INFINITE. This word is the source of muck controversy and difference of opinion. Some hold that there corresponds to infinity a distinct notion, which we are entitled to entertain and reason about, with the same confidence that we discuss measured inter vals, as a yard or mile; while others maintain that the word is a name for a mere negative. Sir W. Hamilton goes so far as to say that " the infinite and the absolute are only the names for two counter-imbecilities of the human mind, transmitted into prop erties of the nature of things—of two subjective negatives converted into objective affirmatives" (Discussions, p. 21). And Mr. J. S. Mill holds a similar view. It„ had also been maintained by Locke that we have no positive idea of the infinite, that it was only the negative of an end or termination (Essay On the Understanding, book ii. chap. 17).
The notion of the infinite has, indeed, been admitted into mathematical reasoning, a circumstance that would seem to imply that we could use it with exactness, and, consequently, it could not be altogether an incompetence or imbecility of the under standing. It appears, however, that mathematicians use the word under peculiar restrictions. They employ it in the two extremes of the infinitely great and the infi nitely little. "If we see a conclusion, which we can nearly attain by the use of a large magnitude, more nearly by the use of a larger, and so on without limit, that is to say, as nearly as we please, if we may use a magnitude as large as we please, but which is never absolutely attained by any magnitude however great, then such conclusion may be said, for abbreviation, to be absolutely true when the magnitude is infinite " (Penny Cyc., art. "Infinite " ). The very same statement might be made regarding the infi nitely small, which is represented in mathematics by the symbol for nothing, although it is not the same as nothing in the strictest sense, namely, the nothing caused by sub tracting a quantity from itself, as two from two. It is nothing in this sense, that if added to a finite quantity, as 10, it produces no augmentation that can be made use of; the quantity for all purposes remains the same. The machinery of infinite quantities plays a large part in the operations of the higher mathematics, and is introduced in order to compare two things naturally incommensurate. Thus, in estimating the area of a curved surface, such as a circle, in straight-lined spaces, such as square inches, the difficulty was got over by a sort of fiction, namely, by supposing the circle to be inscribed by a right-lined figure or polygon, of such a very great number of sides that they coincide to all intents and purposeS with the curved circumference. The coin cidence can never be perfect; but by imagining the sides to be smaller and smaller, and, consequently,. more and more numerous, the difference between the polygon and the circle may become less than any assignable quantity, or, as it may be said, infinitely little, in fact, as good as nothing, so that the estimate of the area of the one will stand for the estimate of the area of the other. This device for overcoming the natural incommensurability of straight and curved, and of number and motion, is the real occa sion of the mathematical use of the term in question. Nor does it give any foundation for the view that would regard the infinite as a positive conception of the mind which we may apply to objects with a conscious meaning.
This will be more apparent when we attend to the difference between two classes of negative notions. The first class includes those whose negative brings something positive; thus, not hot, brings before us a positive experience. namely, cold; not white. according to what is intended, turns up either black or all other colors, which are to us as much a positive, or real, conception as white. Unjust, or not just, is the name for a distinct class of really existing actions, in contrast to the class named just actions, All notions, such as these, which have for opposites really existing things, are real and genuine notions of the mind; they are conceivable by us to the full extent that we are capable of conceiving anything whatsoever. In fact, the highest test of genuine ness, reality, and conceivability, is the existence of a negative, which is also real and positive. body or matter is a real conception by being opposed to space; the one resists our movements, and the other permits them. Body and space together make the extended universe, the world of externality, or objective existence; which has a distinct meaning by contrast to the inextended mind, or the subject universe. But existence, as a whole, is not a real conception, because we have nothing to oppose it to; non-existence is not •ft real opposite, like space to body, or mind to extension; it is only a formal or verbal opposite, made up by using the word for negation to a case that does not admit of the operation. Non-existence is total annihilation. which. of course, we cannot conceive, as we do cold or black, in their opposition to hot and white. so, we have nothing to affirm respecting existence as expressing the absolute totality of things. See EXTENSION.
Now, to which class of notions does infinite belong? Is it a real opposite to the 'finite, like cold to heat, or a verbal and formal opposite, like non-existence? Finite means what has a boundary or termination, and applies strictly to body, which is always conceived by us as bounded and terminating in space. The bounded is, in fact, body (or some analogy of body, as when we fancy an inclosure which we do not actually construct); the absence of bounds is free space, which is a real conception. It means scope for movement., freedom front obstruction, and its opposite is sonic inert matter. standing in our way, to prevent further movement. The unbounded is thus another name for spetec ; and when we arrive at a space with no further prospect of obstruction, we may call that a boundless space, but the only meaning we have thereby is a space which no longer contains material obstruction. And we can conceive of no other end of space. Our whole experience furnishes no other contrast except these two, space and body, and where the one ends, the mind must conceive the other. We may con ceive the not-extended, it is true, by passing to the subject mind, with its feelings and volitions; but within the sphere of the extended we have no choice but between space and body. We cannot conceive the end of space otherwise than by the beginning of resistance; anything else (not being the subject mind) would be non-existence, or annihilation.
The infinite may thus be the name for an abbreviation in mathematics, but as a real notion of the mind, it merely expresses our inability to pass beyond the region of our experience of matter and space.