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Gottfried Leibniz

Gottfried Leibniz

Overview
Gottfried Wilhelm Leibniz (sometimes von Leibniz) (ˈɡɔtfʁiːt ˈvɪlhɛlm fɔn ˈlaɪbnɪts or ˈlaɪpnɪts) (July 1, 1646 – November 14, 1716) was a German
Germans
The Germans are a Germanic ethnic group native to Central Europe. The English term Germans has referred to the German-speaking population of the Holy Roman Empire since the Late Middle Ages....

 philosopher and mathematician
Mathematician
A mathematician is a person whose primary area of study is the field of mathematics. Mathematicians are concerned with quantity, structure, space, and change....

. He wrote in different languages, primarily in Latin
Latin
Latin is an Italic language originally spoken in Latium and Ancient Rome. It, along with most European languages, is a descendant of the ancient Proto-Indo-European language. Although it is considered a dead language, a number of scholars and members of the Christian clergy speak it fluently, and...

 (~40%), French
French language
French is a Romance language spoken as a first language in France, the Romandy region in Switzerland, Wallonia and Brussels in Belgium, Monaco, the regions of Quebec and Acadia in Canada, and by various communities elsewhere. Second-language speakers of French are distributed throughout many parts...

 (~30%) and German
German language
German is a West Germanic language, related to and classified alongside English and Dutch. With an estimated 90 – 98 million native speakers, German is one of the world's major languages and is the most widely-spoken first language in the European Union....

 (~15%).

Leibniz occupies a prominent place in the history of mathematics
History of mathematics
The area of study known as the history of mathematics is primarily an investigation into the origin of discoveries in mathematics and, to a lesser extent, an investigation into the mathematical methods and notation of the past....

 and the history of philosophy
History of philosophy
The history of philosophy is the study of philosophical ideas and concepts through time. Issues specifically related to history of philosophy might include : How can changes in philosophy be accounted for historically? What drives the development of thought in its historical context? To what...

. He developed the infinitesimal calculus
Infinitesimal calculus
Infinitesimal calculus is the part of mathematics concerned with finding slope of curves, areas under curves, minima and maxima, and other geometric and analytic problems. It was independently developed by Gottfried Leibniz and Isaac Newton starting in the 1660s...

 independently of Isaac Newton, and Leibniz's mathematical notation has been widely used ever since it was published.
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Quotations

Omne possibile exigit existere.

Everything that is possible demands to exist.
Encyclopedia
Gottfried Wilhelm Leibniz (sometimes von Leibniz) (ˈɡɔtfʁiːt ˈvɪlhɛlm fɔn ˈlaɪbnɪts or ˈlaɪpnɪts) (July 1, 1646 – November 14, 1716) was a German
Germans
The Germans are a Germanic ethnic group native to Central Europe. The English term Germans has referred to the German-speaking population of the Holy Roman Empire since the Late Middle Ages....

 philosopher and mathematician
Mathematician
A mathematician is a person whose primary area of study is the field of mathematics. Mathematicians are concerned with quantity, structure, space, and change....

. He wrote in different languages, primarily in Latin
Latin
Latin is an Italic language originally spoken in Latium and Ancient Rome. It, along with most European languages, is a descendant of the ancient Proto-Indo-European language. Although it is considered a dead language, a number of scholars and members of the Christian clergy speak it fluently, and...

 (~40%), French
French language
French is a Romance language spoken as a first language in France, the Romandy region in Switzerland, Wallonia and Brussels in Belgium, Monaco, the regions of Quebec and Acadia in Canada, and by various communities elsewhere. Second-language speakers of French are distributed throughout many parts...

 (~30%) and German
German language
German is a West Germanic language, related to and classified alongside English and Dutch. With an estimated 90 – 98 million native speakers, German is one of the world's major languages and is the most widely-spoken first language in the European Union....

 (~15%).

Leibniz occupies a prominent place in the history of mathematics
History of mathematics
The area of study known as the history of mathematics is primarily an investigation into the origin of discoveries in mathematics and, to a lesser extent, an investigation into the mathematical methods and notation of the past....

 and the history of philosophy
History of philosophy
The history of philosophy is the study of philosophical ideas and concepts through time. Issues specifically related to history of philosophy might include : How can changes in philosophy be accounted for historically? What drives the development of thought in its historical context? To what...

. He developed the infinitesimal calculus
Infinitesimal calculus
Infinitesimal calculus is the part of mathematics concerned with finding slope of curves, areas under curves, minima and maxima, and other geometric and analytic problems. It was independently developed by Gottfried Leibniz and Isaac Newton starting in the 1660s...

 independently of Isaac Newton, and Leibniz's mathematical notation has been widely used ever since it was published. He became one of the most prolific inventors in the field of mechanical calculator
Mechanical calculator
A mechanical calculator is a device used to perform the basic operations of arithmetic. Mechanical calculators are comparable in size to small desktop computers and have been rendered obsolete by the advent of the electronic calculator....

s. While working on adding automatic multiplication and division to Pascal's calculator
Pascal's calculator
Blaise Pascal invented the mechanical calculator in 1642. He conceived it while trying to help his father who had been assigned the task of reorganizing the tax revenues of the French province of Haute-Normandie ; first called Arithmetic Machine, Pascal's Calculator and later Pascaline, it could...

, he was the first to describe a pinwheel calculator
Pinwheel calculator
A Pinwheel calculator was a class of mechanical calculator popular in the 19th and 20th century using, for its calculating engine, a set of wheels that had an adjustable number of teeth...

 in 1685 and invented the Leibniz wheel
Leibniz wheel
A Leibniz wheel or stepped drum was a cylinder with a set of teeth of incremental length which, when coupled to a counting wheel, was used in the calculating engine of a class of mechanical calculators...

, used in the arithmometer
Arithmometer
An Arithmometer or Arithmomètre was a mechanical calculator that could add and subtract directly and could perform long multiplications and divisions effectively by using a movable accumulator for the result. Patented in France by Thomas de Colmar in 1820 and manufactured from 1851 to 1915, it...

, the first mass-produced mechanical calculator. He also refined the binary number system, which is at the foundation of virtually all digital computers. In philosophy, Leibniz is mostly noted for his optimism, e.g., his conclusion that our Universe
Universe
The Universe is commonly defined as the totality of everything that exists, including all matter and energy, the planets, stars, galaxies, and the contents of intergalactic space. Definitions and usage vary and similar terms include the cosmos, the world and nature...

 is, in a restricted sense, the best possible one that God
God
God is the English name given to a singular being in theistic and deistic religions who is either the sole deity in monotheism, or a single deity in polytheism....

 could have created. Leibniz, along with René Descartes
René Descartes
René Descartes ; was a French philosopher and writer who spent most of his adult life in the Dutch Republic. He has been dubbed the 'Father of Modern Philosophy', and much subsequent Western philosophy is a response to his writings, which are studied closely to this day...

 and Baruch Spinoza
Baruch Spinoza
Baruch de Spinoza and later Benedict de Spinoza was a Dutch Jewish philosopher. Revealing considerable scientific aptitude, the breadth and importance of Spinoza's work was not fully realized until years after his death...

, was one of the three great 17th century advocates of rationalism
Rationalism
In epistemology and in its modern sense, rationalism is "any view appealing to reason as a source of knowledge or justification" . In more technical terms, it is a method or a theory "in which the criterion of the truth is not sensory but intellectual and deductive"...

. The work of Leibniz anticipated modern logic
Logic
In philosophy, Logic is the formal systematic study of the principles of valid inference and correct reasoning. Logic is used in most intellectual activities, but is studied primarily in the disciplines of philosophy, mathematics, semantics, and computer science...

 and analytic philosophy
Analytic philosophy
Analytic philosophy is a generic term for a style of philosophy that came to dominate English-speaking countries in the 20th century...

, but his philosophy also looks back to the scholastic
Scholasticism
Scholasticism is a method of critical thought which dominated teaching by the academics of medieval universities in Europe from about 1100–1500, and a program of employing that method in articulating and defending orthodoxy in an increasingly pluralistic context...

 tradition, in which conclusions are produced by applying reason to first principles or prior definitions rather than to empirical evidence. Leibniz made major contributions to physics
Physics
Physics is a natural science that involves the study of matter and its motion through spacetime, along with related concepts such as energy and force. More broadly, it is the general analysis of nature, conducted in order to understand how the universe behaves.Physics is one of the oldest academic...

 and technology
Technology
Technology is the making, usage, and knowledge of tools, machines, techniques, crafts, systems or methods of organization in order to solve a problem or perform a specific function. It can also refer to the collection of such tools, machinery, and procedures. The word technology comes ;...

, and anticipated notions that surfaced much later in biology
Biology
Biology is a natural science concerned with the study of life and living organisms, including their structure, function, growth, origin, evolution, distribution, and taxonomy. Biology is a vast subject containing many subdivisions, topics, and disciplines...

, medicine
Medicine
Medicine is the science and art of healing. It encompasses a variety of health care practices evolved to maintain and restore health by the prevention and treatment of illness....

, geology
Geology
Geology is the science comprising the study of solid Earth, the rocks of which it is composed, and the processes by which it evolves. Geology gives insight into the history of the Earth, as it provides the primary evidence for plate tectonics, the evolutionary history of life, and past climates...

, probability theory
Probability theory
Probability theory is the branch of mathematics concerned with analysis of random phenomena. The central objects of probability theory are random variables, stochastic processes, and events: mathematical abstractions of non-deterministic events or measured quantities that may either be single...

, psychology
Psychology
Psychology is the study of the mind and behavior. Its immediate goal is to understand individuals and groups by both establishing general principles and researching specific cases. For many, the ultimate goal of psychology is to benefit society...

, linguistics
Linguistics
Linguistics is the scientific study of human language. Linguistics can be broadly broken into three categories or subfields of study: language form, language meaning, and language in context....

, and information science
Information science
-Introduction:Information science is an interdisciplinary science primarily concerned with the analysis, collection, classification, manipulation, storage, retrieval and dissemination of information...

. He wrote works on politics
Politics
Politics is a process by which groups of people make collective decisions. The term is generally applied to the art or science of running governmental or state affairs, including behavior within civil governments, but also applies to institutions, fields, and special interest groups such as the...

, law
Law
Law is a system of rules and guidelines which are enforced through social institutions to govern behavior, wherever possible. It shapes politics, economics and society in numerous ways and serves as a social mediator of relations between people. Contract law regulates everything from buying a bus...

, ethics
Ethics
Ethics, also known as moral philosophy, is a branch of philosophy that addresses questions about morality—that is, concepts such as good and evil, right and wrong, virtue and vice, justice and crime, etc.Major branches of ethics include:...

, theology
Theology
Theology is the systematic and rational study of religion and its influences and of the nature of religious truths, or the learned profession acquired by completing specialized training in religious studies, usually at a university or school of divinity or seminary.-Definition:Augustine of Hippo...

, history
History
History is the discovery, collection, organization, and presentation of information about past events. History can also mean the period of time after writing was invented. Scholars who write about history are called historians...

, philosophy
Philosophy
Philosophy is the study of general and fundamental problems, such as those connected with existence, knowledge, values, reason, mind, and language. Philosophy is distinguished from other ways of addressing such problems by its critical, generally systematic approach and its reliance on rational...

, and philology
Philology
Philology is the study of language in written historical sources; it is a combination of literary studies, history and linguistics.Classical philology is the philology of Greek and Classical Latin...

. Leibniz's contributions to this vast array of subjects were scattered in various learned journals, in tens of thousands of letters, and in unpublished manuscripts. As of 2011, there is no complete gathering of the writings of Leibniz.

Early life


Gottfried Leibniz was born on July 1, 1646 in Leipzig
Leipzig
Leipzig Leipzig has always been a trade city, situated during the time of the Holy Roman Empire at the intersection of the Via Regia and Via Imperii, two important trade routes. At one time, Leipzig was one of the major European centres of learning and culture in fields such as music and publishing...

, Saxony
Saxony
The Free State of Saxony is a landlocked state of Germany, contingent with Brandenburg, Saxony Anhalt, Thuringia, Bavaria, the Czech Republic and Poland. It is the tenth-largest German state in area, with of Germany's sixteen states....

 (at the end of the Thirty Years' War
Thirty Years' War
The Thirty Years' War was fought primarily in what is now Germany, and at various points involved most countries in Europe. It was one of the most destructive conflicts in European history....

), to Friedrich Leibnütz (Leibniz changed his name) and Catherina Schmuck. Friedrich noted in his family journal: "On Sunday 21 June [NS: 1 July] 1646, my son Gottfried Wilhelm is born into the world after six in the evening, a quarter to seven [ein Viertel uff sieben], Aquarius rising." His father, who was of Sorbian
Sorbs
Sorbs are a Western Slavic people of Central Europe living predominantly in Lusatia, a region on the territory of Germany and Poland. In Germany they live in the states of Brandenburg and Saxony. They speak the Sorbian languages - closely related to Polish and Czech - officially recognized and...

 ancestry, died when Leibniz was six years old, and from that point on he was raised by his mother. Her teachings influenced Leibniz's philosophical thoughts in his later life.

Leibniz's father had been a Professor of Moral Philosophy at the University of Leipzig
University of Leipzig
The University of Leipzig , located in Leipzig in the Free State of Saxony, Germany, is one of the oldest universities in the world and the second-oldest university in Germany...

 and Leibniz inherited his father's personal library. He was given free access to this from the age of seven. While Leibniz's schoolwork focused on a small canon of authorities, his father's library enabled him to study a wide variety of advanced philosophical and theological works – ones that he would not have otherwise been able to read until his college years. Access to his father's library, largely written in Latin, also led to his proficiency in the Latin language. Leibniz was proficient in Latin by the age of 12, and he composed three hundred hexameters of Latin verse in a single morning for a special event at school at the age of 13.

He enrolled in his father's former university at age 15, and he completed his bachelor's degree
Bachelor's degree
A bachelor's degree is usually an academic degree awarded for an undergraduate course or major that generally lasts for three or four years, but can range anywhere from two to six years depending on the region of the world...

 in philosophy in December of 1662. He defended his Disputatio Metaphysica de Principio Individui, which addressed the Principle of individuation
Principle of individuation
The Principle of Individuation is a criterion which supposedly individuates or numerically distinguishes the members of the kind for which it is given, i.e. by which we can supposedly determine, regarding any kind of thing, when we have more than one of them or not. It is also known as a...

, on June 9, 1663. Leibniz earned his master's degree
Master's degree
A master's is an academic degree granted to individuals who have undergone study demonstrating a mastery or high-order overview of a specific field of study or area of professional practice...

 in philosophy on February 7, 1664. He published and defended a dissertation Specimen Quaestionum Philosophicarum ex Jure collectarum, arguing for both a theoretical and a pedagogical relationship between philosophy and law, in December 1664. After one year of legal studies, he was awarded his bachelor's degree in Law on September 28, 1665.

In 1666, (at age 20), Leibniz published his first book, On the Art of Combinations
De Arte Combinatoria
The Dissertatio de arte combinatoria is an early work by Gottfried Leibniz published in 1666 in Leipzig. It is an extended version of his doctoral dissertation, written before the author had seriously undertaken the study of mathematics. The booklet was reissued without Leibniz' consent in 1690,...

, the first part of which was also his habilitation
Habilitation
Habilitation is the highest academic qualification a scholar can achieve by his or her own pursuit in several European and Asian countries. Earned after obtaining a research doctorate, such as a PhD, habilitation requires the candidate to write a professorial thesis based on independent...

 thesis in philosophy. His next goal was to earn his license and doctorate in Law, which normally required three years of study then. In 1666, the University of Leipzig turned down Leibniz's doctoral application and refused to grant him a doctorate in law, most likely due to his relative youth (he was 21 years old at the time). Leibniz subsequently left Leipzig.

