Gottfried Leibniz: Encyclopedia - Gottfried Leibniz

Gottfried Leibniz

Gottfried Wilhelm von Leibniz (also Leibnitz) (July 1 (June 21 Old Style) 1646, Leipzig – November 14, 1716, Hanover) was a German polymath, deemed a genius in his lifetime and since, and the last true polyhistor. Trained as a lawyer and active as a diplomat and librarian, he wrote on philosophy, science, mathematics, theology, history, and comparative philology, even writing verse. Through his service to two major German noble houses, he played a major role in the European politics and diplomacy of his day. He even wrote verse.

Leibniz was, along with Rene Descartes and Baruch Spinoza, one of the three great 17th century rationalists, all versed in mathematics as well as philosophy. He occupies an equally large place in both the history of philosophy and the history of mathematics. In philosophy, he is most remembered for his metaphysical notion of monad. In mathematics, he is credited with coining the term "function". He invented the calculus independently of Newton, and his notation is the one in general use since. He was the greatest logician between the Ancients and Boole and De Morgan, although he published little in the area.

He made major contributions to physics and technology,was the first to conceptualize applied science and his writings anticipate notions that surfaced much later in biology, geology, communication theory, 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. Many of his ideas were too radical for his own age, and were taken up only much later – sometimes not until the present century. As with all great philosophers, his work no doubt contains hitherto unrecognised potential. He was also a prolific inventor of the highest caliber.

Leibniz only published two books in his lifetime. His contributions to a vast array of subjects were scattered in journals, many thousands of letters and memoranda, and in a huge collection of unpublished manuscripts, mostly preserved in the Lower Saxony State Library in Hannover. To date, there is no complete edition of Leibniz's writings.

Leibniz's writings also formed the basis for many ideas of the American revolutionaries, as embodied in the Declaration of Independence and the U.S. Constitution, specifically the notions of "the general welfare" and "the pursuit of happiness."

Gottfried Leibniz - Life

The only biography in English is Aiton (1986). Also see Jolley (2005: chpt. 1), the references therein, and the excerpt from Rouse Ball (1908) bearing on Leibniz.

Gottfried Leibniz - Early life and education

Leibniz's parents were Friedrich Leibniz and Catharina Schmuck. His father, a Professor of Moral Philosophy at the University of Leipzig, died in Leibniz's sixth year. From age 8 on, Leibniz was granted free access to his late father's library. By 12, he had taught himself Latin and had begun Greek. He entered his father's university at 15. By 20, he had also studied at Jena, mastered the standard texts of his day and place on philosophy, theology, and law, and published his first book titled The Combinatorial Art. When Leipzig refused him the doctor of laws degree, allegedly because his teachers were jealous of his youth and genius, Leibniz simply went to the University of Altdorf near Nuremberg,and obtained his degree in five months, submitting the thesis he had intended to submit in Leipzig. Declining the offer of an academic appointment at Altdorf, he spent the rest of his life in the service of two major German noble families.

Gottfried Leibniz - Career

Leibniz's early years post-graduation are a bit confused. He obtained a salaried position as a Nuremberg alchemist, even though he knew nothing about the subject. He met J. C. von Boineburg, the exiled and disgraced former minister of the Elector of Mainz, who soon reconciled with his master and thus introduced Leibniz to the Elector. Leibniz published an essay on the teaching of law, dedicating it to the Elector in the hope of obtaining employment. The stratagem worked; the Elector asked Leibniz to assist with the redrafting of the local legal code. Leibniz remained employed as assistant to von Boineburg in various capacities.

Leibniz's job evolved into a diplomatic one. He published an essay, purported to have been written by a fictitious Polish nobleman, arguing (unsuccessfully) for the German candidate for the Polish crown. Leibniz then drew up a plan urging France to take Egypt and use it as a stepping stone for the conquest of the Dutch East Indies. In return, France would agree to leave Germany undisturbed. In 1672, the French government invited Leibniz to Paris for discussion, but the plan was never adopted.

