Leibniz, Gottfried Wilhelm Von

Born July 1, 1646, in Leipzig; died Nov. 14, 1716, in Hanover. German idealist philosopher, mathematician, physicist, inventor, lawyer, historian, and linguist.

Leibniz studied jurisprudence and philosophy at the universities of Leipzig and Jena. In 1668 he entered the service of the elector of Mainz. In 1672, Leibniz traveled with a diplomatic mission to Paris, where he remained until 1676, studying mathematics and natural science. In December 1676 he returned to Germany and for the next 40 years was in the service of the dukes of Brunswick in Hanover, first as the court librarian and later as the dukes’ historiographer and secret legal counselor. From 1687 to 1690, Leibniz traveled through southern Germany, Austria, and Italy in order to collect materials for a history of the House of Brunswick. In 1700 he founded and became the first president of the Berlin Society of Sciences (later the Prussian Academy of Sciences). In 1711, 1712, and 1716 he met with Peter I and drafted a number of plans for developing education and government in Russia. From 1712 to 1714, Leibniz lived in Vienna. He corresponded extensively with nearly all of the major scholars, scientists, and political figures of his day.

Leibniz’ principal philosophical works were Disputatio metaphysica de principio individui (1685, printed 1846; Russian translation, 1890), Système nouveau de la nature (1695; Russian translation, 1890), Nouveaux Essais sur l’entendement humain (1704, printed 1765; Russian translation, 1936), Essais de Théo-dicée sur la bonté de Dieu, la liberté de l’homme et l’origine du mal (1710; Russian translation, 1887–92) and Monadologie (1714, printed 1720; Russian translation, 1890). His principal mathematical works were De vera proportione circuii ad quadratum (1682), Nova methodus pro maximis et minimis (1684), and De geometria recondita et analysi indivisibilium et infinitorum (1686). Leibniz’ ideas on physics were presented in the works Brevis demonstratio erroris memorabilis Cartesii et aliorum circa legem naturae secundum quam volunt a Deo eandem semper quantitatem motus conservari (1686) and Specimen dynamicum (1695), and his political and legal ideas were set forth in the works Nova methodus discendae docendaeque jurisprudentiae (1667), Mars Christianissimus (1680), and Codex juris gentium diplomaticus (1693).

Philosophy. Leibniz culminated 17th-century philosophy and was the forerunner of German classical philosophy. His philosophical system attained its final form by 1685 after 20 years of evolution, during which he critically analyzed the basic ideas of Democritus, Plato, St. Augustine, Descartes, Hobbes, and Spinoza.

Leibniz sought to synthesize all that was rational in previous philosophy with the latest scientific knowledge, formulating his own methodology, whose most important requirements were universality and rigor of philosophical arguments. The practicability of these requirements is guaranteed, according to Leibniz, by the presence of a priori principles of existence that are independent of experience. Among these principles Leibniz classifies:

(1) the noncontradictory nature of any possible, or conceivable, being (the principle of contradiction);

(2) the logical primacy of the possible over the actual (that which exists); the possibility of an infinite set of noncontradictory “worlds”;

(3) the principle of sufficient reason—there is a sufficient reason that this world and not some other of the possible worlds exists and that a given event and not some other takes place;

(4) the optimality (perfection) of this world as sufficient grounds for its existence.

Leibniz understood the perfection of the real world as the “harmony of essence and existence”: the optimality of relations between the diversity of existing things and actions in nature and their order; and minimum means for maximum result. The consequences of this last ontological principle are a number of other principles: the principle of the uniformity of natural laws, or of universal interrelationships; the law of continuity; the principle of identity of indiscernibles; and the principles of universal change and development, of simplicity, and of completeness.