Leibniz then enrolled in the University of Altdorf
University of Altdorf
The University of Altdorf was a university in Altdorf bei Nürnberg, a small town outside Nuremberg. It was founded in the late 16th century, received university privileges in 1622 and was closed in 1809 by Maximilian I Joseph of Bavaria....

, and almost immediately he submitted a thesis, which he had probably been working on earlier in Leipzig. The title of his thesis was Disputatio de Casibus perplexis in Jure. Leibniz earned his license to practice law and his Doctorate in Law in November of 1666. He next declined the offer of an academic appointment at Altdorf, saying that "my thoughts were turned in an entirely different direction.

As an adult, Leibniz often introduced himself as "Gottfried von Leibniz". Also many posthumously-published editions of his writings presented his name on the title page as "Freiherr
Freiherr
The German titles Freiherr and Freifrau and Freiin are titles of nobility, used preceding a person's given name or, after 1919, before the surname...

 G. W. von Leibniz." However, no document has ever been found from any contemporary government that stated his appointment to any form of nobility
Nobility
Nobility is a social class which possesses more acknowledged privileges or eminence than members of most other classes in a society, membership therein typically being hereditary. The privileges associated with nobility may constitute substantial advantages over or relative to non-nobles, or may be...

.

1666–74


Leibniz's first position was as a salaried alchemist
Alchemy
Alchemy is an influential philosophical tradition whose early practitioners’ claims to profound powers were known from antiquity. The defining objectives of alchemy are varied; these include the creation of the fabled philosopher's stone possessing powers including the capability of turning base...

 in Nuremberg
Nuremberg
Nuremberg[p] is a city in the German state of Bavaria, in the administrative region of Middle Franconia. Situated on the Pegnitz river and the Rhine–Main–Danube Canal, it is located about north of Munich and is Franconia's largest city. The population is 505,664...

, 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
Mainz
Mainz under the Holy Roman Empire, and previously was a Roman fort city which commanded the west bank of the Rhine and formed part of the northernmost frontier of the Roman Empire...

, Johann Philipp von Schönborn
Johann Philipp von Schönborn
Johann Philipp von Schönborn was the Archbishop-Elector of Mainz from 1647 until 1673, the Bishop of Würzburg from 1642 until 1673, and the Bishop of Worms from 1663 until 1673....

. 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 diplomat
Diplomat
A diplomat is a person appointed by a state to conduct diplomacy with another state or international organization. The main functions of diplomats revolve around the representation and protection of the interests and nationals of the sending state, as well as the promotion of information and...

ic role. He published an essay, under the pseudonym of a fictitious Polish
Poland
Poland , officially the Republic of Poland , is a country in Central Europe bordered by Germany to the west; the Czech Republic and Slovakia to the south; Ukraine, Belarus and Lithuania to the east; and the Baltic Sea and Kaliningrad Oblast, a Russian exclave, to the north...

 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
Louis XIV of France
Louis XIV , known as Louis the Great or the Sun King , was a Bourbon monarch who ruled as King of France and Navarre. His reign, from 1643 to his death in 1715, began at the age of four and lasted seventy-two years, three months, and eighteen days...

, backed by French military and economic might. Meanwhile, the Thirty Years' War
Thirty Years' War
The Thirty Years' War was fought primarily in what is now Germany, and at various points involved most countries in Europe. It was one of the most destructive conflicts in European history....

 had left German-speaking Europe exhausted, fragmented, and economically backward. Leibniz proposed to protect German-speaking Europe by distracting Louis as follows. France would be invited to take Egypt
Egypt
Egypt , officially the Arab Republic of Egypt, Arabic: , is a country mainly in North Africa, with the Sinai Peninsula forming a land bridge in Southwest Asia. Egypt is thus a transcontinental country, and a major power in Africa, the Mediterranean Basin, the Middle East and the Muslim world...

 as a stepping stone towards an eventual conquest of the Dutch East Indies
Dutch East Indies
The Dutch East Indies was a Dutch colony that became modern Indonesia following World War II. It was formed from the nationalised colonies of the Dutch East India Company, which came under the administration of the Netherlands government in 1800....

. 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
Paris
Paris is the capital and largest city in France, situated on the river Seine, in northern France, at the heart of the Île-de-France region...

 for discussion, but the plan was soon overtaken by the outbreak of the Franco-Dutch War
Franco-Dutch War
The Franco-Dutch War, often called simply the Dutch War was a war fought by France, Sweden, the Bishopric of Münster, the Archbishopric of Cologne and England against the United Netherlands, which were later joined by the Austrian Habsburg lands, Brandenburg and Spain to form a quadruple alliance...

 and became 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. Soon after arriving, he met Dutch
Netherlands
The Netherlands is a constituent country of the Kingdom of the Netherlands, located mainly in North-West Europe and with several islands in the Caribbean. Mainland Netherlands borders the North Sea to the north and west, Belgium to the south, and Germany to the east, and shares maritime borders...

 physicist and mathematician Christiaan Huygens and realised that his own knowledge of mathematics and physics was patchy. With Huygens as mentor, he began a program of self-study that soon pushed him to making major contributions to both subjects, including inventing his version of the differential and integral calculus
Infinitesimal calculus
Infinitesimal calculus is the part of mathematics concerned with finding slope of curves, areas under curves, minima and maxima, and other geometric and analytic problems. It was independently developed by Gottfried Leibniz and Isaac Newton starting in the 1660s...

. He met Nicolas Malebranche
Nicolas Malebranche
Nicolas Malebranche ; was a French Oratorian and rationalist philosopher. In his works, he sought to synthesize the thought of St. Augustine and Descartes, in order to demonstrate the active role of God in every aspect of the world...

 and Antoine Arnauld
Antoine Arnauld
Antoine Arnauld — le Grand as contemporaries called him, to distinguish him from his father — was a French Roman Catholic theologian, philosopher, and mathematician...

, the leading French philosophers of the day, and studied the writings of Descartes and Pascal
Blaise Pascal
Blaise Pascal , was a French mathematician, physicist, inventor, writer and Catholic philosopher. He was a child prodigy who was educated by his father, a tax collector in Rouen...

, unpublished as well as published. He befriended a German mathematician, Ehrenfried Walther von Tschirnhaus
Ehrenfried Walther von Tschirnhaus
Ehrenfried Walther von Tschirnhaus was a German mathematician, physicist, physician, and philosopher...

; they corresponded for the rest of their lives. In 1675 he was admitted as a foreign honorary member of the French Academy of Sciences
French Academy of Sciences
The French Academy of Sciences is a learned society, founded in 1666 by Louis XIV at the suggestion of Jean-Baptiste Colbert, to encourage and protect the spirit of French scientific research...

, which he continued to follow mostly by correspondence.
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
London
London is the capital city of :England and the :United Kingdom, the largest metropolitan area in the United Kingdom, and the largest urban zone in the European Union by most measures. Located on the River Thames, London has been a major settlement for two millennia, its history going back to its...

, early in 1673. There Leibniz came into acquaintance of Henry Oldenburg
Henry Oldenburg
Henry Oldenburg was a German theologian known as a diplomat and a natural philosopher. He was one of the foremost intelligencers of Europe of the seventeenth century, with a network of correspondents to rival those of Fabri de Peiresc, Marin Mersenne and Ismaël Boulliau...

 and John Collins
John Collins (mathematician)
John Collins was an English mathematician. He is most known for his extensive correspondence with leading scientists and mathematicians such as Giovanni Alfonso Borelli, Gottfried Leibniz, Isaac Newton, and John Wallis...

. After demonstrating a calculating machine he had been designing and building since 1670 to the Royal Society
Royal Society
The Royal Society of London for Improving Natural Knowledge, known simply as the Royal Society, is a learned society for science, and is possibly the oldest such society in existence. Founded in November 1660, it was granted a Royal Charter by King Charles II as the "Royal Society of London"...

, 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
Brunswick-Lüneburg
The Duchy of Brunswick-Lüneburg , or more properly Duchy of Brunswick and Lüneburg, was an historical ducal state from the late Middle Ages until the late Early Modern era within the North-Western domains of the Holy Roman Empire of the German Nation, in what is now northern Germany...

 to visit Hanover proved fateful. Leibniz declined the invitation, but began corresponding with the Duke in 1671. In 1673, the Duke
John Frederick, Duke of Brunswick-Lüneburg
John Frederick was duke of Brunswick-Lüneburg and ruled over the Principality of Calenberg, a subdivision of the duchy, from 1665 until his death....

 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
Habsburg
The House of Habsburg , also found as Hapsburg, and also known as House of Austria is one of the most important royal houses of Europe and is best known for being an origin of all of the formally elected Holy Roman Emperors between 1438 and 1740, as well as rulers of the Austrian Empire and...

 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 was later accused by Newton of being 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
The Hague
The Hague is the capital city of the province of South Holland in the Netherlands. With a population of 500,000 inhabitants , it is the third largest city of the Netherlands, after Amsterdam and Rotterdam...

 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
Ethics (book)
Ethics is a philosophical book written by Benedict de Spinoza. It was written in Latin. Although it was published posthumously in 1677, it is his most famous work, and is considered his magnum opus....

. 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
Duke
A duke or duchess is a member of the nobility, historically of highest rank below the monarch, and historically controlling a duchy...

 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 accept Leibniz were the Electress Sophia of Hanover
Sophia of Hanover
Sophia of the Palatinate was an heiress to the crowns of England and Ireland and later the crown of Great Britain. She was declared heiress presumptive by the Act of Settlement 1701...

 (1630–1714), her daughter Sophia Charlotte of Hanover
Sophia Charlotte of Hanover
Sophia Charlotte of Hanover was the Queen consort of Prussia as wife of Frederick I of Prussia. She was the daughter of Ernst August, Elector of Hanover, and Sophia of the Palatinate...

 (1668–1705), the Queen of Prussia and his avowed disciple, and Caroline of Ansbach
Caroline of Ansbach
Caroline of Brandenburg-Ansbach was the queen consort of King George II of Great Britain.Her father, John Frederick, Margrave of Brandenburg-Ansbach, was the ruler of a small German state...

, the consort of her grandson, the future George II
George II of Great Britain
George II was King of Great Britain and Ireland, Duke of Brunswick-Lüneburg and Archtreasurer and Prince-elector of the Holy Roman Empire from 11 June 1727 until his death.George was the last British monarch born outside Great Britain. He was born and brought up in Northern Germany...

. To each of these women he was correspondent, adviser, and friend. In turn, they all approved of Leibniz more than did their spouses and the future king George I of Great Britain
George I of Great Britain
George I was King of Great Britain and Ireland from 1 August 1714 until his death, and ruler of the Duchy and Electorate of Brunswick-Lüneburg in the Holy Roman Empire from 1698....

.

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
Brunswick-Lüneburg
The Duchy of Brunswick-Lüneburg , or more properly Duchy of Brunswick and Lüneburg, was an historical ducal state from the late Middle Ages until the late Early Modern era within the North-Western domains of the Holy Roman Empire of the German Nation, in what is now northern Germany...

 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
Holy Roman Empire
The Holy Roman Empire was a realm that existed from 962 to 1806 in Central Europe.It was ruled by the Holy Roman Emperor. Its character changed during the Middle Ages and the Early Modern period, when the power of the emperor gradually weakened in favour of the princes...

. The British Act of Settlement 1701
Act of Settlement 1701
The Act of Settlement is an act of the Parliament of England that was passed in 1701 to settle the succession to the English throne on the Electress Sophia of Hanover and her Protestant heirs. The act was later extended to Scotland, as a result of the Treaty of Union , enacted in the Acts of Union...

 designated the Electress Sophia and her descent as the royal family of England, once both King William III
William III of England
William III & II was a sovereign Prince of Orange of the House of Orange-Nassau by birth. From 1672 he governed as Stadtholder William III of Orange over Holland, Zeeland, Utrecht, Guelders, and Overijssel of the Dutch Republic. From 1689 he reigned as William III over England and Ireland...

 and his sister-in-law and successor, Queen Anne
Anne of Great Britain
Anne ascended the thrones of England, Scotland and Ireland on 8 March 1702. On 1 May 1707, under the Act of Union, two of her realms, England and Scotland, were united as a single sovereign state, the Kingdom of Great Britain.Anne's Catholic father, James II and VII, was deposed during the...

, 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
Acta Eruditorum
Acta Eruditorum was the first scientific journal of the German lands, published from 1682 to 1782....

. 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 Ernest Augustus
Ernest Augustus, Elector of Brunswick-Lüneburg
Ernest Augustus was duke of Brunswick-Lüneburg and ruled over the Principality of Calenberg subdivision of the duchy. He was appointed prince-elector, but died before the appointment became effective...

 commissioned Leibniz to write a history of the House of Brunswick, going back to the time of Charlemagne
Charlemagne
Charlemagne was King of the Franks from 768 and Emperor of the Romans from 800 to his death in 814. He expanded the Frankish kingdom into an empire that incorporated much of Western and Central Europe. During his reign, he conquered Italy and was crowned by Pope Leo III on 25 December 800...

 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
Genealogy
Genealogy is the study of families and the tracing of their lineages and history. Genealogists use oral traditions, historical records, genetic analysis, and other records to obtain information about a family and to demonstrate kinship and pedigrees of its members...

 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 1708, John Keill
John Keill
John Keill was born in Edinburgh, Scotland, and was primarilya mathematician and important disciple of Isaac Newton. He studied at Edinburgh University, under David Gregory, and obtained his bachelors degree in 1692 with a distinction in physics and mathematics...

, 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
Newton v. Leibniz calculus controversy
The calculus controversy was an argument between 17th-century mathematicians Isaac Newton and Gottfried Leibniz over who had first invented calculus...

 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
Tsar
Tsar is a title used to designate certain European Slavic monarchs or supreme rulers. As a system of government in the Tsardom of Russia and Russian Empire, it is known as Tsarist autocracy, or Tsarism...

 Peter the Great
Peter I of Russia
Peter the Great, Peter I or Pyotr Alexeyevich Romanov Dates indicated by the letters "O.S." are Old Style. All other dates in this article are New Style. ruled the Tsardom of Russia and later the Russian Empire from until his death, jointly ruling before 1696 with his half-brother, Ivan V...

 stopped in Hanover and met Leibniz, who then took some interest in Russian matters for the rest of his life. In 1712, Leibniz began a two-year residence in Vienna
Vienna
Vienna is the capital and largest city of the Republic of Austria and one of the nine states of Austria. Vienna is Austria's primary city, with a population of about 1.723 million , and is by far the largest city in Austria, as well as its cultural, economic, and political centre...

, where he was appointed Imperial Court Councillor to the Habsburg
Habsburg
The House of Habsburg , also found as Hapsburg, and also known as House of Austria is one of the most important royal houses of Europe and is best known for being an origin of all of the formally elected Holy Roman Emperors between 1438 and 1740, as well as rulers of the Austrian Empire and...

s. On the death of Queen Anne in 1714, Elector George Louis became King George I of Great Britain
George I of Great Britain
George I was King of Great Britain and Ireland from 1 August 1714 until his death, and ruler of the Duchy and Electorate of Brunswick-Lüneburg in the Holy Roman Empire from 1698....

, 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.

