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Gottfried Wilhelm Leibniz

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Gottfried Wilhelm Leibniz
Gottfried Wilhelm Leibniz
(Painting by Bernhard Christoph Francke, c. 1700)

Gottfried Wilhelm Leibniz (also Leibnitz or von Leibniz) (1646 - 1716) was a German philosopher, mathematician, scientist and polymath of the Age of Reason.

As a philosopher, he was, along with René Descartes and Baruch Spinoza, a major figure in the Continental Rationalism movement (the main 17th Century opposition to the British Empiricist school of thought of Hobbes, Locke, Berkeley and Hume). He devised his rather eccentric metaphysical theory of monads operating in a pre-established divine harmony in order to overcome what he saw as some of the drawbacks of the theories of Descartes and Spinoza. He remained a devout Christian throughout his life and his formulation of the Problem of Evil in a world created by a good God was an influential one.

His contributions to Logic were perhaps the most important between Aristotle and the developments in modern formal Logic of the mid-19th Century, and to some extent he anticipated modern Symbolic Logic.

He is equally important in the history of mathematics, as the inventor of calculus (independently of Sir Isaac Newton), and as the discoverer of the binary system (the foundation of virtually all modern computer architectures). He also made major contributions to physics, and anticipated notions that surfaced much later in other sciences biology, medicine, geology, probability theory, psychology, linguistics and information science, as well as writing on politics, law, ethics, theology, history and philology.


Leibniz (pronounced LIBE-nitz) was born on 1 July 1646 in Leipzig in Saxony, eastern Germany. His father, Friedrich Leibniz, died when Gottfried was just six years old, so he mainly learned his religious and moral values from his mother, Catherina Schmuck (the daughter of a lawyer and Friedrich's third wife). However, his father, who had been a Professor of Moral Philosophy at the University of Leipzig, left his personal library to his son, who was granted free access to it from age seven onwards, in addition to attending the regular Nicolai School in Leipzig.

By twelve, he had taught himself Latin, which he used freely all his life, and had begun studying Greek. At 14, he entered the University of Leipzig, mastering the standard university courses in classics, law, Logic, and Scholastic and Aristotelian philosophy under Jakob Thomasius (1622 - 1684), although his education in mathematics was probably not up to the French and British standards.

In 1666, at the age of 20, he published his habilitation thesis and first book, "De Arte Combinatoria" ("On the Art of Combinations"), in which he aimed to reduce all reasoning and discovery to a combination of basic elements (such as numbers, letters, sounds and colours). But, when the University of Leipzig refused to assure him a position teaching law upon graduation, he submitted it to the University of Altdorf instead, and obtained his doctorate in law there within five months. However, he then declined their offer of an academic appointment, and instead spent the rest of his life in the service of two major German noble families.

His first salaried position was as an alchemist in Nuremberg (even though he knew little or nothing about the subject). During this time, he met and was befriended by the wealthy Baron Johann Christian von Boineburg, who employed Leibniz as an assistant and later introduced him to the Elector of Mainz, Johann Philipp von Schönborn. The Elector asked Leibniz to assist with the redrafting of the legal code for his Electorate and, in 1669, appointed him 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.

Leibniz's early service to the Elector was soon followed by a diplomatic role, and his political solution to the increasing ambitions of King Louis XIV of France (principally aimed at diverting him from attacking German areas) received cautious support in Germany and serious attention in France, although it was never actually implemented. However, during his time in Paris, from around 1672, he greatly expanded his knowledge of mathematics and physics, (especially after a stroke of luck in meeting the Dutch physicist and mathematician Christiaan Huygens, who became his mentor for a time), and began contributing to both, including the invention of his version of differential and integral calculus. He also met the leading French philosophers of the day, Nicolas Malebranche and Antoine Arnauld (1612 - 1694), and had the opportunity to study the published and unpublished writings of Descartes and Pascal.

After the French political plan petered out, the Elector of Mainz sent Leibniz on a similar peace mission to the English government in London, early in 1673. The mission ended abruptly with the news of the Elector's death, but not before Leibniz had demonstrated to the Royal Society the first calculating machine that could execute all four basic arithmetical operations (a project he had been working on since 1670). He was made an external member of the prestigious Society, although he was criticized by some of the members, and the experience served to show him that his knowledge of mathematics was less than complete, and encouraged him to redouble his efforts.

With the sudden deaths of both the Elector of Mainz and (a few months earlier) of his other benefactor, von Boineburg, Leibniz returned to Paris, rather than Mainz. Duke Johann Friedrich (the Duke of Brunswick at that time), with whom Leibniz had been in correspondence since 1669, offered him the post of Councillor and librarian at his court in Hanover, and (with no apparent prospects in Paris or the Habsburg imperial court presenting themselves) he reluctantly accepted. He managed to delay his arrival in Hanover until the end of 1676, though, and in the meantime he traveled again to London (where he may have seen some of Sir Isaac Newton's work on calculus, fuelling later claims that he had stolen ideas from Newton), and to The Hague in Holland, where he met Antonie van Leeuwenhoek (1632 - 1723), the discoverer of microorganisms. He also spent several days in intense discussion with the Dutch Rationalist philosopher, Baruch Spinoza, whom Leibniz greatly respected (although dismayed at his apparent contradiction of both Christian and Jewish orthodoxy).

