Dictionary Definition
Leibniz n : German philosopher and mathematician
who thought of the universe as consisting of independent monads and
who devised a system of the calculus independent of Newton
(16461716) [syn: Leibnitz, Gottfried
Wilhelm Leibniz, Gottfried
Wilhelm Leibnitz]
Extensive Definition
Gottfried Wilhelm Leibniz (also Leibnitz or von
Leibniz (July
1 (June
21
Old Style) 1646 – November 14
1716) was a
German
polymath who wrote
primarily in Latin and French.
He occupies an equally grand place in both the
history
of philosophy and the history
of mathematics. He invented calculus independently of
Newton, and
his notation
is the one in general use since then. He also discovered the
binary
system, foundation of virtually all modern computer
architectures. In philosophy, he is mostly remembered for optimism, i.e. his conclusion
that our universe is, in a restricted sense, the best possible one
God could have
made. He was, along with René
Descartes and Baruch
Spinoza, one of the three greatest 17th century rationalists, but his
philosophy also looks back to the scholastic
tradition and anticipates modern logic and analysis.
Leibniz also made major contributions to physics and technology, and anticipated
notions that surfaced much later in biology, medicine, geology, probability
theory, psychology, linguistics, and information
science. He also wrote on politics, law, ethics, theology, history, and philology, even occasional
verse. His contributions to this vast array of subjects are
scattered in journals and in tens of thousands of letters and
unpublished manuscripts. To date, there is no complete edition of
Leibniz's writings.
Biography
The outline of Leibniz's career is as follows: 16461666: Formative years
 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 BrunswickLüneburg, then to his
brother, Duke, then Elector,
Ernst August of Hanover.
 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.
 1698–1716: Courtier to Elector
Georg Ludwig of Hanover.
 1712–14. Resides in Vienna. Appointed Imperial Court Councillor in 1713 by Charles VI, Holy Roman Emperor, at the Habsburg 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.
 1677–98. Courtier, first to
John Frederick, Duke of BrunswickLüneburg, then to his
brother, Duke, then Elector,
Ernst August of Hanover.
Early life
Gottfried Leibniz was born on 1 July (21 June Old Style) 1646 in Leipzig to Friedrich Leibniz and Catherina Schmuck. The name Leibniz was originally Slavonic  Lubeniecz http://www.nndb.com/people/666/000087405/. His father had passed away when he was six, so he learned his religious and moral values from his mother. These would exert a profound influence on his philosophical thought in later life. As an adult, he often styled himself "von Leibniz", and many posthumous editions of his works gave his name on the title page as "Freiherr [Baron] G. W. von Leibniz." However, no document has been found confirming that he was ever granted a patent of nobility.When Leibniz was six years old, his 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 seven onwards. By 12, he had
taught himself Latin,
which he used freely all his life, and had begun studying Greek.
He entered his father's university at age 14, and
completed university studies by 20, specializing in law and
mastering the standard university courses in classics, logic, and
scholastic philosophy. However, his education in mathematics was
not up to the French and British standards. In 1666 (age 20), he
published his first book, also his habilitation thesis in
philosophy, On the
Art of Combinations. When Leipzig declined to
assure him a position teaching law upon graduation, Leibniz
submitted the thesis he had intended to submit at Leipzig to the
University
of Altdorf instead, 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.
1666–74
Leibniz's first position was as a salaried alchemist in Nuremberg, even though he knew nothing about the subject. He soon met Johann Christian von Boineburg (1622–1672), 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 under the employment 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 followed 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 Louis
XIV of France, backed by French military and economic might.
Meanwhile, the Thirty
Years' War had left Germanspeaking Europe exhausted,
fragmented, and economically backward. Leibniz proposed to protect
Germanspeaking 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 gone
irrelevant. 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. Leibniz was extraordinarily fateful when came
into acquaintance of the Dutch physicist
and mathematician Christiaan
Huygens, who was active in Paris then. Soon after arriving in
Paris, Leibniz received a rude awakening; his knowledge of
mathematics and physics was spotty. Finding Huygens as mentor, he
began a program of selfstudy that soon pushed him to making major
contributions to both subjects, including inventing his version of
the differential and integral 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 English
government in London, early in
1673. There Leibniz came into acquaintance of Henry
Oldenburg and
John Collins. After demonstrating a calculating machine to the
Royal
Society 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 Hanover 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 Habsburg imperial
court was forthcoming.
House of Hanover 1676–1716
Leibniz managed to delay his arrival in Hanover until the end of 1676, after making one more short journey to London, where he possibly 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 Hanover, 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 contradicted both Christian and Jewish orthodoxy.In 1677, he was promoted, at his request, to
Privy Counselor 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 the 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 her 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 I of Great Britain.
The population of Hanover 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 British Act
of Settlement 1701 designated the Electress Sophia and her
descent as the royal family of the United Kingdom, once both King
William
III and his sisterinlaw and successor, Queen
Anne, were dead. Leibniz played a role in the initiatives and
negotiations leading up to that Act, but not always an effective
one. For example, something he published anonymously in England,
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 Hanover 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 Habsburgs. 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 the Princess of Wales, Caroline
of Ansbach, George I forbade 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.
Leibniz died in Hanover in 1716: at
the time, he was so out of favor that neither George I (who
happened to be near Hanover 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.
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 wellmannered,
with many friends and admirers all over Europe.