Thus Leibniz began several years in Paris, where he greatly expanded his knowledge of the mathematics and physics of his day, and to begin adding to both subjects. Especially fateful was his meeting Christiaan Huygens, the Dutch physicist and mathematician. Conversations with Huygens moved Leibniz to study geometry. Although Leibniz had previously written on mathematics, especially his 1666 book containing much on combinatorics, and had designed and built a machine for doing arithmetic, he described his study of geometry as having opened a new world to him.

In January, 1673, he was sent on a political mission to London, where he stayed some months, making the acquaintance of, among others, Henry Oldenburg and John Collins. He demonstrated his calculating machine to the Royal Society, whereupon it granted him an external membership.

In 1673 the Elector of Mainz died, and in the following year Leibniz entered the service of the Brunswick family. While the prince sovereign of that family was merely a Duke, who was not elevated to the title of Elector of the Holy Roman Empire until 1692, it was nevertheless quite an honor to serve the Brunswicks as Leibniz did. In 1676, he was appointed Court Councillor and visited London, where he was arguably allowed to read some of Newton's unpublished work. This suppoused favor came back to haunt him in his last years. On his return from London, he spent a few weeks in intense discussion with Spinoza. Leibniz resided in Hannover for the balance of his days, where he was, among other things, the paid librarian of the ducal library. He thenceforth employed his pen on all the various political, historical, and theological matters involving the House of Hanover over the 40 years 1673-1713; the resulting documents form a valuable addition to the historical record of that time.

While serving the Brunswicks, Leibniz was allowed to devote more time to other pursuits. He later asserted that his invention of the differential and integral calculus in 1674 was the first fruit of this decrease in working hours. By 1677, this invention had evolved into a coherent system, although he would not publish it until 1684. (The earliest evidence of its use in his surviving notebooks is 1675.) Most of his mathematical papers were written between 1682 and 1692, many published in the widely read journal Acta Eruditorum, which he and Otto Mencke founded in 1682. His reputation as a mathematician and scientist ("natural philosopher" in those days), on the one hand, and his eminence in diplomacy, history, theology, and philology, on the other, mutually reenforced each other, with the Acta Eruditorum often playing a central role.

Over the period 1687-1690, Leibniz travelled in Germany, Austria, and Italy, seeking archival materials bearing on a big project his master had asked of him, a history of the House of Brunswick going back, ideally, to the fall of the Roman Empire. In 1700 and at Leibniz's suggestion, the Academy of Berlin was created. He served as its first President and drew up its first statutes.

In 1711, John Keill, writing in a British journal, accused Leibniz of having plagiarized Newton's calculus. Thus began the dispute which was to darken the rest of his life. In 1712, he began a two year residence in Vienna, where he was appointed Imperial Court Councillor to the Hapsburgs. His services to the Brunswicks and Hapsburgs were recognized by honours and distinctions of various kinds. In his later years, he often preceded his surname with "von", 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 Leibniz had ever been ennobled in any way (Aiton 1985: 312).

Leibniz nearly accomplished one of the greatest political coups in history. Through meticulous historical researches, Leibniz had established the claim of his student and patroness, the Electress Sophie of Hannover, to the English throne. With the help of Leibniz's political allies in England, led by Robert Harley, Jonathan Swift, Daniel DeFoe, and Anthony Ashley Cooper (the Third Earl of Shaftesbury), Sophie's claim was made law in the 1701 Act of Succession. Because Queen Anne was childless, Sophie was set to become Queen of England at Anne's death, and Leibniz himself was to be the real power behind the throne. Unfortunately for Leibniz, Sophie died and in 1714, the Elector of Hanover succeeded to the throne of England, as George I. Leibniz was not asked to follow the Elector to London, probably because Newton, whose standing in British official circles could not be higher, was seen as having won the priority dispute over the invention of the calculus. Leibniz had evidently fallen out of favour with the Elector, and so spent his final years in neglect. Leibniz, who never married, was survived by his sister's only child.

