Further consequences are (5) that empty space does not exist, since it would have to con sist of points which are all different, and yet all exactly alike. For the same reason, bodies do not consist, as Descartes held, of mere ex tension, since extension, if real, would be merely the infinite repetition of elements which are ex actly alike. (6) Matter is not only infinitely divisible but infinitely divided. There are no atoms, or indivisible parts of matter. If an atom were a true substance or ultimate unit it would be theoretically possible to deduce the whole constitution of the universe from ade quate knowledge of the motion of one atom, whereas in fact many different theories of the universe may agree in yielding the same series of positions for a given atom. All matter is therefore, as we read in the (Monad°logy,' composite, and thus even the smallest particle is a fully organized world; every part of the "divine machine') of Nature is, in its turn, a further machine. In a word, space, motion and body, when body is defined as extended matter in motion, are not substances but merely vera phenomena, orderly and connected appearances of the same kind as, e.g., the rainbow; that is, they are appearances corresponding to com posites or aggregates of individual substances. But, as there are no atoms, no body is really one, and consequently not really .many, since "many° means a collection of ones. Thus the ((corporeal substances° of the physical world must possess a "fortn,° or principle of unity, which is itself immaterial and analogous to our "soul.° The so-called material world must be regarded as an aggregate of immaterial units. This is why perception and volition, the funda mental attributes of the incorporeal elements, are inexplicable on mechanical principles, which only apply to composites. Hence the philo sophical interpretation of the universe has m evitably to be given in terms of final causality, i.e., of the purposive activity of the ultimate simple elements.
Finally (7) since substances are simple units, no substance can begin or cease to exist bit by bit, by gradual development. The simple sub stance or monad, to adopt the technical term borrowed by Leibnitz from the younger Van Helmont, has no parts, and is a kind of spirit ual or metaphysical "point ,° and therefore can only come into existence, if at all, by instan taneous creation, and only perish, if at all, by instantaneous annihilation.
The only important doctrine of Leibnitz which does not figure in the foregoing sketch, taken from the tract on ultimate truths, is also the one doctrine which is most manifestly an excrescence on the system, the theorem of the existence of God. This is, according to Leib nitz, the only "truth of fact° which can be proved by a finite process of logical analysis. Leibnitz attempts to prove it both from a con sideration of the ((eternal truths') or "truths of reason," the universals of pure science, and from the "truths of fact" He holds, as a watheinatician nipst, that thg range of scientific truth is wider than that of actual existence. Besides the existing system of Laws of Nature, other systems are equally conceivable or logic ally possible, and the truths which hold for all possible constitutions of the universe form the body of "eternal truths." But Leibnitz holds
that all possibilities must be founded on some thing actual, otherwise there would be no means of distinguishing the possible from the impos sible, a doctrine which is also fundamental in the philosophy of Lotze. The requisite founda tion in actuality is provided by the existence of God. The "possible worlds ° are just those systems of things which are actually present to the understanding of God. Hence the -ex istence of God is logically necessary, provided only that it is possible, i.e., that the notion of a God or most real being involves no logical contradiction. With the addition of the proof, which Leibnitz believes himself to have given, that the existence of God is logically possible, the "ontological proof° of Descartes becomes completely valid. The existence of God may also be shown by a form of the "cosmological proof," from the "truths of fact" which are concerned with the constitution of the actual world. There must be a sufficient reason why just this one, out of all the logically possible systems, is actual rather than any other, and this reason is found in the will of God. Voli tion, it is assumed, is always directed to the choice of the apparent best, which in the case of an omniscient Got, is also the real best. Now the system of the Pre-established Har mony is the "best° of all possible systems for the universe, precisely because, owing to the complete adaptation of every monad in it to other, the number of "compossibles,) that is, of substances which can co-exist as members of the same system, is greater than on any other constitution of the universe. The famous doctrine that the actual world is the best possible world thus simply means that the quantity of existence in the actual world, as measured by the number of "compossibles° is a maximum. Hence God, in virtue of the principle of the choice of the best, chooses that this system rather than any other pos sible arrangement should exist. It is not clear bow this act of choice is to be understood, nor does Leibnitz help us much by the curious conception that every possible system tends or strives to become actual with a force propor tionate to the amount of reality, i.e., the num ber of ucompossibles" it comprises, and that God's creative activity consists in removing from the actual world the hindrance to exist ence which would otherwise arise from the rival tendencies of the possible worlds to ex istence. If "existence is logically a predicate, it ought, on Leibnitz's principles, to be inherent in the nature of the subject to which it be longs, and the actual world should thus exist eternally and necessarily. If it is not a predicate but an "external° relation imposed on the particular system of the Pre-established Har mony by the will of God, the doctrine of the analytic character of truth, from which the system of Pre-established Harmony is a mere deduction, must be false, and, as a further complication, the °ontological proof" of God's existence must also be worthless. It must be noted that God is not a monad, but stands quite outside the system. The common state ment that Leibnitz spoke of God as the "monad of monads" seems to originate with Hegel and to have no foundation.