without end or limit, the negation of finite: ἄπειρου , "un-endlich."
I. The Indefinite. — Besides the definite consciousness of which logic formulates the laws, there is also an indefinite consciousness which cannot be formulated. Besides complete thoughts, and besides the thoughts which, though incomplete, admit of completion, there are thoughts which it is impossible to complete, and yet which are real, in the sense that they are normal affections of the intellect. Positive knowledge, however extensive it may become, does not and never can fill the whole region of possible thought. At the uttermost reach of discovery there arises, and must ever arise, the question, What lies beyond? Regarding science as a gradually increasing sphere, we may say that every addition to its surface does but bring it into wider contact with surrounding nescience. There is always something which forms alike the raw material of definite thought, and remains after the definiteness which thinking gave to it has been destroyed (H. Spencer, First Principles, p. 21 sq., 88, 90 sq.). This vague element in thought, which is ineradicable, Spencer considers to be the groundwork of the feeling of awe, and-of natural religion. It is the infinite in this sense, the attempt to conceive which involves a contradiction in terms; which can only be believed to exist, but can never become an object to consciousness. "If all thought is limitation; if whatever we conceive is, by the very act of conception, regarded as finite, the infinite, from a human point of view, is merely a name for the absence of those conditions under which thought is possible" (Mansell's Bampton Lectures, p. 48; comp. p. 30, 63, 80, 118; see esp. notes on p. 48 and 51, 4th ed.).
II. The Infinite as an Interminable Series. — Aristotle mentions five ways (Phys. Ausc. 203, b. 15) in which the notion of the ἄπειρον is attained:
(a) From the unlimited duration of time;
(b) from the possibility of perpetually subdividing magnitudes;
(c) from the continuance of growth and decay in nature;
(d) from the fact that limitation is always relative, and never absolute; and
(e), "the strongest proof of all," from the inability to conceive a limit to number, magnitude, and space.
Any given moment of time is both preceded and succeeded by another, and that by another without end. Any magnitude admits of multiplication or division, and the multiples or parts are again capable of multiplication or division, respectively, without limit. Any effect in nature is the result of a cause which, again, is the effect of another cause in an endless regress; and, conversely, every effect is itself the cause of some other effect, and this, in its turn, is the cause of another effect, and so on in an interminable progress. Time, space, and causation thus exhibit infinity in the form of a straight line or series of terms without beginning or end. The characteristics of this mode of the infinite are: (1) that it is purely negative, i.e. is the mere process of passing beyond limitations; (2) that it postulates the perpetual recurrence of limitations as its condition; and (3) that, as an endless series, it is incapable of being thought out, it is always possible and never actual, it cannot be said to exist, but always to be in the act of coming into existence.
It follows from this that, if infinity is an idea realizable 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 both the infinite and the infinitesimal as terms having a definite meaning in calculation. The possibility of conceiving the infinite as complete may be seen more easily from the consideration that any object which we can see, handle, imagine, 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, which 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 formulae 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, infinitum actu, ὅλον τέλειον ) 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 or mathematical infinite is represented by the line, so the rational or spiritual infinite finds its appropriate 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 III may be illustrated 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.
(a) Consciousness, and thought as a mode of consciousness, involve the opposition of the subject which thinks and the object about which it thinks. As a condition 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 constitutes what is called "the finiteness" of the human understanding. 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 present himself as 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-consciousness, 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 PERSONALITY).
(b) The same infinity appears in free will. 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 previous action. The agent has thus his freedom limited and defined, and increasingly 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 infinite (unlimited, unconditioned), because he is self-conditioned. (See LIBERTY).
It is in this sense, rather than in that of infinite magnitude, that infinity is an attribute of God. (See THEISM).
IV. Relation to the Finite. — It follows from what has been said above
(a) that, although the essence of infinity is the transcendence of every limitation, yet that the finite and limited, even when excluded (I and II), is 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 indefinite residuum of thought, definite thinking is presupposed 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, postulates the perpetual recurrence of terms to transcend: ἄπειρον , says Aristotle, μέν ουν ἐστὶν ου κατὰ ποσίν λαμβάνουσιν αἰεί τι λαβεῖν ἔστιν ἔξω (Phys. Ausc. 207, a. 7) — "The quantitative infinite is that which always has something outside it, i.e. a term ‘ not yet reached.'" The spiritual infinite, lastly, as the self-determination of thought and volition, is, ex vi termini, a process of generating at every step the finite and limited.
(b) On the other hand, it would be a reversal of the true order to conceive the infinite to be, as its etymology suggests, the mere negation of the finite, and, as such, a secondary and derived idea. On such a supposition it becomes impossible to explain how we become conscious of limitation at all. How, it may 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 excluded, included, self-imposed by the infinite, posterior and negative.
The relation of GOD, as the Infinite, to the world and the soul, as finite, is considered elsewhere. But, unless (a) be borne in mind, the logical result is deism, and if (b) be neglected, pantheism.
V. Infinity as symbolized in the Imagination. — We find the attempt to picture the infinite to the imagination among non-European nations in the form of a state of vacancy immediately preceding creation. The constituents of the image are generally air and water. The image of mere air or mere water would be no realizable image at all, because involving no distinction. But in the contrast of the two we get that minimum of definiteness which renders the image possible. A beautifully pure representation of the imagined infinite is found in the sacred books of the aborigines of Guatemala (Max Miller's Chips, 1, 333). It is as follows: "There was a time when all that exists in heaven and earth was made. All was then in suspense; all was calm and silent. All was immovable, all peaceful, and the vast space of the heavens was empty. There was no man, no animal, no shore, no trees; heaven alone existed. The face of the earth was not to be seen; there was only the still expanse of the sea and the heaven above. Divine beings were on the waters like a growing light. Their voice was heard as they meditated and consulted, and when the dawn arose man appeared." Here we have as the constituents of the image "empty heaven," or space, and-which is introduced as if not at all contradictory to the statement that "heaven alone existed" the "still expanse of the sea." [Compare this with the account in holy Scripture, where the constituents of the image are (1) "darkness upon the face of the abyss," and (2) the surface of the waters, with the Divine Spirit hovering between the two, and calling light into being.] In the Hindu account the creative spirit is represented as rowing about in a boat upon the ocean.
We. have substantially the same image of the infinite lying at the back of the Greek mind. But there are two differences.
(1) The double image is dismembered. The symbol of Thales is water alone; of Anaximander, the void in suspense; of Anaximenes, the atmosphere of Xenophanes, the globe of the sky. (2) The infinite is not pictured as preceding the emergence of finite things, but as underlying the process of nature, as it is ordinarily known.
The Egyptian symbol of the serpent with his tail in his mouth approaches the mathematical representation of infinite length. — Blunt, Theol. Dict. 1, 346 sq. See Journal of Speculative Philosophy, July, 1870.