LEIBNIZ, GOTTFRIED WILHELM (b. Leipzig, Germany, 1 July 1646; d. Hannover, Germany, 14 November 1716), mathematics, philosophy, metaphysics.

    resides that is truly in motion.54 Now it is evidently the vis primitiva that Leibniz has in mind, since consideration of the vis derivativa (conatus or vis viva) does not serve to identify an absolute motion, so that we may interpret Leibniz as saying that true absolute motion appertains to the metaphysical plane, where it can be perceived only by God. Indeed all bodies have an absolute motion in this sense, for since all monads have activity, all aggregates have vis viva. The world of phenomena is therefore conceived by Leibniz as a world of bodies in absolute motion (rest being a mere abstraction) but a world in which only relative changes of position can be observed.

The theory of monads may be seen as a sustained effort to present, in “cosmological” completeness, a systematic unified structure of knowledge on the basis of a logical reconstruction of the concept of substance. Insofar as the central assertion, namely, that because there are composite “substances,” there must be simple substances, is not only a cosmological and metaphysical statement, but in addition, an assertion of the priority of synthetic over analytic procedures, his efforts retain their original methodological meaning. As revealed in his plan for a characteristica universalis, analysis, for Leibniz, implies synthesis, but a synthesis that must begin with irreducible elements. Where there are no such elements, neither analysis nor synthesis is possible within the framework of Leibniz' constructive methodology. This means that Leibniz, in the course of his protracted efforts to define an individual substance, moved from physical atomism to logical atomism in the modern sense, as represented by Russell. Leibniz believed that he had proved the thesis of an unambiguously defined world, in which the physico-theological theme running through the mechanistic philosophy of his age might once again be seen as a metaphysics in the classical sense.

Influence.

The thought of Leibniz influenced the history of philosophy and science in two ways; first, through the mediocre systematization of Christian Wolff known as the “Leibnizo-Wolffian” philosophy, and secondly, through the significance of particular theories in the history of various sciences. While the tradition of the Leibnizo-Wolffian philosophy ended with Kant, the influence of particular theories of Leibniz lasted through the nineteenth and into the twentieth century.

The controversy with Newton and Clarke was not conducive to the reception of Leibniz' work in physics and hindered the objective evaluation of important contributions such as his law of radial acceleration. The vis viva controversy arose as a direct result of Leibniz' criticism of Descartes and concerned not only the measure of force but also the nature of force itself. While ‘sGravesande and d'Alembert (in his Traité de mécanique) judged the dispute to be merely a semantic argument, Kant in 1747 (Gedanken von der wahren Schätzung der lebendigen Kräfte) made an ineffective attempt at reconciliation. Leibnizian dynamics was developed further by Bošković, who transformed the concept of dynamic force in the direction of a concept of relational force.

Within the framework of rational physics, Kant contributed some essential improvements, such as the completion of the distinction between necessary and contingent propositions by means of the concept of the synthetic a priori and clarification of the principle of causality. Leibniz' protophysical plan, however, remained intact. It was continued later by Whewell, Clifford, Mach, and Dingler, to mention only a few. Insofar as the fundamental concepts of space and time were concerned, the Newtonian ideas of absolute space and time at first prevailed over the Leibnizian ideas of relational space and time. Kant also tried here to mediate between the ideas of Leibniz and Newton, but his own suggestion (space as the origin of the distinction between a nonreflexible figure and its mirror image, such as a pair of gloves) strongly resembled the Newtonian concept of absolute space. Modern relativistic physics has turned the scales in favor of the ideas of Leibniz.

Among the methodological principles of Leibniz, only the principle of sufficient reason has played a prominent role in the history of philosophy. Wolff, disregarding Leibniz' methodological intentions, tried to prove it by ontological means (Philosophia sive ontologia); Kant reduced it essentially to the law of causality and in 1813 Schopenhauer drew on it for the elucidation of his four conditions of verification (Über die vierfache Wurzel des Satzes vom zureichenden Grunde). On the other hand, the methodological project of a characteristica universalis together with the ensuing development of logical calculi has played a most significant role in the history of modern logic. In 1896 Frege, recalling Leibniz, described his Begriffsschrift of 1879 as a lingua characterica (not just a calculus ratiocinator), thus distinguishing it from the parallel efforts of Boole and Peano. De Morgan and Boole tried to carry out what Scholz has described as the “Leibniz program” of the development of a logical algebra of classes. This connection between logic and mathematics, evident also in Peirce and Schröder, was once again weakened by Frege, Peano, and Russell, whose work (especially that of Frege) nevertheless bears the inescapable influence of Leibniz; for even where the differences are greatest,