ORIGIN OF THE VAGUENESS AND APPARENT CONTRADICTIONS IN THE APPLICATION OF THE IDEA OF THE INFINITE.
41. The difficulties in the application of the idea of infinity, seem on the one hand, to prove that either this idea does not exist in us, or is very confused; and on the other hand, that we possess it, and in a very perfect degree. Why do we discover that numbers are not infinite, although at first they seem to be? Why do we deny the infinity of certain dimensions, notwithstanding their infinite prolongation in one sense? Because, on examining these objects, we find that they do not correspond to the type of infinity. If this type did not exist in our mind, how could it be possible for us to make use of it? How could we compare beings with it, if we did not know it? Is it possible to know when any thing arrives at a turn, if we have no idea of that turn? It is comparing without a point of comparison; that is, it is exercising a contradictory act.
42. Although these arguments in favor of the existence of the idea of the infinite, if we examine our own mind, we cannot deny that we find there a certain vagueness and confusion which inspire strong doubts as to the reality of this idea. What is presented to our mind, when we think of the infinite? The imagination abandoned to itself, extends space, expands dimensions, multiplies numbers indefinitely, but it offers nothing to the intellect which has the marks of infinity. If we leave the imagination, and regard the understanding only, it gives a type by which to judge of the infinity or not-infinity of the objects presented to it, but if we reflect on the type itself, it loses the clearness it possessed before, and we even ask if the type really exists.
43. Do we, therefore, deny the existence of this idea? are we going to renounce our intention of explaining it? We do neither. I believe that it is necessary to admit the idea, that it is not impossible to explain it, and that we may even point out the reason of its obscurity.
44. Before passing further, I wish to observe, that one of the causes of the difficulties in the explanation of the idea of the infinite, arises from our not distinguishing the intuitive from the abstract cognition.[36]Many difficulties would be avoided by attending to this distinction. When we say that the idea of the infinite is not intuitive, but abstract, we give the key to the solution of the principal objections brought against it.
45. We have no intuitive idea of infinity; that is to say, this idea does not present to our mind an infinite object; we can have this intuition only when we see the essence of God, which will happen in a future life.
46. If we had now the intuition of an infinite object, we should see its perfections as they are, with their true marks; or rather, we should see how all the perfections dispersed among limited beings, are united in one infinite perfection. We could not refer the idea of the infinite to determinate objects, as, for example, to extension, because these objects contradict the idea. It would be impossible for us to modify the idea in different ways, and apply it first in one sense, and then in another very different sense. The idea is one, and simple; it would, therefore, always relate to an object which is also one and simple, not vague and indeterminate, as now, but with the determination of a necessary existence and an infinite perfection. We should haveintuition of infinite being, as we have intuition of the facts of our consciousness: our cognition of it would be that of an object eminently incommunicable, as predicate to any order of finite beings; and it would be as manifest a contradiction, to apply the idea of this infinity to any number or extension, as it would be to identify an act of our consciousness with external objects.
47. The indeterminate character in which the idea of the infinite is presented us, and the ease with which we modify it in various ways, and apply it to different objects, in different senses, proves that this idea is not intuitive, but abstract and indeterminate, that it is one of those general conceptions, by the aid of which the mind obtains a certain knowledge not afforded by intuition.
This will explain the origin of the vagueness of our idea of infinity. Indeterminate conceptions, and because they are indeterminate, relate to no particular object, or quality, which may be conceived by itself alone, as something which may be realized; they do not contain those determinations which fix our cognition in an absolute manner. The indeterminate manner in which they present any property of beings, causes a difference in the application, accordingly as the particular properties, which are combined with the general, are different. If we take a right-angle triangle, in which we know the measure of all the sides and angles, the determinateness of the idea avoids the vagueness of the intellect, and prevents the application of this idea to cases different from that which is determinate and fixed. But if we take a right-angle, in general, without determining the value of its sides and angles, its applications may be infinite. The more general and indeterminate the idea of a triangle becomes, the greater is the variety of its applications.
48. Indeterminate ideas, in order to represent any thing,must be applied to some property which is the condition of their actual or possible realization. Until this application is made, they are pure intellectual forms, which represent nothing determinate. I do not mean by this, that these ideas are empty conceptions, which cannot be applied outside of the sensible order, as was maintained by Kant;[37]but only that granting them an universal value, I deny that they have by themselves alone a value representative of any thing that can be realized, beyond the property which they express. The idea of apuretriangle can not be realized, for everyrealtriangle would contain something more than is in the idea: it would be a right-angled or oblique-angled, etc., all which, the pure idea abstracts. The object will be indeterminate, in proportion to the indeterminateness of the properties contained in the conception; consequently, that which is presented to the understanding will also be more vague, and the applications which may be made of the idea, will be more varied and numerous, as is the case in the ideas of being, not-being, limit, and the like.