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INFINITE, a term used in mathematics, philosophy and theology with various meanings which are apt to cross each other and cause confusion. It is not possible to do more here than indi cate the chief notions associated with the term.

Negative and Positive.—Speaking broadly, one may distin guish between the negative meanings of the term and its positive meanings, and also between its use in connection with what is quantitative, and its use in connection with what is not quantita tive, but only qualitative. The negative meanings of the term may be expressed variously (according to context) by "indetermi nate or endless in quantity or extent," etc., and "indefinite or inde terminate in quality." The positive meanings may be expressed by the expressions "self-determined," "self-dependent," "complete" or "perfect," as the case may be. The negative use of the term is no doubt the older ; it is the one most obviously suggested by the form and the etymology of the term (Latin in, "not," and finis, "end" or "limit") . The positive meanings were evolved later as the result of philosophical and mathematical reflection on the kind of objects commonly described as infinite. Such reflections have also led to the introduction of numerous distinctions among "infinites," such as "internal" and "external" infinites, "finite" and "infinite" infinites, "absolute" infinite and infinite "of its kind," "single," "double" and "treble," etc., infinite, and, of course, the infinitely great and the infinitely small (or "infinitesimal"). To explain all these distinctions adequately would take up too much space, but the main points should become clear from what follows.

The Infinite and God.—In the history of Western philosophy the term "infinite" (TO air€Cpov) is met with, apparently for the first time, in the teaching of Anaximander (6th cent. B.C.). He used it to describe what he conceived to be the primal matter, "principle," or origin of all things. His teacher Thales had sug gested that it was water; later thinkers identified it with air or fire, etc. What Anaximander probably meant to suggest was that the primal substance must have been something indeterminate or indefinite in quality as compared with such qualitatively definite substances as water, air, fire, etc., which emerged from it in course of time. Whether he conceived his primal substance as indeter minate in quantity as well as in quality is not certain, though highly probable.

At all events the notion of things indeterminate in quantity, or "without end," must have become familiar fairly early, especially in connection with ideas about space, time, number, and the existence and powers of God. Space, even perceived space, seems in a sense to have no boundaries—it seems to fade into a larger space beyond, and the space beyond can only be imagined or con ceived as surrounded by more space, and so on indefinitely. Time too seems to have no limits—whatever the period of time or duration that we live through or imagine or conceive, it is always possible, indeed necessary, to supplement it with more time "before and after." The series of numerals likewise is endless— by adding and subtracting one can extend it both ends indefinitely. Similarly with ideas about God. Once the stage of limited, local deities was passed, God was gradually conceived to be unlimited in power, of infinite duration, and holding sway over endless space. Curiously enough it was not until recent times that there has been a return to some extent to the conception of a Deity of limited power. Deeply impressed with the reality of evil and suffering in the world, some thinkers have found themselves in the dilemma of having to choose between the infinite goodness and the infinite power of God; and they preferred to abandon the belief in His omnipotence. Whether the dilemma is valid, is a question that need not be discussed here. Some find no serious difficulty in reconciling the apparently conflicting alternatives, when due allowance is made for the implications of human free will.

The Infinite and Vastness.--Intimately

connected with the earlier conception of the infinite, especially in its application to space and time, was the idea of immensity or vastness. This was a perfectly natural combination or fusion of ideas, which still clings to the popular use of the term infinite. In a sense the dissociation between "infinite" and "vast" was begun fairly early. Zeno's paradox that motion is impossible, because the moving body would have to pass through an infinite multiplicity of points or positions in order to reach even a near destination, obviously implies or even asserts that a finite line or distance contains some thing infinite, namely, an infinite multitude of points or positions. So that in some way a short finite distance having definite, known limits may still in some way be infinite, that is infinite internally (i.e., infinitely divisible), though not externally (because its limits are fixed).

But the paradoxical method of Zeno failed on the whole to bring out the element of truth in his teaching. And the lesson had to be repeated by Spinoza, among others. His favourite illustration of the internal, finite infinite (or the infinite within a finite whole) is that of a small circle containing a still smaller circle with which it is not concentric. The space between the two circumferences is finite and small. The minimum distance and the maximum distance between them is known. Yet within this small finite space, and within the limits marked by the minimum and the maximum distance there is an infinite variety of distances between the two circumferences. The space is externally finite, internally infinite. So that the infinite is not necessarily "vast" (hence the poet's note about holding infinity in the palm of the hand), nor does it depend on the absence of "limits." Infinity a Quality.—Infinity is rather some kind of positive character or quality. What this quality or character is, Spinoza does not make quite clear except in the case of the "absolutely infinite" and the "infinite of its kind." But what he says in this case is very suggestive, and rather like more recent conceptions of the "infinite" as formulated by Dedekind and various con temporary mathematicians. By "the absolutely infinite" Spinoza meant God as the perfect, complete, self-dependent, self-deter mined ground of all that is. Extension and Thought, which are "attributes" (that is, aspects of God or the Universe) are each "infinite of its kind," that is, each complete in its nature, but not, of course, the whole of Nature (or God). Finite things (or finite modes), on the other hand, are characterized by fragmen tariness, incompleteness, dependence on other things outside them selves, though they are all ultimately parts of the infinite.

Now, this conception of the "infinite" is, from the standpoint or philosophy at all events, the most interesting and essential feature in the newer mathematical conception just referred to. This feature may be stated in the following quotation from A. E. Taylor's Elements of Metaphysics (Book II., Ch. iii.) :—"The infinite must not be confounded with the indefinite or unfinished. Its fundamental property is not the merely negative one of having no end or `last term,' but the positive one of having an internal structure which is the harmonious and complete expression of a single self-consistent principle. The finite, again, is finite not primarily merely because it has a `last term,' i.e., because there is something else outside it, but because its `last term' is arbitrarily determined, i.e., determined by something other than the prin ciple of its internal structure. In other words, the essential defect of the finite is that it is not solely determined by its own structural principle. We can see this even in the simple case of the familiar `infinite series' of arithmetic and algebra. Such a series as I, , a . . . is `infinite' not merely because you never come to the last term, but because its character is determined from within, solely by the principle according to which each term is derived from the one before it ; that the series has no end is a simple consequence of this positive property of self-determination. But suppose I take n terms of this series and no more, where n is a specified number, the resulting series is now finite, not primarily because there are more terms of the same kind outside it, but because the number of terms to be taken is not prescribed by the law of formation of the series, but fixed with reference to some object independent of the principle of the series itself. In other words, only the infinite is in the full sense of the words a com pletely self-determined whole. The finite is the imperfect, not primarily because there is something outside it, but because its contents are not solely prescribed by the principle of structure which they embody." For the special uses of the notions "infinite" and "infinitesimal" see articles SERIES, FUNCTION, GEOMETRY and CALCULUS, Infini tesimal.


J. Keyser, The Human Work of Rigorous ThinkBibliography.-C. J. Keyser, The Human Work of Rigorous Think- ing (1916) ; J. Royce, The World and the Individual, vol. i. (Igo') ; B. Russell, Principles of Mathematics (1903) ; W. R. Sorley, Moral Values and the Idea of God (1918) ; Spinoza, Correspondence trans lated by A. Wolf (1928) ; A. E. Taylor, Elements of Metaphysics (1903) ; S. Alexander, Space, Time and Deity (1928) ; Walter Terrence Stace, The Theory of Knowledge and Existence (1932). (A. Wo.)

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