Leibniz was reportedly overfond of money and personal distinctions, even though he had at some times in his career both. He could be unscrupulous on occasion, as was all too often the case for professional diplomats of the time. On the other hand, he was charming and well-mannered, with many friends and admirers all over Europe.

Gottfried Leibniz - Writings

A good brief annotated partial bibliography in English of Leibniz's published writings is Loemker (1969: 63-69). Leibniz wrote in three languages: the scholastic Latin of his day, French, and (least often) German. Consequently, his fellow Germans do not enjoy a comparative advantage in Leibniz studies, and the French are well represented among Leibniz scholars. During his lifetime, he published many articles in scholarly journals, but only two books, the Combinatorial Art and the Théodicée. The latter and his Nouveaux essais sur l'entendement humain are his only books that are clearly more than mere lengthy essays. Two good collections of English translations are Wiener (1951) and Loemker (1969).

The critical edition of Leibniz's works [1], begun in 1923 and still incomplete, is organized as follows:

• Series 1. Political, Historical, and General Correspondence. 20 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. 5 vols., 1667-98.
• Series 5. Historical and Linguistic Writings. In preparation.
• Series 6. Philosophical Writings. 7 vols., 1663-90, and Nouveaux essais sur l'entendement humain.
• Series 7. Mathematical Writings. 5 vols., 1672-76.
• Series 8. Scientific, Medical, and Technical Writings. In preparation.

Only 22 of these volumes were published before 1990, and only Series 1 saw additions between 1931 and 1962. The table of contents for each volume in Series 1, 3, and 7 is available online. Some of this edition is available online, gratis.

Bodemann completed his catalogs of Leibniz's manuscripts and correspondence in 1895. Only then could one begin to appreciate Leibniz's enormous Nachlass. Only then did it become known that Leibniz wrote about 15,000 letters to more than 1000 correspondents. Moreover, quite a few of Leibniz's "letters" are in fact essays thousands of words long. Much of this vast correspondence was published only in recent decades, and many letters dated later than 1685 remain unpublished.

Gottfried Leibniz - 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 did Leibniz's reputation far more harm than good. 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 publication of the Nouveaux Essais, which Kant read closely, in 1635. 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 that, while debatable in its particulars, drew Anglo-American attention to Leibniz. Shortly thereafter, Louis Couturat published an important study of Leibniz as logician, and edited a volume of Leibniz's heretofore unpublished writings on logic. Russell and Couturat did much to make Leibniz somewhat respectable among 20th century analytical and linguistic philosophers. Nevertheless, the secondary literature on Leibniz did not really blossom until after WWII, and his reputation is perhaps higher now than at any time since he was alive. American Leibniz studies owe much to Leroy Loemker; see, e.g., his (1969).

Gottfried Leibniz - Philosopher

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 (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" (LL 47, W II.4). I 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 lost his desire to publish it upon learning of Locke's 1704 death, so that the New Essays were not published until 1765. The only book on philosophy he published in his lifetime, the Théodicée, appeared in 1710. The Monadologie, composed in 1714 and published posthumously, consists of 90 aphorisms. Opinions vary about the extent to which he may have appropriated the ideas of Spinoza, with whom he had intense discussions in 1676, although most now believe that his views were original. For a good survey of Leibniz's philosophy, see Loemker (1969a).

Gottfried Leibniz - Metaphysics

Leibniz's best known contribution to metaphysics is his theory of monads, as exposited in his Monadologie. Monads are the ultimate elements of the universe, and individual percipient entities. According to Leibniz, monads are centres of force; substance is force, while space, matter, and motion are merely phenomenal. The existence of God is inferred from the harmony prevailing among monads. The monads are "substantial forms of being" with the following properties: they are eternal, indecomposable, individual, following their own laws, un-interacting, and each reflecting the entire universe in pre-established harmony (a historically noteworthy expression of panpsychism). Monads are purported to solve the problem of the interaction between mind and matter that arises in the systems of René Descartes. This notion also solves a problematic individuation the systems of Baruch Spinoza, which represent individual creatures as mere accidental modifications.

Gottfried Leibniz - 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.