Death


Leibniz died in Hanover
Hanover
Hanover or Hannover, on the river Leine, is the capital of the federal state of Lower Saxony , Germany and was once by personal union the family seat of the Hanoverian Kings of Great Britain, under their title as the dukes of Brunswick-Lüneburg...

 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
Prussian Academy of Sciences
The Prussian Academy of Sciences was an academy established in Berlin on 11 July 1700, four years after the Akademie der Künste or "Arts Academy", to which "Berlin Academy" may also refer.-Origins:...

, 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.

Personal life


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 put him in a bad light during the calculus controversy. On the other hand, he was charming, well-mannered, and not without humor and imagination. He had many friends and admirers all over Europe.

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, of which only the Théodicée of 1710 was published in his lifetime.

Leibniz dated his beginning as a philosopher to his Discourse on Metaphysics
Discourse on Metaphysics
The Discourse on Metaphysics is a short treatise by Gottfried Leibniz in which he develops a philosophy concerning physical substance, motion and resistance of bodies, and God's role within the universe...

, which he composed in 1686 as a commentary on a running dispute between Nicolas Malebranche
Nicolas Malebranche
Nicolas Malebranche ; was a French Oratorian and rationalist philosopher. In his works, he sought to synthesize the thought of St. Augustine and Descartes, in order to demonstrate the active role of God in every aspect of the world...

 and Antoine Arnauld
Antoine Arnauld
Antoine Arnauld — le Grand as contemporaries called him, to distinguish him from his father — was a French Roman Catholic theologian, philosopher, and mathematician...

. 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". Between 1695 and 1705, he composed his New Essays on Human Understanding
New Essays on Human Understanding
New Essays on Human Understanding is a chapter-by-chapter rebuttal by Gottfried Leibniz of John Locke's major work, An Essay Concerning Human Understanding. It is one of only two full-length works by Leibniz...

, a lengthy commentary on John Locke
John Locke
John Locke FRS , widely known as the Father of Liberalism, was an English philosopher and physician regarded as one of the most influential of Enlightenment thinkers. Considered one of the first of the British empiricists, following the tradition of Francis Bacon, he is equally important to social...

's 1690 An Essay Concerning Human Understanding
An Essay Concerning Human Understanding
First appearing in 1690 with the printed title An Essay Concerning Humane Understanding, An Essay Concerning Human Understanding by John Locke concerns the foundation of human knowledge and understanding. He describes the mind at birth as a blank slate filled later through experience...

, 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
Monadology
The Monadology is one of Gottfried Leibniz’s best known works representing his later philosophy. It is a short text which sketches in some 90 paragraphs a metaphysics of simple substances, or monads.- Text :...

, 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
Scholasticism
Scholasticism is a method of critical thought which dominated teaching by the academics of medieval universities in Europe from about 1100–1500, and a program of employing that method in articulating and defending orthodoxy in an increasingly pluralistic context...

 and Aristotelian
Aristotelianism
Aristotelianism is a tradition of philosophy that takes its defining inspiration from the work of Aristotle. The works of Aristotle were initially defended by the members of the Peripatetic school, and, later on, by the Neoplatonists, who produced many commentaries on Aristotle's writings...

 turn of mind betrayed the strong influence of one of his Leipzig
Leipzig
Leipzig Leipzig has always been a trade city, situated during the time of the Holy Roman Empire at the intersection of the Via Regia and Via Imperii, two important trade routes. At one time, Leipzig was one of the major European centres of learning and culture in fields such as music and publishing...

 professors, Jakob Thomasius
Jakob Thomasius
Jakob Thomasius was a German academic philosopher and jurist. He is now regarded as an important founding figure in the scholarly study of the history of philosophy. His views were eclectic, and were taken up by his son Christian Thomasius....

, who also supervised his BA thesis in philosophy. Leibniz also eagerly read Francisco Suárez
Francisco Suárez
Francisco Suárez was a Spanish Jesuit priest, philosopher and theologian, one of the leading figures of the School of Salamanca movement, and generally regarded among the greatest scholastics after Thomas Aquinas....

, a Spanish Jesuit
Society of Jesus
The Society of Jesus is a Catholic male religious order that follows the teachings of the Catholic Church. The members are called Jesuits, and are also known colloquially as "God's Army" and as "The Company," these being references to founder Ignatius of Loyola's military background and a...

 respected even in Lutheran
Lutheranism
Lutheranism is a major branch of Western Christianity that identifies with the theology of Martin Luther, a German reformer. Luther's efforts to reform the theology and practice of the church launched the Protestant Reformation...

 universities. Leibniz was deeply interested in the new methods and conclusions of Descartes, Huygens, Newton, and Boyle
Robert Boyle
Robert Boyle FRS was a 17th century natural philosopher, chemist, physicist, and inventor, also noted for his writings in theology. He has been variously described as English, Irish, or Anglo-Irish, his father having come to Ireland from England during the time of the English plantations of...

, 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
Logic
In philosophy, Logic is the formal systematic study of the principles of valid inference and correct reasoning. Logic is used in most intellectual activities, but is studied primarily in the disciplines of philosophy, mathematics, semantics, and computer science...

, and analytic
Analytic philosophy
Analytic philosophy is a generic term for a style of philosophy that came to dominate English-speaking countries in the 20th century...

 and linguistic philosophy
Linguistic philosophy
Linguistic philosophy describes the view that philosophical problems are problems which may be solved either by reforming language, or by understanding more about the language we presently use. The former position is that of ideal language philosophy, the latter the position of ordinary language...

 of the 20th century.

The Principles


Leibniz variously invoked one or another of seven fundamental philosophical Principles:
  • Identity
    Identity (mathematics)
    In mathematics, the term identity has several different important meanings:*An identity is a relation which is tautologically true. This means that whatever the number or value may be, the answer stays the same. For example, algebraically, this occurs if an equation is satisfied for all values of...

    /contradiction
    Contradiction
    In classical logic, a contradiction consists of a logical incompatibility between two or more propositions. It occurs when the propositions, taken together, yield two conclusions which form the logical, usually opposite inversions of each other...

    . If a proposition is true, then its negation is false and vice versa.
  • Identity of indiscernibles
    Identity of indiscernibles
    The identity of indiscernibles is an ontological principle which states that two or more objects or entities are identical if they have all their properties in common. That is, entities x and y are identical if any predicate possessed by x is also possessed by y and vice versa...

    . Two things are identical if and only if they share the same and only the same properties. Frequently invoked in modern logic and philosophy. The "identity of indiscernibles" is often referred to as Leibniz's Law. It has attracted the most controversy and criticism, especially from corpuscular philosophy and quantum mechanics.
  • Sufficient reason
    Principle of sufficient reason
    The principle of sufficient reason states that anything that happens does so for a reason: no state of affairs can obtain, and no statement can be true unless there is sufficient reason why it should not be otherwise...

    . "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."
  • Pre-established harmony
    Pre-established harmony
    Gottfried Leibniz's theory of pre-established harmony is a philosophical theory about causation under which every "substance" only affects itself, but all the substances in the world nevertheless seem to causally interact with each other because they have been programmed by God in advance to...

    . "[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.
  • Law of Continuity
    Law of Continuity
    The Law of Continuity is a heuristic principle introduced by Leibniz based on earlier work by Nicholas of Cusa and Johannes Kepler. It is the principle that "whatever succeeds for the finite, also succeeds for the infinite"...

    . Natura non saltum facit. A mathematical analog to this principle would proceed as follows: if a function
    Function (mathematics)
    In mathematics, a function associates one quantity, the argument of the function, also known as the input, with another quantity, the value of the function, also known as the output. A function assigns exactly one output to each input. The argument and the value may be real numbers, but they can...

     describes a transformation of something to which continuity applies, then its domain
    Domain (mathematics)
    In mathematics, the domain of definition or simply the domain of a function is the set of "input" or argument values for which the function is defined...

     and range
    Range (mathematics)
    In mathematics, the range of a function refers to either the codomain or the image of the function, depending upon usage. This ambiguity is illustrated by the function f that maps real numbers to real numbers with f = x^2. Some books say that range of this function is its codomain, the set of all...

     are both dense set
    Dense set
    In topology and related areas of mathematics, a subset A of a topological space X is called dense if any point x in X belongs to A or is a limit point of A...

    s.
  • Optimism
    Optimism
    The Oxford English Dictionary defines optimism as having "hopefulness and confidence about the future or successful outcome of something; a tendency to take a favourable or hopeful view." The word is originally derived from the Latin optimum, meaning "best." Being optimistic, in the typical sense...

    . "God assuredly always chooses the best."
  • Plenitude
    Plenitude principle
    The plenitude principle or principle of plenitude asserts that everything that can happen will happen.The historian of ideas Arthur Lovejoy was the first to discuss this philosophically important Principle explicitly, it back to Aristotle, who said that no possibilities which remain eternally...

    . "Leibniz believed that the best of all possible worlds would actualize every genuine possibility, and argued in Théodicée
    Théodicée
    Essais de Théodicée sur la bonté de Dieu, la liberté de l'homme et l'origine du mal , more simply known as Théodicée, is a book of philosophy by the famed polymath Gottfried Leibniz...

     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."


Leibniz would on occasion give a speech for a specific principle, but more often took them for granted.

The monads


Leibniz's best known contribution to metaphysics
Metaphysics
Metaphysics is a branch of philosophy concerned with explaining the fundamental nature of being and the world, although the term is not easily defined. Traditionally, metaphysics attempts to answer two basic questions in the broadest possible terms:...

 is his theory of monads
Monad (Greek philosophy)
Monad , according to the Pythagoreans, was a term for Divinity or the first being, or the totality of all beings, Monad being the source or the One meaning without division....

, as exposited in Monadologie
Monadology
The Monadology is one of Gottfried Leibniz’s best known works representing his later philosophy. It is a short text which sketches in some 90 paragraphs a metaphysics of simple substances, or monads.- Text :...

. Monads are to the metaphysical realm what atom
Atom
The atom is a basic unit of matter that consists of a dense central nucleus surrounded by a cloud of negatively charged electrons. The atomic nucleus contains a mix of positively charged protons and electrically neutral neutrons...

s are to the physical/phenomenal. They can also be compared to the corpuscles of the Mechanical Philosophy of René Descartes and others. Monads are the ultimate elements of the universe
Universe
The Universe is commonly defined as the totality of everything that exists, including all matter and energy, the planets, stars, galaxies, and the contents of intergalactic space. Definitions and usage vary and similar terms include the cosmos, the world and nature...

. The monads are "substantial forms of being" with the following properties: they are eternal, indecomposable, individual, subject to their own laws, un-interacting, and each reflecting the entire universe in a pre-established harmony
Pre-established harmony
Gottfried Leibniz's theory of pre-established harmony is a philosophical theory about causation under which every "substance" only affects itself, but all the substances in the world nevertheless seem to causally interact with each other because they have been programmed by God in advance to...

 (a historically important example of panpsychism
Panpsychism
In philosophy, panpsychism is the view that all matter has a mental aspect, or, alternatively, all objects have a unified center of experience or point of view...

). Monads are centers of force
Force
In physics, a force is any influence that causes an object to undergo a change in speed, a change in direction, or a change in shape. In other words, a force is that which can cause an object with mass to change its velocity , i.e., to accelerate, or which can cause a flexible object to deform...

; substance is force, while space
Space
Space is the boundless, three-dimensional extent in which objects and events occur and have relative position and direction. Physical space is often conceived in three linear dimensions, although modern physicists usually consider it, with time, to be part of a boundless four-dimensional continuum...

, matter
Matter
Matter is a general term for the substance of which all physical objects consist. Typically, matter includes atoms and other particles which have mass. A common way of defining matter is as anything that has mass and occupies volume...

, and motion
Motion (physics)
In physics, motion is a change in position of an object with respect to time. Change in action is the result of an unbalanced force. Motion is typically described in terms of velocity, acceleration, displacement and time . An object's velocity cannot change unless it is acted upon by a force, as...

 are merely phenomenal.

The ontological
Ontology
Ontology is the philosophical study of the nature of being, existence or reality as such, as well as the basic categories of being and their relations...

 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 pre-established 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 law
Scientific law
A scientific law is a statement that explains what something does in science just like Newton's law of universal gravitation. A scientific law must always apply under the same conditions, and implies a causal relationship between its elements. The law must be confirmed and broadly agreed upon...

s governing subatomic particle
Subatomic particle
In physics or chemistry, subatomic particles are the smaller particles composing nucleons and atoms. There are two types of subatomic particles: elementary particles, which are not made of other particles, and composite particles...

s.) 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
Free will
"To make my own decisions whether I am successful or not due to uncontrollable forces" -Troy MorrisonA pragmatic definition of free willFree will is the ability of agents to make choices free from certain kinds of constraints. The existence of free will and its exact nature and definition have long...

 is problematic. God
God
God is the English name given to a singular being in theistic and deistic religions who is either the sole deity in monotheism, or a single deity in polytheism....

, too, is a monad, and the existence of God
Existence of God
Arguments for and against the existence of God have been proposed by philosophers, theologians, scientists, and others. In philosophical terms, arguments for and against the existence of God involve primarily the sub-disciplines of epistemology and ontology , but also of the theory of value, since...

 can be inferred from the harmony prevailing among all other monads; God wills the pre-established harmony.

Monads are purported to having gotten rid of the problematic:
  • Interaction between mind
    Mind
    The concept of mind is understood in many different ways by many different traditions, ranging from panpsychism and animism to traditional and organized religious views, as well as secular and materialist philosophies. Most agree that minds are constituted by conscious experience and intelligent...

     and matter arising in the system of Descartes;
  • Lack of individuation
    Principle of individuation
    The Principle of Individuation is a criterion which supposedly individuates or numerically distinguishes the members of the kind for which it is given, i.e. by which we can supposedly determine, regarding any kind of thing, when we have more than one of them or not. It is also known as a...

     inherent to the system of Spinoza, which represents individual creatures as merely accidental.

Theodicy and optimism


(Note that the word "optimism" here is used in the classic sense of optimal, not in the mood-related sense, as being positively hopeful.)

The Theodicy
Théodicée
Essais de Théodicée sur la bonté de Dieu, la liberté de l'homme et l'origine du mal , more simply known as Théodicée, is a book of philosophy by the famed polymath Gottfried Leibniz...

 tries to justify the apparent imperfections of the world by claiming that it is optimal among all possible worlds
Best of all possible worlds
The phrase "the best of all possible worlds" was coined by the German polymath Gottfried Leibniz in his 1710 work Essais de Théodicée sur la bonté de Dieu, la liberté de l'homme et l'origine du mal...

. It must be the best possible and most balanced world, because it was created by an all powerful and all knowing God, who would not choose to create an imperfect world if a better world could be known to him or possible to exist. In effect, apparent flaws that can be identified in this world must exist in every possible world, because otherwise God would have chosen to create the world that excluded those flaws.

Leibniz asserted that the truths of theology (religion) and philosophy cannot contradict each other, since reason and faith are both "gifts of God" so that their conflict would imply God contending against himself. The Theodicy is Leibniz's attempt to reconcile his personal philosophical system with his interpretation of the tenets of Christianity. This project was motivated in part by Leibniz's belief, shared by many conservative philosophers and theologians during the Enlightenment
Age of Enlightenment
The Age of Enlightenment was an elite cultural movement of intellectuals in 18th century Europe that sought to mobilize the power of reason in order to reform society and advance knowledge. It promoted intellectual interchange and opposed intolerance and abuses in church and state...

