Gottfried Wilhelm Leibniz

Gottfried Wilhelm Leibniz
Gottfried Wilhelm Leibniz
Philosophical work
Era	17th-century philosophy (Modern Philosophy)
Region	Western Philosophers
School	Continental rationalism
Main interests	Metaphysics, Epistemology, Science, Mathematics, theodicy
Notable ideas	Calculus we all use, innate knowledge, optimism, monad

Gottfried Wilhelm Leibniz (also von Leibni(t)z) (July 1 (June 21 Old Style) 1646, Leipzig – November 14 1716, Hannover) was a German polymath/polyhistor, deemed a genius in his day and since.

Educated in law and serving as factotum to two major German noble houses--one becoming the British royal family while he served it--he played a major role in the European politics and diplomacy of his day. He wrote on philosophy, science, technology, mathematics, theology, history, law, ethics, politics, and philology, even occasional verse. His contributions to this vast array of subjects are scattered in journals, many thousands of letters and memoranda, and in a huge collection of unpublished manuscripts. To date, there is no complete edition of Leibniz's writings, and a complete account of his accomplishments is not yet possible.

Leibniz occupies an equally large place in both the history of philosophy and the history of mathematics. In philosophy, he was, along with Rene Descartes and Baruch Spinoza, one of the three great 17th century rationalists, who were all versed in mathematics as well as philosophy. His philosophy looks back to the Scholastic tradition, and forward to modern logic and analysis. He is most remembered for the metaphysical notion of monad. In mathematics, he invented the calculus independently of Newton, and his notation is the one in general use since. He was the first to employ the term function.

He made major contributions to physics and technology, anticipating notions that surfaced much later in biology, medicine,geology, psychology, knowledge engineering, and information science. Key figures in their fields have argued that his writings contain anticipations of relativity, fractal geometry, topology, and even quantum mechanics. Much of his thinking was not only too radical for his day, but was not understood until as recently as the 20th century.

Life

The only biography in English is Aiton (1986). The most lively short account of Leibniz’s life, one also doing fair justice to the breadth of his interests and activities, is Mates (1986: 14-35), who cites the German biographies extensively. Also see MacDonald Ross (1984: chpt. 1), the chapter by Ariew in Jolley (1995), and Jolley (2005: chpt. 1). For a biographical glossary of Leibniz's intellectual contemporaries, see AG 350.

Coming of age

Leibniz's parents were Friedrich Leibnütz and Catherina Schmuck. He began spelling his name "Leibniz" early in adult life, but others often referred to him as "Leibnitz," a spelling which persisted until the 20th century. In later life, he often signed himself "von Leibniz", and many posthumous editions of his works gave his name on the title page as "Freiherr [Baron] G. W. von Leibniz." But no document has been found confirming that he was ever granted a patent of nobility (Aiton 1985: 312). In the 17th and 18th centuries, it was not unusual for the ambitious to insert, starting in midlife, a "de" or "von" before their surnames, to suggest a nobility which they in fact did not possess; cases in point include Voltaire and Beaumarchais.

At age 6, Leibniz's father, a Professor of Moral Philosophy at the University of Leipzig, died, leaving a personal library to which Leibniz was granted free access from age 8 onwards. By 12, he had taught himself Latin, a language he employed freely all his life, and had begun Greek. He entered his father's university at 15, and completed his university studies by age 20, specializing in law and mastering the standard university course of his day and place in classics, logic, and scholastic philosophy. His mathematical education was not up to the French and British standard of the time. In 1666, he published his habilitation thesis in philosophy and first book, Dissertation on the Art of Combinations. When Leipzig declined to assure him a position teaching law upon graduation, Leibniz submitted to the University of Altdorf near Nuremberg the thesis he had intended to submit at Leipzig, and obtained his doctorate in law in five months. He then declined an offer of academic appointment at Altdorf, and spent the rest of his life in the service of two major German noble families.

Career

Leibniz's career can be outlined as follows:

• 1666-74: Mainly in service to the Elector of Mainz, Johann Philipp von Schönborn, and his minister, Baron von Boineburg.
  • 1672-76. Resides in Paris, making two important sojourns to London.
• 1676-1716. In service to the House of Hanover.
  • 1677-98. Courtier, first to John Frederick, Duke of Brunswick-Lüneburg, then to his son, Elector Ernst August of Hannover.
    • 1687-90. Travels extensively in Germany, Austria, and Italy, researching a book the Elector has commissioned him to write on the history of the House of Brunswick.
  • 1698-1714: Courtier to Elector Georg Ludwig of Hannover.
    • 1712-14: Imperial Court Councillor at the Hapsburg court in Vienna.
  • 1714-16: Georg Ludwig, upon becoming George I of Great Britain, forbids Leibniz to follow him to London. Leibniz ends his days in relative neglect.

Leibniz's first position was as a salaried alchemist in Nuremberg, even though he knew nothing about the subject. He soon met J. C. von Boineburg, the dismissed chief minister of the Elector of Mainz, Johann Philipp von Schönborn. Von Boineburg hired Leibniz as an assistant, and shortly thereafter reconciled with the Elector and introduced Leibniz to him. Leibniz then dedicated an essay on law to the Elector in the hope of obtaining employment. The stratagem worked; the Elector asked Leibniz to assist with the redrafting of the legal code for his Electorate. In 1669, Leibniz was appointed Assessor in the Court of Appeal. Although von Boineburg died late in 1672, Leibniz remained in the employ of his widow until she dismissed him in 1674.

Von Boineburg did much to promote Leibniz's reputation, and the latter's memoranda and letters began to attract favorable notice. Leibniz's service to the Elector soon took on a diplomatic role. He published an essay, under the pseudonym of a fictitious Polish nobleman, arguing (unsuccessfully) for the German candidate for the Polish crown. The main European geopolitical reality during Leibniz's adult life was the ambition of the French king, Louis XIV, backed by French military and economic might. Meanwhile, the Thirty Years War had left German-speaking Europe exhausted, fragmented, and economically backward. Leibniz proposed to protect German-speaking Europe by distracting Louis as follows. France would be invited to take Egypt as a stepping stone towards an eventual conquest of the Dutch East Indies. In return, France would agree to leave Germany and the Netherlands undisturbed. This plan obtained the Elector's cautious support. In 1672, the French government invited Leibniz to Paris for discussion, but the plan was soon overtaken by events and became moot. Napoleon's failed invasion of Egypt in 1798 can be seen as an unwitting implementation of Leibniz's plan.

Thus Leibniz began several years in Paris, during which he greatly expanded his knowledge, of mathematics and physics, and began contributing to both. He met Malebranche and Antoine Arnauld, the leading French philosophers of the day, and studied the writings of Descartes and Pascal, unpublished as well as published. He befriended a German mathematician, Ehrenfried Walther von Tschirnhaus; they corresponded for the rest of their lives. Especially fateful was Leibniz's making the acquaintance of the Dutch physicist and mathematician Christiaan Huygens, then active in Paris. Soon after arriving in Paris, Leibniz received a rude awakening; his knowledge of mathematics and physics was spotty. Under Huygen's guidance, he began a program of self-study that soon resulted in his making major contributions to both subjects, including inventing the differential calculus.

When it became clear that France would not implement its part of Leibniz's Egyptian plan, the Elector sent his nephew, escorted by Leibniz, on a related mission to the British government in London, early in 1673. There Leibniz made the acquaintance of Henry Oldenburg and John Collins. After demonstrating to the Royal Society a calculating machine he had been designing and building since 1670, the first such machine that could execute all four basic arithmetical operations, the Society made him an external member. The mission ended abruptly when news reached it of the Elector's death, whereupon Leibniz promptly returned to Paris and not, as had been planned, to Mainz.

The sudden deaths of Leibniz's two patrons in the same winter meant that Leibniz had to find a new basis for his career. In this regard, a 1669 invitation from the Duke of Brunswick to visit Hannover proved fateful. Leibniz declined the invitation, but began corresponding with the Duke in 1671. In 1673, the Duke offered him the post of Counsellor which Leibniz very reluctantly accepted two years later, only after it became clear that no employment in Paris, whose intellectual stimulation he relished, or with the Hapsburg imperial court was forthcoming.