Gottfried Leibniz - 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.

Gottfried Leibniz - 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 that has been prevalent in academic philosophy for most of the twentieth century came about from work in analytical philosophy and mathematical logic. In these traditions Leibniz's Calculus Ratiocinator is usually called "symbolic logic". In symbolic logic 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 own symbolism was meant to be a calculus ratiocinator as well as a lingua characteristica. Traditions associated with Frege's work tend to hold to this view of Leibout bniz's Calculus Ratiocinator.

In contrast is a view that has little prevalence in academic philosophy and came about from work in synthetic philosophy and electronic engineering. This view sees Leibniz's Calculus Ratiocinator as a computing machine. From this perspective the Calculus Ratiocinator is a central processing unit, an actual physical mechanism used to calculate the various ratios of integral and differential calculus. As a consequence we might view the Universal Characteristic as a universal symbolism helping us depict the mathematics of the qualitative flows and transformations of our cosmos, and the Calculus Ratiocinator provides the means of calculating 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.

Gottfried Leibniz - 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 the first principle, the number of simple ideas is much greater than Leibniz thought. As for the second principle, logic can indeed be grounded in a symmetrical combining operation, but that operation is analogous to one of addition or multiplication. Logic also requires unary negation.

Leibniz published little on formal logic in his lifetime, and nearly everything he wrote on the topic consists of working drafts found in his Nachlass. The subsequent logical work 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, some of which Clarence Irving Lewis (1918) and Parkinson (1966) translated into English. Despite this attention directed to Leibniz's logical work (also see Nicholas Rescher 1954), we owe our present understanding of Leibniz as logician mainly to the work of Wolfgang Lenzen, beginning around 1980. For a summary, see his (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 Lewis began.

Gottfried Leibniz - Mathematician

To Leibniz we owe the term "function", which he first employed in 1694 to describe a quantity related to a curve, such as a curve's slope or a specific point on the curve.

Leibniz's approach to the calculus was far from rigorous (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. Moreover, Leibniz's calculus 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, Dedekind and others, who based their work on a rigorous notion of limit and on a precise understanding of the real numbers. Their work banished the notion of infinitesimal into the wilderness of obsolete mathematics (although engineers, physicists, and economists continued to use it). But beginning in 1960, Abraham Robinson showed how to make rigorour sense of Leibniz's notion of infinitesimal, and how they could be given algebraic properties free of paradox. The resulting nonstandard analysis can be seen as a great belated triumph of Leibniz's joint mathematical and metaphysical intuition.

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 above under "Philosopher".

Gottfried Leibniz - 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) argues that there is no relation between what Leibniz meant by "analysis situs" and what we now know as topology. Hirano (1997) argues differently, quoting Mandelbrot 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." (Mandelbrot, B., 1977. The Fractal Geometry of Nature. Freeman: 419)

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.

Gottfried Leibniz - The dispute over who first invented the calculus

A recent thorough scholarly discussion of the calculus priority dispute is Hall (1980). The principle source for the balance of this section is Rouse Ball (1908).

Leibniz is credited along with Isaac Newton with inventing the infinitesimal calculus in the 1670s. According to Leibniz's notebooks, a critical breakthrough in his work occurred on November 11, 1675, when he demonstrated integral calculus for the first time to find the area under the function y = x. He introduced several notations used in calculus 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.

The last years of his life — from 1709 to 1716 — were embittered by a long controversy with John Keill, Newton, and others. The question was whether Leibniz had discovered differential calculus independently of Newton's previous investigations, or whether he had derived the fundamental idea from Newton and merely invented another notation for it. As history shows, the quarrel was manipulated by Newton. The most remarkable aspect of the whole barren struggle was this: no participant doubted for a moment that Newton had already developed his method of fluxions when Leibnitz began work 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 Leibnitz's textbooks were widely circulated. His own manuscripts came to light only after his death, and then they could no longer be dated.

