Leibniz, Gottfried Wilhelm

I INTRODUCTION

Leibniz, Gottfried Wilhelm, also Leibnitz, Baron Gottfried Wilhelm von (1646-1716), German philosopher, mathematician, and statesman, regarded as one of the supreme intellects of the 17th century.

Leibniz was born in Leipzig. He was educated at the universities of Leipzig, Jena, and Altdorf. Beginning in 1666, the year in which he was awarded a doctorate in law, he served Johann Philipp von Schönborn, archbishop elector of Mainz, in a variety of legal, political, and diplomatic capacities. In 1673, when the elector's reign ended, Leibniz went to Paris. He remained there for three years and also visited Amsterdam and London, devoting his time to the study of mathematics, science, and philosophy. In 1676 he was appointed librarian and privy councillor at the court of Hannover. For the 40 years until his death, he served Ernest Augustus, duke of Brunswick-Lüneburg, later elector of Hannover, and George Louis, elector of Hannover, later George I, king of Great Britain and Ireland.

Leibniz was considered a universal genius by his contemporaries. His work encompasses not only mathematics and philosophy but also theology, law, diplomacy, politics, history, philology, and physics.

II MATHEMATICS

Leibniz's contribution in mathematics was to discover, in 1675, the fundamental principles of infinitesimal calculus. This discovery was arrived at independently of the discoveries of the English scientist Sir Isaac Newton, whose system of calculus was invented in 1666. Leibniz's system was published in 1684, Newton's in 1687, and the method of notation devised by Leibniz was universally adopted (see Mathematical Symbols). In 1672 he also invented a calculating machine capable of multiplying, dividing, and extracting square roots, and he is considered a pioneer in the development of mathematical logic.

III PHILOSOPHY

In the philosophy expounded by Leibniz, the universe is composed of countless conscious centers of spiritual force or energy, known as monads. Each monad represents an individual microcosm, mirroring the universe in varying degrees of perfection and developing independently of all other monads. The universe that these monads constitute is the harmonious result of a divine plan. Humans, however, with their limited vision, cannot accept such evils as disease and death as part of a universal harmony. This Leibnizian universe, “the best of all possible worlds,” is satirized as a utopia by the French author Voltaire in his novel Candide (1759).

Important philosophical works by Leibniz include Essays in Theodicy on the Goodness of God, the Liberty of Man, and the Origin of Evil (2 volumes, 1710; translated in Philosophical Works,1890), Monadology (1714; published in Latin as Principia Philosophiae,1721; translated 1890), and New Essays Concerning Human Understanding (1703; published 1765; translated 1916). The latter two greatly influenced German philosophers of the 18th century, including Christian von Wolff and Immanuel Kant.

Leibniz, Gottfried Wilhelm

b. July 1 [June 21, old style], 1646, Leipzig

d. Nov. 14, 1716, Hannover, Hanover

German philosopher, mathematician, and political adviser, important both as a metaphysician and as a logician and distinguished also for his independent invention of the differential and integral calculus.

Early life and education

Leibniz was born into a pious Lutheran family near the end of the Thirty Years' War, which had laid Germany in ruins. As a child, he was educated in the Nicolai School but was largely self-taught in the library of his father, who had died in 1652. At Easter time in 1661, he entered the University of Leipzig as a law student; there he came into contact with the thought of men who had revolutionized science and philosophy--men such as Galileo, Francis Bacon, Thomas Hobbes, and René Descartes. Leibniz dreamed of reconciling--a verb that he did not hesitate to use time and again throughout his career--these modern thinkers with the Aristotle of the Scholastics. His baccalaureate thesis, De Principio Individui ("On the Principle of the Individual"), which appeared in May 1663, was inspired partly by Lutheran nominalism (the theory that universals have no reality but are mere names) and emphasized the existential value of the individual, who is not to be explained either by matter alone or by form alone but rather by his whole being (entitate tota). This notion was the first germ of the future "monad." In 1666 he wrote De Arte Combinatoria ("On the Art of Combination"), in which he formulated a model that is the theoretical ancestor of some modern computers: all reasoning, all discovery, verbal or not, is reducible to an ordered combination of elements, such as numbers, words, sounds, or colours.

