Gottfried Wilhelm Leibniz

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Gottfried Wilhelm von Leibniz
Era17th-century philosophy
(Modern Philosophy)
RegionWestern Philosophers
SchoolContinental rationalism
Main interests
Metaphysics, Epistemology, Science, Mathematics
Notable ideas
Calculus (along with Newton), innate knowledge, monads

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. He is considered the last true polyhistor, as he mastered nearly all knowledge extant in his time. Trained as a lawyer and active as a diplomat and librarian, he wrote on philosophy, science, mathematics, theology and wrote poetry. He occupies an equally large place in both the history of philosophy and the history of mathematics. Leibniz was instrumental in establishing major libraries in Hanover and Wolfenbttel and scientific societies in Berlin and Vienna. He was fascinated by the application of technology to the solution of both practical and theoretical problems. His working model of an arithmetical calculating machine attracted widespread attention and was the basis of his election to the Royal Society in 1673. He worked on design of hydraulic presses, windmills, lamps, submarines, clocks etc. He even proposed a method for desalination of water. He is credited with coining the term "function" (1694), which he used to describe a quantity related to a curve, such as a curve's slope or a specific point on the curve. He invented the calculus independently of Newton and is credited with the integral and the product rule. Leibniz's notation for the calculus is the one now in general use. As a child he discovered the binary number system (base 2) the one now used on computers. He proposed the basis for the branch of mathematics now known as general topology and the basis for the modern branch of mathematics known as symbolic logic.

His concept of vis viva anticipated the later idea of conservation of energy. He developed a new classical theory of motion based on kinetic and potential energy (dynamics). He is one of the founders of modern science of geology by proposing that the earth had a molten core and made major discoveries in historiography, legal theory, and comparative philology. He anticipated Einstein by arguing against Newton that space, time and motion were not absolute, but relative. Leibniz rule in interacting theories is considered important in supersymmetry on lattices (quantum mechanics).His principle of "sufficient reason" is discussed even today in cosmology and his principle of the identity of indiscernibles in quantum mechanics (he is credited by some with anticipating this field). His contributions on a vast array of subjects are found scattered in obscure journals, in hundreds of correspondences, and in a huge collection of unpublished manuscripts, the majority of which are preserved in the Lower Saxony State Library in Hanover. To date, there has been 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."

Life

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. The only biography in English is Aiton (1986). Also see excerpt from Rouse Ball (1908) [1] bearing on Leibniz. Leibniz appears as a character in Neal Stephenson's novel Quicksilver.

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 father's library. By 12, he had taught himself Latin and had begun Greek. He entered his late father's university at 15. By 20, he had also studied at Jena, mastered the standard texts of his day on mathematics, 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 a thesis he had written in Lepzig. He then declined an academic appointment at Altdorf, and spent the rest of his life in the service of two Electors, first that of Mainz then of Hannover.

Career

At Altdorf, Leibniz wrote an essay on the study of law, dedicated to the Elector of Mainz. That essay led to his appointment by the Elector to a commission for the revision of some statutes.

Leibniz subsequently joined the Elector's diplomatic service. While working there, he supported (unsuccessfully) the German candidate for the Polish crown. Leibniz then drew up a plan to offer German co-operation with France, whereby France would take Egypt and use it as a basis for attack against Holland in Asia. In return, France would agree to leave Germany undisturbed. In 1672, Leibniz was invited to Paris by the French government to explain the details of the deal, but the plan was never adopted.

In Paris, Leibniz met Christiaan Huygens, a Dutch mathematician. His conversation with Huygens inspired Leibniz to study geometry. Although Leibniz had previously written some tracts on various minor points in mathematics, the most important being a paper on combinations written in 1668, and had designed and built a machine for doing arithmetic, he described his study of geometry as opening a new world to him.

In January, 1673, he was sent on a political mission to London, where he stayed some months and made the acquaintance of Henry Oldenburg, John Collins, and others. It was at this time that he communicated the memoir to the Royal Society in which he was found to have been forestalled by Gabriel Mouton.

In 1673 the Elector of Mainz died, and in the following year Leibniz entered the service of the Brunswick family. In 1676, he again visited London, and then moved to Hanover, where, until his death, he occupied the well-paid post of librarian in the ducal library. His pen was thenceforth employed in all the political matters which affected the Hanoverian family, and his services were recognized by honours and distinctions of various kinds. His memoranda on the various political, historical, and theological questions which concerned the dynasty during the forty years from 1673 to 1713 form a valuable contribution to the history of that time.

Serving the Brunswicks gave Leibniz more time for other pursuits. He later asserted that his invention of the differential and integral calculus in 1674 was the first fruit of this increased leisure. By 1677, this invention had evolved into a coherent system, although he would not publish it until 1684. (The earliest traces of its use in his surviving notebooks is in 1675.) Most of his mathematical papers were written between 1682 and 1692, many of them in the widely read journal Acta Eruditorum, which he and Otto Mencke founded in 1682.

In 1700 and at his suggestion, the Academy of Berlin was created, and he drew up its first statutes. 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. Evidently, Leibniz had fallen out of favour with the Elector, and so spent his final years in neglect. Leibniz never married, and was survived by his sister's only child.

Leibniz was reportedly overfond of money and personal distinctions, even though he never lacked for either. He could be unscrupulous on occasion (as might be expected of a professional diplomat of the time). On the other hand, he was charming and well-mannered, with many friends and admirers all over Europe. His mathematical reputation was enhanced by his eminence in diplomacy, philosophy, and literature, one considerably enhanced by his editorship of the Acta Eruditorum.