The ideas of infinitesimal calculus can be expressed either in the notation of fluxions or in that of differentials. The former was used by Newton in 1666, but no distinct account of fluxions was printed until 1693. The earliest use of differentials in the notebooks of Leibniz may be traced to 1675. This notation was employed in the letter sent to Newton in 1677; the differential notation also appears in the memoir of 1684 described below.

From the point of view of Newton's supporters, the case in favour of the independent invention by Leibniz rested on the fact that he published a description of his method some years before Newton printed anything on fluxions, that he always alluded to the discovery as being his own invention, and that for some years this statement was unchallenged; while of course there must be a strong presumption that he acted in good faith. According to them, to rebut this case it is necessary to show (i) that he saw some of Newton's papers on the subject in or before 1675, or at least 1677, and (ii) that he thence derived the fundamental ideas of the calculus. The fact that his claim was unchallenged for some years is, in the particular circumstances of the case, immaterial.

That Leibniz saw some of Newton's manuscripts was always intrinsically probable; but when, in 1849, C. J. Gerhardt examined Leibniz's papers he found among them a manuscript copy of extracts from Newton's De Analysi per Equationes Numero Terminorum Infinitas (which was printed in the De Quadratura Curvarum in 1704) in Leibniz's handwriting, the existence of which had been previously unsuspected, together with the notes on their expression in the differential notation. The question of the date at which these extracts were made is therefore all important. It is known that a copy of Newton's manuscript had been sent to Tschirnhausen in May, 1675, and as in that year he and Leibniz were engaged together on a piece of work, it is not impossible that these extracts were made then. It is also possible that they may have been made in 1676, as Leibniz discussed the question of analysis by infinite series with Collins and Oldenburg in that year. It is a priori probable that they would have then shown him the manuscript of Newton on that subject, a copy of which was possessed by one or both of them. On the other hand it may be supposed that Leibniz made the extracts from the printed copy in or after 1704. Leibniz, shortly before his death, admitted in a letter to Abbot Antonio Conti, that in 1676 Collins had shown him some of Newton's papers, but implied that they were of little or no value. Presumably he referred 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 no use of the manuscript from which he had copied extracts, or whether he had previously invented the calculus, are questions on which at this time no direct evidence is available. It is, however, worth noting that the unpublished Portsmouth Papers show that when, in 1711, Newton went carefully (and with an obvious bias favoring him) into the whole dispute, he picked out this manuscript as the one which had probably somehow fallen into the hands of Leibniz. At that time there was no direct evidence that Leibniz had seen this manuscript before it was printed in 1704, and accordingly Newton's conjecture was not published; but Gerhardt's discovery of the copy made by Leibniz tends to confirm the accuracy of Newton's judgment in the matter. It is said by those who question Leibniz's good faith that to a man of his ability, the manuscript, especially if supplemented by the letter of 10 December 1672, would supply sufficient hints to give him a clue as to the methods of the calculus. Though as the fluxional notation is not employed in it, anyone who used it would have to invent a notation; but this is denied by others.

There was at first no reason to suspect the good faith of Leibniz. It was not until the appearance in 1704 of an anonymous review of Newton's tract on quadrature, in which it was implied that Newton had borrowed the idea of the fluxional calculus from Leibniz, that any responsible mathematician questioned the statement that Leibniz had invented the calculus independently of Newton. While Duillier had accused Leibniz, in 1699, of plagiarism from Newton, Duillier was not a person of consequence. With respect to the review of Newton's quadrature work, it is universally admitted that there was no justification or authority for the statements made in the review, which was rightly attributed to Leibniz. But the subsequent discussion led to a critical examination of the whole question, and doubt was expressed. 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 --which was thoroughly machined by Newton, as we shall see -- issued in 1712, and references are given for all the allegations made.

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, which are interesting as giving Newton's account of suppousedly why he was induced to take any 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's defense or explanation of his silence is given in the following 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 published works against me, I would have to investigate in great detail the past thirty to forty years, of which I remember little: I would have to search my old letters, of which many are lost, furthermore I mostly didn't regard the moment in time: the others are buried in a great heap of papers, which I could unravel only with patience and time: but I don't have enough leisure time, since I have been entrusted at present with an occupation of a totally different kind."]