, in the rational and enlightened nature of the Christian religion, at least as this was defined in tendentious comparisons between Christian and non Western or "primitive" religious practices and beliefs. It was also shaped by Leibniz's belief in the perfectibility of human nature (if humanity relied on correct philosophy and religion as a guide), and by his belief that metaphysical necessity must have a rational or logical foundation, even if this metaphysical causality seemed inexplicable in terms of physical necessity (the natural laws identified by science).

Because reason and faith must be entirely reconciled, any tenet of faith which could not be defended by reason must be rejected. Leibniz then approached one of the central criticisms of Christian theism: if God is all good, all wise and all powerful, how did evil come into the world? The answer (according to Leibniz) is that, while God is indeed unlimited in wisdom and power, his human creations, as creations, are limited both in their wisdom and in their will (power to act). This predisposes humans to false beliefs, wrong decisions and ineffective actions in the exercise of their free will. God does not arbitrarily inflict pain and suffering on humans; rather he permits both moral evil (sin) and physical evil (pain and suffering) as the necessary consequences of metaphysical evil (imperfection), as a means by which humans can identify and correct their erroneous decisions, and as a contrast to true good.

Further, although human actions flow from prior causes that ultimately arise in God, and therefore are known as a metaphysical certainty to God, an individual's free will is exercised within natural laws, where choices are merely contingently necessary, to be decided in the event by a "wonderful spontaneity" that provides individuals an escape from rigorous predestination.

The Theodicy was deemed illogical by the philosopher Bertrand Russell
Bertrand Russell
Bertrand Arthur William Russell, 3rd Earl Russell, OM, FRS was a British philosopher, logician, mathematician, historian, and social critic. At various points in his life he considered himself a liberal, a socialist, and a pacifist, but he also admitted that he had never been any of these things...

. Russell points out that moral and physical evil must result from metaphysical evil (imperfection). But imperfection is merely finitude or limitation; if existence is good, as Leibniz maintains, then the mere existence of evil requires that evil also be good. In addition, libertarian
Libertarianism (metaphysics)
Libertarianism is one of the main philosophical positions related to the problems of free will and determinism, which are part of the larger domain of metaphysics. In particular, libertarianism, which is an incompatibilist position, argues that free will is logically incompatible with a...

 Christian theology defines sin as not necessary but contingent, the result of free will. Russell maintains that Leibniz failed logically to show that metaphysical necessity (divine will) and human free will are not incompatible or contradictory.

The mathematician Paul du Bois-Reymond, in his "Leibnizian Thoughts in Modern Science", wrote that Leibniz thought of God as a mathematician
Mathematician
A mathematician is a person whose primary area of study is the field of mathematics. Mathematicians are concerned with quantity, structure, space, and change....

:

As is well known, the theory of the maxima and minima
Maxima and minima
In mathematics, the maximum and minimum of a function, known collectively as extrema , are the largest and smallest value that the function takes at a point either within a given neighborhood or on the function domain in its entirety .More generally, the...

 of functions
Function (mathematics)
In mathematics, a function associates one quantity, the argument of the function, also known as the input, with another quantity, the value of the function, also known as the output. A function assigns exactly one output to each input. The argument and the value may be real numbers, but they can...

 was indebted to him for the greatest progress through the discovery of the method of tangent
Tangent
In geometry, the tangent line to a plane curve at a given point is the straight line that "just touches" the curve at that point. More precisely, a straight line is said to be a tangent of a curve at a point on the curve if the line passes through the point on the curve and has slope where f...

s. 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
Calculus of variations
Calculus of variations is a field of mathematics that deals with extremizing functionals, as opposed to ordinary calculus which deals with functions. A functional is usually a mapping from a set of functions to the real numbers. Functionals are often formed as definite integrals involving unknown...

the question being to determine among an infinite number of possible worlds, that for which the sum of necessary evil is a minimum.


The statement that "we live in the best of all possible worlds" drew scorn, most notably from Voltaire
Voltaire
François-Marie Arouet , better known by the pen name Voltaire , was a French Enlightenment writer, historian and philosopher famous for his wit and for his advocacy of civil liberties, including freedom of religion, free trade and separation of church and state...

, who lampooned it in his comic novella Candide
Candide
Candide, ou l'Optimisme is a French satire first published in 1759 by Voltaire, a philosopher of the Age of Enlightenment. The novella has been widely translated, with English versions titled Candide: or, All for the Best ; Candide: or, The Optimist ; and Candide: or, Optimism...

 by having the character Dr. Pangloss (a parody of Leibniz and Maupertuis
Pierre Louis Maupertuis
Pierre-Louis Moreau de Maupertuis was a French mathematician, philosopher and man of letters. He became the Director of the Académie des Sciences, and the first President of the Berlin Academy of Science, at the invitation of Frederick the Great....

) repeat it like a mantra
Mantra
A mantra is a sound, syllable, word, or group of words that is considered capable of "creating transformation"...

. From this, the adjective "Panglossian" describes a person who believes that the world about us is the best possible one.

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.


Leibniz's calculus ratiocinator
Calculus ratiocinator
The Calculus Ratiocinator is a theoretical universal logical calculation framework, a concept described in the writings of Gottfried Leibniz, usually paired with his more frequently mentioned characteristica universalis, a universal conceptual language....

, which resembles symbolic logic
Mathematical logic
Mathematical logic is a subfield of mathematics with close connections to foundations of mathematics, theoretical computer science and philosophical logic. The field includes both the mathematical study of logic and the applications of formal logic to other areas of mathematics...

, can be viewed as a way of making such calculations feasible. Leibniz wrote memoranda 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
Gottlob Frege
Friedrich Ludwig Gottlob Frege was a German mathematician, logician and philosopher. He is considered to be one of the founders of modern logic, and made major contributions to the foundations of mathematics. He is generally considered to be the father of analytic philosophy, for his writings on...

 Begriffsschrift
Begriffsschrift
Begriffsschrift is a book on logic by Gottlob Frege, published in 1879, and the formal system set out in that book...

 and in writings by Charles Sanders Peirce and his students in the 1880s, and hence well after Boole
George Boole
George Boole was an English mathematician and philosopher.As the inventor of Boolean logic—the basis of modern digital computer logic—Boole is regarded in hindsight as a founder of the field of computer science. Boole said,...

 and De Morgan
Augustus De Morgan
Augustus De Morgan was a British mathematician and logician. He formulated De Morgan's laws and introduced the term mathematical induction, making its idea rigorous. The crater De Morgan on the Moon is named after him....

 began that logic in 1847.

Leibniz thought symbol
Symbol
A symbol is something which represents an idea, a physical entity or a process but is distinct from it. The purpose of a symbol is to communicate meaning. For example, a red octagon may be a symbol for "STOP". On a map, a picture of a tent might represent a campsite. Numerals are symbols for...

s 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
Infinitesimal calculus
Infinitesimal calculus is the part of mathematics concerned with finding slope of curves, areas under curves, minima and maxima, and other geometric and analytic problems. It was independently developed by Gottfried Leibniz and Isaac Newton starting in the 1660s...

 is an example of his skill in this regard. C.S. Peirce, a 19th-century pioneer of semiotics
Semiotics
Semiotics, also called semiotic studies or semiology, is the study of signs and sign processes , indication, designation, likeness, analogy, metaphor, symbolism, signification, and communication...

, shared Leibniz's passion for symbols and notation, and his belief that these are essential to a well-running 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 well known in his day, including Egyptian hieroglyphics, Chinese character
Chinese character
Chinese characters are logograms used in the writing of Chinese and Japanese , less frequently Korean , formerly Vietnamese , or other languages...

s, and the symbols of astronomy
Astronomy
Astronomy is a natural science that deals with the study of celestial objects and phenomena that originate outside the atmosphere of Earth...

 and chemistry
Chemistry
Chemistry is the science of matter, especially its chemical reactions, but also its composition, structure and properties. Chemistry is concerned with atoms and their interactions with other atoms, and particularly with the properties of chemical bonds....

, he deemed not real. Instead, he proposed the creation of a characteristica universalis
Characteristica universalis
The Latin term characteristica universalis, commonly interpreted as universal characteristic, or universal character in English, is a universal and formal language imagined by the German philosopher Gottfried Leibniz able to express mathematical, scientific, and metaphysical concepts...

 or "universal characteristic", built on an alphabet of human thought
Alphabet of human thought
The alphabet of human thought is a concept originally proposed by Gottfried Leibniz that provides a universal way to represent and analyze ideas and relationships, no matter how complicated, by breaking down their component pieces...

 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.


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
Mnemonic
A mnemonic , or mnemonic device, is any learning technique that aids memory. To improve long term memory, mnemonic systems are used to make memorization easier. Commonly encountered mnemonics are often verbal, such as a very short poem or a special word used to help a person remember something,...

 way to number any set of elementary concepts using the prime numbers. Leibniz's idea of reasoning through a universal language of symbols and calculations however remarkably foreshadows great 20th century developments in formal systems, such as Turing completeness
Turing completeness
In computability theory, a system of data-manipulation rules is said to be Turing complete or computationally universal if and only if it can be used to simulate any single-taped Turing machine and thus in principle any computer. A classic example is the lambda calculus...

, where computation was used to define equivalent universal languages (see Turing degree
Turing degree
In computer science and mathematical logic the Turing degree or degree of unsolvability of a set of natural numbers measures the level of algorithmic unsolvability of the set...

).

Because Leibniz was a mathematical novice when he first wrote about the characteristic, at first he did not conceive it as an algebra
Algebra
Algebra is the branch of mathematics concerning the study of the rules of operations and relations, and the constructions and concepts arising from them, including terms, polynomials, equations and algebraic structures...

 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), 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.

Formal logic



Leibniz is the most important logician between Aristotle and 1847, when George Boole
George Boole
George Boole was an English mathematician and philosopher.As the inventor of Boolean logic—the basis of modern digital computer logic—Boole is regarded in hindsight as a founder of the field of computer science. Boole said,...

 and Augustus De Morgan
Augustus De Morgan
Augustus De Morgan was a British mathematician and logician. He formulated De Morgan's laws and introduced the term mathematical induction, making its idea rigorous. The crater De Morgan on the Moon is named after him....

 each published books that began modern formal logic. Leibniz enunciated the principal properties of what we now call conjunction
Logical conjunction
In logic and mathematics, a two-place logical operator and, also known as logical conjunction, results in true if both of its operands are true, otherwise the value of false....

, disjunction, negation
Negation
In logic and mathematics, negation, also called logical complement, is an operation on propositions, truth values, or semantic values more generally. Intuitively, the negation of a proposition is true when that proposition is false, and vice versa. In classical logic negation is normally identified...

, identity
Identity (mathematics)
In mathematics, the term identity has several different important meanings:*An identity is a relation which is tautologically true. This means that whatever the number or value may be, the answer stays the same. For example, algebraically, this occurs if an equation is satisfied for all values of...

, set inclusion
Subset
In mathematics, especially in set theory, a set A is a subset of a set B if A is "contained" inside B. A and B may coincide. The relationship of one set being a subset of another is called inclusion or sometimes containment...

, and the empty set
Empty set
In mathematics, and more specifically set theory, the empty set is the unique set having no elements; its size or cardinality is zero. Some axiomatic set theories assure that the empty set exists by including an axiom of empty set; in other theories, its existence can be deduced...

. The principles of Leibniz's logic and, arguably, of his whole philosophy, reduce to two:
  1. All our ideas are compounded from a very small number of simple ideas, which form the alphabet of human thought
    Alphabet of human thought
    The alphabet of human thought is a concept originally proposed by Gottfried Leibniz that provides a universal way to represent and analyze ideas and relationships, no matter how complicated, by breaking down their component pieces...

    .
  2. Complex ideas proceed from these simple ideas by a uniform and symmetrical combination, analogous to arithmetical multiplication.


With regard to the first point, the number of simple ideas is much greater than Leibniz thought. As for the second, 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
Quantification
Quantification has several distinct senses. In mathematics and empirical science, it is the act of counting and measuring that maps human sense observations and experiences into members of some set of numbers. Quantification in this sense is fundamental to the scientific method.In logic,...

 variables
Variable (mathematics)
In mathematics, a variable is a value that may change within the scope of a given problem or set of operations. In contrast, a constant is a value that remains unchanged, though often unknown or undetermined. The concepts of constants and variables are fundamental to many areas of mathematics and...

 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
History of Western Philosophy (Russell)
A History of Western Philosophy by the philosopher Bertrand Russell is a conspectus of Western philosophy from the pre-Socratic philosophers to the early 20th century. Although criticised for its over-generalization and its omissions, particularly from the post-Cartesian period, it was a popular...

, Bertrand Russell
Bertrand Russell
Bertrand Arthur William Russell, 3rd Earl Russell, OM, FRS was a British philosopher, logician, mathematician, historian, and social critic. At various points in his life he considered himself a liberal, a socialist, and a pacifist, but he also admitted that he had never been any of these things...

 went so far as to claim 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
Function (mathematics)
In mathematics, a function associates one quantity, the argument of the function, also known as the input, with another quantity, the value of the function, also known as the output. A function assigns exactly one output to each input. The argument and the value may be real numbers, but they can...

 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
Abscissa
In mathematics, abscissa refers to that element of an ordered pair which is plotted on the horizontal axis of a two-dimensional Cartesian coordinate system, as opposed to the ordinate...

, ordinate
Ordinate
In mathematics, ordinate refers to that element of an ordered pair which is plotted on the vertical axis of a two-dimensional Cartesian coordinate system, as opposed to the abscissa...

, tangent
Tangent
In geometry, the tangent line to a plane curve at a given point is the straight line that "just touches" the curve at that point. More precisely, a straight line is said to be a tangent of a curve at a point on the curve if the line passes through the point on the curve and has slope where f...

, chord
Chord (geometry)
A chord of a circle is a geometric line segment whose endpoints both lie on the circumference of the circle.A secant or a secant line is the line extension of a chord. More generally, a chord is a line segment joining two points on any curve, such as but not limited to an ellipse...

, and the perpendicular
Surface normal
A surface normal, or simply normal, to a flat surface is a vector that is perpendicular to that surface. A normal to a non-flat surface at a point P on the surface is a vector perpendicular to the tangent plane to that surface at P. The word "normal" is also used as an adjective: a line normal to a...

. In the 18th century, "function" lost these geometrical associations.

Leibniz was the first to see that the coefficients of a system of linear equation
Linear equation
A linear equation is an algebraic equation in which each term is either a constant or the product of a constant and a single variable....

s could be arranged into an array, now called a matrix
Matrix (mathematics)
In mathematics, a matrix is a rectangular array of numbers, symbols, or expressions. The individual items in a matrix are called its elements or entries. An example of a matrix with six elements isMatrices of the same size can be added or subtracted element by element...

, which can be manipulated to find the solution of the system, if any. This method was later called Gaussian elimination
Gaussian elimination
In linear algebra, Gaussian elimination is an algorithm for solving systems of linear equations. It can also be used to find the rank of a matrix, to calculate the determinant of a matrix, and to calculate the inverse of an invertible square matrix...

. Leibniz's discoveries of Boolean algebra and of symbolic logic
Mathematical logic
Mathematical logic is a subfield of mathematics with close connections to foundations of mathematics, theoretical computer science and philosophical logic. The field includes both the mathematical study of logic and the applications of formal logic to other areas of mathematics...

, also relevant to mathematics, are discussed in the preceding section. A detailed treatment of Leibniz's writings on the calculus may be found in Bos (1974).

Calculus


Leibniz is credited, along with Sir Isaac Newton
Isaac Newton
Sir Isaac Newton PRS was an English physicist, mathematician, astronomer, natural philosopher, alchemist, and theologian, who has been "considered by many to be the greatest and most influential scientist who ever lived."...

