It follows from this that, if infinity is an idea realiza- ble by the mind, it must be conceived in some other way than as a linear series; it must be capable of an expression which is at once definite, and yet preserves the true character of infinity. Mathematical science does this by the summation of an infinite series in a finite expression, and manipulates botli the infinite and the infinitesimal as terms having a definite meaning in calculation. The possibility of conceiving tlie infinite as complete may be seen more easily from the consider- ation that any object which we can see, handle, imag- ine, conceive, without any difficulty, e. g. a fruit, or a stone, is really the sum of an infinite number of parts into which it may be divided, an infinite, therefore, wliich is not merely coming into existence, but actually exists here and now. Regarded, too, under the aspect of a term in the line of causation, any object in nature sums up an infinite series in itself. For, as an effect, it is the result of all previous causes, and, as a cause, the germ of all succeeding effects.
These summations of the serial infinite, whether achieved by the formuhe of mathematics or presented as complete, in every portion of space, in every period of time, and in every object in nature, are anticipations of a higher form of infinity which is revealed by the mind of man.
III. The Spiritual Infinite (infinitum rationis, infini- tum actu, 'itXov TtXtiov) differs from the former, not so much in excluding as including the limit or boundary of which it is the negation, i. e. as not limited from without and perpetually passing beyond the limit, but as limiting itself. As the natural dr mathematical in- finite is represented by the line, so the rational or spir- itual infinite finds its apiiropriate symbol in the circle, i. e. the line which is without beginning or end, and at the same time is limited at every point by itself. It is thus at once absolutely unlimited, and yet absolutely definite. The transition from II to Illmay be iUus- tratcd by the mathematical definition of a straight line as the chord of an infinite circle. Such is the infinite
as exhibited in (a) the thought and (b) the volition of man.
(rt) Consciousness, and thought as a mode of con- sciousness, involve the opposition of the subject which thinks and the object about which it thinks. As a con- dition of thinking at all, the mind must set its thought over against itself as not itself, and conversely, as the condition of an object being thought of at all, it must be presented as distinct from the mind which thinks of it. Here, then, is a limitation or barrier which consti- tutes what is called " the finiteness" of the human un- derstanding. The thinker is limited and conditioned by his thought, the thought is limited and conditioned by the thinker. But, as it is possible to present any object to thought, it is competent for the thinker to pre- sent himself && the object about which he thinks, i. e. to be at once the subject which thinks and the object which is thought about. This capability of self-con- sciousness, of which, so far as can be ascertained, the lower animals are destitute, constitutes at once the pride and the degradation of man, is a source at once of his best and his worst actions. Here we have the analogue of the line returning, as the circumference of a circle, into itself. The limitation of the thinker by the object thought of is as real as before, only it is a limitation of himself by himself: he is conditioned, as before, but self-conditioned, i. e. infinite. See Personalty.
(b) The same infinity appears in free wiU. As free, a man does an action which originates absolutely with himself. But this action has a permanent effect on his character, and thus determines the quality of the next action. This new action is also originated absolutely by the free agent, but the agent himself is modified, conditioned, limited, by the previons action. The agent has thus his freedom limited and defined, and increas- ingly so with every fresh action, but he is limited by that of which he is himself the absolute originator. He is finite (limited, conditioned) and at the same time in-< finite (unlimited, unconditioned), because he is self-con-, ditioned. See Liberty.
It is in this sense, rather than in that of infinite mag- nitude, that infinity is an attribute of God, See The- ism.
lY. Relation to the Finite. — It follows from what has been said above (a) that, although the essence of infin- ity is the transcendence of every limitation, yet that the finite and limited, even when excluded (I and II), Js postulated as a condition of infinity, and that in the higher forms of infinity the limit is included, or, rather, imposed from within. Even in the sense of the indefi- nite residuum of thought, definite thinking is presup- posed as the condition of our becoming conscious of the vague element beyond. The serial infinite, again, as the mere process of transcending every given term, pos- tulates the perpetual recurrence of terms to transcend : UTnipov, says Aristotle, fxlv ovv tariv ov Kara Troalv Xafiliavovciv, ahi tl XajSuv i'cTiv tlio {Phys. Ausc. 207, a. 7) — " The quantitative infinite is that which al- viaxs has something outside it, i. e. a term 'not yet reached.' " The spiritual infinite, lastly, as the self-de- termination of thought and volition, is, ex vi termini, a process of generating at every step the finite and limit- ed, {h) On the other hand, it -would be a reversal of the true order to conceive the infinite to be, as its etj-- mology suggests, the mere negation of the finite, and, as such, a secondarj' and deriveii idea. On such a sup- position it becomes impossible to explain how we be- come conscious of limitation at all. How, it maj' be asked, do we know that thought is finite if we know nothing first of the infinite? How is the consciousness of limitation possible except as the negation of what is unlimited? The infinite is thus, as the condition of the finite, prior and positive; the finite, as the limit ex- cluded, included, self-imposed by the infinite, posterior and negative.