In the spirit of 17th-century rationalism, Leibniz distinguished between the intelligible world, or the world of the truly real (metaphysical reality), and the sensory world, or the merely apparent (phenomenal) physical world. The real world, according to Leibniz, consists of infinite active psychic substances, the indivisible primal elements of being—monads, which are in a state of preestablished harmony. The harmony (one-to-one correspondence) of monads was originally established by god when he chose to create this “best of all possible worlds.” Because of this harmony, even though no monad can affect the others (monads as substances are independent of each other), nonetheless the development of each conforms fully to the development of the others and of the world as a whole. This interdependence is a result of the monads’ capacity, implanted in them by god, to represent, perceive, or express and reflect all other monads and the entire world (“the monad is a mirror of the universe”). The monad’s activity consists in the succession of perceptions and is determined by the monad’s individual “aspiration” (appetite) for new perceptions. Although all this activity arises immanently from the monad itself, at the same time it is a development of the individual program, implanted in the monad at the beginning, of the “full individual concept” that god envisioned in all its details before creating the world. Thus, all of the monad’s actions are fully interrelated and predetermined. The monads form an ascending hierarchy: their position depends on the clarity and distinctness of their representation of the world. A special place in this hierarchy is occupied by monads capable not only of perception but also of self-awareness or apperception (among which Leibniz classified human souls).

The physical world, Leibniz believed, exists only as an imperfect, sensory expression of the true world of monads, as a phenomenon of man in his cognition of the objective world. However, insofar as physical phenomena are ultimately produced by the real monads that stand behind them, Leibniz considered them “well-founded,” thus justifying the significance of the physical sciences. Leibniz considered space, matter, time, mass, motion, causality, and interaction as they were construed in the physics and mechanics of his day to be such well-founded phenomena.

In the theory of knowledge, Leibniz tried to find a compromise between Cartesian rationalism and Lockean empiricism and sensualism. Believing that without sensory experience no intellectual activity is possible, Leibniz at the same time strongly opposed Locke’s idea of the soul as a tabula rasa and accepted Locke’s statement that there is “nothing in the intellect that was not previously in the senses” only with the proviso “except the intellect itself.” Leibniz taught of the mind’s innate ability to know a number of ideas and truths: he classified the higher categories of being such as “I,” “identity,” “being,” and “perception” as innate ideas and the universal and necessary truths of logic and mathematics as innate truths. This innate ability, however, is not given in finished form but only as a “predisposition,” a rudiment. In contrast to Locke, Leibniz ascribed much greater importance to probabilistic knowledge, pointing out the necessity of working out probability theory and game theory. Leibniz introduced the division of all truths according to their source and special role into the truths of reason and the truths of fact, assigning to the former the properties of necessity and to the latter the properties of chance.

Physics. Leibniz developed the theory of the relativity of space, time, and motion. He established as a quantitative measure of motion the “vital force” (kinetic energy)—the product of the mass of a body and the square of its velocity (in contrast to Descartes, who considered the measure of motion to be the product of mass and velocity—the “dead force” as Leibniz called it). Using in part the results obtained by Huygens, Leibniz discovered the law of conservation of vital forces, which was the first statement of the law of the conservation of energy, and also advanced the idea of the conversion of some types of energy into others. On the basis of the philosophical principle of the optimality of all actions in nature, Leibniz formulated one of the most important variational principles of physics—the principle of least action, which later was named the Maupertuis principle. Leibniz made a number of discoveries in specialized branches of physics: in the theory of elasticity and the theory of oscillations, and, in particular, the discovery of the formula for calculating the strength of beams (Leibniz’ formula).

Logic. Leibniz developed a theory of analysis and synthesis and was the first to formulate the law of sufficient reason. He also gave a definition of the law of identity that is now accepted in modern logic. Leibniz created a classification of definitions that was the most complete for his era and elaborated the theory of constructive definitions, as well as other theories. In Leibniz’ work Dissertatio de arte combinatoria, written in 1666, some aspects of modern mathematical logic were anticipated. Leibniz advanced the idea of using mathematical symbols in logic and constructs of logical calculi, posed the problem of constructing a logical foundation for mathematics, and proposed the use of the binary system of calculation for the purposes of computational mathematics. Leibniz first expressed the idea that machine modeling of human functions might be possible and introduced the term “model.”