Writings and edition
Leibniz mainly wrote in three languages: scholastic Latin (ca. 40%), French (ca. 35%), and German (less than 25%). During his lifetime, he published many pamphlets and scholarly articles, but only two "philosophical" books, the Combinatorial Art and the Théodicée. (He published numerous pamphlets, often anonymous, on behalf of the House of BrunswickLüneburg, most notably the "De jure suprematum" a major consideration of the nature of sovereignty.) One substantial book appeared posthumously, his Nouveaux essais sur l'entendement humain, which Leibniz had withheld from publication after the death of John Locke. Only in 1895, when Bodemann completed his catalogues of Leibniz's manuscripts and correspondence, 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 of essay length. Much of his vast correspondence, especially the letters dated after 1685, remains unpublished, and much of what is published has been so only in recent decades. 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)The extant parts of the critical edition of Leibniz's
writings (see photograph there) 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.
 Series 7. Mathematical Writings. 3 vols., 1672–76.
 Series 8. Scientific, Medical, and Technical Writings''. In preparation.
The systematic cataloguing of all of Leibniz's
Nachlass
was begun in 1901. Two World wars, the NS dictatorship (with Jewish
genocide, including an employee of the project, and other personal
consequences), and decades of German division (two states with the
cold war's "iron curtain" in between, separating scholars and also
scattered portions of his literary estates), greatly hampered the
ambitious edition project which had and has to deal with seven
languages used on ca. 200 000 pages of written and printed paper.
In 1985 it was reorganized and included in a joint program of
German federal and state ("Länder") academies. Since then the
branches in Potsdam, Münster,
Hannover
and Berlin
have jointly published 25 volumes of the critical edition (until
2006) with an average of 870 pages (compared to only 19 volumes
since 1923), plus preparing index and concordance works (so, had
that "speed" of work been possible from the beginning, the project
would already be completed).
Posthumous reputation
When Leibniz died, his reputation was in decline. He was remembered for only one book, the Théodicée, whose supposed central argument Voltaire lampooned in his Candide. Voltaire's depiction of Leibniz's ideas was so influential that many believed it to be an accurate description (this misapprehension may still be the case among certain lay people). Thus Voltaire and his Candide bear some of the blame for the lingering failure to appreciate and understand Leibniz's ideas. Leibniz had an ardent disciple, Christian Wolff, whose dogmatic and facile outlook did Leibniz's reputation much harm. 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. His work on law, diplomacy, and history was seen as of ephemeral interest. The vastness and richness of his correspondence went unrecognized.Much of Europe came to doubt that Leibniz had
discovered the calculus independently of Newton, and hence his
whole work in mathematics and physics was neglected. Voltaire, an
admirer of Newton, also wrote Candide at least in part to discredit
Leibniz's claim to having discovered the calculus and Leibniz's
charge that Newton's theory of universal gravitation was incorrect.
The rise of relativity and subsequent work in the history of
mathematics has put Leibniz's stance in a more favorable
light.
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 multivolume edition of Leibniz's
writings, followed in the 19th century by a number of editions,
including those edited by Erdmann, Foucher de Careil, Gerhardt,
Gerland, Klopp, and Mollat. 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 critical study of Leibniz's metaphysics.
Shortly thereafter, Louis
Couturat published an important study
of Leibniz, and edited a volume of Leibniz's heretofore unpublished
writings, mainly on logic. While their conclusions, especially
Russell's, were subsequently challenged and often dismissed, they
made Leibniz somewhat respectable among 20th century analytical and
linguistic philosophers in the English speaking world (Leibniz had
already been of great influence to many Germans such as Bernhard
Riemann). For example, Leibniz's phrase salva
veritate, meaning interchangeability without loss of or
compromising the truth, recurs in Willard
Quine's writings. Nevertheless, the secondary Englishlanguage
literature on Leibniz did not really blossom until after World War
II. This is especially true of English speaking countries; in
Gregory Brown's bibliography fewer than 30 of the English language
entries were published before 1946. American Leibniz studies owe
much to Leroy Loemker (1904–85) through his translations and his
interpretive essays in .
Nicholas Jolley has surmised that Leibniz's
reputation as a philosopher is now perhaps higher than at any time
since he was alive because:
 Work in the history of 17th and 18th century ideas has revealed more clearly the 17th century "Intellectual Revolution" that preceded the better known Industrial and commercial revolutions of the 18th and 19th centuries.
 The doctrinaire contempt for metaphysics, characteristic of analytic and linguistic philosophy, has faded;
 Analytic and contemporary philosophy continue to invoke his notions of identity, individuation, and possible worlds;
 The 17th and 18th century belief that natural science, especially physics, differs from philosophy mainly in degree and not in kind, is no longer dismissed out of hand. That modern science includes a "scholastic" as well as a "radical empiricist" element is more accepted now than in the early 20th century;
 He is now seen as a major prolongation of the mighty endeavor begun by Plato and Aristotle: the universe and man's place in it are amenable to human reason.
The University of Hannover (German spelling) is
named after him.
In 1985, the German government
created the
Leibniz Prize, annual awards of 1.55 million Euros for
experimental results, and 770,000 Euros for theoretical ones. It is
the world's largest prize for scientific achievement.
Philosopher
Leibniz's philosophical thinking appears fragmented, 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 wrote only two philosophical treatises, and the 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 a
running dispute between Malebranche and
Antoine
Arnauld. This led to an extensive and valuable correspondence
with Arnauld (, ); it and the Discourse were not published until
the 19th century. In 1695, Leibniz made his public entrée into
European philosophy with a journal article titled "New System of
the Nature and Communication of Substances" (, , ). 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, (, , ) especially when these were
inconsistent with Christian orthodoxy.
Unlike Descartes and Spinoza, Leibniz had a
thorough university education in philosophy. His lifelong scholastic
and Aristotelian
turn of mind betrayed the strong influence of one of his Leipzig professors,
Jakob
Thomasius, who also supervised his BA thesis in philosophy.
