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“infinite” to God, so far as concerns the attributes of duration and ubiquity; and as applied to his other attributes the term is figurative, signifying that they are incomprehensible and inexhaustible. Such being the idea of the infinite, it is attained as follows: There is no difficulty, says Locke, as to the way in which we come by the idea of the finite. Every obvious portion of extension and period of succession which affects us is bounded. If we take one of these periods or portions, we find that we can double it, or “otherwise multiply it,” as often as we wish, and that there is no reason to stop, nor are we one jot nearer the end at any point of the multiplication than when we set out. “By repeating as often as we will any idea of space, we get the idea of infinity; by being able to repeat the idea of any length of duration, we come by the idea of eternity.” There is a difference, then, between the ideas of the infinity of space, time, and number, and of an infinite space, time, and number. The former idea we have; it is the idea that we can continue without end the process of multiplication or progression. The latter we have not; it would be the idea of having completed the infinite multiplication, it would be the result of the never-ending progression. And this is evidently a contradiction in terms. To sum the matter up, the term “infinite” always relates to the notion of quantity. Quantity is that which is essentially capable of increase or decrease. There is then an infinity of quantity; there is no quantity which is the absolute limit to quantity. Such a quantity would be incapable of increase, and hence contradictory to quantity. But an actual infinite quantity (whether of space, time, or number) would be one than which there could be no greater; and hence the impossibility of our having a positive idea of an actual or completed infinite.
Leibniz’s reply consists simply in carrying out this same thought somewhat further. It is granted that the idea of an infinite quantity of any kind is absurd and self-contradictory. But what does this prove, except that the notions of quantity and infinity are incompatible with each other, that they contradict each other? Hence, instead of the infinite being a mode of quantity, it must be conceived as essentially distinct from and even opposed to quantity. Locke’s argument is virtually a reductio ad absurdum of the notion that the infinite is capable of parts. In the few pages of comment which Leibniz in 1696 wrote upon Locke, this topic of the infinite is one of the few touched upon. His words upon that occasion were as follows: “I agree with Mr. Locke that, properly speaking, there is no space, time, nor number which is infinite; and that it is only true that however great be a space, a time, or a number, there is always another which is still greater, and this without end; and that, therefore, the infinite is not to be found in a whole made up of parts. But it does not cease to exist: it is found in the absolute, which is without parts, and of which compound things [phenomena in space and time, or facts which may be numbered] are only limitations. The positive infinite being nothing else than the absolute, it may be said that there is, in this sense, a positive idea of the infinite, and that it is anterior to the idea of the finite.” In other words, while the infinite is to Locke an indefinite extension of the finite, which alone is positively “given,” to Leibniz the infinite is the positive and real, and the finite is only in and by it. The finite is the negative.
Leibniz amplifies this thought upon other occasions, as in his present more extended examination. “There is no infinite number, line, or quantity, if they are taken as true wholes.” “We deceive ourselves in trying to imagine an absolute space which should be an infinite whole, composed of parts. There is none such. It is an idea which implies contradiction; and all these ‘infinites’ and ‘infinitesimals’ are of use only in geometry, as imaginary roots are in algebra.” That which is ordinarily called the infinite, that is, the quantitative infinite, is in reality only the indefinite. “We involve ourselves in difficulty when we talk about a series of numbers extending to infinity; we imagine a last term, an infinite number, or one infinitely little. But these are only fictions. All number is finite and assignable, [that is, of a certain definite quantity]; every line is the same. ‘Infinites’ and ‘infinitesimals’ signify only quantities which can be taken as large or as small as one wishes, simply for the purpose of showing that there is no error which can be assigned. Or we are to understand by the infinitely little, the state of vanishing or commencing of a quantum after the analogy of a quantum already formed.” On the other hand, the true infinite “is not an aggregate, nor a whole of parts; it is not clothed with magnitude, nor does it consist in number. . . . The Absolute alone, the indivisible infinite, has true unity,—I mean God.” And as he sums up the matter: “The infinite, consisting of parts, is neither one nor a whole; it cannot be brought under any notion of the mind except that of quantity. Only the infinite without parts is one, and this is not a whole [of parts]: this infinite is God.”
It cannot be admitted, however, that Locke has given a correct account of the origin of the notion of the quantitative infinite, or—to speak philosophically, and not after the use of terms convenient in mathematics—the indefinite. According to him, its origin is the mere empirical repeating of a sensuous datum of time and space. According to Leibniz, this repetition, however long continued, can give no idea beyond itself; it can never generate the idea that the process of repetition may be continued without a limit. Here, as elsewhere, he objects that experience cannot guarantee notions beyond the limits of experience. Locke’s process of repetition could tell us that a number had been extended up to a given point; not that it could be extended without limit. The source of this latter idea must be found, therefore, where we find the origin of all extra-empirical notions,—in reason. “Its origin is the same as that of universal and necessary truths.” It is not the empirical process of multiplying, but the fact that the same reason for multiplying always exists, that originates and guarantees the idea. “Take a straight line and prolong it in such a way that it is double the first. It is evident that the second, being perfectly similar to the first, can be itself doubled; and we have a third, which in turn is similar to the preceding. The same reason always being present, it is not possible that the process should ever be brought to a stop. Thus the line can be prolonged ‘to infinity.’ Therefore the idea of ‘infinity’ comes from the consideration of the identity of relation or of reason.”
The considerations which we have grouped together in this chapter serve to show the fundamental philosophical difference between Locke and Leibniz. Although, taken in detail, they are self-explanatory, a few words may be permitted upon their unity and ultimate bearing. It is characteristic of Locke that he uses the same principle of explanation with reference to the conceptions of substance, identity and diversity, and infinity, and that this principle is that of spatial and temporal relation. Infinity is conceived as quantitative, as the successive addition of times and spaces; identity and diversity are oneness and difference of existence as determined by space and time; substance is the underlying static substratum of qualities, and, as such, is considered after the analogy of things existing in space and through time. It must not be forgotten that Locke believed as thoroughly as Leibniz in the substantial existence of the world, of the human soul, and of God; in the objective continuity of the world, and the personal identity of man, and in the true infinity of God. Whatever negative or sceptical inferences may have afterwards been drawn from Locke’s premises were neither drawn nor dreamed of by him. His purpose was in essence one with that of Leibniz.
But the contention of Leibniz is that when substance, identity, and infinity are conceived of by mechanical categories, or measured by the sensible standard of space and time, they lose their meaning and their validity. According to him such notions are spiritual in their nature, and to be spiritually conceived of. “Spiritual,” however, does not mean opposed to the sensible; it does not mean something to be known by a peculiar kind of intuition unlike our knowledge of anything else. It means the active and organic basis of the sensible, its significance and ideal purpose. It is known by knowing the sensible or mechanical as it really is; that is, as it is completely, as a concretum, in Leibniz’s phrase. Leibniz saw clearly that to make the infinite something at one end of the finite, as its mere external limit, or something miraculously intercalated into the finite, was to deprive it of meaning, and, by making it unknowable, to open the way for its denial. To make identity consist in the removal of all diversity (as must be done if it be thought after the manner of external relations), is to reduce it to nothing,—as Hume, indeed, afterwards showed. Substance, which is merely a support behind qualities, is unknowable, and hence unverifiable. While, then, the aim of both Locke and Leibniz