Leibniz managed to delay his arrival in Hannover until the end of 1676, after making one more short journey to London, where he was shown some of Newton's unpublished work on the calculus. This fact was deemed evidence supporting the accusation, made decades later, that he had stolen the calculus from Newton. On the journey from London to Hannover, Leibniz stopped in The Hague where he met Leeuwenhoek, the discoverer of microorganisms. He also spent several days in intense discussion with Spinoza, who had just completed his masterwork, the Ethics. Leibniz respected Spinoza's powerful intellect, but was dismayed by his conclusions that (in Leibniz's view) contradicted Christian orthodoxy.

In 1677, he was promoted, at his request, to Privy Counsellor of Justice, a post he held for the rest of his life. Leibniz served three consecutive rulers of the House of Brunswick as historian, political adviser, and most consequentially, as librarian of the ducal library. He thenceforth employed his pen on all the various political, historical, and theological matters involving the House of Brunswick; the resulting documents form a valuable part of the historical record for that period.

Among the few people in north Germany to warm to Leibniz were the Electress Sophia of Hanover (1630-1714), her daughter Sophia Charlotte of Hanover (1668-1705), the Queen of Prussia and his avowed disciple, and Caroline of Ansbach, the consort of her grandson, the future George II. To each of these women he was correspondent, adviser, and friend. In turn, they all warmed to him more than did their spouses and the future king George 1. For a recent study of Leibniz's correspondence with Sophia Charlotte, see MacDonald Ross (1998).

The population of Hannover was only about 10,000, and its provinciality eventually grated on Leibniz. Nevertheless, to be a major courtier to the House of Brunswick was quite an honor, especially in light of the meteoric rise in the prestige of that House during Leibniz's association with it. In 1692, the Duke of Brunswick became a hereditary Elector of the Holy Roman Empire. The Electress Sophia was the granddaughter of James 1 of Great Britain and hence in the line of succession to the British throne. Moreover she was neither Catholic nor married to one. Invoking these facts, the British Act of Settlement of 1701 designated her and her descent as the royal family of the United Kingdom, once both King William I and his sister and successor, Queen Anne, were dead. Leibniz played a role in the initiatives and negotiations that culminated in that Act. He was not necessarily effective; e.g., a document he wrote and published anonymously, thinking to promote the Brunswick cause, was formally censured by the British Parliament.

The Brunswicks tolerated the enormous effort Leibniz devoted to intellectual pursuits unrelated to his duties as a courtier, pursuits such as perfecting the calculus, writing about other mathematics, logic, physics, and philosophy, and keeping up a vast correspondence. He began working on the calculus in 1674; the earliest evidence of its use in his surviving notebooks is 1675. By 1677 he had a coherent system in hand, but did not publish it until 1684. Leibniz's most important mathematical papers were published between 1682 and 1692, usually in a journal which he and Otto Mencke founded in 1682, the Acta Eruditorum. That journal played a key role in advancing his mathematical and scientific reputation, which in turn enhanced his eminence in diplomacy, history, theology, and philosophy.

The Elector Ernst August commissioned Leibniz to write a history of the House of Brunswick, going back to the time of Charlemagne or earlier, hoping that the resulting book would advance his dynastic ambitions. From 1687 to 1690, Leibniz traveled extensively in Germany, Austria, and Italy, seeking and finding archival materials bearing on this project. Decades went by but no history appeared; the next Elector became quite annoyed at Leibniz's apparent dilatoriness. Leibniz never finished the project, in part because of his huge output on many other fronts, but also because he insisted on writing a meticulously researched and erudite book based on archival sources, when his patrons would have been quite happy with a short popular book, one perhaps little more than a genealogy with commentary, to be completed in three years or less. They never knew that he had in fact carried out a fair part of his assigned task: when the material Leibniz had written and collected for his history of the House of Brunswick was finally published in the 19th century, it filled three volumes.

In 1711, John Keill, writing in the journal of the Royal Society and with Newton's presumed blessing, accused Leibniz of having plagiarized Newton's calculus. Thus began the calculus priority dispute which darkened the remainder of Leibniz's life. A formal investigation by the Royal Society (in which Newton was an unacknowledged participant), undertaken in response to Leibniz's demand for a retraction, upheld Keill's charge. Historians of mathematics writing since 1900 or so have tended to acquit Leibniz, pointing to important differences between Leibniz's and Newton's versions of the calculus.

In 1711, while traveling in northern Europe, the Russian Tsar Peter the Great stopped in Hannover and met Leibniz, who then took some interest in matters Russian over the rest of his life. In 1712, Leibniz began a two year residence in Vienna, where he was appointed Imperial Court Councillor to the Hapsburgs. On the death of Queen Anne in 1714, Elector Georg Ludwig became King George I of Great Britain, under the terms of the 1701 Act of Settlement. Even though Leibniz had done much to bring about this happy event, it was not to be his hour of glory. Despite the intercession of Caroline of Ansbach, the Princess of Wales, George I forbad Leibniz to join him in London until he completed at least one volume of the history of the Brunswick family his father had commissioned nearly 30 years earlier. Moreover, for George I to include Leibniz in his London court would have been deemed insulting to Newton, who was seen as having won the calculus priority dispute and whose standing in British official circles could not have been higher. Finally, his dear friend and defender, the dowager Electress Sophia, died in 1714.

When Leibniz died in 1716, he was so out of favor that neither George I (who happened to be near Hannover at the time) nor any fellow courtier other than his personal secretary attended the funeral. Even though Leibniz was a life member of the Royal Society and the Berlin Academy of Sciences, neither organization saw fit to honor his passing. His grave went unmarked for more than 50 years. Thus the indifference of official Germany and England to the passing of the most accomplished European mind since Aristotle. Leibniz was eulogized by Fontenelle, before the Academie des Sciences in Paris, which had admitted him as a foreign member in 1700. The eulogy was composed at the behest of the Duchess of Orleans, a niece of the Electress Sophia.

Leibniz never married. He complained on occasion about money, but the fair sum he left to his sole heir, his sister's stepson, proved that the Brunswicks had, by and large, paid him well. In his diplomatic endeavors, he at times verged on the unscrupulous, as was all too often the case with professional diplomats of his day. On several occasions, Leibniz backdated and altered personal manuscripts, actions which cannot be excused or defended and which put him in a bad light during the calculus controversy. On the other hand, he was charming and well-mannered, with many friends and admirers all over Europe.

Writings

Leibniz wrote in three languages: scholastic Latin, French, and (least often) German. During his lifetime, he published many pamphlets and scholarly articles, but only two books, the Combinatorial Art and the Théodicée. Only one substantial book appeared posthumously, his Nouveaux essais sur l'entendement humain. Bodemann completed his catalogs of Leibniz's manuscripts and correspondence in 1895. Only then did the enormous extent of Leibniz's Nachlass become clear: about 15,000 letters to more than 1000 recipients plus more than 40,000 other items. Moreover, quite a few of these letters are as long as essays. Much of his vast correspondence was published only in recent decades, and many letters dated later than 1685 remain unpublished. The amount, variety, and disorder of Leibniz's writings are a predictable result of a situation he described as follows:

"I cannot tell you how extraordinarily distracted and spread out I am. I am trying to find various things in the archives; I look at old papers and hunt up unpublished documents. From these I hope to shed some light on the history of the [House of] Brunswick. I receive and answer a huge number of letters. At the same time, I have so many mathematical results, philosophical thoughts, and other literary innovations that should not be allowed to vanish that I often do not know where to begin." (1695 letter to Vincent Placcius in Gerhardt, Philosophical Writings of Leibniz III 194. Revision of translation in Mates 1986.)

The extant part of the critical edition of Leibniz's entire writings are organized as follows:

• Series 1. Political, Historical, and General Correspondence. 21 vols., 1666-1701.
• Series 2. Philosophical Correspondence. 1 vol., 1663-85.
• Series 3. Mathematical, Scientific, and Technical Correspondence. 6 vols., 1672-96.
• Series 4. Political Writings. 6 vols., 1667-98.
• Series 5. Historical and Linguistic Writings. Inactive.
• Series 6. Philosophical Writings. 7 vols., 1663-90, and Nouveaux essais sur l'entendement humain. Inactive.
• Series 7. Mathematical Writings. 3 vols., 1672-76.
• Series 8. Scientific, Medical, and Technical Writings. In preparation.