, with the inventing of infinitesimal calculus
Infinitesimal calculus
Infinitesimal calculus is the part of mathematics concerned with finding slope of curves, areas under curves, minima and maxima, and other geometric and analytic problems. It was independently developed by Gottfried Leibniz and Isaac Newton starting in the 1660s...

 (that comprises differential and integral 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 graph of a function y = ƒ(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 cleverly suggestive notation for the calculus is probably his most enduring mathematical legacy. Leibniz did not publish anything about his calculus until 1684. The product rule
Product rule
In calculus, the product rule is a formula used to find the derivatives of products of two or more functions. It may be stated thus:'=f'\cdot g+f\cdot g' \,\! or in the Leibniz notation thus:...

 of differential calculus
Differential calculus
In mathematics, differential calculus is a subfield of calculus concerned with the study of the rates at which quantities change. It is one of the two traditional divisions of calculus, the other being integral calculus....

 is still called "Leibniz's law". In addition, the theorem that tells how and when to differentiate under the integral sign is called the Leibniz integral rule
Leibniz integral rule
In mathematics, Leibniz's rule for differentiation under the integral sign, named after Gottfried Leibniz, tells us that if we have an integral of the formthen for x \in the derivative of this integral is thus expressible...

.

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
Heuristic
Heuristic refers to experience-based techniques for problem solving, learning, and discovery. Heuristic methods are used to speed up the process of finding a satisfactory solution, where an exhaustive search is impractical...

 argument mainly grounded in geometric intuition. Leibniz also freely invoked mathematical entities he called infinitesimal
Infinitesimal
Infinitesimals have been used to express the idea of objects so small that there is no way to see them or to measure them. The word infinitesimal comes from a 17th century Modern Latin coinage infinitesimus, which originally referred to the "infinite-th" item in a series.In common speech, an...

s, manipulating them in ways suggesting that they had paradox
Paradox
Similar to Circular reasoning, A paradox is a seemingly true statement or group of statements that lead to a contradiction or a situation which seems to defy logic or intuition...

ical algebra
Algebra
Algebra is the branch of mathematics concerning the study of the rules of operations and relations, and the constructions and concepts arising from them, including terms, polynomials, equations and algebraic structures...

ic properties. George Berkeley
George Berkeley
George Berkeley , also known as Bishop Berkeley , was an Irish philosopher whose primary achievement was the advancement of a theory he called "immaterialism"...

, 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
Faith
Faith is confidence or trust in a person or thing, or a belief that is not based on proof. In religion, faith is a belief in a transcendent reality, a religious teacher, a set of teachings or a Supreme Being. Generally speaking, it is offered as a means by which the truth of the proposition,...

 as theology
Theology
Theology is the systematic and rational study of religion and its influences and of the nature of religious truths, or the learned profession acquired by completing specialized training in religious studies, usually at a university or school of divinity or seminary.-Definition:Augustine of Hippo...

 grounded in Christian
Christianity
Christianity is a monotheistic religion based on the life and teachings of Jesus as presented in canonical gospels and other New Testament writings...

 revelation
Revelation
In religion and theology, revelation is the revealing or disclosing, through active or passive communication with a supernatural or a divine entity...

.

From 1711 until his death, Leibniz's life was envenomed by a long dispute
Newton v. Leibniz calculus controversy
The calculus controversy was an argument between 17th-century mathematicians Isaac Newton and Gottfried Leibniz over who had first invented calculus...

 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.

Modern, rigorous calculus emerged in the 19th century, thanks to the efforts of Augustin Louis Cauchy
Augustin Louis Cauchy
Baron Augustin-Louis Cauchy was a French mathematician who was an early pioneer of analysis. He started the project of formulating and proving the theorems of infinitesimal calculus in a rigorous manner, rejecting the heuristic principle of the generality of algebra exploited by earlier authors...

, Bernhard Riemann
Bernhard Riemann
Georg Friedrich Bernhard Riemann was an influential German mathematician who made lasting contributions to analysis and differential geometry, some of them enabling the later development of general relativity....

, Karl Weierstrass
Karl Weierstrass
Karl Theodor Wilhelm Weierstrass was a German mathematician who is often cited as the "father of modern analysis".- Biography :Weierstrass was born in Ostenfelde, part of Ennigerloh, Province of Westphalia....

, and others, who based their work on the definition of a limit
Limit of a function
In mathematics, the limit of a function is a fundamental concept in calculus and analysis concerning the behavior of that function near a particular input....

 and on a precise understanding of real number
Real number
In mathematics, a real number is a value that represents a quantity along a continuum, such as -5 , 4/3 , 8.6 , √2 and π...

s. While Cauchy still used infinitesimals as a foundational concept for the calculus, following Weierstrass they were gradually eliminated from calculus, though continued to be studied outside of analysis. Infinitesimals survived in science and engineering, and even in rigorous mathematics, via the fundamental computational device known as the differential
Differential (mathematics)
In mathematics, the term differential has several meanings.-Basic notions:* In calculus, the differential represents a change in the linearization of a function....

. Beginning in 1960, Abraham Robinson
Abraham Robinson
Abraham Robinson was a mathematician who is most widely known for development of non-standard analysis, a mathematically rigorous system whereby infinitesimal and infinite numbers were incorporated into mathematics....

 worked out a rigorous foundation for Leibniz's infinitesimals, using model theory
Model theory
In mathematics, model theory is the study of mathematical structures using tools from mathematical logic....

. The resulting non-standard analysis
Non-standard analysis
Non-standard analysis is a branch of mathematics that formulates analysis using a rigorous notion of an infinitesimal number.Non-standard analysis was introduced in the early 1960s by the mathematician Abraham Robinson. He wrote:...

 can be seen as a belated vindication of Leibniz's mathematical reasoning.

Topology


Leibniz was the first to use the term analysis situs, later used in the 19th century to refer to what is now known as topology
Topology
Topology is a major area of mathematics concerned with properties that are preserved under continuous deformations of objects, such as deformations that involve stretching, but no tearing or gluing...

. There are two takes on this situation. On the one hand, Mates, citing a 1954 paper in German by Jacob Freudenthal
Jacob Freudenthal
Jacob Freudenthal was a German philosopher.Freudenthal received his education at the universities of Breslau and Göttingen, and at the rabbinical seminary of Breslau...

, 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
Seven Bridges of Königsberg
The Seven Bridges of Königsberg is a historically notable problem in mathematics. Its negative resolution by Leonhard Euler in 1735 laid the foundations of graph theory and prefigured the idea of topology....

 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 Hideaki Hirano argues differently, quoting Mandelbrot
Benoît Mandelbrot
Benoît B. Mandelbrot was a French American mathematician. Born in Poland, he moved to France with his family when he was a child...

:

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
Fractal
A fractal has been defined as "a rough or fragmented geometric shape that can be split into parts, each of which is a reduced-size copy of the whole," a property called self-similarity...

 promoted by Mandelbrot drew on Leibniz's notions of self-similarity
Self-similarity
In mathematics, a self-similar object is exactly or approximately similar to a part of itself . Many objects in the real world, such as coastlines, are statistically self-similar: parts of them show the same statistical properties at many scales...

 and the principle of continuity: natura non facit saltus
Natura non facit saltus
Natura non facit saltum has been a principle of natural philosophy since at least Aristotle's time. It appears as an axiom in the works of Gottfried Leibniz and Isaac Newton, the co-inventors of the infinitesimal calculus, see Law of Continuity...

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

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.

Physics



Leibniz contributed a fair amount to the statics and dynamics emerging about him, often disagreeing with Descartes and Newton
Isaac Newton
Sir Isaac Newton PRS was an English physicist, mathematician, astronomer, natural philosopher, alchemist, and theologian, who has been "considered by many to be the greatest and most influential scientist who ever lived."...

. He devised a new theory of motion
Motion (physics)
In physics, motion is a change in position of an object with respect to time. Change in action is the result of an unbalanced force. Motion is typically described in terms of velocity, acceleration, displacement and time . An object's velocity cannot change unless it is acted upon by a force, as...

 (dynamics
Dynamics (mechanics)
In the field of physics, the study of the causes of motion and changes in motion is dynamics. In other words the study of forces and why objects are in motion. Dynamics includes the study of the effect of torques on motion...

) based on kinetic energy
Kinetic energy
The kinetic energy of an object is the energy which it possesses due to its motion.It is defined as the work needed to accelerate a body of a given mass from rest to its stated velocity. Having gained this energy during its acceleration, the body maintains this kinetic energy unless its speed changes...

 and potential energy
Potential energy
In physics, potential energy is the energy stored in a body or in a system due to its position in a force field or due to its configuration. The SI unit of measure for energy and work is the Joule...

, 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.

Until the discovery of subatomic particles and the quantum mechanics
Quantum mechanics
Quantum mechanics, also known as quantum physics or quantum theory, is a branch of physics providing a mathematical description of much of the dual particle-like and wave-like behavior and interactions of energy and matter. It departs from classical mechanics primarily at the atomic and subatomic...

 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
Albert Einstein
Albert Einstein was a German-born theoretical physicist who developed the theory of general relativity, effecting a revolution in physics. For this achievement, Einstein is often regarded as the father of modern physics and one of the most prolific intellects in human history...

 by arguing, against Newton, that space
Space
Space is the boundless, three-dimensional extent in which objects and events occur and have relative position and direction. Physical space is often conceived in three linear dimensions, although modern physicists usually consider it, with time, to be part of a boundless four-dimensional continuum...

, time
Time
Time is a part of the measuring system used to sequence events, to compare the durations of events and the intervals between them, and to quantify rates of change such as the motions of objects....

 and motion are relative, not absolute. Leibniz's rule
Leibniz's rule
Leibniz's rule is either of the following:* The General Leibniz rule, a form of the product rule in differential calculus* The Leibniz integral rule...

 is an important, if often overlooked, step in many proofs in diverse fields of physics. The principle of sufficient reason
Principle of sufficient reason
The principle of sufficient reason states that anything that happens does so for a reason: no state of affairs can obtain, and no statement can be true unless there is sufficient reason why it should not be otherwise...

 has been invoked in recent cosmology
Cosmology
Cosmology is the discipline that deals with the nature of the Universe as a whole. Cosmologists seek to understand the origin, evolution, structure, and ultimate fate of the Universe at large, as well as the natural laws that keep it in order...

, and his identity of indiscernibles
Identity of indiscernibles
The identity of indiscernibles is an ontological principle which states that two or more objects or entities are identical if they have all their properties in common. That is, entities x and y are identical if any predicate possessed by x is also possessed by y and vice versa...

 in quantum mechanics, a field some even credit him with having anticipated in some sense. Those who advocate digital philosophy
Digital philosophy
Digital philosophy is a direction in philosophy and cosmology advocated by certain mathematicians and theoretical physicists, e.g., Gregory Chaitin, Edward Fredkin, Stephen Wolfram, and Konrad Zuse ....

, a recent direction in cosmology, claim Leibniz as a precursor.

The vis viva


Leibniz's vis viva
Vis viva
In the history of science, vis viva is an obsolete scientific theory that served as an elementary and limited early formulation of the principle of conservation of energy...

 (Latin for living force) is mv2, twice the modern kinetic energy
Kinetic energy
The kinetic energy of an object is the energy which it possesses due to its motion.It is defined as the work needed to accelerate a body of a given mass from rest to its stated velocity. Having gained this energy during its acceleration, the body maintains this kinetic energy unless its speed changes...

. He realized that the total energy would be conserved in certain mechanical systems, so he considered it an innate motive characteristic of matter. 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. In reality, both energy
Energy
In physics, energy is an indirectly observed quantity. It is often understood as the ability a physical system has to do work on other physical systems...

 and momentum
Momentum
In classical mechanics, linear momentum or translational momentum is the product of the mass and velocity of an object...

 are conserved, so the two approaches are equally valid.

Other natural science


By proposing that the earth has a molten core, he anticipated modern geology
Geology
Geology is the science comprising the study of solid Earth, the rocks of which it is composed, and the processes by which it evolves. Geology gives insight into the history of the Earth, as it provides the primary evidence for plate tectonics, the evolutionary history of life, and past climates...

. In embryology
Embryology
Embryology is a science which is about the development of an embryo from the fertilization of the ovum to the fetus stage...

, 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
Life sciences
The life sciences comprise the fields of science that involve the scientific study of living organisms, like plants, animals, and human beings. While biology remains the centerpiece of the life sciences, technological advances in molecular biology and biotechnology have led to a burgeoning of...

 and paleontology
Paleontology
Paleontology "old, ancient", ὄν, ὀντ- "being, creature", and λόγος "speech, thought") is the study of prehistoric life. It includes the study of fossils to determine organisms' evolution and interactions with each other and their environments...

, he revealed an amazing transformist intuition, fueled by his study of comparative anatomy and fossils. One of his principal works on this subject, Protogaea, unpublished in his lifetime, has recently been published in English for the first time. He worked out a primal organismic theory
Organismic theory
Organismic theories in psychology are a family of holistic psychological theories which tend to stress the organization, unity, and integration of human beings expressed through each individual's inherent growth or developmental tendency...

. 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
Psychology
Psychology is the study of the mind and behavior. Its immediate goal is to understand individuals and groups by both establishing general principles and researching specific cases. For many, the ultimate goal of psychology is to benefit society...

, he anticipated the distinction between conscious and unconscious
Unconscious mind
The unconscious mind is a term coined by the 18th century German romantic philosopher Friedrich Schelling and later introduced into English by the poet and essayist Samuel Taylor Coleridge...

 states. In public health, he advocated establishing a medical administrative authority, with powers over epidemiology
Epidemiology
Epidemiology is the study of health-event, health-characteristic, or health-determinant patterns in a population. It is the cornerstone method of public health research, and helps inform policy decisions and evidence-based medicine by identifying risk factors for disease and targets for preventive...

 and veterinary medicine
Veterinary medicine
Veterinary Medicine is the branch of science that deals with the prevention, diagnosis and treatment of disease, disorder and injury in non-human animals...

. 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
Balance of trade
The balance of trade is the difference between the monetary value of exports and imports of output in an economy over a certain period. It is the relationship between a nation's imports and exports...

. He even proposed something akin to what much later emerged as game theory
Game theory
Game theory is a mathematical method for analyzing calculated circumstances, such as in games, where a person’s success is based upon the choices of others...

. In sociology
Sociology
Sociology is the study of society. It is a social science—a term with which it is sometimes synonymous—which uses various methods of empirical investigation and critical analysis to develop a body of knowledge about human social activity...

 he laid the ground for communication theory
Communication theory
Communication theory is a field of information and mathematics that studies the technical process of information and the human process of human communication.- History :- Origins :...

.

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
Applied science
Applied science is the application of scientific knowledge transferred into a physical environment. Examples include testing a theoretical model through the use of formal science or solving a practical problem through the use of natural science....

. He designed wind-driven propellers and water pumps, mining machines to extract ore, hydraulic presses, lamps, submarines, clocks, etc. With Denis Papin
Denis Papin
Denis Papin was a French physicist, mathematician and inventor, best known for his pioneering invention of the steam digester, the forerunner of the steam engine and of the pressure cooker.-Life in France:...

, he invented a steam engine
Steam engine
A steam engine is a heat engine that performs mechanical work using steam as its working fluid.Steam engines are external combustion engines, where the working fluid is separate from the combustion products. Non-combustion heat sources such as solar power, nuclear power or geothermal energy may be...

