(4) Still another meaning of the infinite is the unattainable limit of an unending prows of construction. 'This is well illustrated in the view taken by some idealists that lime and space are constructed in the act of thinking them. This is conceived, not as eomplete, but its having no limit to the possibility of its con tinuance. Infinite space is thus the unattainable result of tile unending iiro•css of eonstrueting space. It is the fact that you can never come to the end of possible space construction. One species of the mathematieal infinite seems to fall under this head. Thus, in the improper fraction 1. as the value of a:. decreases by oile r half the value of the fraction is doubled. If the decrease is continued without end. the fraction is conceived as approaching infinity. Ifere again infinity is the unattainable result of the unend ing process of halving the denominator. This infinite differs from the infinite of space and time considered under (a) in that space and time are often regarded as existing apart from any construction on our part, while the former is con sidered simply as the algebraic expression of the fact that our construetion can continue without end. It is not considered as an actual reality, tint as an unattainable ideal, while infinite space and infinite time are by some realists considered to be actual facts.
(5) Infinite in another sense is any member of an all-inclusive system of reciprocally deter mining members. Thus, according to Hegel. all reality is infinite. Hegel justifies the use of the term by pointing out that in a true system any member in being determined by the other mem ber, is really self-determined, because its sys tematic relation to the other members is an in tegral constituent of its own nature. These other members exercise no foreign compulsion upon it to make it what it is. They determine it only because it is part of its own being to be so deter mined. They do not set limits to it which it may not pass, but they give it opportunity to be itself. To take one of the apparently most re fractory instances, the billiard-ball is not infinite in sense la, because though spatial its place is only a part of a larger space from which it is marked off by definite boundaries. But the ball is something more than so much space. It is an object that undergoes various changes under certain conditions—e.g. when struck it rolls. This dependence of its changes upon the action of other objects is often considered another evi dence of its finitude. But it must be remembered that it is only because it is the nature of the ball to roll when struck that it rolls at all. In
rolling it does not succumb to external force, but it acts its own part. Though determined, it is self-determined. Though compelled, it is free. This free, self-determined nature of every mem ber of any free system is what Hegel calls infi nite. He considers the term infinite appropriate because the systematic conditions which deter mine the ball's action do not limit it in the sense of repressing its spontaneous tendency to act out its own nature. They simply give it a chance to show one side of its nature in a cer tain definite movement. There is no reason why the term infinite should not he employed in this case. But there is every reason to remember that this sort of infinitude is thoroughly compatible with h finitude in another sense—e.g. When Hegel says that the finite is essentially infinite. he means that that which is conditioned ( first im m t ) is conditioned because it is its ‘ery nature to act in response to these condi tions; that condition is not restraint (Sehranke) of an inherent tendency by an external limit (Creme). A reader in philosophy must keep himself always on the alert to detect the various meanings of the word infinite, and then lie will be able to understand many paradoxes that at first appear to he illogical contradictions. Wheth er the infinite really exists depends on the kind of infinite you mean. The existence of infinite la is discussed elsewhere (see NUMBER; SPACE; TIME) ; infinites lb, 2 in at least certain eases, and 5 are unquestionable. Infinite 3 is, if it does exist, unknowable, and there is lint the slightest reason to assert its existence. Infinite 4 does not exist at any particular time from the very nature of the ease. because it is an unattain able ideal. The infinite process. is real at least in the case of time in the sense that the flow of time never ceases: hut at no moment has it completed its unending course, and in this sense infinite time does not now exist, nor ever has existed. nor ever will. But when the present tense is mused, not specifically. but universally, it is then true that there is infinite time. Any further treatment of the question of infinity would be out of place here.
Consult: Royce, The World and the Individual, vol. i., supplementary essay (New York, 1900) ; Bradley, Appearance and Reality (2d ed., Lon don, 1897) ; Couturat, De l'infini mathematique (Paris, 1896) ; Bosanquet, Logic (Oxford. 1888) ; Rolzano, Paradosrien des Unendliehen (Leipzig, 1851) ; Hegel. Wissensehaft der Loyik ( Berlin, 1841); J. Cohn, Gesehichte des Unendliehkeits problems im abendhindischen Dcnken Lis Kant (Leipzig, 1896).