Fundamental Philosophy, Vol. 2 (of 2)

by Jaime Luciano Balmes

93. Neither can an infinite number be represented in geometrical values.

Let us take a line one foot long. It is evident that if we produce this line infinitely in opposite directions, the number of feet will be in some sense infinite, since the foot is supposed to be repeated infinite times: the expression of the number of the feet will be the expression of an infinite value. Now, I say that this number is not infinite, because there are other numbers still greater. In each foot there are twelve inches; therefore, the number of inches contained in the line will be twelve times as great as the number of feet; consequently the number of feet is not infinite. Neither is the number of inches infinite; for they in their turn may be divided into lines, the lines into points; and it is evident that the number of the smaller quantities will be proportionally greater than the number of the greater quantities. There will be twelve times as many inches as feet, twelve times as many lines as inches, and twelve times as many points as lines; and this progression can never end, because the value of a line is infinitely divisible.

94. Pushing to infinity the divisibility of an infinite line, we seem to have an infinite number in the elements which constitute it; but a slight reflection will dissipate this illusion. For it is evident that we can draw other infinite lines by the side of the supposed infinite line; and since according to the supposition, each of them may be infinitely divided, it follows that the sum of the elements of all the lines will give a greater number than the sum of the elements of any one of them.

95. If we wish to find an infinite number of parts in values of extension, we must suppose a solid infinite in all its dimensions, with all its parts infinitely divided. But not even then should we have an absolutely infinite number, although we should have the greatest which can be represented in values of extension.

Conceding that an infinite extension existed which is infinitely divisible, the number of its parts would not be absolutely infinite; for we can conceive other beings besides extended beings, and considering both under the general idea of being, we might unite them in a number which would be greater than that of extended beings alone.

96. No imaginable species of beings infinitely multiplied, can give an absolutely infinite number. The reason is the same as that given in the last paragraph: the existence of beings of one species does not render the existence of beings of another species impossible. Therefore, besides the supposed infinity of the number of beings of a determinate species, there are other numbers which, united with this, produce a number greater than the pretended infinity.

97. The existence of an absolutely infinite number requires: first, the existence of infinite species of beings; and secondly, the existence of infinite individuals of each species. Let us see if these conditions can be realized.

98. There seems to be no doubt of the intrinsic possibility of infinite species. The scale of beings is between two extremes, nothing and infinite perfection: the space between these extremes is infinite; and beings may be distributed on it in an infinite gradation.

99. Admitting the intrinsic possibility of an infinite gradation in the scale of beings, the question occurs, whether their possibility is only ideal, or also real, that is, may be realized. God is infinitely powerful; if the infinite gradation is intrinsically possible, God can produce it; for whatever is intrinsically possible falls within the reach of divine omnipotence. On the other hand, supposing, as we must, the liberty of God, there is no doubt but God is free to create all that he can create. If then there is nothing repugnant in an infinity of the species of beings distributed in an infinite gradation, these beings may exist if God will it. Therefore denying all limit to the number of species and of individuals of each species, it seems that the infinite number would exist, since it is impossible to imagine any increase or limitation in the collection of all beings.