Leibniz also eagerly read Francisco
Suárez, a Spanish Jesuit
respected even in Lutheran
universities. Leibniz was deeply interested in the new methods and
conclusions of Descartes, Huygens, Newton, and Boyle, but
viewed their work through a lens heavily tinted by scholastic
notions. Yet it remains the case that Leibniz's methods and
concerns often anticipate the logic, and analytic
and linguistic
philosophy of the 20th century.
The Principles
Leibniz variously invoked one or another of seven fundamental philosophical Principles (Mates 1986: chpts. 7.3, 9): Identity/contradiction. If a proposition is true, then its negation is false and vice versa.
 Identity of indiscernibles. Two things are identical if and only if they share the same properties. Frequently invoked in modern logic and philosophy.
 Sufficient reason. "There must be a sufficient reason [often known only to God] for anything to exist, for any event to occur, for any truth to obtain." (LL 717).
 Preestablished harmony. See Jolley (1995: 129–31), Woolhouse and Francks (1998), and Mercer (2001). "[T]he appropriate nature of each substance brings it about that what happens to one corresponds to what happens to all the others, without, however, their acting upon one another directly." (Discourse on Metaphysics, XIV) A dropped glass shatters because it "knows" it has hit the ground, and not because the impact with the ground "compels" the glass to split.
 Continuity. Natura non saltum facit. A mathematical analog to this principle would go as follows. If a function describes a transformation of something to which continuity applies, then its domain and range are both dense sets.
 Optimism. "God assuredly always chooses the best." (LL 311).
 Plenitude. "Leibniz believed that the best of all possible worlds would actualize every genuine possibility, and argued in Théodicée that this best of all possible worlds will contain all possibilities, with our finite experience of eternity giving no reason to dispute nature's perfection." (From Plenitude.)
The second principle here is often referred to as
Leibniz's
Law http://plato.stanford.edu/entries/identityindiscernible/.
The Identity of Indiscernibles has attracted the most controversy
and criticism, especially from corpuscular philosophy and quantum
mechanics.
Leibniz would on occasion give a speech for a
specific principle, but more often took them for granted. For a
precis of what Leibniz meant by these and other Principles, see
Mercer (2001: 473–84). For a classic discussion of
Sufficient Reason and Plenitude,
see Lovejoy (1957).
The monads
Leibniz's best known contribution to metaphysics is his theory of monads, as exposited in Monadologie. Monads are to the metaphysical realm what atoms are to the physical/phenomenal. Monads are the ultimate elements of the universe. The monads are "substantial forms of being" with the following properties: they are eternal, indecomposable, individual, subject to their own laws, uninteracting, and each reflecting the entire universe in a preestablished 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 possess no
material or spatial character. They also differ from atoms by their
complete mutual independence, so that interactions among monads are
only apparent. Instead, by virtue of the principle of
preestablished harmony, each monad follows a preprogrammed set of
"instructions" peculiar to itself, so that a monad "knows" what to
do at each moment. (These "instructions" may be seen as analogs of
the scientific
laws governing subatomic
particles.) By virtue of these intrinsic instructions, each
monad is like a little mirror of the universe. Monads need not be
"small"; e.g., each human being constitutes a monad, in which case
free
will is problematic. God, too, is a monad,
and the existence
of God can be inferred from the harmony prevailing among all
other monads; God wills the preestablished harmony.
Monads are purported to having gotten rid of the
problematic:
The monadology was thought arbitrary, even
eccentric, in Leibniz's day and since.
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. Rutherford (1998) is a detailed scholarly study of Leibniz's theodicy.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 BoisReymond, 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. In addition, the modern observations that lead to
the Finetuned
Universe arguments seem to support his view:
 The 3+1 dimensional structure of spacetime may be ideal. In order to sustain complexity such as life, a universe probably requires three spatial and one temporal dimension. Most universes deviating from 3+1 either violate some fundamental physical laws, or are impossible. The mathematically richest number of spatial dimensions is also 3 (in the sense of topological nontriviality).
 The universe, solar system, and Earth are the "best possible" in that they enable intelligent life to exist. Such life has evolved on Earth only because the Earth, solar system, and Milky Way possess a number of unusual characteristics; see Ward & Brownlee (2000), Morris (2003: chpts. 5,6).
 The most sweeping form of optimism derives from the Anthropic Principle (Barrow and Tipler 1986). Physical reality 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. Were the numerical values of these constants to differ by a few percent from their observed values, it is unlikely that the resulting universe would contain complex structures.
Our physical
laws, universe,
solar
system, and home planet are all
"best" in the sense that they enable complex structures such as
galaxies, stars, and, ultimately, intelligent
life. On the other hand, it is also reasonable to believe that
life might be more intelligent given some other set of
circumstances. Further, some modern proponents of Leibnizian
optimism seem to confuse the term "best" with terms like "good" or
"better"; by definition, there can be only one "best".
Symbolic thought
Leibniz believed that much of human reasoning could be reduced to calculations of a sort, and that such calculations could resolve many differences of opinion: "The only way to rectify our reasonings is to make them as tangible as those of the Mathematicians, so that we can find our error at a glance, and when there are disputes among persons, we can simply say: Let us calculate [calculemus], without further ado, to see who is right." (The Art of Discovery 1685, W 51) Leibniz's calculus ratiocinator, which resembles symbolic logic, can be viewed as a way of making such calculations feasible. Leibniz wrote memoranda (many of which are translated in Parkinson 1966) that can now be read as groping attempts to get symbolic logic—and thus his calculus—off the ground. But Gerhard and Couturat did not publish these writings until modern formal logic had emerged in Frege's Begriffsschrift and in writings by Charles Peirce and his students in the 1880s, and hence well after Boole and De Morgan began that logic in 1847.Leibniz thought symbols were important for human
understanding. He attached so much importance to the invention of
good notations that he attributed all his discoveries in
mathematics to this. His notation for the infinitesimal
calculus is an example of his skill in this regard. Charles
Peirce, a 19th century pioneer of semiotics, shared Leibniz's
passion for symbols and notation, and his belief that these are
essential to a wellrunning logic and mathematics.