Some of these volumes, along with work in progress, are available online, gratis. Even though work on this edition began in 1901, only 22 volumes had appeared by 1990, in part because the only additions between 1931 and 1962 were four volumes in Series 1.

Three important collections of English translations are W (Wiener 1951), LL (Loemker 1969), and AG (Ariew and Garber 1989).

Posthumous reputation

When Leibniz died, his reputation was in decline. He was remembered for only one book, the Theodicee, whose supposed central argument Voltaire was to lampoon in his Candide. Leibniz had an ardent disciple, Christian Wolff, whose dogmatic and simplistic outlook much harmed Leibniz's reputation. In any event, philosophical fashion was moving away from the rationalism and system building of the 17th century, of which Leibniz had been such an ardent exponent. Much of Europe came to doubt that he had invented the calculus independently of Newton, and hence his whole work in mathematics and physics was neglected. His work on law, diplomacy, and history was seen as of ephemeral interest. No one suspected the vastness and richness of his correspondence.

Leibniz's long march to his present glory began with the 1765 publication of the Nouveaux Essais, which Kant read closely. In 1768, Dutens edited the first multi-volume edition of Leibniz's writings, followed in the 19th century by similar editions put together by Erdmann, Foucher de Careil, Gerhardt, Gerland, and Klopp. Publication of Leibniz's correspondence with notables such as Antoine Arnauld, Samuel Clarke, Sophia of Hanover, and her daughter Sophia Charlotte of Hanover, began.

In 1900, Bertrand Russell published a study of Leibniz's metaphysics. Shortly thereafter, Louis Couturat published an important study of Leibniz as logician, and edited a volume of Leibniz's heretofore unpublished writings on logic. While the particulars of their studies can be debated, Russell and Couturat made Leibniz somewhat respectable among 20th century analytical and linguistic philosophers. (E.g., Leibniz's phrase salva veritate recurs in Willard Quine's writings.) Nevertheless, the secondary literature on Leibniz did not really blossom until after WWII; in Gregory Brown's online bibliography, fewer than 30 English language entries were published before 1946. American Leibniz studies owe much to Leroy Loemker (1904-85); see, e.g., his (1969) annotated translations and his interpretive essays included in LeClerc (1973).

Leibniz's reputation as a philosopher is now perhaps higher than at any time since he was alive because (Jolley 2005: 217-19):

• Doctrinaire contempt for metaphysics, derived from analytic and linguistic philosophy, has faded;
• Analytic philosophy continues to profit from his notions of identity, individuation, and possible worlds;
• His belief that natural science, especially physics, differs from philosophy mainly in degree and not in kind, is back in fashion;
• He is now seen as a major prolongation of the mighty endeavor begun by Plato and Aristotle: humans can learn about the universe and man's place therein by exercising their reason.

Philosopher

It is very difficult to grasp Leibniz's philosophical thinking, because his philosophical writings consist mainly of a multitude of short pieces: journal articles, manuscripts published long after his death, and many letters to many correspondents. He only wrote two philosophical treatises, and the only one he published in his lifetime, the Théodicée of 1710, is as much theological as philosophical. Leibniz dated his beginning as a philosopher to his Discourse on Metaphysics, which he composed in 1686 as a commentary on an ongoing dispute between Malebranche and Antoine Arnauld. This led to an extensive and valuable correspondence with Arnauld (AG 69, LL §§36,38); it and the Discourse were not published until the 19th century. In 1695, Leibniz made his public entrée into European philosophy by publishing a journal article titled "New System of the Nature and Communication of Substances" (AG 138, LL §47, W II.4). Over 1695-1705, he composed his New Essays on Human Understanding, a lengthy commentary on John Locke's 1690 An Essay Concerning Human Understanding, but upon learning of Locke's 1704 death, lost the desire to publish it, so that the New Essays were not published until 1765. The Monadologie, composed in 1714 and published posthumously, consists of 90 aphorisms.

Leibniz met Spinoza in 1676, read some of his unpublished writings, and has since been suspected of appropriating some of Spinoza's ideas. While Leibniz admired Spinoza's powerful intellect, he was also forthrightly dismayed by Spinoza's conclusions (AG 272-84, W III.8, LL §§14,20), especially when these were inconsistent with Christian orthodoxy.

Introductions to Leibniz's philosophy include Loemker (1969a), MacDonald Ross (1984), and Jolley (2005). For an introduction to Leibniz's metaphysics, see the chapters by Mercer, Rutherford, and Sleigh in Jolley (1995); see Mercer (2001) for an advanced study. For an introduction to those aspects of Leibniz of most value to the philosophy of logic and of language, see Jolley (1995, chpts. 7,8); Mates (1986) is more advanced. MacRae (Jolley 1995: chpt. 6) discusses Leibniz's theory of knowledge. For a glossary of the philosophical terminology recurring in Leibniz's writings and the secondary literature, see Jolley (2005: 223-29).

The Principles

Leibniz variously invoked one or another of seven fundamental philosophical Principles (Mates 1986: chpts. 7.3, 9). He would on occasion propose a rationale for a specific principle, but more often took them as self-evident:

• Identity / Contradiction. Given a proposition and its negation, one is true and the other is false.
• Identity of indiscernibles.
• Sufficient Reason. "There must be a sufficient reason for anything to exist, for any event to occur, for any truth to obtain." (LL 717). However, the sufficient reason is often known only to God.
• Pre-established harmony. See Jolley (1995: 129-31).
• Continuity. Natura non saltum facit. Mathematically, if the domain of a function is a dense set, the same will be true of its range.
• Optimism. "God assuredly always chooses the best." (LL 311).
• Plenitude.

The Monads

Leibniz's best known contribution to metaphysics is his theory of monads, as exposited in his Monadologie. Monads are to the mental realm what atoms are to the physical. (Curiously, Leibniz argued that matter was infinitely divisible, in which case atoms do not exist.) Monads are the ultimate elements of the universe, and are also entities capable of perception. The monads are "substantial forms of being" with the following properties: they are eternal, indecomposable, individual, subject to their own laws, un-interacting, and each reflecting the entire universe in a pre-established harmony (a historically important example of panpsychism). Monads are centers of force; substance is force, while space, matter, and motion are merely phenomenal.

The ontological essence of a monad is its irreducible simplicity. Unlike atoms, monads posses no material or spatial character. They also differ from atoms by their complete mutual independence, so that interactions among monads are only apparent. Instead, each monad follows a preprogrammed set of "instructions" peculiar to itself, so that a monad "knows" what to do at each moment. (These "instructions" can be seen as analogous to the scientific laws governing subatomic particles.) By virtue of these instrinsic instructions, each monad is like a little mirror of the universe. Monads need not be "small"; e.g., each human being constitutes a monad, which makes free will problematic. God, too, is a monad, and His existence is inferred from the harmony prevailing among all other monads; He wills the instructions. Monads purportedly solve the problematic:

• Interaction between mind and matter arising in the system of Descartes;
• Lack of individuation inherent to the system of Spinoza, which represent individual creatures as mere accidental.

The monadology was thought arbitrary, even eccentric, in Leibniz's day and since. It now seems less so, in light of quantum mechanics with its "action at a distance" and entanglement.

Theodicy and optimism

The Théodicée tries to justify the apparent imperfections of the world by claiming that it is optimal among all possible worlds. It must be the best possible and most balanced world, because it was created by a perfect God.

The statement that "we live in the best of all possible worlds" drew scorn, most notably from Voltaire, who lampooned it in his comic novel Candide by having the character Dr. Pangloss (a parody of Leibniz) repeat it like a mantra. Thus the adjective "panglossian", describing one so naive as to believe that the world about us is the best possible one.

The mathematician Paul du Bois-Reymond, in his "Leibnizian Thoughts in Modern Science," wrote that Leibniz thought of God as a mathematician. "As is well known, the theory of the maxima and minima of functions was indebted to him for the greatest progress through the discovery of the method of tangents. Well, he conceives God in the creation of the world like a mathematician who is solving a minimum problem, or rather, in our modern phraseology, a problem in the calculus of variations --- the question being to determine among an infinite number of possible worlds, that for which the sum of necessary evil is a minimum."

