ABSTRACT
1. General Statement.—The conception of Infinity presents itself in philosophy primarily as a mode of escape from the difficulties involved in the existence of finite things. It was used in this way, more particularly, by the Cartesians. When any problem was too hard for them (such as the interaction of Mind and Body), they took refuge in the Infinite. Descartes himself led the way by contending that the conception of the Infinite is essentially prior to that of the Finite; and this doctrine was more fully developed by Spinoza. In general, this view was illustrated by means of spatial conceptions. A particular extent of Space, it was urged, can only be definitely conceived by being thought of as cut off from a larger whole, which, in the end, cannot be supposed to be itself limited. It must be regarded as being infinite in its kind; and this way of thinking of it leads very naturally to the supposition that other things also are to be supposed to be infinite in their kind. A stretch of time may be regarded as cut off from the infinite time, and a process of thought from the infinite thought. This way of thinking leads pretty directly to the doctrine that all determination is, in its essence, negation. To say that a thing is blue, is to say that it is not of any other colour. To say that a man has particular characteristics, is to say that he lacks the other characteristics that might belong to a human being. Thus any definite apprehension of a finite being implies the conception of the infinite from which it is distinguished. The limited implies the unlimited: the imperfect implies the perfect. Now, if we consider the three types of instances that have here been referred to—space, colour, character—it seems clear that there are great differences between them. In the case of spatial extent, the portion that is divided off from the rest—the extent of a country, for instance—is definitely bounded, and we can clearly understand its limitations only by taking note of what is beyond the boundary; and, however far we may go in the extension of its bounds (unless we are dealing with a whole, such as the surface of our earth, which is itself definitely bounded), we can always think of some further extent beyond the boundary that we have reached. The whole, in this case, has to be thought of as being endless, or without limit, and is thus to be regarded as being, in the strictest sense, infinite. The colour blue, on the other hand, can be apprehended without definite reference to any other colour, though it may be true that its special character cannot be distinctly noted without distinguishing it from other colours. At any rate, it is not, in any definite way, bounded by the others; nor is there any necessity to assume that there is an endless series of other possible colours. Blue is, however, a limited colour; and, when we regard it from the point of view of colour in general, it may be said to be incomplete—i.e. it is not the whole of colour. It is not necessarily imperfect. A blue sky would not be made more perfect by the addition of all the hues of the rainbow. The character of a person, again, has a similar incompleteness. Julius Cæsar had a certain incompleteness, in so far as he lacked the characteristics of Plato; and Plato, in so far as he lacked those of Julius Cæsar. Here also the one character can hardly be regarded as being bounded by the other. But it may be said that they are both imperfect, when we compare them with an ideal human type. They fall short of a standard that we have before our minds. Thus it would seem that there are three distinguishable sorts of finitude—the bounded, the incomplete, and the imperfect; and three corresponding types of infinity—the boundless, the complete, and the perfect. It is necessary to bear in mind also the distinction between the indefinite and the infinite. The indefinite is not actually boundless, but is that to which no precise limits can be assigned. That all these conceptions are distinct seems clear. In some cases, indeed, they are somewhat sharply opposed to one another. The boundless means almost that which cannot be completed (though sometimes what can be represented as boundless can also, from a different point of view, be represented as complete). What is complete can hardly be boundless, and need not be perfect. The visible circumference of a circle is complete when it is fully filled in; but it is not infinitely extended, nor does it consist of an infinite number of visible parts; and it may not be perfect—its curvature may not be strictly uniform. Or, to take a more trivial instance, this book will be complete when all the problems included in its scope have been dealt with; but I trust it will not be endless, and I know that it will be very imperfect. Even what is perfect need not perhaps be complete in itself. A perfect hero may be complete only in his self-renunciation.