While Leibniz's death put a temporary stop to the controversy, bitter 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 allegedly 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 that a material date in a manuscript was allegedly falsified (1675 being altered to 1673), casts doubt on his testimony. Several points should be noted: what Leibniz is alleged to have received was a number of suggestions rather than an account of the calculus; it is possible that since Leibniz did not publish his results of 1677 until 1684 and since the differential notation and its subsequent development were all of his own invention, Leibniz may have been led, thirty years later, to minimize any assistance which he had obtained originally, and finally to recognize the question is somewhat immaterial when set against the expressive power of calculus itself. Nevertheless, it is important to remember that the whole dispute was tainted with a bias for Newton, for example: In response to a letter the Royal Society set up a committee to pronounce on the priority dispute. It was totally biased, not asking 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 (as already mentioned) near the beginning of 1713 but not seen by Leibniz until the autumn of 1714. If science (natural philosophy) then were handled like now, Leibniz would be considered the sole inventor of the calculus since he published first. The ideological bias favoring England made Newton's notation standard in his country an error that cost them almost a century and a half of virtual stagnation in mathematics. Considering Leibniz intellectual prowess (as proven by his other accomplishments) he had a vastly higher potential than that necessary to invent the calculus (which many consider to have been more than ready to be invented). While this controversy has been overanalized, Newton's proven sins are just surfacing, for example, John Flamsteed had helped Newton out on his Principia Mathematica, but then witheld information from him. Newton then seized all of Flamsteed's work and had it published by Flamsteed's mortal enemy, Edmond Halley. But in the nick of time Flamsteed won in court, against the publication of the information. Because of this Newton had him excised from future editions of the Principia. Even Robert Hooke, secretary to the Royal Society, contended that Newton did not himself invent the notion that an inverse-square force law governs planetary motion. "Newton stole the idea from me," he said. Also the article "Newton's Debt to Cudworth" by Danton B. Sailor in Journal of the History of Ideas, 1988, proves 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 made people think he did. Unfortunately, there are many instances like this.

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.

Gottfried Leibniz - Science and technology

Leibniz's contributions are currently discussed seriously in many fields at the forefront of science not only as anticipations and possible discoveries not yet recognized, but as useful intellectual tools for advancing knowledge. Until the discovery of subatomic particles and the quantum mechanics governing them, many of Leibniz's speculative ideas made no sense. Leibniz devised a new theory of motion (dynamics) based on kinetic and potential energy. He anticipated Einstein by arguing, against Newton, that space, time and motion were relative, not absolute. Leibniz's rule in interacting theories plays a role in supersymmetry on lattices (quantum mechanics). His principle of "sufficient reason" has been invoked in recent discussions of cosmology, and his "identity of indiscernibles" in quantum mechanics (Some key figures even credit him with having anticipated this field.) By proposing that the earth has a molten core, he anticipated modern geology.

In medicine, he exhorted the physicians of his time - with some results - to ground their theories in detailed comparative observations and verified experiments, firmly distinguishing scientific and metaphysical points of view. In embryology, he proposed that organisms are the outcome of the combination of infinite series of microstructures and of their powers. In the life sciences, he revealed an amazing transformist and paleontological intuition, fueled by his study of comparative anatomy and fossils. He worked out a primal organismic theory. In psychology he anticipated the distinction between conscious and unconscious states. In public health, he worked to establish 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.

He was fascinated by the application of technology to solve practical problems. Following the motto "theoria cum praxis", Leibniz pleaded for combining theory with its practical application applied science. He designed wind-driven propellers, mining machines to extract ore, hydraulic presses, windmills, lamps, submarines, clocks etc., and with Denis Papin, even invented a steam engine. He even proposed a method for desalinating water. He was thought eccentric when he called for the invention of a form of public transport that would reduce the travel time from Hannover to Amsterdam, a distance of about 300 km, to six hours. He served as technical advisor to the ducal silver mines in the Harz Mountains (Aiton 1985: 107-114, 136), which were subject to chronic flooding. His efforts here were not crowned with success.