. He even proposed a method for desalinating water. From 1680 to 1685, he struggled to overcome the chronic flooding that afflicted the ducal silver
Silver
Silver is a metallic chemical element with the chemical symbol Ag and atomic number 47. A soft, white, lustrous transition metal, it has the highest electrical conductivity of any element and the highest thermal conductivity of any metal...

 mines in the Harz Mountains, but did not succeed.

Computation


Leibniz may have been the first computer scientist and information theorist. Early in life, he documented the binary numeral system
Binary numeral system
The binary numeral system, or base-2 number system, represents numeric values using two symbols, 0 and 1. More specifically, the usual base-2 system is a positional notation with a radix of 2...

 (base
Radix
In mathematical numeral systems, the base or radix for the simplest case is the number of unique digits, including zero, that a positional numeral system uses to represent numbers. For example, for the decimal system the radix is ten, because it uses the ten digits from 0 through 9.In any numeral...

 2), then revisited that system throughout his career. He anticipated Lagrangian interpolation
Lagrange polynomial
In numerical analysis, Lagrange polynomials are used for polynomial interpolation. For a given set of distinct points x_j and numbers y_j, the Lagrange polynomial is the polynomial of the least degree that at each point x_j assumes the corresponding value y_j...

 and algorithmic information theory
Algorithmic information theory
Algorithmic information theory is a subfield of information theory and computer science that concerns itself with the relationship between computation and information...

. His calculus ratiocinator
Calculus ratiocinator
The Calculus Ratiocinator is a theoretical universal logical calculation framework, a concept described in the writings of Gottfried Leibniz, usually paired with his more frequently mentioned characteristica universalis, a universal conceptual language....

 anticipated aspects of the universal Turing machine
Universal Turing machine
In computer science, a universal Turing machine is a Turing machine that can simulate an arbitrary Turing machine on arbitrary input. The universal machine essentially achieves this by reading both the description of the machine to be simulated as well as the input thereof from its own tape. Alan...

. In 1934, Norbert Wiener
Norbert Wiener
Norbert Wiener was an American mathematician.A famous child prodigy, Wiener later became an early researcher in stochastic and noise processes, contributing work relevant to electronic engineering, electronic communication, and control systems.Wiener is regarded as the originator of cybernetics, a...

 claimed to have found in Leibniz's writings a mention of the concept of feedback
Feedback
Feedback describes the situation when output from an event or phenomenon in the past will influence an occurrence or occurrences of the same Feedback describes the situation when output from (or information about the result of) an event or phenomenon in the past will influence an occurrence or...

, central to Wiener's later cybernetic
Cybernetics
Cybernetics is the interdisciplinary study of the structure of regulatory systems. Cybernetics is closely related to information theory, control theory and systems theory, at least in its first-order form...

 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
Stepped Reckoner
The Step Reckoner was a digital mechanical calculator invented by German mathematician Gottfried Wilhelm Leibniz around 1672 and completed in 1694. The name comes from the translation of the German term for its operating mechanism; staffelwalze meaning 'stepped drum'...

" attracted fair attention and was the basis of his election to the Royal Society
Royal Society
The Royal Society of London for Improving Natural Knowledge, known simply as the Royal Society, is a learned society for science, and is possibly the oldest such society in existence. Founded in November 1660, it was granted a Royal Charter by King Charles II as the "Royal Society of London"...

 in 1673. A number of such machines were made during his years in Hanover
Hanover
Hanover or Hannover, on the river Leine, is the capital of the federal state of Lower Saxony , Germany and was once by personal union the family seat of the Hanoverian Kings of Great Britain, under their title as the dukes of Brunswick-Lüneburg...

, 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 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 by Charles Babbage
Charles Babbage
Charles Babbage, FRS was an English mathematician, philosopher, inventor and mechanical engineer who originated the concept of a programmable computer...

 and Ada Lovelace
Ada Lovelace
Augusta Ada King, Countess of Lovelace , born Augusta Ada Byron, was an English writer chiefly known for her work on Charles Babbage's early mechanical general-purpose computer, the analytical engine...

. 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. 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.

Librarian


While serving as librarian of the ducal libraries in Hanover
Hanover
Hanover or Hannover, on the river Leine, is the capital of the federal state of Lower Saxony , Germany and was once by personal union the family seat of the Hanoverian Kings of Great Britain, under their title as the dukes of Brunswick-Lüneburg...

 and Wolfenbuettel, Leibniz effectively became one of the founders of library science
Library science
Library science is an interdisciplinary or multidisciplinary field that applies the practices, perspectives, and tools of management, information technology, education, and other areas to libraries; the collection, organization, preservation, and dissemination of information resources; and the...

. 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
Library classification
A library classification is a system of coding and organizing documents or library materials according to their subject and allocating a call number to that information resource...

 in ignorance of the only other such system then extant, that of the Bodleian Library
Bodleian Library
The Bodleian Library , the main research library of the University of Oxford, is one of the oldest libraries in Europe, and in Britain is second in size only to the British 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
Library of Congress
The Library of Congress is the research library of the United States Congress, de facto national library of the United States, and the oldest federal cultural institution in the United States. Located in three buildings in Washington, D.C., it is the largest library in the world by shelf space and...

 and the British Library
British Library
The British Library is the national library of the United Kingdom, and is the world's largest library in terms of total number of items. The library is a major research library, holding over 150 million items from every country in the world, in virtually all known languages and in many formats,...

.

He called for the creation of an empirical
Empirical
The word empirical denotes information gained by means of observation or experimentation. Empirical data are data produced by an experiment or observation....

 database
Database
A database is an organized collection of data for one or more purposes, usually in digital form. The data are typically organized to model relevant aspects of reality , in a way that supports processes requiring this information...

 as a way to further all sciences. His characteristica universalis
Characteristica universalis
The Latin term characteristica universalis, commonly interpreted as universal characteristic, or universal character in English, is a universal and formal language imagined by the German philosopher Gottfried Leibniz able to express mathematical, scientific, and metaphysical concepts...

, calculus ratiocinator
Calculus ratiocinator
The Calculus Ratiocinator is a theoretical universal logical calculation framework, a concept described in the writings of Gottfried Leibniz, usually paired with his more frequently mentioned characteristica universalis, a universal conceptual language....

, 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
Esperanto
is the most widely spoken constructed international auxiliary language. Its name derives from Doktoro Esperanto , the pseudonym under which L. L. Zamenhof published the first book detailing Esperanto, the Unua Libro, in 1887...

 and its rivals), symbolic logic
Mathematical logic
Mathematical logic is a subfield of mathematics with close connections to foundations of mathematics, theoretical computer science and philosophical logic. The field includes both the mathematical study of logic and the applications of formal logic to other areas of mathematics...

, even the World Wide Web
World Wide Web
The World Wide Web is a system of interlinked hypertext documents accessed via the Internet...

.

Advocate of scientific societies


Leibniz emphasized that research
Research
Research can be defined as the scientific search for knowledge, or as any systematic investigation, to establish novel facts, solve new or existing problems, prove new ideas, or develop new theories, usually using a scientific method...

 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
Prussian Academy of Sciences
The Prussian Academy of Sciences was an academy established in Berlin on 11 July 1700, four years after the Akademie der Künste or "Arts Academy", to which "Berlin Academy" may also refer.-Origins:...

 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.

Lawyer, moralist


With the possible exception of Marcus Aurelius, no philosopher has ever had as much experience with practical affairs of state as Leibniz. Leibniz's writings on law, ethics, and politics were long overlooked by English-speaking scholars, but this has changed of late.

While Leibniz was no apologist for absolute monarchy
Absolute monarchy
Absolute monarchy is a monarchical form of government in which the monarch exercises ultimate governing authority as head of state and head of government, his or her power not being limited by a constitution or by the law. An absolute monarch thus wields unrestricted political power over the...

 like Hobbes, or for tyranny in any form, neither did he echo the political and constitutional views of his contemporary John Locke
John Locke
John Locke FRS , widely known as the Father of Liberalism, was an English philosopher and physician regarded as one of the most influential of Enlightenment thinkers. Considered one of the first of the British empiricists, following the tradition of Francis Bacon, he is equally important to social...

, 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 well-being 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.


In 1677, Leibniz 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; in doing so, he anticipated the European Union
European Union
The European Union is an economic and political union of 27 independent member states which are located primarily in Europe. The EU traces its origins from the European Coal and Steel Community and the European Economic Community , formed by six countries in 1958...

. He believed that 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
Ecumenism
Ecumenism or oecumenism mainly refers to initiatives aimed at greater Christian unity or cooperation. It is used predominantly by and with reference to Christian denominations and Christian Churches separated by doctrine, history, and practice...

 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
John Frederick, Duke of Brunswick-Lüneburg
John Frederick was duke of Brunswick-Lüneburg and ruled over the Principality of Calenberg, a subdivision of the duchy, from 1665 until his death....

—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
Brunswick-Lüneburg
The Duchy of Brunswick-Lüneburg , or more properly Duchy of Brunswick and Lüneburg, was an historical ducal state from the late Middle Ages until the late Early Modern era within the North-Western domains of the Holy Roman Empire of the German Nation, in what is now northern Germany...

 remained Lutheran because the Duke's children did not follow their father.) These efforts included corresponding with the French bishop Jacques-Bénigne Bossuet
Jacques-Bénigne Bossuet
Jacques-Bénigne Bossuet was a French bishop and theologian, renowned for his sermons and other addresses. He has been considered by many to be one of the most brilliant orators of all time and a masterly French stylist....

, 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
Protestant Reformation
The Protestant Reformation was a 16th-century split within Western Christianity initiated by Martin Luther, John Calvin and other early Protestants. The efforts of the self-described "reformers", who objected to the doctrines, rituals and ecclesiastical structure of the Roman Catholic Church, led...

.

Philologist


Leibniz the philologist 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
Hebrew language
Hebrew is a Semitic language of the Afroasiatic language family. Culturally, is it considered by Jews and other religious groups as the language of the Jewish people, though other Jewish languages had originated among diaspora Jews, and the Hebrew language is also used by non-Jewish groups, such...

 was the primeval language of the human race. He also refuted the argument, advanced by Swedish scholars in his day, that a form of proto-Swedish
Swedish language
Swedish is a North Germanic language, spoken by approximately 10 million people, predominantly in Sweden and parts of Finland, especially along its coast and on the Åland islands. It is largely mutually intelligible with Norwegian and Danish...

 was the ancestor of the Germanic languages
Germanic languages
The Germanic languages constitute a sub-branch of the Indo-European language family. The common ancestor of all of the languages in this branch is called Proto-Germanic , which was spoken in approximately the mid-1st millennium BC in Iron Age northern Europe...

. He puzzled over the origins of the Slavic languages
Slavic languages
The Slavic languages , a group of closely related languages of the Slavic peoples and a subgroup of Indo-European languages, have speakers in most of Eastern Europe, in much of the Balkans, in parts of Central Europe, and in the northern part of Asia.-Branches:Scholars traditionally divide Slavic...

, was aware of the existence of Sanskrit
Sanskrit
Sanskrit , is a historical Indo-Aryan language and the primary liturgical language of Hinduism, Jainism and Buddhism.Buddhism: besides Pali, see Buddhist Hybrid Sanskrit Today, it is listed as one of the 22 scheduled languages of India and is an official language of the state of Uttarakhand...

, and was fascinated by classical Chinese
Classical Chinese
Classical Chinese or Literary Chinese is a traditional style of written Chinese based on the grammar and vocabulary of ancient Chinese, making it different from any modern spoken form of Chinese...

.

He published the princeps editio (first modern edition) of the late medieval
Late Middle Ages
The Late Middle Ages was the period of European history generally comprising the 14th to the 16th century . The Late Middle Ages followed the High Middle Ages and preceded the onset of the early modern era ....

 Chronicon Holtzatiae
Chronicon Holtzatiae
The Chronicon Holtzatiae auctore presbytero Bremensi is a Latin universal chronicle from the year 1448, but concentrating on the County of Holstein and written by an anonymous presbyter of Bremen originally from Holstein. It has received three modern editions, the first by the famous Gottfried...

, a Latin chronicle of the County of Holstein.

Sinophile


Leibniz was perhaps the first major European intellect to take a close interest in Chinese
China
Chinese civilization may refer to:* China for more general discussion of the country.* Chinese culture* Greater China, the transnational community of ethnic Chinese.* History of China* Sinosphere, the area historically affected by Chinese culture...

 civilization, which he knew by corresponding with, and reading other works by, European Christian missionaries posted in China. Having read Confucius Sinicus Philosophus on the first year of its publication, he concluded that Europeans could learn much from the Confucian
Confucianism
Confucianism is a Chinese ethical and philosophical system developed from the teachings of the Chinese philosopher Confucius . Confucianism originated as an "ethical-sociopolitical teaching" during the Spring and Autumn Period, but later developed metaphysical and cosmological elements in the Han...

 ethical tradition. He mulled over the possibility that the Chinese character
Chinese character
Chinese characters are logograms used in the writing of Chinese and Japanese , less frequently Korean , formerly Vietnamese , or other languages...

s were an unwitting form of his universal characteristic
Characteristica universalis
The Latin term characteristica universalis, commonly interpreted as universal characteristic, or universal character in English, is a universal and formal language imagined by the German philosopher Gottfried Leibniz able to express mathematical, scientific, and metaphysical concepts...

. He noted with fascination how the I Ching
I Ching
The I Ching or "Yì Jīng" , also known as the Classic of Changes, Book of Changes and Zhouyi, is one of the oldest of the Chinese classic texts...

 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.

Leibniz's attraction to Chinese philosophy
Chinese philosophy
Chinese philosophy is philosophy written in the Chinese tradition of thought. The majority of traditional Chinese philosophy originates in the Spring and Autumn and Warring States era, during a period known as the "Hundred Schools of Thought", which was characterized by significant intellectual and...

 originates from his perception that Chinese philosophy was similar to his own. The historian E.R. Hughes suggests that Leibniz's ideas of "simple substance" and "pre-established harmony" were directly influenced by Confucianism
Confucianism
Confucianism is a Chinese ethical and philosophical system developed from the teachings of the Chinese philosopher Confucius . Confucianism originated as an "ethical-sociopolitical teaching" during the Spring and Autumn Period, but later developed metaphysical and cosmological elements in the Han...

, pointing to the fact that they were conceived during the period that he was reading Confucius Sinicus Philosophus.

As polymath


While making his grand tour of European archives to research the Brunswick family history that he never completed, Leibniz stopped in Vienna
Vienna
Vienna is the capital and largest city of the Republic of Austria and one of the nine states of Austria. Vienna is Austria's primary city, with a population of about 1.723 million , and is by far the largest city in Austria, as well as its cultural, economic, and political centre...

 between May 1688 and 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
Holy Roman Emperor
The Holy Roman Emperor is a term used by historians to denote a medieval ruler who, as German King, had also received the title of "Emperor of the Romans" from the Pope...

 and in subsequent memoranda, he advocated reorganizing the Austrian economy, reforming the coinage of much of central Europe, negotiating a Concordat
Concordat
A concordat is an agreement between the Holy See of the Catholic Church and a sovereign state on religious matters. Legally, they are international treaties. They often includes both recognition and privileges for the Catholic Church in a particular country...

 between the Habsburg
Habsburg
The House of Habsburg , also found as Hapsburg, and also known as House of Austria is one of the most important royal houses of Europe and is best known for being an origin of all of the formally elected Holy Roman Emperors between 1438 and 1740, as well as rulers of the Austrian Empire and...

s and the Vatican
Holy See
The Holy See is the episcopal jurisdiction of the Catholic Church in Rome, in which its Bishop is commonly known as the Pope. It is the preeminent episcopal see of the Catholic Church, forming the central government of the Church. As such, diplomatically, and in other spheres the Holy See acts and...