After completing his legal studies in 1666, Leibniz applied for the degree of doctor of law. He was refused because of his age and consequently left his native city forever. At Altdorf--the university town of the free city of Nürnberg--his dissertation De Casibus Perplexis ("On Perplexing Cases") procured him the doctor's degree at once, as well as the immediate offer of a professor's chair, which, however, he declined. During his stay in Nürnberg, he met Johann Christian, Freiherr von Boyneburg, one of the most distinguished German statesmen of the day. Boyneburg took him into his service and introduced him to the court of the prince elector, the archbishop of Mainz, Johann Philipp von Schönborn, where he was concerned with questions of law and politics.

King Louis XIV of France was a growing threat to the German Holy Roman Empire. To ward off this danger and divert the King's interests elsewhere, the Archbishop hoped to propose to Louis a project for an expedition into Egypt; because he was using religion as a pretext, he expressed the hope that the project would promote the reunion of the church. Leibniz, with a view toward this reunion, worked on the Demonstrationes Catholicae. His research led him to situate the soul in a point--this was new progress toward the monad--and to develop the principle of sufficient reason (nothing occurs without a reason). His meditations on the difficult theory of the point were related to problems encountered in optics, space, and movement; they were published in 1671 under the general title Hypothesis Physica Nova ("New Physical Hypothesis"). He asserted that movement depends, as in the theory of the German astronomer Johannes Kepler, on the action of a spirit (God).

In 1672 the Elector sent the young jurist on a mission to Paris, where he arrived at the end of March. In September, Leibniz met with Antoine Arnauld, a Jansenist theologian (Jansenism was a nonorthodox Roman Catholic movement that spawned a rigoristic form of morality) known for his writings against the Jesuits. Leibniz sought Arnauld's help for the reunion of the church. He was soon left without protectors by the deaths of Freiherr von Boyneburg in December 1672 and of the Elector of Mainz in February 1673; he was now, however, free to pursue his scientific studies. In search of financial support, he constructed a calculating machine and presented it to the Royal Society during his first journey to London, in 1673.

Late in 1675 Leibniz laid the foundations of both integral and differential calculus. With this discovery, he ceased to consider time and space as substances--another step closer to monadology. He began to develop the notion that the concepts of extension and motion contained an element of the imaginary, so that the basic laws of motion could not be discovered merely from a study of their nature. Nevertheless, he continued to hold that extension and motion could provide a means for explaining and predicting the course of phenomena. Thus, contrary to Descartes, Leibniz held that it would not be contradictory to posit that this world is a well-related dream. If visible movement depends on the imaginary element found in the concept of extension, it can no longer be defined by simple local movement; it must be the result of a force. In criticizing the Cartesian formulation of the laws of motion, known as mechanics, Leibniz became, in 1676, the founder of a new formulation, known as dynamics, which substituted kinetic energy for the conservation of movement. At the same time, beginning with the principle that light follows the path of least resistance, he believed that he could demonstrate the ordering of nature toward a final goal or cause.

The Hanoverian period

Leibniz continued his work but was still without an income-producing position. By October 1676, however, he had accepted a position in the employment of John Frederick, the duke of Braunschweig-Lüneburg. John Frederick, a convert to Catholicism from Lutheranism in 1651, had become duke of Hanover in 1665. He appointed Leibniz librarian, but, beginning in February 1677, Leibniz solicited the post of councillor, which he was finally granted in 1678. It should be noted that, among the great philosophers of his time, he was the only one who had to earn a living. As a result, he was always a jack-of-all-trades to royalty.

Trying to make himself useful in all ways, Leibniz proposed that education be made more practical, that academies be founded; he worked on hydraulic presses, windmills, lamps, submarines, clocks, and a wide variety of mechanical devices; he devised a means of perfecting carriages and experimented with phosphorus. He also developed a water pump run by windmills, which ameliorated the exploitation of the mines of the Harz Mountains, and he worked in these mines as an engineer frequently from 1680 to 1685. Leibniz is considered to be among the creators of geology because of the observations he compiled there, including the hypothesis that the Earth was at first molten. These many occupations did not stop his work in mathematics: In March 1679 he perfected the binary system of numeration (i.e., using two as a base), and at the end of the same year he proposed the basis for analysis situs, now known as general topology, a branch of mathematics that deals with selected properties of collections of related physical or abstract elements. He was also working on his dynamics and his philosophy, which was becoming increasingly anti-Cartesian. At this point, Duke John Frederick died on Jan. 7, 1680, and his brother, Ernest Augustus I, succeeded him.