In 1677, he was promoted to Privy Counsellor of Justice of the House of Hanover, a post he held for the rest of his life, serving three consecutive rulers as historian and political adviser, and as librarian of the ducal library. He was also correspondent, adviser and friend to several influential women in the Hanover court, including the Electress Sophia of Hanover, her daughter Sophia Charlotte of Hanover (the Queen of Prussia) and Caroline of Ansbach (consort of the future King George II of England) and, despite the rather provincial nature of Hanover itself, Leibniz was to some extent involved in European politics of the period.

He was also fortunate in that the Brunswicks allowed, and even encouraged, his intellectual pursuits unrelated to his duties as a courtier, as well as the vast correspondence he kept up. Assigned projects, like a commissioned history of the Guelf family of the House of Brunswick, however, often tended to take a back seat to his personal scientific and philosophical work, and his dilatoriness led to a certain amount of bad feeling.

Leibniz's most important mathematical papers were published between 1682 and 1692, particularly in the "Acta Eruditorum", a journal which he co-founded with Otto Mencke in 1682, and which played a key role in advancing his mathematical and scientific reputation. During the 1690s, he put much energy into promoting scientific societies, and he was involved in moves to set up academies in Berlin, Dresden, Vienna, and St. Petersburg. He was elected to the Paris Academy in 1701, after publication of his paper on the binary system of arithmetic. He kept up regular correspondence with over 600 correspondents, including most of the great scholars of Europe, in a great variety of different fields. He often styled himself "von Leibniz", even though we have no confirmation that he was ever granted a patent of nobility.

In 1712, Leibniz began a two year residence in Vienna, where he was appointed Imperial Court Councillor to the Habsburgs. When Elector Georg Ludwig became King George I of Great Britain in 1714, under the terms of the 1701 Act of Settlement (which Leibniz himself had done much to bring about), he had hoped to be brought into the British court, but his hopes were soon dashed (especially as Newton was seen as having "won" the calculus priority dispute). Finally, his dear friend and defender, the dowager Electress Sophia, died in 1714.

Leibniz himself died in Hanover, Germany, on 14 November 1716, and his funeral was shunned by the court of George I, the Royal Society and the Berlin Academy of Sciences, his grave remaining unmarked for more than 50 years. He had never married or raised a family, and he left a fair sum (earned from his well-paid position with the Brunswicks) to his sole heir, his sister's stepson.

Work Back to Top

Leibniz wrote a great deal, but published hardly any of it during his life, his philosophy seeming to take a back seat to his many other interests and duties. Other than the "Théodicée" of 1710 (the only treatise Leibniz published in his lifetime), 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), giving his philosophical thinking a fragmented appearance.

His first philosophical work, the "Discours de métaphysique" ("Discourse on Metaphysics"), written in 1686 but not actually published until the 19th Century, was a commentary on a running dispute between the French Rationalists Nicolas Malebranche and Antoine Arnauld (1612 - 1694), and led to an extensive and valuable correspondence with Arnauld (covering, among other matters, one of Leibniz's lifelong aims, the reunification of the Christian Churches). Between 1695 and 1705, he wrote his "Nouveaux essais sur l'entendement humain" ("New Essays on Human Understanding"), a lengthy commentary on John Locke's 1690 "An Essay Concerning Human Understanding", although it too was not published until 1765, long after his death.

The "Théodicée", written in 1710, is as much theological as philosophical. It tries to justify the apparent imperfections of the world and to tackle the Problem of Evil in a world created by a good God by claiming that our world is optimal among all possible worlds, and that it must be the best possible and most balanced world, simply because it was created by a perfect God. The optimism of this idea was lampooned by Voltaire in his comic novella, "Candide", although the modern observations that led to the "fine-tuned Universe" argument may seem to support his view.

Perhaps Leibniz's best known contribution to Metaphysics (as exposited in the "Monadologie" of 1714), is his theory of monads, a form of dualistic Idealism sometimes known as Panpsychism or Parallelism. Monads are the ultimate elements of the universe: eternal, immaterial, indecomposable, individual, non-interacting, subject to their own laws, and each reflecting the entire universe in a pre-established harmony (effectively "programmed" in advance by God, whom he called the "central monad", to "harmonize" with each other). Unlike atoms, monads possess no material or spatial character, and are completely mutually independent, so that interactions among monads are only apparent. For Leibniz, then, everything in the material world that we see and touch are actually only phenomena, merely appearances or by-products of the real world, which is in fact an infinite array of these non-material monads. In some ways, this is not unlike the very modern idea of the universe being ultimately composed of energy.