A cautious defense of Leibnizian optimism would invoke certain scientific principles that emerged in the two centuries since his death and that are now thoroughly established: the principle of least action, the conservation of mass, and the conservation of energy. Recent scientific developments enable the following bolder defense. Contemporary physics can be seen as grounded in the numerical values of a handful of dimensionless constants, the best known of which are the fine structure constant and the ratio of the rest mass of the proton to the electron. Small changes in the numerical values of these constants are likely to result in a universe incapable of harboring complexity. Our universe is "best" because it is capable of supporting complex structures such as galaxies, stars, and, ultimately, life on Earth.

It has also been argued that the solar system has a number of fortuitous characteristics that support Earth's long lived and melioristic biosphere: the Earth is rich in metals; it is of the right size and distance from the sun, and has the right rotation period and axis tilt; the Moon and Jupiter have the right sizes and orbits to enable them to shield the Earth from bolide impacts; for whatever reason, such impacts have been happily rare since life emerged; and so on (Ward & Brownlee, 2000; Morris 2003: chpts. 5,6).

Symbolic thought

Leibniz thought symbols to be very important for the understanding of things. He also tried to develop an alphabet of human thought, in which he tried to represent all fundamental concepts using symbols and combined these symbols to represent more complex thoughts, a project which he never completed. A related concept is mathesis universalis. Toki Pona is an example of a modern constructed language with the same idea.

Leibniz defined characters as any written signs, and "real" characters were those which represent ideas directly—as the Chinese ideography was thought to do—and not the words for them. Among real characters, some simply serve to represent ideas, and some serve for reasoning. Egyptian hieroglyphics, Chinese ideograms, and the symbols of astronomy and chemistry belong to the first category, but Leibniz declared them to be imperfect, and desired the second category of characters for what he called his universal characteristic. Leibniz's characteristic, as first conceived, did not take the form of an algebra, probably because he was then a novice in mathematics, but the form of a universal language or script. Only in 1676 did he conceive of a kind of algebra of thought, modelled on conventional algebra and its notation.

Leibniz attached so much importance to the invention of good notation that he attributed to this alone the whole of his discoveries in mathematics. His notation for the infinitesimal calculus affords a most splendid example of his skill in this regard. Leibniz's passion for symbols and notation, and his belief that these are at the core of logic and mathematics, foretells in some respects Charles Peirce's writings on semiotics.

Characteristica Universalis (Universal characteristic) and Calculus Ratiocinator

Leibniz's project to develop the Characteristica universalis and Calculus Ratiocinator have become critically important to recent philosophy and the history of ideas. The importance is not only for our understanding of Leibniz's legacy, but also for those traditions that locate their origins in his work, such as mathematics, modernity, the European Enlightenment, and the many controversial offshoots including postmodern theory. However the Characteristica Universalis and Calculus Ratiocinator also appear to hold great significance for understanding Leibniz's relation to contemporary issues in biology, climate change and resource policy, and consequently how ethics and metaphysics are able to meaningfully engage with these pressing matters.

A central issue concerns our interpretation of the Calculus Ratiocinator. Two different perspectives have now become apparent on what Leibniz meant to refer to by this term. It seems that the perspective one takes on this matter will also influence the way one views the connection between the Calculus Ratiocinator and Characteristica Universalis, and one's subsequent understanding of the goals of modernity and connected projects.

The received view in academic philosophy for most of the twentieth century emerged from analytical philosophy and mathematical logic. In these traditions Leibniz's Calculus Ratiocinator is usually called "symbolic logic". In his logical writings, Leibniz enunciated the principal properties of what we now call conjunction, disjunction, negation, identity, set inclusion, and the empty set. From this perspective the Calculus Ratiocinator is only a part (or a subset) of the Universal Characteristic. A perfect Universal Characteristic would therefore comprise a "logical calculus". Gottlob Frege remarked that his symbolism was meant to be a calculus ratiocinator as well as a lingua characteristica. Frege's ample intellectual descent tends to view the Calculus Ratiocinator in this light.

A contrasting view, little known in academic philosophy and that emerged from synthetic philosophy and electronic engineering, sees Leibniz's Calculus Ratiocinator as a computing machine. From this perspective the Calculus Ratiocinator is a central processing unit, an actual physical mechanism that calculates the various ratios of integral and differential calculus. Consequently, one might view the Universal Characteristic as a universal symbolism that helps depict the mathematics of the qualitative flows and transformations of our cosmos. The Calculus Ratiocinator then is how one calculates the large scale quantities of such flows.

Leibniz fixed the time necessary to form his project: "I think that some chosen men could finish the matter within five years"; and finally remarked: "And so I repeat, what I have often said, that a man who is neither a prophet nor a prince can never undertake any thing more conducive to the good of the human race and the glory of God".

But in a 1714 letter to Nicholas Remond, he remarked:

"...if I had been less distracted, or if I were younger or had talented young men to help me, I should still hope to create a kind of universal symbolistic in which all truths of reason would be reduced to a kind of calculus... It would be very difficult to form or invent this... characteristic, but very easy to learn it without dictionaries." (LL §68.I)

In another letter later that year, again to Remond, he further stated:

"I have spoken to the Marquis de l'Hôpital and others about my universal symbolistic, but they have paid no more attention to it than if I had told them about a dream of mine. I should have to support it too by some obvious application, but to achieve this it would be necessary to work out at least a part of my characteristic, a task which is not easy, espeically in my present condition and without the advantage of dicussions with men who could stimulate and help me..." (LL §68.II)

What Leibniz actually meant by these terms may forever remain moot. However, it is worth considering that current software programs that use networks of block diagrams and pictograms to generate the mathematics and kinetics of ecological-physical-chemistry and dynamic socioeconomic systems all appear to aim at the kind of systems simulation which constituted Leibniz's unfinished Enlightenment project.

Formal logic

Leibniz is the most important logician between Aristotle and 1847, when George Boole and Augustus De Morgan each published books that began modern formal logic. The principles of Leibniz's logic and, arguably, of his whole philosophy, reduce to two:

1. All our ideas are compounded from a very small number of simple ideas, which form the alphabet of human thought.
2. Complex ideas proceed from these simple ideas by a uniform and symmetrical combination, analogous to arithmetical multiplication.

With regard to (1), the number of simple ideas is much greater than Leibniz thought. As for (2), logic can indeed be grounded in a symmetrical combining operation, but that operation is analogous to either of addition or multiplication. Logic also requires, at minimum, unary negation, the notion of universe of discourse, and quantified variables ranging over that universe.

Leibniz published little on formal logic in his lifetime; nearly everything he wrote on the topic consists of working drafts found in his Nachlass. The subsequent work on logic by the fellow Germans Johann Heinrich Lambert and Ploucquet also had no issue. The world slumbered on as if Leibniz had never been a logician until Louis Couturat published the relevant manuscripts in 1903. Selections from this volume have been translated into English; see Clarence Irving Lewis (1918), Wiener (1951), Parkinson (1966), Loemker (1969), and Ariew and Garber (1989). Our present understanding of Leibniz the logician emerged mainly from the work of Wolfgang Lenzen, beginning around 1980; for a summary, see Lenzen (2004).

Charles Peirce, Hugh MacColl, Frege, and Bertrand Russell all shared Leibniz's dream of combining symbolic logic, mathematics, and philosophy. The culmination of Leibniz's approach to logic is, arguably, the algebraic logic of Ernst Schröder and the modal logic founded by Clarence Irving Lewis. For an example of how a highly formal contemporary logician draws inspiration from, and sheds light on, Leibniz's work, see Zalta (2000).

Mathematician

Although the mathematical notion of function was implicit in trigonometric and logarithmic tables, which existed in his day, Leibniz was the first, in 1692 and 1694, to employ it explicitly, to denote any of several geometric concepts derived from a curve, such as abscissa, ordinate, tangent, chord, and the perpendicular (Struik 1969: 367). Only later did "function" lose these geometrical associations.