But Leibniz took his speculations much further.
Defining a character as any
written sign, he then defined a "real" character as one that
represents an idea directly and not simply as the word embodying
the idea. Some real characters, such as the notation of logic,
serve only to facilitate reasoning. Many characters wellknown in
his day, including Egyptian
hieroglyphics, Chinese
characters, and the symbols of astronomy and chemistry, he deemed not real.
(Loemker, however, who translated some of Leibniz's works into
English, said that the symbols of chemistry were real characters so
there is disagreement among Leibniz scholars on this point.)
Instead, he proposed the creation of a characteristica
universalis or "universal characteristic," built on an
alphabet of human thought in which each fundamental concept
would be represented by a unique "real" character. "It is obvious
that if we could find characters or signs suited for expressing all
our thoughts as clearly and as exactly as arithmetic expresses
numbers or geometry expresses lines, we could do in all matters
insofar as they are subject to reasoning all that we can do in
arithmetic and geometry. For all investigations which depend on
reasoning would be carried out by transposing these characters and
by a species of calculus." (Preface to the General Science, 1677.
Revision of Rutherford's translation in Jolley 1995: 234. Also W
I.4) Complex thoughts would be represented by combining characters
for simpler thoughts. Leibniz saw that the uniqueness of prime
factorization suggests a central role for prime
numbers in the universal characteristic, a striking
anticipation of Gödel
numbering. Granted, there is no intuitive or mnemonic way to number any set
of elementary concepts using the prime numbers.
Because Leibniz was a mathematical novice when he
first wrote about the characteristic, at first he did not conceive
it as an algebra but
rather as a universal
language or script. Only in 1676 did he conceive of a kind of
"algebra of thought," modeled on and including conventional algebra
and its notation. The resulting characteristic included a logical
calculus, some combinatorics, algebra, his analysis situs (geometry
of situation) discussed in 3.2, a universal concept language, and
more.
What Leibniz actually intended by his characteristica
universalis and calculus
ratiocinator, and the extent to which modern formal logic does justice to the
calculus, may never be established. A good introductory discussion
of the "characteristic" is Jolley (1995: 226–40). An early, yet
still classic, discussion of the "characteristic" and "calculus" is
Couturat (1901: chpts. 3,4).
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. Leibniz enunciated the principal properties of what we now call conjunction, disjunction, negation, identity, set inclusion, and the empty set. The principles of Leibniz's logic and, arguably, of his whole philosophy, reduce to two: All our ideas are compounded from a very small number of simple ideas, which form the alphabet of human thought.
 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. The formal
logic that emerged early in the 20th century also requires, at
minimum, unary negation
and quantified
variables ranging over
some universe
of discourse.
Leibniz published nothing on formal logic in his
lifetime; most of what he wrote on the subject consists of working
drafts.
In his book
History of Western Philosophy, Bertrand
Russell went as far as claiming that Leibniz had developed
logic in his unpublished writings to a level which was reached only
200 years later.
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). In the 18th century, "function" lost these geometrical associations.Leibniz was the first to see that the
coefficients of a system of linear
equations could be arranged into an array, now called a
matrix,
which can be manipulated to find the solution of the system, if
any. This method was later called Gaussian
elimination. Leibniz's discoveries of Boolean
algebra and of symbolic
logic, also relevant to mathematics, are 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.
Calculus
Leibniz is credited, along with Isaac Newton, with the discovery of 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. This ingenious and suggestive notation for the calculus is probably his most enduring mathematical legacy. Leibniz did not publish anything about his calculus 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 law." In addition, the theorem that tells how and when to differentiate under the integral sign is called Leibniz integral 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. Leibniz also freely invoked
mathematical entities he called infinitesimals,
manipulating them 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.
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 fundamentally Newton's. Hall (1980) gives a
thorough scholarly discussion of the
calculus priority dispute.
Modern, rigorous calculus emerged in the 19th
century, thanks to the efforts of Augustin
Louis Cauchy, Bernhard
Riemann, Karl
Weierstrass, and others, who based their work on the definition
of a limit
and on a precise understanding of real numbers.
Their work discredited the use of infinitesimals to justify
calculus. Yet, infinitesimals survived in science and engineering,
and even in rigorous mathematics, via the fundamental computational
device known as the differential.
Beginning in 1960, Abraham
Robinson worked out a rigorous foundation for Leibniz's
infinitesimals, using model
theory. The resulting nonstandard
analysis can be seen as a belated vindication of Leibniz's
mathematical reasoning.
Topology
Leibniz was the first to use the term analysis situs (LL §27), later used 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 Jacob Freudenthal, argues:"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
Königsberg 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."
But Hirano (1997) argues differently, quoting
Mandelbrot (1977: 419):
"...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 the fractal geometry promoted by Mandelbrot
drew on Leibniz's notions of selfsimilarity 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
selfsimilarity. Leibniz's improvement of Euclid's axiom contains
the same concept.
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, often disagreeing with Descartes and Newton. He devised a new theory of motion (dynamics) based on kinetic energy and potential energy, which posited space as relative, whereas Newton felt strongly space was absolute. 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) and Wilson (1989).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
little sense. For instance, he anticipated Albert
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. The
principle of sufficient reason has been invoked in recent
cosmology, and his
identity
of indiscernibles in quantum
mechanics, a field some even credit him with having anticipated
in some sense. Those who advocate digital
philosophy, a recent direction in cosmology, claim Leibniz as a
precursor.
