THE LIMIT.
25. The wordinfiniteis equivalent tonot finite, and seems to express a negation. But negations are not always truly such, although the terms imply it; for if that which is denied, be a negation, the denial of it is an affirmation. This is the reason why two negatives are said to be equivalent to an affirmative. If I say, it has not varied, and you deny it, you deny my negation; for it is the same thing to deny that it has not varied, as to affirm that it has varied. In order, therefore, to determine whether the wordinfiniteexpresses a true negative, we must know what is meant by the wordfinite.
26. The finite is that which has a limit. A limit is the term beyond which there is nothing of the object limited. The limits of a line, are the points beyond which the line does not extend; the limit of a number, is the extreme where the number stops; the limit of human knowledge, is the point to which we may arrive, but which we cannot go beyond. A limit being a negation, to deny a limit, is to deny a negation, and is consequently an affirmation.
27. It is easy to see from these examples, that a limit in the ordinary sense, expresses an idea distinct from what mathematicians define it. They call a limit every expression, whether finite, infinite, or a nullity, which a quantity may continually approach without ever reaching. Thus, the value 0/a is the limit of the decrement of a fraction, the numerator of which is variable x/a; because, if we suppose X to be constantly diminishing, the fraction will approach the expression 0/a, without ever being confounded with it, so long as X does not entirely disappear. If we suppose (b + x)/a an expression in which X is decreasing, the expression will continually approach (b + 0)/a = b/a, which will be the limit of the fraction. If we suppose the expression a/x, in which X is decreasing, we shall continually approach the expression a/0 = ∞, an infinite value which the fraction can never attain, until X becomes 0, which cannot happen, because X is a true quantity. These examples show that mathematicians admit limits which are finite, infinite, or a nullity, and prove that mathematicians employ the word limit in a different sense from its ordinary as well as philosophical meaning.
28. A limit, therefore, expresses a true negation, and theword finite, or limited, necessarily involves a negative idea. That which is not, is not limited; therefore the finite is not an absolute negation. An absolute negation is nothing, and we do not call the finite nothing. Therefore, in the idea of finite are contained being, and a negation of another being. A line one foot in length, involves the positive value of one foot, and the negation of all value of more than a foot. Therefore, the finite, in so far as finite, involves a negation relatively to a being. If we could express this idea in the abstract, using the word finity, as we have the word infinity, we should say that finity in itself expresses only the negation of being relatively to a being.
29. Hence, the word infinite is not negative; for it is the negation of a negation. The infinite is the not-finite; it is that which has no negation of being, consequently that which possesses all being.
30. We have, therefore, an idea of the infinite, and this idea is not a pure negation. But it must not be supposed that we have arrived at the last term of the analysis of the infinite. We are still far from it, and it is even doubtful whether we shall obtain any satisfactory result after long investigations.