France was growing more intolerant at home--from 1680 to 1682 there were harsh persecutions of the Protestants that paved the way for the revocation of the Edict of Nantes on Oct. 18, 1685--and increasingly menacing on its frontiers, for as early as 1681, despite the reigning peace, Louis XIV took Strasbourg and laid claim to 10 cities in Alsace. France was thus becoming a real danger to the empire, which had already been shaken on the east by a Hungarian revolt and by the advance of the Turks, who had been stopped only by the victory of John III Sobieski, king of Poland, at the siege of Vienna in 1683. Leibniz served both his prince and the empire as a patriot. He suggested to his prince a means of increasing the production of linen and proposed a process for the desalinization of water; he recommended classifying the archives and wrote, in both French and Latin, a violent pamphlet against Louis XIV.

During this same period Leibniz continued to perfect his metaphysical system through research into the notion of a universal cause of all being, attempting to arrive at a starting point that would reduce reasoning to an algebra of thought. He also continued his developments in mathematics; in 1681 he was concerned with the proportion between a circle and a circumscribed square and, in 1684, with the resistance of solids. In the latter year he published Nova Methodus pro Maximis et Minimis ("New Method for the Greatest and the Least"), which was an exposition of his differential calculus.

Leibniz' noted Meditationes de Cognitione, Veritate et Ideis (Reflections on Knowledge, Truth, and Ideas) appeared at this time and defined his theory of knowledge: things are not seen in God--as Nicolas Malebranche suggested--but rather there is an analogy, a strict relation, between God's ideas and man's, an identity between God's logic and man's. In February 1686, Leibniz wrote his Discours de métaphysique (Discourse on Metaphysics). In the March publication of Acta, he disclosed his dynamics in a piece entitled Brevis Demonstratio Erroris Memorabilis Cartesii et Aliorum Circa Legem Naturae ("Brief Demonstration of the Memorable Error of Descartes and Others About the Law of Nature"). A further development of Leibniz' views, revealed in a text written in 1686 but long unpublished, was his generalization concerning propositions that in every true affirmative proposition, whether necessary or contingent, the predicate is contained in the notion of the subject. It can be said that, at this time, with the exception of the word monad (which did not appear until 1695), his philosophy of monadology was defined.

In 1685 Leibniz was named historian for the House of Brunswick and, on this occasion, Hofrat ("court adviser"). His job was to prove, by means of genealogy, that the princely house had its origins in the House of Este, an Italian princely family, which would allow Hanover to lay claim to a ninth electorate. In search of these documents, Leibniz began travelling in November 1687. Going by way of southern Germany, he arrived in Austria, where he learned that Louis XIV had once again declared a state of war; in Vienna, he was well received by the Emperor; he then went to Italy. Everywhere he went, he met scientists and continued his scholarly work, publishing essays on the movement of celestial bodies and on the duration of things. He returned to Hanover in mid-July 1690. His efforts had not been in vain. In October 1692 Ernest Augustus obtained the electoral investiture.

Until the end of his life, Leibniz continued his duties as historian. He did not, however, restrict himself to a genealogy of the House of Brunswick; he enlarged his goal to a history of the Earth, which included such matters as geological events and descriptions of fossils. He searched by way of monuments and linguistics for the origins and migrations of peoples; then for the birth and progress of the sciences, ethics, and politics; and, finally, for the elements of a historia sacra. In this project of a universal history, Leibniz never lost sight of the fact that everything interlocks. Even though he did not succeed in writing this history, his effort was influential because he devised new combinations of old ideas and invented totally new ones.

In 1691 Leibniz was named librarian at Wolfenbüttel and propagated his discoveries by means of articles in scientific journals. In 1695 he explained a portion of his dynamic theory of motion in the Système nouveau ("New System"), which treated the relationship of substances and the preestablished harmony between the soul and the body: God does not need to bring about man's action by means of his thoughts, as Malebranche asserted, or to wind some sort of watch in order to reconcile the two; rather, the Supreme Watchmaker has so exactly matched body and soul that they correspond--they give meaning to each other--from the beginning. In 1697, De Rerum Originatione (On the Ultimate Origin of Things) tried to prove that the ultimate origin of things can be none other than God. In 1698, De Ipsa Natura ("On Nature Itself") explained the internal activity of nature in terms of Leibniz' theory of dynamics.