The vis viva
Leibniz 's vis viva (Latin for living force) is
mv2, twice the modern Kinetic
energy. He realized that the total energy would be conserved in
certain mechanical systems, so he considered it an innate motive
characteristic of matter (see AG 155–86, LL §§53–55, W II.6–7a).
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, 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 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, Garland 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 winddriven 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. From 1680 to 1685, he struggled to overcome the chronic flooding that afflicted the ducal silver mines in the Harz Mountains, but did not succeed. (Aiton 1985: 107–114, 136)Information technology
Leibniz may have been the first computer scientist and information theorist. Early in life, he discovered the binary number system (base 2), which was later (and is now) used on most computers, then revisited that system throughout his career. (See Couturat, 1901: 473–78.) He anticipated Lagrangian interpolation and algorithmic information theory. His calculus ratiocinator anticipated aspects of the universal Turing machine. In 1934, Norbert Wiener claimed to have found in Leibniz's writings a mention of the concept of feedback, central to Wiener's later cybernetic theory.In 1671, Leibniz began to invent a machine that
could execute all four arithmetical operations, gradually improving
it over a number of years. This 'Stepped
Reckoner' 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 Hanover, by a
craftsman working under Leibniz's supervision. It was not an
unambiguous success because it did not fully mechanize the
operation of carrying. Couturat (1901: 115) reported finding an
unpublished note by Leibniz, dated 1674, describing a machine
capable of performing some algebraic operations.
Leibniz was groping towards hardware and software
concepts worked out much later in 18301845 by Charles
Babbage and Ada
Lovelace. 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.http://www.edge.org/discourse/schirrmacher_eurotech.html
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. Davis (2000) discusses Leibniz's prophetic role
in the emergence of calculating machines and of formal
languages.
Librarian
While serving as librarian of the ducal libraries in Hanover and Wolfenbuettel, Leibniz effectively became one of the founders of library science.http://members.tripod.com/ClintonGreen/universal.html#6 The latter library was enormous for its day, as it contained more than 100,000 volumes, and Leibniz helped design a new building for it, believed to be the first building explicitly designed to be a library. He also designed a book indexing system in ignorance of the only other such system then extant, that of the Bodleian Library at Oxford University. He also called on publishers to distribute abstracts of all new titles they produced each year, in a standard form that would facilitate indexing. He hoped that this abstracting project would eventually include everything printed from his day back to Gutenberg. Neither proposal met with success at the time, but something like them became standard practice among English language publishers during the 20th century, under the aegis of the Library of Congress and the British Library.He called for the creation of an empirical database as a way to further
all sciences. His characteristica
universalis, calculus
ratiocinator, and a "community of minds"—intended,
among other things, to bring political and religious unity to
Europe—can be seen as distant unwitting anticipations of
artificial languages (e.g., Esperanto and its
rivals), symbolic
logic, even the World Wide
Web.
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 drew up its first statutes, and served as its first President for the remainder of his life. That Academy evolved into the German Academy of Sciences, the publisher of the ongoing critical edition of his works. On Leibniz’s projects for scientific societies, see Couturat (1901: App. IV).Lawyer, moralist
No philosopher has ever had as much experience with practical affairs of state as Leibniz, except possibly Marcus Aurelius. 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 (in order of difficulty) Jolley (2005: chpt. 7), Gregory Brown's chapter in Jolley (1995), Hostler (1975), and Riley (1996).While Leibniz was no apologist for absolute
monarchy like Hobbes, or for
tyranny in any form, neither did he echo the political and
constitutional views of his contemporary John Locke,
views invoked in support of democracy, in 18th century America and
later elsewhere. The following excerpt from a 1695 letter to Baron
J. C. Boineburg's son Philipp is very revealing of Leibniz's
political sentiments:
"As for.. the great question of the power of
sovereigns and the obedience their peoples owe them, I usually say
that it would be good for princes to be persuaded that their people
have the right to resist them, and for the people, on the other
hand, to be persuaded to obey them passively. I am, however, quite
of the opinion of Grotius, that one
ought to obey as a rule, the evil of revolution being greater
beyond comparison than the evils causing it. Yet I recognize that a
prince can go to such excess, and place the wellbeing of the state
in such danger, that the obligation to endure ceases. This is most
rare, however, and the theologian who authorizes violence under
this pretext should take care against excess; excess being
infinitely more dangerous than deficiency." (LL: 59, fn 16.
Translation revised.)
Leibniz foresaw the European
Union. In 1677, he (LL: 58, fn 9) called for a European
confederation, governed by a council or senate, whose members would
represent entire nations and would be free to vote their
consciences. Europe would adopt a uniform religion. He reiterated
these proposals in 1715.
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 protoSwedish 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 no volume of the planned Academy edition series "Historical and Linguistic Writings" has appeared.Sinophile
Leibniz was perhaps the first major European intellect 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 characters were an unwitting form of his universal characteristic. He noted with fascination how the I Ching hexagrams correspond to the binary numbers from 0 to 111111, and concluded that this mapping was evidence of major Chinese accomplishments in the sort of philosophical mathematics he admired.On Leibniz, the I Ching, and binary numbers, see
Aiton (1985: 245–48). Leibniz's writings on Chinese civilization
are collected and translated in Cook and Rosemont (1994), and
discussed in Perkins (2004).
