POSSIBILITY OF AN ACTUAL INFINITE NUMBER.
91. Is an infinite number possible? Does the union of the idea of number with the idea of the absolute negation of limit, involve any contradiction which prevents the realization of the conception?
Whatever number we may conceive, we can always conceive one still greater: this seems to show that no existing number can be absolutely infinite. If we suppose this number to be realized, an intelligence may know it, and may multiply it by two, three, or any other number; therefore the number may be increased, and consequently it is not infinite.
This difficulty is far from being conclusive, if we examine it carefully. The intellectual act of which it speaks, would be impossible on the supposition of the existence of an infinite number. If the intelligence should not know the infinity of the number, it might make the multiplication, but it would fall into a contradiction through its ignorance; for the number being absolutely infinite, could not be increased; its multiplication would be an absurdity, and the intelligence making it, would combine two ideas which would still be repugnant, although not known to be so by the intelligence. If the absolute infinity of the existing number were known to the intelligence, the idea of multiplication could never be associated with it; for the intelligence would know that all possible products already exist.
92. An absolutely infinite number cannot be expressed in the algebraic or geometrical values; the attempt so to express it limits it in a certain sense, and therefore destroys its absolute infinity. If the expression ∞, represented an absolutely infinite number, it would not be susceptible of any combination which would increase it: to suppose that it may be multiplied by other numbers, finite or infinite, is to take its infinity in another than an absolute sense.
The fraction a/0 does not express an infinite value in all the strictness of the word; for it is evident that whatever be the value of a/0 it will always be less than 2a/0 or, in general, less than na/0 n representing a value greater than unity.
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.
On this supposition the most perfect created beings possible would exist, and no more perfect being in the sphere of creatures could be conceived. All that can be imagined would already exist, from nothing to infinite perfection.
100. Still it must be observed that the collection of created beings, whatever be their perfection, are necessarily subject to the condition of dependence on another being; a condition from which the infinite being above is essentially exempt. This condition involves limitation; therefore, all created beings must be finite.
101. Does the character of finite, which is met with in all created beings, involve a determinate limit beyond which they cannot pass? If this limit exists, is not the number of possible species also limited? And if these species are not infinite, is not an infinite number an illusion?
Although the intrinsic possibility of the infinite scale of beings seems beyond a doubt, we must beware of solving too quickly the present question. With respect to indeterminate conceptions, we see no possible limit; but would this still be so, if we had an intuitive knowledge of the species? Are we sure that in the particular qualities of beings, combined with limitation and dependence, which are essential to them, we should not discover a term beyond which they cannot go, by reason of the constitution of their nature? How impotent philosophy is to solve such questions!
102. Whatever may be concluded as to this infinity of species and their respective perfection, I do not believe that an actually infinite number can exist. Among these species must be counted intelligences which exercise their acts in succession. This is evidently so; for in this number are included human minds which think and wish in asuccessivemanner. The acts of these intelligences may be numbered: this we know from consciousness. Therefore there would never be an infinite number, because these acts, being successive, can never be all at the same time.
103. It may be answered that in this case we might suppose that spirits, including our own, have only one act of intelligence and will. To this I reply, that besides contradicting the nature of created beings, which, because they are finite, must be subject to change, it is also open to another objection, inasmuch as it eliminates at once many species of beings, and thus, instead of preserving the infinity, renders it impossible. Who can deny the possibility of that which exists? If, as our experience informs us, there now exist beings of successive activity, why would not these beings be possible on the supposition that the divine omnipotence had exerted all its infinite creative power?
104. This difficulty, which is founded on the nature of finite intelligences, seems to render the existence of an infinite number impossible, and it becomes still stronger if we examine the question under a more general aspect.
The existence of an absolutely infinite number excludes the existence of any other number. That which is numbered is not substance alone, but its modifications also. This has already been demonstrated with regard to intelligences, and is true in general of all finite beings. Every finite being is changeable, and its changes may be counted. The modifications produced by the changes cannot all exist at once, for some of them exclude others. Therefore, an actual infinite number is never possible.
105. Let us apply these considerations to the sensible world. Motion is a modification to which bodies are subject. This modification is essentially successive. A motion, the parts of which co-exist, is absurd. The co-existence of different states, which result from different motions, is also absurd: things that are contradictory cannot exist at the same time, and many of these situations are contradictory, because one of them necessarily involves the negation of others. If a line falling on another line revolve around a point, it will successively describe different angles. When it forms an angle of 45 degrees, it will not form an angle of 30 degrees, nor of 40, nor 70, nor 80; these angles mutually exclude one another. A portion of matter will form different figures, according to the arrangement which is given to the parts of which it is composed. When these parts form a globe, they will not form a cube; these two solids cannot exist at the same time, formed of the same portion of matter.
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 bethe 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 theconception 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.