possibility of the process itself, by exhausting the continuum, time, which is the object in and upon which the act of numbering is supposed to operate. The very meaning of their infinity is, that no process, either of adding portion to portion, or of successively dividing any portion of them into smaller and smaller portions, can ever exhaust their capacity for increase, or their capacity of being divided.1 Time and space, as those inseparable elements in concrete perceptions which are the source of their continuity, are therefore antecedent conditions of number and of geometrical figure respectively; these being definite limitations introduced, by means of thought attending to differences in the senseelement, into the pure and unlimited continuity of the abstract elements of time and space. Wherever you perceive a limit, either in time or in space, you also always perceive time or space on both sides of it, and not on one only. Their limits are differences or distinctions in their content, which are always found within, never beyond, their content in its entirety. Consequently, whenever you think of a limit ideally, you also think of a time or a space, as yet unlimited by thought,-beyond it if your thought is advancing in the direction of increase, within it if in the direction of decrease. All thought, it has been well said, is limitation; but it does not follow 1 Among all the services which M. Renouvier has rendered to philosophy, there is perhaps none greater than his exposure of the fallacy of what he calls "l'infini actuel," or the thought that infinity can be actually realised, that is, represented as possibly finite. See his Traité de Logique Générale &c., 2nd Edition 1875, Vol. I., pp. 44-67. Also his Principes de la Nature, 2nd Edit., 1892, Vol. I., pp. 80-84. Also his Philosophie de la Règle et du Compas, in L'Année Philosophique 1891, pp. 27-28. The last named work especially should be familiar to all students of the philosophy of geometry, notwithstanding that Kant's distinction between analytic and synthetic judgments is made one of the corner stones of the argument. BOOK II. § 6. The Conception Infinity. BOOK II. & 6. that it can include either time or space as a whole within its limiting boundary. This is in fact preThe vented by the relation in which thought stands to Conception perception, perception being one of its essential constituents, and time and space being, each in its kind, inseparable elements of perception. of Infinity. From these facts it results, that, when we come to conceive time and space as abstract percepts, we must conceive them as infinite, because we perceive them only as inseparable elements in perceptions, and therein as existing beyond as well as within any limit or difference which we perceive in them, or any boundary which we can impose upon them by thought. As inseparable elements in perception, their essential character is continuity; and this perceptual character is the basis of their infinity, when they are taken as abstract percepts. Thus the perception of their transcending any limitation, whenever we bring it to the test either of presentation or of representation, is the fact which, when we consciously attend to it, becomes the conception of their infinity; the infinity of time, in the direction of extensibility, being called by the particular name of eternity. Those things and those only which have no possible or conceivable final limits are conceived as infinite. Nothing to which we can conceive a final limit can be conceived as necessarily, or in its nature, infinite, though it may be conceived as indefinite, that is, as becoming either indefinitely great or indefinitely small. Time and Space it is beyond our power to conceive as having such a final limit. We conceive them as both unlimited and unfigured continuity. It is a simple logical blunder to suppose, that time or space, when conceived as infinite, are conceived as completed totalities, (which would be to conceive them eo ipso as finite), on the ground that all conception is limitation. The perceptual fact that time and space escape or transcend limitation by conception, not the perception of time and space as totalities, is indeed itself a limitation; not however a limitation which conception imposes upon time and space, but one which they, as perceptually given data, impose upon our power of conception. The fact that they escape limitation by conception is the very fact by which our conception of them as infinite is itself constituted, or by which it becomes a definite conception. number, on the other With closed figure and hand, the case is different. by limitation, the limitation of conscious attention and thought. Their nature and being is to have a limit or a boundary. Consequently, though we may conceive them as magnitudes capable of indefinite expansion or indefinite contraction, we cannot conceive them as unlimited in magnitude, in either direction, without conceiving them abolished altogether as closed figures, or as continua expressed by numbers. There is no such thing as an infinitely great or infinitely small closed figure; there is no such thing as an infinitely great or infinitely small continuum of time expressed by number. Changes in their magnitude are capable only of an indefinite progression. Time and Space, as the formal element in perception, being continua occupied by feeling, the material element, and distinguished into portions by differences in feeling, are the ultimate founda BOOK II. CH. I. $ 6. The Conception of Infinity. BOOK II. § 6. The Conception of Infinity. tion of every conception of Quantity or Magnitude. Similarly the material element, with its countless differences, is the ultimate foundation of every conception of Quality. Quantity and Quality are perceptually contrasted, but are not logical opposites. This follows from their nature as conceptions, framed by thought, of what, as data, are ultimate and inseparable elements in all empirical percepts. The logical opposites of quantity and quality in general are not-quantity and not-quality, either of which, if affirmed of anything concrete, would be equivalent to affirming its not-being, that is denying its existence; since both alike are conceptions of elements which are essential, as well as inseparable, in the composition of perceptual realities. Quantity again, when taken as we are now taking it, in connection with the analysis of consciousness, and not only in respect of its place in mathematical theory, is exhaustively divisible into the logical opposites, finite and infinite quantity. Every quantity which is arrived at or described by numerical or geometrical limitation is a finite quantity; every quantity which continues beyond every such limitation, and in virtue of that continuation, is an infinite quantity. But this beyond is of two kinds. If the successive limitations are in the direction of division or decrease, karȧ Siaiper, the infinity of the quantity which escapes them is expressed by saying, that it is infinitely divisible, or divisible in infinitum, without ceasing to exist as quantity. If the limitations are in the direction of addition or increase, кarà πρόσθεσιν, then the quantity which escapes them is called infinite simply, or in the case of time, eternal. We may, perhaps, tabulate quantity BOOK II. taken in this sense as follows: Finite. Сн. І. § 6. The Conception of Infinity. Quantity in Infinite in order Infinite Infinite in order No piling up of finite quantities, no process in indefinitum, can ever result in infinite quantity, for nfinite quantity is essentially continuation beyond any conceivable limit whatever. Such is our conception of time and space, owing to their perceptual origin, as the formal element in perception, in which they appear as the durational and extensional coelements with feelings of any and every kind. Such is our conception of them when they make part of the one real world, the objective panorama of real existence, necessarily imparting to it their own infinity, and as it were incorporating the positively knowable world with the infinite universe of which it is a portion. To conceive time, space, and the universe, as infinite, is to conceive the fact, that, as percepts, they transcend conception.2 I pass now to another branch of the subject. We have seen in the foregoing Section, that there is no science of absolute time taken alone, but that any measurement of time which is generally applicable depends upon some previous measurement of 2 For a fuller discussion of this subject I would refer to my Philosophy of Reflection, Chapter VIII. (Vol. II., pp. 67-121), and also to my Aristotelian Address for Nov. 1893, The Conception of Infinity, published in the Proceedings of the Aristotelian Society, Vol. II., No. 3, 1894, though there are some statements in the latter, to which I should now perhaps not be inclined to adhere. |