Leibniz was the first to see that the coefficients of a system of linear equations could be arranged into arrays, now called determinants, which can be manipulated to find the solution of the system, if any. This method was later called Cramer's Rule. Leibniz's discovery of Boolean algebra and of symbolic logic was discussed in the preceding section.

A comprehensive scholarly treatment of Leibniz's mathematical writings has yet to be written, perhaps because Series 7 of the Academy edition is very far from complete.

The calculus

Leibniz is credited, along with Isaac Newton, with inventing the infinitesimal calculus. According to Leibniz's notebooks, a critical breakthrough occurred on November 11, 1675, when he employed integral calculus for the first time to find the area under the function y = x. He introduced several notations used to this day, for instance the integral sign ∫ representing an elongated S, from the Latin word summa and the d used for differentials, from the Latin word differentia. Leibniz did not publish any of his results until 1684. For an English translation of this paper, see Struik (1969: 271-84), who also translates parts of two other key papers by Leibniz on the calculus. The product rule of differential calculus is still called "Leibniz's rule."

Leibniz's approach to the calculus fell well short of later standards of rigor (the same can be said of Newton's). We now see a Leibniz "proof" as being in truth mostly a heuristic hodgepodge, mainly grounded in geometric intuition and an intuitive understanding of differentials. Leibniz also freely invoked mathematical entities he called infinitesimals, manipulating them freely in ways suggesting that they had paradoxical algebraic properties. George Berkeley, in a tract called The Analyst and elsewhere, ridiculed this and other aspects of the early calculus, pointing out that natural science grounded in the calculus required just as big of a leap of faith as theology grounded in Christian revelation. Some of Berkeley's arguments are now seen as well taken.

The calculus as we now know it emerged in the 19th century, thanks to the efforts of Cauchy, Riemann, Weierstrass, and others, who based their work on a rigorous notion of limit and on a precise understanding of the real numbers. Their work banished infinitesimals into the wilderness of obsolete mathematics (although engineers, physicists, and economists continued to use them). But beginning in 1960, Abraham Robinson showed how to make sense of Leibniz's infinitesimals, and how to give them algebraic properties free of paradox. The resulting nonstandard analysis can be seen as a great belated triumph of Leibniz's mathematical and ontological intuition.

From 1711 until his death, Leibniz's life was envenomed by a long dispute with John Keill, Newton, and others, over whether Leibniz had invented the calculus independently of Newton, or whether he had merely invented another notation for ideas that were fundamental Newton's. Hall (1980) gives a thorough scholarly discussion of the calculus priority dispute.

Topology

Leibniz was the first to employ the term analysis situs (LL §27), later employed in the 19th century to refer to what is now known as topology. There are two takes on this situation. On the one hand, Mates (1986: 240), citing a 1954 paper in German by Freudenthal, argues as follows:

"Although for [Leibniz] the situs of a sequence of points is completely determined by the distance between them and is altered if those distances are altered, his admirer Euler, in the famous 1736 paper solving the Konigsberg Bridge Problem and its generalizations, used the term geometria situs in such a sense that the situs remains unchanged under topological deformations. He mistakenly credits Leibniz with originating this concept. ...it is sometimes not realized that Leibniz used the term in an entirely different sense and hence can hardly be considered the founder of that part of mathematics."

Hirano (1997) argues differently, quoting Mandelbrot (1977: 419) as follows:

"...To sample Leibniz' scientific works is a sobering experience. Next to calculus, and to other thoughts that have been carried out to completion, the number and variety of premonitory thrusts is overwhelming. We saw examples in 'packing,'... My Leibniz mania is further reinforced by finding that for one moment its hero attached importance to geometric scaling. In "Euclidis Prota"..., which is an attempt to tighten Euclid's axioms, he states,...: 'I have diverse definitions for the straight line. The straight line is a curve, any part of which is similar to the whole, and it alone has this property, not only among curves but among sets.' This claim can be proved today."

Thus Mandelbrot's well-known fractal geometry drew on Leibniz's notions of self-similarity and the principle of continuity: natura non facit saltus. We also see that when Leibniz wrote, in a metaphysical vein, that "the straight line is a curve, any part of which is similar to the whole..." he was anticipating topology by more than two centuries. As for "packing," Leibniz told to his friend and correspondent Des Bosses to imagine a circle, then to inscribe within it three congruent circles with maximum radius; the latter smaller circles could be filled with three even smaller circles by the same procedure. This process can be continued infinitely, from which arises a good idea of self-similarity. Leibniz's improvement of Euclid's axiom contains the same concept.

The dispute over who first invented the calculus

As history shows, Newton manipulated the quarrel. The most remarkable aspect of this barren struggle was that no participant doubted for a moment that Newton had already developed his method of fluxions when Leibniz began working on the differential calculus. Yet there was no proof, only Newton's word. He had published nothing but a calculation of a tangent, and the note: "This is only a special case of a general method whereby I can calculate curves and determine maxima, minima, and centers of gravity." How this was done he explained to a pupil a full twenty years later, when Leibniz's articles were already well-read. Newton's manuscripts came to light only after his death, by which time they could no longer be dated.

The infinitesimal calculus can be expressed either in the notation of fluxions or in that of differentials. Newton employed fluxions as early as 1666, but did not publish an account thereof until 1693. The earliest use of differentials in Leibniz's notebooks may be traced to 1675. He employed this notation in a 1677 letter to Newton. The differential notation also appeared in Leibniz's memoir of 1684, described below.

From the point of view of Newton's supporters, the claim that Leibniz invented the calculus independently of Newton rested on the fact that Leibniz:

• Published a description of his method some years before Newton printed anything on fluxions;
• Always alluded to the discovery as being his own invention. Moreover, this statement went unchallenged some years;
• Rightly enjoys the strong presumption that he acted in good faith.

According to them, to rebut this case it is necessary to show that he (i) saw some of Newton's papers on the subject in or before 1675 or at least 1677, and (ii) obtained the fundamental ideas of the calculus from those papers. They see the fact that Leibniz's claim went unchallenged for some years as immaterial.

That Leibniz saw some of Newton's manuscripts had always been likely. In 1849, C. J. Gerhardt, while going through Leibniz's manuscripts, found extracts from Newton's De Analysi per Equationes Numero Terminorum Infinitas (published in 1704 as part of the De Quadratura Curvarum ) in Leibniz's handwriting, the existence of which had been previously unsuspected, along with notes re-expressing the content of these extracts in Leibniz's differential notation. Hence when these extracts were made becomes all-important. It is known that a copy of Newton's manuscript had been sent to Tschirnhaus in May 1675, a time when he and Leibniz were collaborating; it is not impossible that these extracts were made then. It is also possible that they may have been made in 1676, when Leibniz discussed analysis by infinite series with Collins and Oldenburg. It is a priori probable that they would have then shown him the manuscript of Newton on that subject, a copy of which one or both of them surely possessed. On the other hand it may be supposed that Leibniz made the extracts from the printed copy in or after 1704. Shortly before his death, Leibniz admitted in a letter to Abbot Antonio Conti, that in 1676 Collins had shown him some of Newton's papers, but Leibniz also implied that they were of little or no value. Presumably he was referring to Newton's letters of 13 June and 24 October 1676, and to the letter of 10 December, 1672, on the method of tangents, extracts from which accompanied the letter of 13 June.

Whether Leibniz made use of the manuscript from which he had copied extracts, or whether he had previously invented the calculus, are questions on which no direct evidence is available at present. It is, however, worth noting that the unpublished Portsmouth Papers show that when Newton went carefully (but with an obvious bias) into the whole dispute in 1711, he picked out this manuscript as the one which had probably somehow fallen into Leibniz's hands. At that time there was no direct evidence that Leibniz had seen this manuscript before it was printed in 1704; hence Newton's conjecture was not published. But Gerhardt's discovery of the copy made by Leibniz tends to confirm the accuracy of Newton's conjecture. Those who question Leibniz's good faith allege that to a man of his ability, the manuscript, especially if supplemented by the letter of 10 December 1672, sufficed to give him a clue as to the methods of the calculus. Since Newton's work at issue did employ the fluxional notation, anyone building on that work would have to invent a notation, but some deny this.