All of these writings opposed Cartesianism, which was judged to be damaging to faith. Plans for the creation of German academies followed in rapid succession. With the help of the electress Sophia Charlotte, daughter of Ernest Augustus and soon to become the first queen of Prussia (January 1701), the German Academy of Sciences in Berlin was founded on July 11, 1700.

On Jan. 23, 1698, Ernest Augustus died, and his son, George Louis, succeeded him. Leibniz found himself confronted with an uneducated, boorish prince, a reveller who kept him in the background. Leibniz took advantage of every pretext to leave Hanover; he was constantly on the move; his only comfort lay in his friendship with Sophia Charlotte and her mother, Princess Sophia. Once again, he set to work on the reunion of the church: in Berlin, it was a question of uniting the Lutherans and the Calvinists; in Paris, he had to subdue Bishop Bénigne Bossuet's opposition; in Vienna (to which Leibniz returned in 1700) he enlisted the support of the Emperor, which carried great weight; in England, it was the Anglicans who needed convincing.

The death in England of William, duke of Gloucester, in 1700 made George Louis, great-grandson of James I, a possible heir to the throne. It fell to Leibniz, jurist and historian, to develop his arguments concerning the rights of the House of Braunschweig-Lüneburg with respect to this succession.

The War of the Spanish Succession began in March 1701 and did not come to a close until September 1714, with the Treaty of Baden. Leibniz followed its episodes as a patriot hostile to Louis XIV. His fame as a philosopher and scientist had by this time spread all over Europe; he was named a foreign member by the Academy of Sciences of Paris in 1700 and was in correspondence with most of the important European scholars of the day. If he was publishing little at this point, it was because he was writing Théodicée, which was published in 1710. In this work he set down his ideas on divine justice.

Leibniz was impressed with the qualities of the Russian tsar Peter the Great, and in October 1711 the ruler received him for the first time. Following this, he stayed in Vienna until September 1714, and during this time the Emperor promoted him to the post of Reichhofrat ("adviser to the empire") and gave him the title of Freiherr ("baron"). About this time he wrote the Principes de la nature et de la Grâce fondés en raison, which inaugurated a kind of preestablished harmony between these two orders. Further, in 1714 he wrote the Monadologia, which synthesized the philosophy of the Théodicée. In August 1714, the death of Queen Anne brought George Louis to the English throne under the name of George I. Returning to Hanover, where he was virtually placed under house arrest, Leibniz set to work once again on the Annales Imperii Occidentis Brunsvicenses (1843-46; "Braunschweig Annals of the Western Empire"). At Bad-Pyrmont, he met with Peter the Great for the last time in June 1716. From that point on, he suffered greatly from gout and was confined to his bed until his death.

Leibniz was a man of medium height with a stoop, broad-shouldered but bandy-legged, as capable of thinking for several days sitting in the same chair as of travelling the roads of Europe summer and winter. He was an indefatigable worker, a universal letter writer (he had more than 600 correspondents), a patriot and cosmopolitan, a great scientist, and one of the most powerful spirits of Western civilization.

Leibniz

With the logical work of the German mathematician, philosopher, and diplomat Gottfried Wilhelm Leibniz, we encounter one of the great triumphs, and tragedies, in the history of logic. He created in the 1680s a symbolic logic (see illustration

illustration: Representations of the universal affirmative, "All A's are B's" in modern logic.

) that is remarkably similar to George Boole's system of 1847--and Boole is widely regarded as the initiator of mathematical or symbolic logic. But nothing other than vague generalities about Leibniz' goals for logic was published until 1903--well after symbolic logic was in full blossom. Thus one could say that, great though Leibniz' discoveries were, they were virtually without influence in the history of logic. (There remains some slight possibility that Lambert or Boole may have been directly or indirectly influenced by Leibniz' logical system.)