As polymath
The following episode from the life of Leibniz illustrates the breadth of his genius, and the difficulties awaiting those who try to come to terms with it. While making his grand tour of European archives to research the Brunswick family history he never completed, Leibniz stopped in Vienna, May 1688 – February 1689, where he did much legal and diplomatic work for the Brunswicks. He visited mines, talked with mine engineers, and tried to negotiate export contracts for lead from the ducal mines in the Harz mountains. His proposal that the streets of Vienna be lit with lamps burning rapeseed oil was implemented. During a formal audience with the Austrian Emperor and in subsequent memoranda, he advocated reorganizing the Austrian economy, reforming the coinage of much of central Europe, negotiating a Concordat between the Habsburgs and the Vatican, and creating an imperial research library, official archive, and public insurance fund. He wrote and published an important paper on mechanics. Leibniz also wrote a short paper, first published by Louis Couturat in 1903, later translated as LL 267 and WF 30, summarizing his views on metaphysics. The paper is undated; that he wrote it while in Vienna was determined only in 1999, when the ongoing critical edition finally published Leibniz's philosophical writings for the period 1677–90. Couturat's reading of this paper was the launching point for much 20th century thinking about Leibniz, especially among analytic philosophers. But after a meticulous study of all of Leibniz's philosophical writings up to 1688—a study the 1999 additions to the critical edition made possible—Mercer (2001) begged to differ with Couturat's reading; the jury is still out.Leibniz was not devoid of humor and imagination;
see W IV.6 and LL § 40. Also see a curious passage titled
"Leibniz's Philosophical Dream," first published by Bodemann in
1895 and translated on p. 253 of Morris, Mary, ed. and trans.,
1934. Philosophical Writings. Dent & Sons Ltd.
References
Works
Four important collections of English translations are W (Wiener 1951), LL (Loemker 1969), AG (Ariew and Garber 1989), and WF (Woolhouse and Francks, 1998).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). Partially translated in LL §1 and Parkinson (1966).
 1671. Hypothesis Physica Nova (New Physical Hypothesis). LL §8.I (part)
 1673 Confessio philosophi (A Philosopher's Creed, English translation)
 1684. Nova methodus pro maximis et minimis (New Method for maximums and minimums). Translation in Struik, D. J., 1969. A Source Book in Mathematics, 1200–1800. Harvard Uni. Press: 271–81.
 1686. Discours de métaphysique. Martin and Brown (1988). Jonathan Bennett's translation. AG 35, LL §35, W III.3, WF 1.
 1703. Explication de l'Arithmétique Binaire (Explanation of Binary Arithmetic). Gerhardt, Mathematical Writings VII.223. Lloyd Strickland's translation.
 1710. Théodicée. Farrer, A.M., and Huggard, E.M., trans., 1985 (1952). Theodicy. Open Court. W III.11 (part).
 1714. Monadologie. Nicholas Rescher, trans., 1991. The Monadology: An Edition for Students. Uni. of Pittsburg Press. Jonathan Bennett's translation. Latta's translation. AG 213, LL §67, W III.13, WF 19. French, latin and spanish edition, with facsimil of Leibniz's manuscript.
 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). Jonathan Bennett's translation.
Collections of shorter works in translation:
 Bennett, Jonathan. Various texts.
 Cook, Daniel, and Rosemont, Henry Jr., 1994. Leibniz: Writings on China. Open Court.
 Dascal, Marcelo, 1987. Leibniz: Language, Signs and Thought. John Benjamins.
 Martin, R.N.D., and Brown, Stuart, 1988. Discourse on Metaphysics and Related Writings. St. Martin's Press.
 Parkinson, G.H.R., 1966. Leibniz: Logical Papers. Oxford Uni. Press.
 ———, and Morris, Mary, 1973. Leibniz: Philosophical Writings. London: J M Dent & Sons.
 Riley, Patrick, 1988 (1972). Leibniz: Political Writings. Cambridge Uni. Press.
 Rutherford, Donald. Various texts.
 Strickland, Lloyd, 2006. Shorter Leibniz Texts. Continuum Books. Online.
 Regrettably out of print and lacks index.
 Woolhouse, R.S., and Francks, R., 1998. Leibniz: Philosophical Texts. Oxford Uni. Press.
Donald Rutherford's online
bibliography.
Secondary literature
A modern biography in English is Aiton (1985). An 1845 English biography by John M. Mackie is available on Google Books here. A lively short account of Leibniz’s life, one also taking a critical approach to his philosophy, 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.For a first introduction to Leibniz's philosophy,
turn to the Introduction of an anthology of his writings in English
translation, e.g., Wiener (1951), Loemker (1969a), Woolhouse and
Francks (1998). Then turn to the monographs 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's thought 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 glossaries of the
philosophical terminology recurring in Leibniz's writings and the
secondary literature, see Woolhouse and Francks (1998: 285–93) and
Jolley (2005: 223–29).
Introductory:
 Jolley, Nicholas, 2005. Leibniz. Routledge.
 MacDonald Ross, George, 1984. Leibniz. Oxford Univ. Press.
 W. W. Rouse Ball, 1908. A Short Account of the History of Mathematics, 4th ed. (see Discussion)
Intermediate:
 Aiton, Eric J., 1985. Leibniz: A Biography. Hilger (UK).
 Brown, Gregory, 2004, "Leibniz's Endgame and the Ladies of the Courts," Journal of the History of Ideas 65: 75–100.
 Hall, A. R., 1980. Philosophers at War: The Quarrel between Newton and Leibniz. Cambridge Univ. Press.
 Hostler, J., 1975. Leibniz's Moral Philosophy. UK: Duckworth.
 Jolley, Nicholas, ed., 1995. The Cambridge Companion to Leibniz. Cambridge Univ. Press.