At first, there was no reason to suspect Leibniz's good faith. True, in 1699 Duillier had accused Leibniz of plagiarizing Newton, but Duillier was not a person of consequence. It was not until the 1704 publication of an anonymous review of Newton's tract on quadrature, a review implying that Newton had borrowed the idea of the fluxional calculus from Leibniz, that any responsible mathematician doubted that Leibniz had invented the calculus independently of Newton. With respect to the review of Newton's quadrature work, all admit that there was no justification or authority for the statements made in the review, which were rightly attributed to Leibniz. But the subsequent discussion led to a critical examination of the whole question, and doubts emerged. Had Leibniz derived the fundamental idea of the calculus from Newton? The case against Leibniz, as it appeared to Newton's friends, was summed up in the Commercium Epistolicum of 1712, which referenced all allegations. That document was thoroughly machined by Newton.

No such summary (with facts, dates, and references) of the case for Leibniz was issued by his friends; but Johann Bernoulli attempted to indirectly weaken the evidence by attacking the personal character of Newton in a letter dated 7 June 1713. The charges were false. When pressed for an explanation, Bernoulli most solemnly denied having written the letter. In accepting the denial, Newton added in a private letter to Bernoulli the following remarks, Newton's claimed reasons for why he took part in the controversy. "I have never," he said, "grasped at fame among foreign nations, but I am very desirous to preserve my character for honesty, which the author of that epistle, as if by the authority of a great judge, had endeavoured to wrest from me. Now that I am old, I have little pleasure in mathematical studies, and I have never tried to propagate my opinions over the world, but I have rather taken care not to involve myself in disputes on account of them."

Leibniz explained his silence as follows, in a letter to Conti dated 9 April 1716:

"Pour répondre de point en point à l'ouvrage publié contre moi, il falloit entrer dans un grand détail de quantité de minutiés passées il y a trente à quarante ans, dont je ne me souvenois guère: il me falloit chercher mes vieilles lettres, dont plusiers se sont perdus, outre que le plus souvent je n'ai point gardé les minutes des miennes: et les autres sont ensevelies dans un grand tas de papiers, que je ne pouvois débrouiller qu'avec du temps et de la patience; mais je n'en avois guère le loisir, étant chargé présentement d'occupations d'une toute autre nature."

["In order to respond point by point to all the work published against me, I would have to go into much minutiae that occurred thirty, forty years ago, of which I remember little: I would have to search my old letters, of which many are lost. Moreover, in most cases I did not keep a copy, and when I did, the copy is buried in a great heap of papers, which I could sort through only with time and patience. I have enjoyed little leisure, being so weighted down of late with occupations of a totally different nature."]

While Leibniz's death put a temporary stop to the controversy, the debate persisted for many years.

To Newton's staunch supporters this was a case of Leibniz's word against a number of contrary, suspicious details. His unacknowledged possession of a copy of part of one of Newton's manuscripts may be explicable; but it appears that on more than one occasion, Leibniz deliberately altered or added to important documents (e.g., the letter of June 7, 1713, in the Charta Volans, and that of April 8, 1716, in the Acta Eruditorum), before publishing them, and falsified a date on a manuscript (1675 being altered to 1673). All this casts doubt on his testimony.

Several points should be noted. Considering Leibniz intellectual prowess, as demonstrated by his other accomplishments, he had more than the requisite ability to invent the calculus (which was more than ready to be invented in any case). What he is alleged to have received was a number of suggestions rather than an account of the calculus; it is possible that since he did not publish his results of 1677 until 1684 and since the differential notation was his invention, Leibniz may have minimized, 30 years later, any benefit he may have enjoyed from reading Newton's work in manuscript. Moreover, he may have seen the question of who originated the calculus as immaterial when set against the expressive power of his notation.

In any event, a bias favoring Newton tainted the whole affair from the outset. The Royal Society set up a committee to pronounce on the priority dispute, in response to a letter it had received from Leibniz. That committee never asked Leibniz to give his version of the events. The report of the committee, finding in favor of Newton, was written by Newton himself and published as "Commercium Epistolicum" (mentioned above) early in 1713. But Leibniz did not see it until the autumn of 1714. If science worked then as it does now, Leibniz would be considered the sole inventor of the calculus since he published first.

There is a sense in which Newton's "victory" was a hollow one. That victory plus nationalist bias resulted in Newton's notation standard in his country, an error that led to almost a century and a half of virtual stagnation in British mathematics.

While this controversy has been overanalyzed, Newton's proven sins have gradually come to light. For example, John Flamsteed had helped Newton with his Principia, but then witheld information from him. Newton thereupon seized all of Flamsteed's work and sought to have it published by Flamsteed's mortal enemy, Edmond Halley. But Flamsteed asked a court to block the publication of the information, and won in the nick of time. Newton then removed all mention of Flamsteed from future editions of the Principia. Even Robert Hooke, secretary to the Royal Society, contended that Newton did not invent the inverse-square force law governing planetary motion: "Newton stole the idea from me." Danton B. Sailor, in his "Newton's Debt to Cudworth" in the 1988 Journal of the History of Ideas, showed that Newton stole a theory of the origins of atomism from a Cambridge Platonist, Ralph Cudworth, instead of "turning to Nature for truth" as Newton claimed. Newton was guilty of other deplorable incidents of this nature.

The prevailing opinion in the eighteenth century was against Leibniz. Today the consensus is that Leibniz and Newton independently invented and described the calculus, and the glory of that achievement rightly redounds to both of them. "Despite... points of resemblance, the methods [of Newton and Leibniz] are profoundly different, so making the priority row a nonsense." (Grattan-Guinness 1997: 247).

Scientist and engineer

Leibniz's writings are currently discussed, not only for their anticipations and possible discoveries not yet recognized, but as ways of advancing present knowledge. Much of his writing on physics is included in Gerhardt's Mathematical Writings. His writings on other scientific and technical subjects are mostly scattered and relatively little known, because the Academy edition has yet to publish any volume in its Series Scientific, Medical, and Technical Writings .

Physics

Leibniz contributed a fair amount to the statics and dynamics emerging about him, oftten disagreeing with Descartes, Newton and their followers. He devised a new theory of motion (dynamics) based on kinetic and potential energy. While he may have been Newton's peer as co-discoverer of the calculus, he was not in Newton's league as a physicist and may even deserve to be ranked below his mentor Huygens. An important example of Leibniz's mature physical thinking is his Specimen Dynamicum of 1695. (AG 117, LL §46, W II.5) On Leibniz and physics, see the chapter by Garber in Jolley (1995).

Until the discovery of subatomic particles and the quantum mechanics governing them, many of Leibniz's speculative ideas about aspects of nature not reducible to statics and dynamics made no sense. For instance, he anticipated Einstein by arguing, against Newton, that space, time and motion are relative, not absolute. Leibniz's rule in interacting theories plays a role in supersymmetry and in the lattices of quantum mechanics. His "principle of sufficient reason" has been invoked in recent cosmology, and his "identity of indiscernibles" in quantum mechanics (Some even credit him with having anticipated this field.) A recent direction in cosmology whose advocates invoke Leibniz as a precursor is digital philosophy.

The vis viva

See main article: Conservation of energy: Historical development.

Leibniz 's vis viva (Latin for living force) is an invariant mathematical characteristic of certain mechanical systems (see AG 155-86, LL §§53-55, W II.6-7a). It can be seen as a special case of the conservation of energy. Here too his thinking gave rise to another regrettable nationalistic dispute. His "vis viva" was seen as rivaling the conservation of momentum championed by Newton in England and by Descartes in France; hence academics in those countries tended to neglect Leibniz's idea. Engineers eventually found "vis viva" useful when making certain calculations, so that the two approaches eventually were seen as complementary.

Other natural science

By proposing that the earth has a molten core, he anticipated modern geology. In embryology, he was a preformationist, but also proposed that organisms are the outcome of a combination of an infinite number of possible microstructures and of their powers. In the life sciences and paleontology, he revealed an amazing transformist and intuition, fueled by his study of comparative anatomy and fossils. He worked out a primal organismic theory. On Leibniz and biology, see Loemker (1969a: VIII). In medicine, he exhorted the physicians of his time -- with some results -- to ground their theories in detailed comparative observations and verified experiments, and to distinguish firmly scientific and metaphysical points of view.