Writings

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. With the exception of Nouveaux essais sur l'entendement humain (1765), his extended response to John Locke's An Essay Concerning Human Understanding (1690), his books are short, arguably more in the nature of extended essays. Leibniz's correspondance consists of about 15,000 letters addressed to about 600 correspondents all over Europe. Many 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.

Leibniz left an enormous Nachlass, not cataloged until 1895. The standard annotated bibliography of Leibniz's writings is Ravier(1937), in French. The critical edition of Leibniz's works [2], begun in 1923 and still incomplete, is organized as follows:

  • Series 1. Political and Historical 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. [3]. Some of this edition is available online, gratis.

Two good anthologies of English translations are Wiener (1951) and Loemker (1969).

Posthumous reputation

When Leibniz died, his reputation was in decline. He was remembered for only one book, whose central argument Voltaire was to lampoon in his Candide. Leibniz had an ardent disciple, Christian Wolff, but his dogmatic and simplistic outlook did Leibniz's ideas 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 doubted that the 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 in 1765, with the publication of the Nouveaux Essais, which Kant read closely. The 19th century saw the publication of several multi-volume editions of Leibniz's writings, edited by Erdmann, Foucher de Careil, Gerhard, and Gerland, along single volumes containing Leibniz's complete correspondence with a given person. But no comprehensive bibliography of Leibniz's published writings was available before Ravier (1937).

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 [4] 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 get off the ground until after WWII. In any event, 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).

The dispute over who first invented the calculus

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.

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: it is a difficult question of conflicting and circumstantial evidence.

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 during the eighteenth century the prevalent opinion was against Leibniz, today the majority of those concerned are inclined to believe the two men, Leibniz and Newton, discovered and described the calculus independently.

Science and technology

The vis viva

See main article: Conservation of energy: Historical development.

During 1676 to 1689, Leibniz noticed what he called the vis viva (Latin for living force), a mathematical characteristic of certain mechanical systems that stayed constant even when the system changed. Though it is now recognized that Leibniz had discovered a limited case of the conservation of energy, his ideas unfortunately led him into another nationalistic dispute.

It appeared at the time that his principle 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 until eventually practical engineers demonstrated its usefulness in calculation.

It was subsequently appreciated that the two approaches are complementary.

Information technology

In the 1670s, building on a related simpler machine invented by Pascal, Leibniz built the first machine that could carry out all four arithmetic operations.

Norbert Wiener, writing in 1934, claimed that Leibniz was the first to describe the concept of feedback, central to Wiener's later cybernetic theory.

Philosophy

Most of Leibniz's philosophical writings were composed in the last twenty to twenty-five years of his life. Whether his views were original or whether they were appropriated from Spinoza, whom he visited in 1676, is still in question among philosophers, though the evidence seems to suggest that his views were original. As to Leibniz's system on philosophy, he regarded the ultimate elements of the universe as individual percipient beings whom he called monads. According to Leibniz, monads are centres of force; substance is force, while space, matter, and motion are merely phenomenal; finally, the existence of God is inferred from the existing harmony among the monads. His contributions to literature were almost as considerable as his contributions to philosophy; in particular, Leibniz successfully refuted the prevalent belief that Hebrew was the primeval language of the human race.

Metaphysics

Leibniz's best known contribution to metaphysics is his theory of monads, as exposited in his Monadology. 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 René Descartes' systems. This notion also solves a problematic individuation the systems of Baruch Spinoza, which represent individual creatures as mere accidental modifications.

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.

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.

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 Leibniz'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".

In his last letters he remarked: "If I had been less busy, or if I were younger or helped by well-intentioned young people, I would have hoped to have evolved a characteristic of this kind"; and: "I have spoken of my general characteristic to the Marquis de l'Hôpital and others; but they paid no more attention than if I had been telling them a dream. It would be necessary to support it by some obvious use; but, for this purpose, it would be necessary to construct a part at least of my characteristic; -- and this is not easy, above all to one situated as I am".

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 Boole. The principles of his 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.

For Leibniz, characters were 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.

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 1901, some of which Clarence Irving Lewis (1918) and Parkinson (1966) translated into English. Although Lewis and Nicholas Rescher in 1954 both drew attention to Leibniz's logical work, 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).

Modern logic began in 1847 in the UK, when George Boole and Augustus De Morgan independently published short books, in complete ignorance of Leibniz's partial anticipation of their results. Charles Peirce and Hugh MacColl shared Leibniz's dream of combining symbolic logic, mathematics, and philosophy. The culmination of Leibniz's work approach to logic was, arguably, the algebraic logic of Ernst Schröder and the modal logic founded by Lewis.


Works

Ravier, E., 1966 (1937). Bibliographie des oeuvres de Leibniz. Germany: Georg Olms.

Major works are in bold. The year shown is usually the year in which the work was completed, not of its eventual publication.

Secondary literature

  • Aiton, Eric J., 1985. Leibniz: A Biography. Hilger (UK).
  • Burkhardt, Hans, 1980. Logik und Semiotik in der Philosophie von Leibniz. Philosophia Verlag.
  • Jolley, Nicholas, ed., 1995. The Cambridge Companion to Leibniz. Cambridge Uni. Press.
  • Jolley, Nicholas, 2005. Leibniz. Routledge.
  • Ishiguro, Hide, 1990. Leibniz's Philosophy of Logic and Language. Cambridge 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.
  • 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.
  • Ross, George MacDonald, 1982.
  • ------, 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.
  • W. W. Rouse Ball, 1908. A Short Account of the History of Mathematics, 4th ed. (see Discussion)

Quotes

"[Monads are] simple substances without parts and without windows through which anything could come in or go out"
-Monadology
"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

Wikiquote has quotations related to Gottfried Wilhelm Leibniz.
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