Leibniz' logical research was not entirely symbolic, however, nor was he without influence in the history of (nonsymbolic) logic. Early in his life, Leibniz was strongly interested in the program of Lull, and he wrote the De arte combinatoria (1666); this work followed the general Lullian goal of discovering truths by combining concepts into judgments in exhaustive ways and then methodically assessing their truth. Leibniz later developed a goal of devising what he called a "universally characteristic language" (lingua characteristica universalis) that would, first, notationally represent concepts by displaying the more basic concepts of which they were composed, and second, naturally represent (in the manner of graphs or pictures, "iconically") the concept in a way that could be easily grasped by readers, no matter what their native tongue. Leibniz studied and was impressed by the method of the Egyptians and Chinese in using picturelike expressions for concepts. The goal of a universal language had already been suggested by Descartes for mathematics as a "universal mathematics"; it had also been discussed extensively by the English philologist George Dalgarno (c. 1626-87) and, for mathematical language and communication, by the French algebraist François Viète (1540-1603). The search for a universal language to replace Latin was seriously taken up again in the late 19th century, first by Giuseppe Peano--whose work on Interlingua, an uninflected form of Latin, was directly inspired by Leibniz' conception--and then with Esperanto. The goal of a logical language also inspired Gottlob Frege, and in the 20th century it prompted the development of the logical language Loglan and the computer language Prolog.

Another and distinct goal Leibniz proposed for logic was a "calculus of reason" (calculus ratiocinator). This would naturally first require a symbolism but would then involve explicit manipulations of the symbols according to established rules by which either new truths could be discovered or proposed conclusions could be checked to see if they could indeed be derived from the premises. Reasoning could then take place in the way large sums are done--that is, mechanically or algorithmically--and thus not be subject to individual mistakes and failures of ingenuity. Such derivations could be checked by others or performed by machines, a possibility that Leibniz seriously contemplated. Leibniz' suggestion that machines could be constructed to draw valid inferences or to check the deductions of others was followed up by Charles Babbage, William Stanley Jevons, and Charles Sanders Peirce and his student Allan Marquand in the 19th century, and with wide success on modern computers after World War II.

The symbolic calculus that Leibniz devised seems to have been more of a calculus of reason than a "characteristic" language. It was motivated by his view that most concepts were "composite": they were collections or conjunctions of other more basic concepts. Symbols (letters, lines, or circles) were then used to stand for concepts and their relationships. This resulted in what is called an "intensional" rather than an "extensional" logic--one whose terms stand for properties or concepts rather than for the things having these properties. Leibniz' basic notion of the truth of a judgment was that the concepts making up the predicate were "included in" the concept of the subject. What Leibniz symbolized as "A ," or what we might write as "A = B" was that all the concepts making up concept A also are contained in concept B, and vice versa.

Leibniz used two further notions to expand the basic logical calculus. In his notation, "A B C" indicates that the concepts in A and those in B wholly constitute those in C. We might write this as "A + B = C" or "A B = C"--if we keep in mind that A, B, and C stand for concepts or properties, not for individual things. Leibniz also used the juxtaposition of terms in the following way: "AB C," which we might write as "A B = C" or "A B = C," signifies in his system that all the concepts in both A and B wholly constitute the concept C.

A universal affirmative judgment, such as "All A's are B's," becomes in Leibniz' notation "A AB." This equation states that the concepts included in the concepts of both A and B are the same as those in A. A syllogism, "All A's are B's; all B's are C's; therefore all A's are C's," becomes the sequence of equations "A = AB; B =BC; therefore A =AC." This conclusion can be derived from the premises by two simple algebraic substitutions and the associativity of logical multiplication. Leibniz' interpretation of particular and negative statements was more problematic. Although he later seemed to prefer an algebraic, equational symbolic logic, he experimented with many alternative techniques, including graphs.

As with many early symbolic logics, including many developed in the 19th century, Leibniz' system had difficulties with particular and negative statements, and it included little discussion of propositional logic and no formal treatment of quantified relational statements. (Leibniz later became keenly aware of the importance of relations and relational inferences.) Although Leibniz might seem to deserve to be credited with great originality in his symbolic logic--especially in his equational, algebraic logic--it turns out that such insights were relatively common to mathematicians of the 17th and 18th centuries who had a knowledge of traditional syllogistic logic. In 1685 Jakob Bernoulli published a pamphlet on the parallels of logic and algebra and gave some algebraic renderings of categorical statements. Later the symbolic work of Lambert, Ploucquet, Euler, and even Boole--all apparently uninfluenced by Leibniz' or even Bernoulli's work--seems to show the extent to which these ideas were apparent to the best mathematical minds of the day.