Gottfried Leibniz - The vis viva

During 1676 to 1689, Leibniz noticed what he called the vis viva (Latin for living force), an invariant mathematical characteristic of certain mechanical systems. Though it is now recognized that Leibniz had discovered a special case of the conservation of energy, his thinking here led him into another regrettable nationalistic dispute. It appeared at the time that "vis viva" was at variance with the conservation of momentum championed by Newton in England and by René Descartes in France. This led to the neglect of his idea by academics in those countries. Engineers eventually demonstrated the usefulness of "vis viva" in making their calculations. It was subsequently appreciated that the two approaches are complementary.

Gottfried Leibniz - 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. Around 1670, he began to invent and improve a machine that could execute all four arithmetical operations. This machine attracted fair attention and was the basis of his election to the Royal Society in 1673.

It can be argued that Leibniz was groping towards hardware and software concepts worked out by Charles Babbage and Ada Lovelace, 1830-45. In 1679, while mulling over his of binary arithmetic, Leibniz imagined a machine in which binary numbers were represented by spherical pellets, governed by a rudimentary sort of punched cards.

"This [binary] calculus could be implemented by a machine (without wheels), in the following manner, easily to be sure and without effort. A container shall be provided with holes in such a way that they can be opened and closed. They are to be open at those places that correspond to a 1 and remain closed at those that correspond to a 0. Through the opened gates small cubes or marbles are to fall into tracks, through the others nothing. It [the gate array] is to be shifted from column to column as required."(De Progressione Dyadica, Pars I, ms dated 15 March 1679. Published in facsimile, with German translation, in Erich Hochstetter and Hermann-Josef Greve, eds., 1966. Herrn von Leibniz: Rechnung mit Null und Einz. Berlin: Siemens Aktiengesellschaft: 46-47. English translation by Verena Huber-Dyson, 1995.)

In the shift registers at the heart of all electronic digital computers, voltage gradients and pulses of electrons have taken the place of gravity and marbles, but otherwise things run roughly as Leibniz envisioned in 1679.

Leibniz anticipated Lagrangian interpolation, algorithmic information theory and digital philosophy . 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.

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.

Leibniz warmly advocated the formation of national scientific societies not only along the lines of the two of which he was a member, the British Royal Society and the French Academie Royale des Sciences but with many grandiose ideas like the use of theoretical science for the creation of technology (he was the first to propose applied science ). 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 and drew up its first statutes.

Gottfried Leibniz - Philology

Leibniz refuted the belief, widely held at the time, that Hebrew was the primeval language of the human race.

Gottfried Leibniz - 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), for an interpretation, see Perkins (2004).

Gottfried Leibniz - Works

LL = Loemker (1969). W = Wiener (1951).

The ongoing complete critical edition 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).
• 1684. Nova methodus pro maximis et minimis (New Method for maximums and minimums).
• 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. 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. 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.
• Strickland, Lloyd, 2006. Shorter Leibniz Texts. Continuum Books. Online.
• Wiener, Philip, 1951. Leibniz: Selections. Scribner. Regrettably out of print.
• Woolhouse, R.S., and Francks, R., trans. and eds., 1997. Leibniz's New System and Associated Texts. Oxford Uni. Press.

Gottfried Leibniz - Secondary literature

• 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.
• 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.
• 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.
• W. W. Rouse Ball, 1908. A Short Account of the History of Mathematics, 4th ed. (see Discussion)

Gottfried Leibniz - Quotes

"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 go out." Monadology (LL 67.1,7) "In whatever manner God created the world, it would always have been regular and in a certain general order. God, however, has chosen the most perfect, that is to say, the one which is at the same time the simplest in hypothesis and the richest in phenomena."

See also

• Leibniz-Gemeinschaft
• Leibniz formula
• digital philosophy

Adapted from the Wikipedia article "Gottfried Leibniz", under the G.N U Free Docmentation License. Please also see http://en.wikipedia.org/wiki