Social science

In psychology he anticipated the distinction between conscious and unconscious states. On Leibniz and psychology, see Loemker (1969a: IX). In public health, he advocated establishing a medical administrative authority, with powers over epidemiology and veterinary medicine. He worked to set up a coherent medical training programme, oriented towards public health and preventive measures. In economic policy, he proposed tax reforms and a national insurance scheme, and discussed the balance of trade. He even proposed something akin to what much later emerged as game theory. In sociology he laid the ground for communication theory.

Technology

In 1906, Gerland published a volume of Leibniz's writings bearing on his many practical inventions and engineering work. To date, few of these writings have been translated into English. Nevertheless, it is well understood that Leibniz was a serious inventor, engineer, and applied scientist, with great respect for practical life. Following the motto theoria cum praxis, he urged that theory be combined with practical application, and thus has been claimed as the father of applied science. He designed wind-driven propellers and water pumps, mining machines to extract ore, hydraulic presses, lamps, submarines, clocks, etc. With Denis Papin, he invented a steam engine. He even proposed a method for desalinating water. He was deemed eccentric when he said that it should be possible to travel from Hannover to Amsterdam, a distance of about 300 km, in six hours. For a number of years, he struggled to overcome the chronic flooding that afflicted several ducal silver mines in the Harz Mountains, but his efforts were not crowned with success. (Aiton 1985: 107-114, 136)

Information technology

Leibniz may have been the first computer scientist and information theorist. Early in his career, he discovered the binary number system (base 2), the one subsequently employed on all computers. In 1671, he began to invent and improve a machine that could execute all four arithmetical operations, gradually improving it over a number of years. This machine attracted fair attention and was the basis of his election to the Royal Society in 1673. A number of such machines were made during his years in Hannover, by a craftsman working under Leibniz's supervision. It was not an unambiguous success, because he failed to mechanize fully the operation of carrying.

Leibniz groped towards hardware and software concepts worked out much later by Charles Babbage and Ada Lovelace, 1830-45. In 1679, while mulling over his binary arithmetic, Leibniz imagined a machine in which binary numbers were represented by marbles, governed by a rudimentary sort of punched cards.[1] Modern electronic digital computers replace Leibniz's marbles moving by gravity with shift registers, voltage gradients, and pulses of electrons, but otherwise they run roughly as Leibniz envisioned in 1679.

Leibniz anticipated Lagrangian interpolation and algorithmic information theory. His "Characteristica Universalis" anticipated the universal Turing machine. Norbert Wiener, writing in 1934, claimed that Leibniz was the first to describe the concept of feedback, central to Wiener's later cybernetic theory.

The librarian

Leibniz is one of the founding figures in library science, and was instrumental in establishing the major libraries in Hannover and Wolfenbuettel. The latter is believed to be the first building explicitly designed to be a library, and Leibniz helped design it. In his role as ducal Librarian, he is believed to have created the first index system. He also called on publishers to distribute abstracts of all new titles they produced, in a standard form that would facilitate indexing. He hoped that this abstracting project could be extended back to the dawn of printing in the 15th century. Neither proposal met with success. See here.

He called for the creation of an empirical database as a means of furthering all the sciences. Leibniz's ideas of "universal language", "reasoning calculus", and a "community of minds", which he intended primarily as means to bring political and religious unity to Europe, can be seen as a distant unwitting anticipation of artificial languages, such as Esperanto and its rivals, modern fomal logic, even the world wide web. If there is one major figure in the western tradition such that Wikipedia fulfills his fondest hopes, that figure is Leibniz.

Advocate of scientific societies

Leibniz emphasized that research was a collaborative endeavor. Hence he warmly advocated the formation of national scientific societies along the lines of the British Royal Society and the French Academie Royale des Sciences. More specifically, in his correspondence and travels he urged the creation of such societies in Dresden, Saint Petersburg, Vienna, and Berlin. Only one such project came to fruition; in 1700, the Berlin Academy of Sciences was created. Leibniz served as its first President , for life,and drew up its first statutes. That Academy evolved into the German Academy of Sciences, the publisher of the ongoing critical edition of his works.

Lawyer, Moralist, Theologian

No philosopher has ever had as much experience with practical affairs of state as Leibniz, Marcus Aurelius possibly excepted. Leibniz's writings on law, ethics, and politics (e.g., AG 19, 94, 111, 193; Riley 1988; LL §§2, 7, 20, 29, 44, 59, 62, 65; W I.1, IV.1-3) were long overlooked by English speaking scholars but this has changed of late; see Riley (1996), Jolley (2005: chpt. 7), and Gregory Brown's chapter in Jolley (1995).

Most of the secondary literature on Leibniz the theologian is in French. On Leibniz and the concept of God, see Jolley (1995: chpt. 10), by Blumenfeld. Although Leibniz's writings very freely invoked God, and discussed Christian theology with great assurance, he was never seen at Sunday services during the last two decades of his life, and declined to take Communion on his deathbed. Consequently, he was suspected of being a secret atheist by his fellow Hannoverians while alive, and by much of Europe after his death. In fact, Leibniz the Lutheran comes off as far more religious than Descartes the tepid Catholic, Spinoza the Jew expelled from his synagogue, Locke the suspected Socinian, and Hobbes the near-atheist.

Ecumenism

Leibniz devoted considerable intellectual and diplomatic effort to what would now be called ecumenical endeavor, seeking to reconcile first the Roman Catholic and Lutheran churches, later the Lutheran and Reformed churches. In this respect, he followed the example of his early patrons, Baron von Boineburg and the Duke John Frederick, both cradle Lutherans who converted to Catholicism as adults, who did what they could to encourage the reunion of the two faiths, and who warmly welcomed such endeavors by others. (The House of Brunswick remained Lutheran because the Duke's children did not follow their father.) These efforts included corresponding with the French bishop Bossuet, and involved Leibniz in a fair bit of theological controversy. He evidently thought that the thoroughgoing application of reason would suffice to heal the breach caused by the Reformation.

Philologist

Leibniz was an avid student of languages, eagerly latching on to any information about vocabulary and grammar that came his way. He refuted the belief, widely held by Christian scholars in his day, that Hebrew was the primeval language of the human race. He also refuted the argument, advanced by Swedish scholars in his day, that some sort of proto-Swedish was the ancestor of the Germanic languages. He puzzled over the origins of the Slavic languages, was aware of the existence of Sanskrit, and was fascinated by classical Chinese. Scholarly appreciation of Leibniz the philologist is hampered by the fact that the first volume of the Academy edition series "Historical and Linguistic Writings" has yet to appear.

The Sinophile

In his later years, Leibniz became perhaps the first European intellect of the first rank to take a close interest in Chinese civilization, which he knew by corresponding with, and reading other work by, European Christian missionaries posted in China. He concluded that Europeans could learn much from the Confucian ethical tradition. He mulled over the possibility that the Chinese ideograms were an unwitting form of his universal characteristic. He noted with fascination how the I Ching mapped into the binary numbers he had invented, and wrongly concluded that this mapping was evidence of major Chinese accomplishments in the sort of philosophical mathematics he admired. Leibniz's writings on Chinese civilization are collected and translated in Cook and Rosemont (1994), and studied in Perkins (2004).

Works

AG = Ariew & Garber (1989). LL = Loemker (1969). W = Wiener (1951).

The ongoing critical edition of all of Leibniz's writings is Sämtliche Schriften und Briefe.

Selected works; major ones in bold. The year shown is usually the year in which the work was completed, not of its eventual publication.