 LeClerc, Ivor, ed., 1973. The Philosophy of Leibniz and the Modern World. Vanderbilt Univ. Press.
 Loemker, Leroy, 1969a, "Introduction" to his Leibniz: Philosophical Papers and Letters. Reidel: 1–62.
 Luchte, James, 2006, 'Mathesis and Analysis: Finitude and the Infinite in the Monadology of Leibniz,' London: Heythrop Journal.
 Arthur O. Lovejoy, 1957 (1936). "Plenitude and Sufficient Reason in Leibniz and Spinoza" in his The Great Chain of Being. Harvard Uni. Press: 144–82. Reprinted in Frankfurt, H. G., ed., 1972. Leibniz: A Collection of Critical Essays. Anchor Books.
 MacDonald Ross, George, 1999, "Leibniz and SophieCharlotte" in Herz, S., Vogtherr, C.M., Windt, F., eds., Sophie Charlotte und ihr Schloß. München: Prestel: 95–105. English translation.
 Perkins, Franklin, 2004. Leibniz and China: A Commerce of Light. Cambridge Univ. Press.
 Riley, Patrick, 1996. Leibniz's Universal Jurisprudence: Justice as the Charity of the Wise. Harvard Univ. Press.
 Strickland, Lloyd, 2006. Leibniz Reinterpreted. Continuum: London and New York
Advanced
 Adams, Robert M., 1994. Leibniz: Determinist, Theist, Idealist. Oxford Uni. Press.
 Bueno, Gustavo, 1981. Introducción a la Monadología de Leibniz. Oviedo: Pentalfa.
 Louis Couturat, 1901. La Logique de Leibniz. Paris: Felix Alcan. Donald Rutherford's English translation in progress.
 Ishiguro, Hide, 1990 (1972). Leibniz's Philosophy of Logic and Language. Cambridge Univ. 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.
 Mates, Benson, 1986. The Philosophy of Leibniz: Metaphysics and Language. Oxford Univ. Press.
 Mercer, Christia, 2001. Leibniz's metaphysics: Its Origins and Development. Cambridge Univ. Press.
 Robinet, André, 2000. Architectonique disjonctive, automates systémiques et idéalité transcendantale dans l'oeuvre de G.W. Leibniz: Nombreux textes inédits. Vrin
 Rutherford, Donald, 1998. Leibniz and the Rational Order of Nature. Cambridge Univ. Press.
 Wilson, Catherine, 1989. Leibniz's Metaphysics. Princeton Univ. Press.
 Woolhouse, R. S., ed., 1993. G. W. Leibniz: Critical Assessments, 4 vols. Routledge. A remarkable onestop collection of many valuable articles.
Online
bibliography by Gregory Brown.
Other works cited
 John D. Barrow and Frank J. Tipler, 1986. The Anthropic Cosmological Principle. Oxford Univ. Press.
 Martin Davis, 2000. The Universal Computer: The Road from Leibniz to Turing. W W Norton.
 Du BoisReymond, Paul, 18nn, "Leibnizian Thoughts in Modern Science," ???.
 Ivor GrattanGuinness, 1997. The Norton History of the Mathematical Sciences. W W Norton.
 Hirano, Hideaki, 1997, "Cultural Pluralism And Natural Law." Unpublished.
 Reinhard Finster, Gerd van den Heuvel: Gottfried Wilhelm Leibniz. Mit Selbstzeugnissen und Bilddokumenten. 4. Auflage. Rowohlt, Reinbek bei Hamburg 2000 (Rowohlts Monographien, 50481), ISBN 3499504812
 Benoît 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.
 Zalta, E. N., 2000, "A (Leibnizian) Theory of Concepts," Philosophiegeschichte und logische Analyse / Logical Analysis and History of Philosophy 3: 137–183.
Quotations
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.
 "God is the final reason of salvation, of grace, of faith and of election in Jesus Christ." (Theodicy: Essays on the Justice of God and the Freedom of Man in the Origin of Evil, Part I, 126)
 "With every lost hour, a part of life perishes." "Deeds make people." Loemker's (1969: 58) translation of other Leibniz mottoes.
 "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)
 "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)
 "It is unworthy of excellent men to lose hours like slaves in the labour of calculation which could safely be relegated to anyone else if machines were used."
 "Truths of reason are necessary and their opposite is impossible: truths of fact are contingent and their opposite possible."
 "It is one of my most important and very best verified maxims that nature makes no leaps. This I have called the law of continuity."
 "Why is there something, rather than nothing?"
 "There are two kinds of truths: truths of reasoning and truths of fact."
 "The soul is the mirror of an indestructible universe."
See also
 monadology
 characteristica universalis
 universal language
 Calculus ratiocinator
 alphabet of human thought
 Newton v. Leibniz calculus controversy
 LeibnizGemeinschaft
 Gottfried Wilhelm Leibniz Prize
 Leibniz formula
 Leibniz integral rule for differentiation under the integral sign
 Leibniz test
 digital philosophy
 Anthropic Principle
 Stepped Reckoner
 Problem of the futures contingents
 LeibnizKeks a biscuit named for Leibniz
 Leibniz harmonic triangle
External links
wikisource author Gottfried Leibniz The Life of Godfrey William von Leibnitz Full text of John M. Mackie's 1845 biography of Leibniz from Google Books.
 Online texts, including New Essays, the correspondence with Clarke, and many others in easiertoread versions.
 Leibnitiana — Gregory Brown.
 Lloyd Strickland's web page. Scroll down for many Leibniz links.
 Leibniztranslations.com — Original Leibniz translations of many works including many never before translated into English
 Leibnizmenu: useful links
 Internet Encyclopedia of Philosophy: "Leibniz" — Douglas Burnham.