, and creating an imperial research library, official archive, and public insurance fund. He wrote and published an important paper on mechanics
Mechanics
Mechanics is the branch of physics concerned with the behavior of physical bodies when subjected to forces or displacements, and the subsequent effects of the bodies on their environment....

.

Leibniz also wrote a short paper, first published by Louis Couturat
Louis Couturat
Louis Couturat was a French logician, mathematician, philosopher, and linguist.-Life:Born in Ris-Orangis, Essonne, France, he was educated in philosophy and mathematics at the École Normale Supérieure...

 in 1903, summarizing his views on metaphysics
Metaphysics
Metaphysics is a branch of philosophy concerned with explaining the fundamental nature of being and the world, although the term is not easily defined. Traditionally, metaphysics attempts to answer two basic questions in the broadest possible terms:...

. 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
Analytic philosophy
Analytic philosophy is a generic term for a style of philosophy that came to dominate English-speaking countries in the 20th century...

. 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.

As a mathematician and philosopher


When Leibniz died, his reputation was in decline. He was remembered for only one book, the Théodicée
Théodicée
Essais de Théodicée sur la bonté de Dieu, la liberté de l'homme et l'origine du mal , more simply known as Théodicée, is a book of philosophy by the famed polymath Gottfried Leibniz...

, whose supposed central argument Voltaire
Voltaire
François-Marie Arouet , better known by the pen name Voltaire , was a French Enlightenment writer, historian and philosopher famous for his wit and for his advocacy of civil liberties, including freedom of religion, free trade and separation of church and state...

 lampooned in his Candide
Candide
Candide, ou l'Optimisme is a French satire first published in 1759 by Voltaire, a philosopher of the Age of Enlightenment. The novella has been widely translated, with English versions titled Candide: or, All for the Best ; Candide: or, The Optimist ; and Candide: or, Optimism...

. Voltaire's depiction of Leibniz's ideas was so influential that many believed it to be an accurate description. 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
Christian Wolff (philosopher)
Christian Wolff was a German philosopher.He was the most eminent German philosopher between Leibniz and Kant...

, whose dogmatic and facile outlook did Leibniz's reputation much harm. He also influenced David Hume
David Hume
David Hume was a Scottish philosopher, historian, economist, and essayist, known especially for his philosophical empiricism and skepticism. He was one of the most important figures in the history of Western philosophy and the Scottish Enlightenment...

 who read his Théodicée
Théodicée
Essais de Théodicée sur la bonté de Dieu, la liberté de l'homme et l'origine du mal , more simply known as Théodicée, is a book of philosophy by the famed polymath Gottfried Leibniz...

 and used some of his ideas. 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 proponent. 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
KANT
KANT is a computer algebra system for mathematicians interested in algebraic number theory, performing sophisticated computations in algebraic number fields, in global function fields, and in local fields. KASH is the associated command line interface...

 read closely. In 1768, Dutens edited the first multi-volume 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
Antoine Arnauld
Antoine Arnauld — le Grand as contemporaries called him, to distinguish him from his father — was a French Roman Catholic theologian, philosopher, and mathematician...

, Samuel Clarke
Samuel Clarke
thumb|right|200px|Samuel ClarkeSamuel Clarke was an English philosopher and Anglican clergyman.-Early life and studies:...

, Sophia of Hanover
Sophia of Hanover
Sophia of the Palatinate was an heiress to the crowns of England and Ireland and later the crown of Great Britain. She was declared heiress presumptive by the Act of Settlement 1701...

, and her daughter Sophia Charlotte of Hanover
Sophia Charlotte of Hanover
Sophia Charlotte of Hanover was the Queen consort of Prussia as wife of Frederick I of Prussia. She was the daughter of Ernst August, Elector of Hanover, and Sophia of the Palatinate...

, began.

In 1900, Bertrand Russell
Bertrand Russell
Bertrand Arthur William Russell, 3rd Earl Russell, OM, FRS was a British philosopher, logician, mathematician, historian, and social critic. At various points in his life he considered himself a liberal, a socialist, and a pacifist, but he also admitted that he had never been any of these things...

 published a critical study of Leibniz's metaphysics
Metaphysics
Metaphysics is a branch of philosophy concerned with explaining the fundamental nature of being and the world, although the term is not easily defined. Traditionally, metaphysics attempts to answer two basic questions in the broadest possible terms:...

. Shortly thereafter, Louis Couturat
Louis Couturat
Louis Couturat was a French logician, mathematician, philosopher, and linguist.-Life:Born in Ris-Orangis, Essonne, France, he was educated in philosophy and mathematics at the École Normale Supérieure...

 published an important study of Leibniz, and edited a volume of Leibniz's heretofore unpublished writings, mainly on logic. They made Leibniz somewhat respectable among 20th-century analytical
Analytic philosophy
Analytic philosophy is a generic term for a style of philosophy that came to dominate English-speaking countries in the 20th century...

 and linguistic
Linguistic philosophy
Linguistic philosophy describes the view that philosophical problems are problems which may be solved either by reforming language, or by understanding more about the language we presently use. The former position is that of ideal language philosophy, the latter the position of ordinary language...

 philosophers in the English-speaking world (Leibniz had already been of great influence to many Germans such as Bernhard Riemann
Bernhard Riemann
Georg Friedrich Bernhard Riemann was an influential German mathematician who made lasting contributions to analysis and differential geometry, some of them enabling the later development of general relativity....

). For example, Leibniz's phrase salva veritate
Salva veritate
Salva veritate is the logical condition in virtue of which interchanging two expressions may be done without changing the truth-value of statements in which the expressions occur. The phrase occurs in two fragments from Gottfried Leibniz's General Science...

, meaning interchangeability without loss of or compromising the truth, recurs in Willard Quine's writings. Nevertheless, the secondary English-language 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 LeClerc (1973).

Nicholas Jolley has surmised that Leibniz's reputation as a philosopher is now perhaps higher than at any time since he was alive. Analytic and contemporary philosophy continue to invoke his notions of identity
Identity (philosophy)
In philosophy, identity, from , is the relation each thing bears just to itself. According to Leibniz's law two things sharing every attribute are not only similar, but are the same thing. The concept of sameness has given rise to the general concept of identity, as in personal identity and...

, individuation
Principle of individuation
The Principle of Individuation is a criterion which supposedly individuates or numerically distinguishes the members of the kind for which it is given, i.e. by which we can supposedly determine, regarding any kind of thing, when we have more than one of them or not. It is also known as a...

, and possible worlds
Possible Worlds
Possible Worlds may refer to:* Possible worlds, a concept in philosophy* Possible Worlds , by John Mighton** Possible Worlds , by Robert Lepage, based on the Mighton play* Possible Worlds , by Peter Porter...

, while the doctrinaire contempt for metaphysics, characteristic of analytic and linguistic philosophy, has faded. Work in the history of 17th- and 18th-century ideas
History of ideas
The history of ideas is a field of research in history that deals with the expression, preservation, and change of human ideas over time. The history of ideas is a sister-discipline to, or a particular approach within, intellectual history...

 has revealed more clearly the 17th-century "Intellectual Revolution" that preceded the better-known Industrial
Industrial Revolution
The Industrial Revolution was a period from the 18th to the 19th century where major changes in agriculture, manufacturing, mining, transportation, and technology had a profound effect on the social, economic and cultural conditions of the times...

 and commercial revolutions of the 18th and 19th centuries. 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
Scholasticism
Scholasticism is a method of critical thought which dominated teaching by the academics of medieval universities in Europe from about 1100–1500, and a program of employing that method in articulating and defending orthodoxy in an increasingly pluralistic context...

" as well as a "radical empiricist" element is more accepted now than in the early 20th century. Leibniz's thought is now seen as a major prolongation of the mighty endeavor begun by Plato
Plato
Plato , was a Classical Greek philosopher, mathematician, student of Socrates, writer of philosophical dialogues, and founder of the Academy in Athens, the first institution of higher learning in the Western world. Along with his mentor, Socrates, and his student, Aristotle, Plato helped to lay the...

 and Aristotle
Aristotle
Aristotle was a Greek philosopher and polymath, a student of Plato and teacher of Alexander the Great. His writings cover many subjects, including physics, metaphysics, poetry, theater, music, logic, rhetoric, linguistics, politics, government, ethics, biology, and zoology...

: the universe and man's place in it are amenable to human reason
Reason
Reason is a term that refers to the capacity human beings have to make sense of things, to establish and verify facts, and to change or justify practices, institutions, and beliefs. It is closely associated with such characteristically human activities as philosophy, science, language, ...

.

In 1985, the German government created the Leibniz Prize
Gottfried Wilhelm Leibniz Prize
The Gottfried Wilhelm Leibniz Prize is a research prize awarded by the Deutsche Forschungsgemeinschaft every year since 1985 to scientists working in Germany. This highest German research prize consists of a research grant of 2.5 million euro, to be used within seven years...

, offering an annual award of 1.55 million euro
Euro
The euro is the official currency of the eurozone: 17 of the 27 member states of the European Union. It is also the currency used by the Institutions of the European Union. The eurozone consists of Austria, Belgium, Cyprus, Estonia, Finland, France, Germany, Greece, Ireland, Italy, Luxembourg,...

s for experimental results and 770,000 euros for theoretical ones. It is the world's largest prize for scientific achievement.

The collection of manuscript papers of Leibniz at the Gottfried Wilhelm Leibniz Bibliothek – Niedersächische Landesbibliothek were inscribed on UNESCO
UNESCO
The United Nations Educational, Scientific and Cultural Organization is a specialized agency of the United Nations...

’s Memory of the World Register in 2007.

Leibniz biscuits


Leibniz-Keks
Leibniz-Keks
The Leibniz-Keks or Choco Leibniz is a German brand of biscuit, produced by the Bahlsen food company since 1891.- Name :The brand name Leibniz comes from the philosopher and mathematician Gottfried Wilhelm Leibniz. The only connection between man and biscuit is that Leibniz was one of the more...

, a popular brand of biscuits in Germany, are named after Gottfried Leibniz. These biscuits honour Leibniz because he was a resident of Hanover, where the company is based.

Writings and edition


Leibniz mainly wrote in three languages: scholastic Latin
Latin
Latin is an Italic language originally spoken in Latium and Ancient Rome. It, along with most European languages, is a descendant of the ancient Proto-Indo-European language. Although it is considered a dead language, a number of scholars and members of the Christian clergy speak it fluently, and...

, French
French language
French is a Romance language spoken as a first language in France, the Romandy region in Switzerland, Wallonia and Brussels in Belgium, Monaco, the regions of Quebec and Acadia in Canada, and by various communities elsewhere. Second-language speakers of French are distributed throughout many parts...

 and German
German language
German is a West Germanic language, related to and classified alongside English and Dutch. With an estimated 90 – 98 million native speakers, German is one of the world's major languages and is the most widely-spoken first language in the European Union....

. During his lifetime, he published many pamphlets and scholarly articles, but only two "philosophical" books, the Combinatorial Art and the Théodicée
Théodicée
Essais de Théodicée sur la bonté de Dieu, la liberté de l'homme et l'origine du mal , more simply known as Théodicée, is a book of philosophy by the famed polymath Gottfried Leibniz...

. (He published numerous pamphlets, often anonymous, on behalf of the House of Brunswick-Lüneburg
Brunswick-Lüneburg
The Duchy of Brunswick-Lüneburg , or more properly Duchy of Brunswick and Lüneburg, was an historical ducal state from the late Middle Ages until the late Early Modern era within the North-Western domains of the Holy Roman Empire of the German Nation, in what is now northern Germany...

, most notably the "De jure suprematum" a major consideration of the nature of sovereignty
Sovereignty
Sovereignty is the quality of having supreme, independent authority over a geographic area, such as a territory. It can be found in a power to rule and make law that rests on a political fact for which no purely legal explanation can be provided...

.) One substantial book appeared posthumously, his Nouveaux essais sur l'entendement humain, which Leibniz had withheld from publication after the death of John Locke
John Locke
John Locke FRS , widely known as the Father of Liberalism, was an English philosopher and physician regarded as one of the most influential of Enlightenment thinkers. Considered one of the first of the British empiricists, following the tradition of Francis Bacon, he is equally important to social...

. Only in 1895, when Bodemann completed his catalogues of Leibniz's manuscripts and correspondence, did the enormous extent of Leibniz's Nachlass
Nachlass
Nachlass is a German word, used in academia to describe the collection of manuscripts, notes, correspondence, and so on left behind when a scholar dies. The word is a compound in German: nach means 'after', and the verb lassen means 'leave'. The plural can be either Nachlasse or Nachlässe...

 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 in a letter 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.


The extant parts of the critical edition of Leibniz's writings are organized as follows:
  • Series 1. Political, Historical, and General Correspondence. 21 vols., 1666–1701.
  • Series 2. Philosophical Correspondence. 1 vol., 1663–85.
  • Series 3. Mathematical, Scientific, and Technical Correspondence. 6 vols., 1672–96.
  • Series 4. Political Writings. 6 vols., 1667–98.
  • Series 5. Historical and Linguistic Writings. Inactive.
  • Series 6. Philosophical Writings. 7 vols., 1663–90, and Nouveaux essais sur l'entendement humain.
  • 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 began in 1901. It was hampered by two world wars, the Nazi dictatorship (with the Holocaust, which affected a Jewish 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 scattering portions of his literary estates). The ambitious project has had to deal with seven languages contained in some 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
Potsdam
Potsdam is the capital city of the German federal state of Brandenburg and part of the Berlin/Brandenburg Metropolitan Region. It is situated on the River Havel, southwest of Berlin city centre....

, Münster
Münster
Münster is an independent city in North Rhine-Westphalia, Germany. It is located in the northern part of the state and is considered to be the cultural centre of the Westphalia region. It is also capital of the local government region Münsterland...

, Hannover and Berlin
Berlin
Berlin is the capital city of Germany and is one of the 16 states of Germany. With a population of 3.45 million people, Berlin is Germany's largest city. It is the second most populous city proper and the seventh most populous urban area in the European Union...

 have jointly published 25 volumes of the critical edition, with an average of 870 pages, and prepared index and concordance
Concordance (publishing)
A concordance is an alphabetical list of the principal words used in a book or body of work, with their immediate contexts. Because of the time and difficulty and expense involved in creating a concordance in the pre-computer era, only works of special importance, such as the Vedas, Bible, Qur'an...

 works.

Selected works


The year given is usually that in which the work was completed, not of its eventual publication.
  • 1666. De Arte Combinatoria
    De Arte Combinatoria
    The Dissertatio de arte combinatoria is an early work by Gottfried Leibniz published in 1666 in Leipzig. It is an extended version of his doctoral dissertation, written before the author had seriously undertaken the study of mathematics. The booklet was reissued without Leibniz' consent in 1690,...