• 1666. De Arte Combinatoria (On the Art of Combination). LL §1 (part).
• 1671. Hypothesis Physica Nova (New Physical Hypothesis). LL §8.I (part)
• 1684. Nova methodus pro maximis et minimis (New Method for maximums and minimums). Struik (1969: 271-81).
• 1686. Discours de métaphysique. Martin, R.N.D., and Brown, Stuart, eds. and trans.,1988. Discourse on metaphysics and related writings. With an introduction, notes, and glossary. St. Martin's Press. Jonathan Bennett's translation. AG 35, LL §35, W III.3.
• 1705. Explication de l'Arithmétique Binaire (Explanation of Binary Arithmetic).
• 1710. Théodicée. Farrer, A.M., and Huggard, E.M., trans. and eds., 1985 (1952). Theodicy. Open Court. Project Gutenberg. W III.11 (part).
• 1714. Monadologie. Nicholas Rescher, trans., 1991. The Monadology: An Edition for Students. Uni. of Pittsburg Press. Jonathan Bennett's translation. AG 213, LL §67, W III.13.
• 1765. Nouveaux essais sur l'entendement humain. Completed 1704. Remnant, Peter, and Bennett, Jonathan, trans., 1996. New Essays on Human Understanding. Cambridge Uni. Press. W III.6 (part).

Collections of shorter works in translation:

• Ariew, R., and Garber, D., 1989. Leibniz: Philosophical Essays. Hackett.
• Cook, Daniel, and Rosemont, Henry Jr., 1994. Leibniz: Writings on China. Open Court.
• Dascal, Marcelo, 1987. Leibniz: Language, Signs and Thought. John Benjamins.
• Loemker, Leroy E., 1969 (1956). Leibniz: Philosophical Papers and Letters. Reidel.
• Parkinson, G.H.R., 1966. Leibniz: Logical Papers. Oxford Uni. Press.
• Riley, Patrick, 1988 (1972). Leibniz: Political Writings. Cambridge Uni. Press.
• Rutherford, Donald. Online.
• Strickland, Lloyd, 2006. Shorter Leibniz Texts. Continuum Books. Online.
• Struik, D. J., 1969. A Source Book in Mathematics, 1200-1800. Harvard Uni. Press.
• Wiener, Philip, 1951. Leibniz: Selections. Scribner. Regrettably out of print.
• Woolhouse, R.S., and Francks, R., 1997. Leibniz's New System and Associated Texts. Oxford Uni. Press.

Donald Rutherford's online bibliography.

Secondary literature

• Adams, Robert M., 1994. Leibniz: Determinist, Theist, Idealist. Oxford Uni. Press.
• Aiton, Eric J., 1985. Leibniz: A Biography. Hilger (UK).
• Burkhardt, Hans, 1980. Logik und Semiotik in der Philosophie von Leibniz. Philosophia Verlag.
• Hall, A. R., 1980. Philosophers at War: The Quarrel between Newton and Leibniz. Cambridge Uni. Press.
• Hirano, Hideaki, 1997, "Cultural Pluralism And Natural Law," Self Published.
• Hostler, J., 1975. Leibniz's Moral Philosophy. UK: Duckworth.
• Ishiguro, Hide, 1990 (1972). Leibniz's Philosophy of Logic and Language. Cambridge Uni. Press.
• Jolley, Nicholas, ed., 1995. The Cambridge Companion to Leibniz. Cambridge Uni. Press.
• Jolley, Nicholas, 2005. Leibniz. Routledge.
• LeClerc, Ivor, ed., 1973. The Philosophy of Leibniz and the Modern World. Vanderbilt Uni. Press.
• Lenzen, Wolfgang, 2004. "Leibniz's Logic," in Gabbay, D., and Woods, J., eds., Handbook of the History of Logic, Vol. 3. North Holland: 1-84. Online.
• Loemker, Leroy, 1969a, "Introduction" to his Leibniz: Philosophical Papers and Letters. Reidel: 1-62.
• MacDonald Ross, George, 1984. Leibniz. Oxford Uni. Press.
• ------, 1999, "Leibniz and Sophie-Charlotte" in Herz, S., Vogtherr, C.M., Windt, F., eds., Sophie Charlotte und ihr Schloß. München: Prestel: 95–105. English translation.
• Mates, Benson, 1986. The Philosophy of Leibniz : Metaphysics and Language. Oxford Uni. Press.
• Mercer, Christia, 2001. Leibniz's metaphysics : Its Origins and Development. Cambridge Uni. Press.
• Perkins, Franklin, 2004. Leibniz and China: A Commerce of Light. Cambridge Uni. Press.
• Riley, Patrick, 1996. Leibniz's Universal Jurisprudence: Justice as the Charity of the Wise. Harvard Uni. Press.
• W. W. Rouse Ball, 1908. A Short Account of the History of Mathematics, 4th ed. (see Discussion)
• Zalta, E. N., 2000, "A (Leibnizian) Theory of Concepts," Philosophiegeschichte und logische Analyse / Logical Analysis and History of Philosophy'3: 137-183.

Gregory Brown's online bibliography.

Other works cited

• Ivor Grattan-Guinness, 1997. The Norton History of the Mathematical Sciences. W W Norton.
• Benoit Mandelbrot, 1977. The Fractal Geometry of Nature. Freeman.
• Simon Conway Morris, 2003. Life's Solution: Inevitable Humans in a Lonely Universe. Cambridge Uni. Press.
• Ward, P. D., and Brownlee, D., 2000. Rare Earth: Why Complex Life is Uncommon in the Universe. Springer Verlag.

Quotes

More quotes. Wiener (1951: 567-70) lists 44 quotable "proverbs" beginning with "Justice is the charity of the wise."

• "In the realm of spirit, seek clarity; in the material world, seek utility." Mates's (1986: 15) translation of Leibniz's motto.
• "The monad... is nothing but a simple substance which enters into compounds. Simple means without parts... Monads have no windows through which anything could enter or leave." Monadology (LL §67.1,7)
• "...no matter how God might have created the world, it would always have been regular and in a certain general order. But God has chosen that world which is the most perfect, that is to say, which is at the same time the simplest in its hypotheses and the richest in phenomena..." Discourse on Metaphysics (LL §35.6)
• "I am convinced that the unwritten knowledge scattered among men of different callings surpasses in quantity and in importance anything we find in books, and that the greater part of our wealth has yet to be recorded.... The chief defect of many scholars is that they occupy themselves only with vague and well-worn arguments when there are so many opportunities for exercising their minds on solid and real objectives, to the advantage of the public. Hunters, fishermen, merchants, sea voyagers, and even games of skill as well as of chance, furnish material with which to augment substantially the useful sciences.Even in the games of children there are things to interest the greatest mathematician." (W I.7. Translation revised.)
• "As to that great question of the power of sovereigns and the obedience the people owe them, I am wont to say that it would be good for princes to be convinced that the people have a right to resist them. The people, on the other hand, should be persuaded to obey passively. I share enough of Grotius's opinions, however, to believe that the people should obey as a rule, the harm stemming from revolution being incomparably greater than what provokes it." (1695 letter to von Boineburg, quoted in Wiener 1951: xlviii. Translation revised.)
• "I maintain that men could be incomparably happier than they are, and that they could, in a short time, make great progress in increasing their happiness, if they were willing to set about it as they should. We have in hand excellent means to do in 10 years more than could be done in several centuries without them, if we apply ourselves to making the most of them, and do nothing else except what must be done." (Translated in Riley 1972: 104, and quoted in Mates 1986: 120)

See also

• monad
• characteristica universalis
• Calculus ratiocinator
• Leibniz-Gemeinschaft
• Leibniz formula
• digital philosophy

External links

• Online translations of a number of Leibniz texts, by Jonathan Bennett.
• Monadologie, in German.
• Donald Rutherford's work in progess for his translation of Couturat's The Logic of Leibniz.
• Leibnitiana -- Gregory Brown.
• Lloyd Strickland's web page. Scroll down for many Leibniz links.
• Table of contents for the Leibniz Review, 1998-.
• Leibniz and the English-Speaking World (list of abstracts)
• European Graduate School - Gottfried Leibniz.
• O'Connor, John J.; Robertson, Edmund F., "Gottfried Wilhelm Leibniz", MacTutor History of Mathematics Archive, University of St Andrews
• Leibniz biography and bibliography.
• The Internet Encyclopedia of Philosophy: Leibniz -- Douglas Burnham.
• Stanford Encyclopedia of Philosophy:
  • Leibniz: Ethics -- Andrew Youpa.
  • Leibniz: Causation -- Mark Bobro.
  • Leibniz: Problem of evil -- Michael Murrary.
  • Leibniz: Philosophy of mind -- Kulstad and Carlin.
• Encyclopedia Britannica, 11th ed.