected to me that contradictory modifications are not a part of the infinite number, which only relates to the possible: but this does not destroy my demonstration; it rather confirms it. For as the absolute infinite number implies the absolute negation of all limit, when, in treating of the realization of this conception, I meet with things that are contradictory, I say that the realization of the conception is contradictory, because the general and indeterminate conception is more extended than all possible number.
109. The origin of their greater conception is, that the indeterminate conception abstracts all conditions, that of time included; but the reality does not and cannot abstract these conditions. Hence arises the conflict between the conception and its realization, and this explains why the conception is not contradictory, although its realization is impossible.
Let us suppose a number realized containing all the species and individuals possible, we may reflect on the conception of the infinite number, and say that the true infinity of the number requires the absolute negation of all limit; but thinking of the collection of things which exists, we can find it a limit, for concerning this collection of units in general, we may add to it another number expressing the new modifications which may be produced. At the instant A, the number of units may be expressed by M. At the instant B, there will be a new collection of units which may be expressed by N. The sum of M + N will be greater than either M or N alone. Therefore, neither M nor N will be absolutely infinite. The indeterminate conception abstracts instants and relates to the sum above; hence it includes things which cannot co-exist.
[CHAPTER XV.]
IDEA OF ABSOLUTELY INFINITE BEING.
110. We are entering on a difficult question. Serious difficulties are found in the idea of the infinite in general; the idea of absolutely infinite being is not less difficult. We have seen that there are different orders of infinities, each one of which is a conception formed by the association of the two ideas of a particular being and the negation of limit. But it is easy to see that none of the infinities hitherto examined can be called infinite in the strict sense of the term: they are all limited under many aspects,—none of them is an infinitely perfect being. The idea of this being is not fully possessed by us while in this life; still it may be analyzed and explained with more clearness than it is by most authors. The great difficulties, which we meet with in this attempt, show the necessity of deep meditation, and the transcendency of the errors which originate in a wrong understanding of the word infinite when applied to God.
111. What is an absolutely infinite being? It might seem that we had said all that is necessary in defining the absolutely infinite being to be that which has no negation of being: but this is a common notion which leaves much to be desired. It is an indisputable truth that the infinite being has no negation of being; but it is a truth so far beyond our reach that it presents to our weak understanding only a gloomy confusion, as soon as we attempt to determine exactly its true sense.