     (On the Art of Combination); partially translated in Loemker §1 and Parkinson (1966).
  • 1671. Hypothesis Physica Nova (New Physical Hypothesis); Loemker §8.I (partial).
  • 1673 Confessio philosophi (A Philosopher's Creed); an English translation is available.
  • 1684. Nova methodus pro maximis et minimis (New method for maximums and minimums); translated in Struik, D. J., 1969. A Source Book in Mathematics, 1200–1800. Harvard University Press: 271–81.
  • 1686. Discours de métaphysique; Martin and Brown (1988), Ariew and Garber 35, Loemker §35, Wiener III.3, Woolhouse and Francks 1. An online translation by Jonathan Bennett is available.
  • 1703. Explication de l'Arithmétique Binaire (Explanation of Binary Arithmetic); Gerhardt, Mathematical Writings VII.223. An online translation by Lloyd Strickland is available.
  • 1710. Théodicée
    Théodicée
    Essais de Théodicée sur la bonté de Dieu, la liberté de l'homme et l'origine du mal , more simply known as Théodicée, is a book of philosophy by the famed polymath Gottfried Leibniz...

    ; Farrer, A.M., and Huggard, E.M., trans., 1985 (1952). Wiener III.11 (part). An online translation is available at Project Gutenberg
    Project Gutenberg
    Project Gutenberg is a volunteer effort to digitize and archive cultural works, to "encourage the creation and distribution of eBooks". Founded in 1971 by Michael S. Hart, it is the oldest digital library. Most of the items in its collection are the full texts of public domain books...

    .
  • 1714. Monadologie; translated by Nicholas Rescher
    Nicholas Rescher
    Nicholas Rescher is an American philosopher at the University of Pittsburgh. In a productive research career extending over six decades, Rescher has established himself as a systematic philosopher of the old style and author of a system of pragmatic idealism which weaves together threads of...

    , 1991. The Monadology: An Edition for Students. University of Pittsburg Press. Ariew and Garber 213, Loemker §67, Wiener III.13, Woolhouse and Francks 19. Online translations: Jonathan Bennett's translation; Latta's translation; French, Latin and Spanish edition, with facsimile of Leibniz's manuscript.
  • 1765. Nouveaux essais sur l'entendement humain; completed in 1704. Remnant, Peter, and Bennett, Jonathan, trans., 1996. New Essays on Human Understanding. Cambridge University Press. Wiener III.6 (part). An online translation by Jonathan Bennett is available.

Collections


Five important collections of English translations are Wiener (1951), Loemker (1969), Ariew and Garber (1989), Woolhouse and Francks (1998), and Strickland (2006). The ongoing critical edition of all of Leibniz's writings is Sämtliche Schriften und Briefe.

See also

  • General Leibniz rule
  • German inventors and discoverers
    German inventors and discoverers
    This is a list of German inventors and discoverers. The following list comprises people from Germany or German-speaking Europe, also of people of predominantly German heritage, in alphabetical order of the surname. The main section includes existing articles, indicated by blue links, and possibly...

  • Gottfried Wilhelm Leibniz Scientific Community
  • Leibniz formula for π
  • Leibniz formula for determinants
  • Leibniz harmonic triangle
    Leibniz harmonic triangle
    The Leibniz harmonic triangle is a triangular arrangement of fractions in which the outermost diagonals consist of the reciprocals of the row numbers and each inner cell is the absolute value of the cell above minus the cell to the left...

  • Leibniz integral rule
    Leibniz integral rule
    In mathematics, Leibniz's rule for differentiation under the integral sign, named after Gottfried Leibniz, tells us that if we have an integral of the formthen for x \in the derivative of this integral is thus expressible...

     for differentiation under the integral sign
  • Leibniz test
  • Leibniz's notation
  • Leibniz operator
  • Leibniz–Clarke correspondence
  • Newton v. Leibniz calculus controversy
    Newton v. Leibniz calculus controversy
    The calculus controversy was an argument between 17th-century mathematicians Isaac Newton and Gottfried Leibniz over who had first invented calculus...

  • Scientific Revolution
    Scientific revolution
    The Scientific Revolution is an era associated primarily with the 16th and 17th centuries during which new ideas and knowledge in physics, astronomy, biology, medicine and chemistry transformed medieval and ancient views of nature and laid the foundations for modern science...

  • Stepped Reckoner
    Stepped Reckoner
    The Step Reckoner was a digital mechanical calculator invented by German mathematician Gottfried Wilhelm Leibniz around 1672 and completed in 1694. The name comes from the translation of the German term for its operating mechanism; staffelwalze meaning 'stepped drum'...

     – mechanical calculator

Primary literature

  • Alexander, H G (ed) The Leibniz-Clarke Correspondence. Manchester: Manchester University Press
    Manchester University Press
    Manchester University Press is the university press of the University of Manchester, England and a publisher of academic books and journals. Manchester University Press has developed into an international publisher...

    , 1956.
  • Ariew, R & D Garber, 1989. Leibniz: Philosophical Essays. Hackett.
  • Arthur, Richard, 2001. The Labyrinth of the Continuum: Writings on the Continuum Problem, 1672–1686. Yale University Press.
  • Cohen, Claudine and Wakefield, Andre, 2008. Protogaea. University of Chicago Press.
  • Cook, Daniel, and Rosemont, Henry Jr., 1994. Leibniz: Writings on China. Open Court.
  • Loemker, Leroy, 1969 (1956). Leibniz: Philosophical Papers and Letters. Reidel.
  • Remnant, Peter, and Bennett, Jonathan, 1996 (1981). Leibniz: New Essays on Human Understanding. Cambridge University Press.
  • Riley, Patrick, 1988. Leibniz: Political Writings. Cambridge University Press.
  • Sleigh, Robert C., Look, Brandon, and Stam, James, 2005. Confessio Philosophi: Papers Concerning the Problem of Evil, 1671–1678. Yale University Press.
  • Strickland, Lloyd, 2006. The Shorter Leibniz Texts: A Collection of New Translations. Continuum.
  • Ward, A. W.
    Adolphus William Ward
    Sir Adolphus William Ward was an English historian and man of letters.He was born at Hampstead, London, and was educated in Germany and at Peterhouse, Cambridge....

     Leibniz as a Politician  (lecture, 1911)
  • Wiener, Philip, 1951. Leibniz: Selections. Scribner.
  • Woolhouse, R.S., and Francks, R., 1998. Leibniz: Philosophical Texts. Oxford University Press.

Secondary literature

  • Adams, Robert Merrihew. Lebniz: Determinist, Theist, Idealist. New York: Oxford, Oxford University Press, 1994.
  • Aiton, Eric J., 1985. Leibniz: A Biography. Hilger (UK).
  • Antognazza, M.R.(2008) Leibniz: An Intellectual Biography. Cambridge Univ. Press.
  • Albeck-Gidron, Rachel, The Century of the Monads: Leibniz's Metaphysics and 20th-Century Modernity, Bar-Ilan University Press.
  • Bos, H. J. M. (1974) "Differentials, higher-order differentials and the derivative in the Leibnizian calculus," Arch. History Exact Sci. 14: 1—90.
  • Couturat, Louis
    Louis Couturat
    Louis Couturat was a French logician, mathematician, philosopher, and linguist.-Life:Born in Ris-Orangis, Essonne, France, he was educated in philosophy and mathematics at the École Normale Supérieure...

    , 1901. La Logique de Leibniz. Paris: Felix Alcan.
  • Davis, Martin
    Martin Davis
    Martin David Davis, is an American mathematician, known for his work on Hilbert's tenth problem . He received his Ph.D. from Princeton University in 1950, where his adviser was Alonzo Church . He is Professor Emeritus at New York University. He is the co-inventor of the Davis-Putnam and the DPLL...

    , 2000. The Universal Computer: The Road from Leibniz to Turing. WW Norton.
  • Deleuze, Gilles
    Gilles Deleuze
    Gilles Deleuze , was a French philosopher who, from the early 1960s until his death, wrote influentially on philosophy, literature, film, and fine art. His most popular works were the two volumes of Capitalism and Schizophrenia: Anti-Oedipus and A Thousand Plateaus , both co-written with Félix...

    , 1993. The Fold: Leibniz and the Baroque. University of Minnesota Press.
  • Du Bois-Reymond, Paul, 18nn. "Leibnizian Thoughts in Modern Science".
  • Finster, Reinhard & Gerd van den Heuvel. Gottfried Wilhelm Leibniz. Mit Selbstzeugnissen und Bilddokumenten. 4. Auflage. Rowohlt, Reinbek bei Hamburg 2000 (Rowohlts Monographien, 50481), ISBN 3-499-50481-2.
  • Grattan-Guinness, Ivor
    Ivor Grattan-Guinness
    Ivor Grattan-Guinness, born 23 June 1941, in Bakewell, in England, is a historian of mathematics and logic.He gained his Bachelor degree as a Mathematics Scholar at Wadham College, Oxford, got an M.Sc in Mathematical Logic and the Philosophy of Science at the London School of Economics in 1966...

    , 1997. The Norton History of the Mathematical Sciences. W W Norton.
  • Hall, A. R., 1980. Philosophers at War: The Quarrel between Newton and Leibniz. Cambridge University Press
    Cambridge University Press
    Cambridge University Press is the publishing business of the University of Cambridge. Granted letters patent by Henry VIII in 1534, it is the world's oldest publishing house, and the second largest university press in the world...

    .
  • Heidegger, Martin
    Martin Heidegger
    Martin Heidegger was a German philosopher known for his existential and phenomenological explorations of the "question of Being."...

    , 1983. The Metaphysical Foundations of Logic. Indiana University Press.
  • Hirano, Hideaki, 1997. "Cultural Pluralism And Natural Law." Unpublished.
  • Hostler, J., 1975. Leibniz's Moral Philosophy. UK: Duckworth.
  • Jolley, Nicholas, ed., 1995. The Cambridge Companion to Leibniz. Cambridge University Press.
  • LeClerc, Ivor, ed., 1973. The Philosophy of Leibniz and the Modern World. Vanderbilt University Press
    Vanderbilt University Press
    Vanderbilt University Press is a university press that is part of Vanderbilt University in Nashville, Tennessee.Vanderbilt University Press is the principal publishing arm of Vanderbilt University. The Press selects, edits, and markets scholarly texts, especially in the areas of the humanities and...

    .
  • Lovejoy, Arthur O., 1957 (1936) "Plenitude and Sufficient Reason in Leibniz and Spinoza" in his The Great Chain of Being. Harvard University Press: 144–82. Reprinted in Frankfurt, H. G., ed., 1972. Leibniz: A Collection of Critical Essays. Anchor Books.
  • Mandelbrot, Benoît
    Benoît Mandelbrot
    Benoît B. Mandelbrot was a French American mathematician. Born in Poland, he moved to France with his family when he was a child...

    , 1977. The Fractal Geometry of Nature. Freeman.
  • Mackie, John Milton; Guhrauer, Gottschalk Eduard
    Gottschalk Eduard Guhrauer
    Gottschalk Eduard Guhrauer was a German philologist and biographer. He is known principally for his 1842 biography of G. W. Leibniz and his completion of T. W. Danzel's life of Lessing....

    , 1845. Life of Godfrey William von Leibnitz. Gould, Kendall and Lincoln.
  • Mates, Benson, 1986. The Philosophy of Leibniz: Metaphysics and Language. Oxford University Press.
  • Mercer, Christia, 2001. Leibniz's metaphysics: Its Origins and Development. Cambridge University Press.
  • Morris, Simon Conway
    Simon Conway Morris
    Simon Conway Morris FRS is an English paleontologist made known by his detailed and careful study of the Burgess Shale fossils, an exploit celebrated in Wonderful Life by Stephen Jay Gould...

    , 2003. Life's Solution: Inevitable Humans in a Lonely Universe. Cambridge University Press.
  • Perkins, Franklin, 2004. Leibniz and China: A Commerce of Light. Cambridge University Press.
  • Rensoli, Lourdes, 2002. El problema antropologico en la concepcion filosofica de G. W. Leibniz. Leibnitius Politechnicus. Universidad Politecnica de Valencia.
  • Riley, Patrick
    Patrick T. Riley
    Patrick Thomas Riley is Michael Oakeshott Professor Emeritus of Political Science at the University of Wisconsin-Madison. He is notable for his translations of the political writings of Gottfried Leibniz and his research on social contract theory, as well as the history of universal jurisprudence...

    , 1996. Leibniz's Universal Jurisprudence: Justice as the Charity of the Wise. Harvard University Press
    Harvard University Press
    Harvard University Press is a publishing house established on January 13, 1913, as a division of Harvard University, and focused on academic publishing. In 2005, it published 220 new titles. It is a member of the Association of American University Presses. Its current director is William P...

    .
  • Rutherford, Donald, 1998. Leibniz and the Rational Order of Nature. Cambridge University Press.
  • Struik, D. J., 1969. A Source Book in Mathematics, 1200–1800. Harvard University Press.
  • Ward, P. D.
    Peter Ward (paleontologist)
    Peter Douglas Ward is a paleontologist and professor of Biology and of Earth and Space Sciences at the University of Washington, Seattle, and has written popular science works for a general audience. He is also an adviser to the Microbes Mind Forum....

    , and Brownlee, D., 2000. Rare Earth: Why Complex Life is Uncommon in the Universe. Springer Verlag.
  • Wilson, Catherine, 1989. 'Leibniz's Metaphysics. Princeton University Press
    Princeton University Press
    -Further reading:* "". Artforum International, 2005.-External links:* * * * *...

    .
  • Zalta, E. N., 2000. "A (Leibnizian) Theory of Concepts", Philosophiegeschichte und logische Analyse / Logical Analysis and History of Philosophy 3: 137–183.

External links


  • An extensive bibliography
  • Internet Encyclopedia of Philosophy
    Internet Encyclopedia of Philosophy
    The Internet Encyclopedia of Philosophy is a free online encyclopedia on philosophical topics and philosophers founded by James Fieser in 1995. The current general editors are James Fieser and Bradley Dowden...

    : "Leibniz" Douglas Burnham.
  • Stanford Encyclopedia of Philosophy
    Stanford Encyclopedia of Philosophy
    The Stanford Encyclopedia of Philosophy is a freely-accessible online encyclopedia of philosophy maintained by Stanford University. Each entry is written and maintained by an expert in the field, including professors from over 65 academic institutions worldwide...

    . Articles on Leibniz.
  • George MacDonald Ross, Leibniz, Originally published: Oxford University Press (Past Masters) 1984; Electronic edition: Leeds Electronic Text Centre July 2000
  • translations by Jonathan Bennett
    Jonathan Bennett (philosopher)
    Jonathan Francis Bennett is a British philosopher of language and metaphysics, and a historian of early modern philosophy.Born in Greymouth, New Zealand, Bennett was educated at the University of Oxford. He has taught at the University of Cambridge , Simon Fraser University , the University of...

    , of the New Essays, the exchanges with Bayle, Arnauld and Clarke, and about 15 shorter works.
  • Leibnitiana Gregory Brown.
  • Gottfried Wilhelm Leibniz: Texts and Translations, compiled by Donald Rutherford, UCSD
  • Leibniz-translations.com Scroll down for many Leibniz links.
  • Leibniz Prize.
  • Philosophical Works of Leibniz translated by G.M. Duncan
  • Leibnitiana, links and resources compiled by Gregory Brown, University of Houston
    University of Houston
    The University of Houston is a state research university, and is the flagship institution of the University of Houston System. Founded in 1927, it is Texas's third-largest university with nearly 40,000 students. Its campus spans 667 acres in southeast Houston, and was known as University of...

    .
  • Leibnizian Resources, many links organized by Markku Roinila, University of Helsinki
    University of Helsinki
    The University of Helsinki is a university located in Helsinki, Finland since 1829, but was founded in the city of Turku in 1640 as The Royal Academy of Turku, at that time part of the Swedish Empire. It is the oldest and largest university in Finland with the widest range of disciplines available...

    .