Stanford Encyclopedia of Philosophy. Leibniz on:
 Ethics — Andrew Youpa.
 Causation — Marc Bobro.
 Problem of evil — Michael Murray.
 Philosophy of mind — Kulstad and Carlin.
 Encyclopædia Britannica, 11th ed.: "Leibniz."
 Leibniz Prize.
 Table of contents for the Leibniz Review, 1998–.
 European Graduate School — Gottfried Leibniz.
 Books and Writers: Brief Leibniz biography and bibliography.
 Sundry comments,
often mentioning Leibniz, prompted by:
 Schirrmacher, Frank, "WakeUp Call for Europe Tech," Frankfurter Allgemeine, 10.07.00.
 Harry Maugan's blog: Leibniz compared to Voltaire via Candide.
 Monadology: text with concordances and frequency list
 WebDeleuze:Leibniz: Lecture series by French philosopher Gilles Deleuze on Leibniz from 1980 in English and other languages.
Leibniz in Afrikaans: Gottfried Wilhelm
Leibniz
Leibniz in Arabic: غوتفريد لايبنتز
Leibniz in Azerbaijani: Qotfrid Leybnits
Leibniz in Bengali: গটফ্রিড লাইবনিৎস
Leibniz in Min Nan: Gottfried Leibniz
Leibniz in Bosnian: Gottfried Wilhelm
Leibniz
Leibniz in Breton: Gottfried Leibniz
Leibniz in Bulgarian: Готфрид Лайбниц
Leibniz in Catalan: Gottfried Wilhelm
Leibniz
Leibniz in Czech: Gottfried Wilhelm
Leibniz
Leibniz in Welsh: Gottfried Wilhelm von
Leibniz
Leibniz in Danish: Gottfried Wilhelm
Leibniz
Leibniz in German: Gottfried Wilhelm
Leibniz
Leibniz in Lower Sorbian: Gottfried Wilhelm
Leibniz
Leibniz in Estonian: Gottfried Wilhelm
Leibniz
Leibniz in Modern Greek (1453): Γκότφριντ
Βίλχελμ Λάιμπνιτς
Leibniz in Spanish: Gottfried Leibniz
Leibniz in Esperanto: Gottfried Wilhelm
Leibniz
Leibniz in Basque: Gottfried Wilhelm
Leibniz
Leibniz in Persian: گوتفرید لایبنیتز
Leibniz in French: Gottfried Wilhelm von
Leibniz
Leibniz in Friulian: Gottfried Leibniz
Leibniz in Galician: Gottfried Wilhelm
Leibniz
Leibniz in Korean: 고트프리트 라이프니츠
Leibniz in Hindi: गाटफ्रीड लैबनिट्ज़
Leibniz in Croatian: Gottfried Leibniz
Leibniz in Ido: Gottfried Wilhelm Leibniz
Leibniz in Indonesian: Gottfried Leibniz
Leibniz in Interlingua (International Auxiliary
Language Association): Gottfried Wilhelm von Leibniz
Leibniz in Icelandic: Gottfried Wilhelm von
Leibniz
Leibniz in Italian: Gottfried Leibniz
Leibniz in Hebrew: גוטפריד וילהלם לייבניץ
Leibniz in Georgian: გოტფრიდ ლაიბნიცი
Leibniz in Latin: Godefridus Guilielmus
Leibnitius
Leibniz in Luxembourgish: Gottfried Wilhelm
Leibniz
Leibniz in Lithuanian: Gottfried Leibniz
Leibniz in Hungarian: Gottfried Wilhelm
Leibniz
Leibniz in Macedonian: Готфрид Лајбниц
Leibniz in Marathi: गॉटफ्रीड लाइब्नित्स
Leibniz in Dutch: Gottfried Wilhelm
Leibniz
Leibniz in Japanese: ゴットフリート・ライプニッツ
Leibniz in Norwegian: Gottfried Leibniz
Leibniz in Norwegian Nynorsk: Gottfried
Leibniz
Leibniz in Uighur: ﻟﯧﻴﺒﯩﻨﯩﺰ
Leibniz in Piemontese: Gottfried Wilhelm von
Leibniz
Leibniz in Polish: Gottfried Wilhelm
Leibniz
Leibniz in Portuguese: Gottfried Leibniz
Leibniz in Romanian: Gottfried Wilhelm von
Leibniz
Leibniz in Russian: Лейбниц, Готфрид
Вильгельм
Leibniz in Sardinian: Gottfried Leibniz
Leibniz in Scots: Gottfried Leibniz
Leibniz in Sicilian: Gottfried Leibniz
Leibniz in Simple English: Gottfried
Leibniz
Leibniz in Slovak: Gottfried Wilhelm
Leibniz
Leibniz in Slovenian: Gottfried Wilhelm
Leibniz
Leibniz in Serbian: Готфрид Вилхелм
Лајбниц
Leibniz in SerboCroatian: Gottfried
Leibniz
Leibniz in Finnish: Gottfried Leibniz
Leibniz in Swedish: Gottfried Wilhelm von
Leibniz
Leibniz in Tagalog: Gottfried Leibniz
Leibniz in Tamil: கோட்பிரீட் லைப்னிட்ஸ்
Leibniz in Thai: กอทท์ฟรีด วิลเฮล์ม
ไลบ์นิซ
Leibniz in Vietnamese: Gottfried Leibniz
Leibniz in Turkish: Gottfried Leibniz
Leibniz in Ukrainian: Лейбніц Ґотфрід
Вільгельм
Leibniz in Volapük: Gottfried Leibniz
Leibniz in Chinese: 戈特弗里德·莱布尼茨