Mathematics. Leibniz’ most important contribution in mathematics was the development, at the same time as and independently of Newton, of the differential and integral calculus. Leibniz obtained his first results in 1675 under the influence of Huygens and based on his own mastery of Pascal’s triangle, the algebraic methods of Descartes, and the works of Wallis and Mercator. A systematic essay on the differential calculus was first published in 1684 and on the integral calculus in 1686. In these essays, definitions of the differential and integral were given and the symbols d and for the differential and the integral, respectively, were introduced. Leibniz presented the rules for differentiating sums, products, quotients, any constant power, and the function of a function (the invariance of the first differential), as well as the rules for finding and distinguishing (by using the second differential) the extrema of curves and for finding inflection points. He also established in these essays the inverse nature of differentiation and integration.

Applying his calculus to a number of problems of mechanics (such as the cycloid, catenary, and brachistochrone), Leibniz, together with Huygens and Jakob and Johann Bernoulli, came quite close to creating the calculus of variations (1686–96). In subsequent works Leibniz indicated (1695) the formula for repeated differentiation of a product (Leibniz’ formula) and rules for differentiating a number of the most important transcendental functions, and in the years 1702–03 he developed the integration of rational fractions. Leibniz made extensive use of the expansion of functions into infinite power series, established the test for convergence of an alternating series, and gave the solutions of certain types of ordinary differential equations in quadrature. Leibniz introduced such terms as “differential,” “differential calculus,” “differential equation,” “function,” “variable,” “constant,” “coordinates,” “abscissa,” “algebraic and transcendental curves,” and “algorithm” (in a sense close to the modern).

Although the attempts made by Leibniz to provide a logical foundation for the differential calculus cannot be considered successful, his clear understanding of the essence of the new analytic methods and his comprehensive treatment of the apparatus of the calculus contributed to the fact that it was his version of the calculus that largely determined the subsequent development of mathematical analysis.

In addition to analysis, Leibniz made a number of important discoveries in other fields of mathematics: combinatorics, algebra (the principles of the theory of determinants), and geometry, where he laid the foundations for the theory of the contiguity of curves (1686), worked out, together with Huygens, the theory of the envelopes of a family of curves (1692–94), and proposed the idea of geometric calculations.

Other fields. In his work Protogaea (1693), Leibniz advanced the idea that the earth had evolved and summarized the material that he had collected in the field of paleontology. Leibniz introduced into biology the idea of the integral nature of organic systems and the principle of the irreducibility of the organic to the mechanical; he construed evolution as a continuous unfolding of preformed germs.

In psychology, Leibniz advanced the idea of unconscious “small perceptions,” and developed a theory of unconscious mental activity.

In linguistics, Leibniz elaborated a theory of the historical origin of languages, created a system for classifying languages by families, and developed the study of the origin of names. Leibniz made significant contributions to the development of German philosophic and scientific vocabulary.

In politics and law, Leibniz upheld the concept of natural law and the theory of the social contract, wrote a number of proposals for church unification (consolidation of the Catholic and Protestant churches, unification of the Lutheran and Reformed churches), and advocated an alliance among the princes of all Germany and peaceful cooperation in Europe.

Leibniz was a talented inventor. He designed optical instruments and hydraulic machines, worked on the creation of a “pneumatic engine,” and invented the first integrating mechanism and a calculator that was unique for the time.

Influence. Leibniz had a significant influence on the subsequent development of philosophy and science. C. Wolff—a student of Leibniz and a systematizer of his philosophy—and his school contributed to the spread of Leibniz’ ideas in Germany, where prior to Kant he was the leading philosophical authority. During his lifetime Leibniz founded a mathematical school (the Bernoulli brothers, L’Hôpital, and Tschirnhaus) from which Euler emerged in the 18th century. Many of Leibniz’ ideas were accepted by classical German philosophy. In the 20th century the ideas of “monadology” were developed by representatives of personalism and other idealistic schools (such as Husserl and Whitehead).