According to Leibniz's theory, human beings and even God himself are monads, and the existence of God can be inferred from the harmony prevailing among all other monads (because it is God who wills the pre-established harmony). Thus, the external world is ideal (in the philosophical sense of Idealism) and phenomenal (in the philosophical sense of Phenomenology), and its motion is the result of a dynamic force (the pre-established harmony or, effectively, God) on these simple and immaterial monads. Although Leibniz claimed to believe in the existence of free will, his programme is essentially a deterministic one.

Leibniz also used his theory of monads in an attempt to overcome the problematic interaction between mind and matter arising in the system of Descartes (the so-called mind-body problem in Philosophy of Mind). Indeed, he developed the theory largely to address it, unimpressed as he was with Spinoza's earlier solution to this problem, with its lack of individuation and its representation of individual creatures as merely accidental. In Leibniz's conception of things, there is really no need for the concept of causality at all, as everything is pre-arranged and organized by the omnipotent God himself. Leibniz's theory was, however, considered somewhat arbitrary and eccentric, even in his own day.

Towards the end of his life, Leibniz published an essay called "Principles of Nature and Grace, Based on Reason" (1714), in which he put forth his Principle of Sufficient Reason, which states that there is an explanation for every fact and an answer to every question, however intractable they may appear. Following from this, he was perhaps the first philosopher to explicitly ask the question "Why is there something rather than nothing?", a fundamental and apparently intractable question that others had shied away from. His answer to this question, perhaps disappointingly, was the rather formulaic one of God, and when asked for an explanation for God's existence his equally unsatisfactory answer was that God is a necessary being, such that his non-existence would be logically impossible, a response that Hume and others took immediate issue with.

Although he published nothing on formal Logic in his lifetime, Leibniz can be considered the most important logician between Aristotle and the mid-19th Century developments in modern formal Logic of George Boole and Augustus De Morgan. He enunciated the principal properties of what we now call conjunction, disjunction, negation, identity, set inclusion and the empty set.

Quite early in his career, Leibniz's developed a calculus ratiocinator, which resembles modern Symbolic Logic to some extent, based on his belief that much of human reasoning could be reduced to calculations of a sort, and out of his passion for symbols and notation. He defining a "real" character as a written sign that represents an idea directly, and not simply as the word embodying an idea (like Egyptian hieroglyphics, Chinese characters, and the symbols of astronomy and chemistry), and proposed a "universal characteristic", built on an alphabet of human thought, in which each fundamental concept would be represented by a unique "real" character, with more complex thoughts represented by combining characters. Later, in 1676, when he had a better grounding in mathematics, he conceives of a kind of "algebra of thought", modeled on and including conventional algebra and its notation.

In mathematics, Leibniz was the first (in 1692 and 1694) to employ the mathematical notion of a function explicitly to denote any of several geometric concepts derived from a curve, as well as the first to see that the coefficients of a system of linear equations could be arranged into an array (now called a matrix) which can be manipulated to find the solution of the system.

He is credited, along with Sir Isaac Newton, with the discovery of infinitesimal calculus, which, according to his notebooks, he employed for the first time in November 1675 to find the area under the function y = x. He introduced several notations used to this day (e.g. the elongated S for the integral sign, and the d used for differentials) and, in the autumn of 1676, he discovered the familiar d(xn) = nxn-1dx.

However, he did not publish anything about his calculus until 1684 and, from 1711 until his death, Leibniz's life was envenomed by a long and antagonistic dispute with John Keill, Newton and others, over whether Leibniz had invented the calculus independently of Newton, or whether he had merely invented another notation for ideas that were fundamentally Newton's. Although his approach to the calculus fell well short of later standards of rigor (as did Newton's), and later work discredited the use of infinitesimals to justify calculus, his work marked an important start in the discipline, and much of his analysis has been vindicated.

As early as 1671, Leibniz began to invent a calculating machine, the first that could execute all four arithmetical operations, gradually improving it over a number of years (he was elected to the Royal Society in 1673 on the strength of it). By 1679, he had perfected his binary system of arithmetic (base 2), which was later used in most computers, although he did not publish anything until 1701. He imagined a machine in which binary numbers were represented by marbles, governed by a rudimentary sort of punched card system, groping towards hardware and software concepts worked out much later by Charles Babbage and Ada Lovelace.

In physics, Leibniz often disagreed with both Descartes and Newton. He devised a new theory of motion (dynamics) based on kinetic energy and potential energy, which posited space, time and motion as relative (whereas Newton felt strongly that space was absolute), anticipating by over 200 years Albert Einstein and recent cosmology and quantum mechanics. His "vis viva" (living force) represented an alternative viewpoint to the conservation of momentum championed by Newton in England and by Descartes in France.

He also dabbled in other sciences, often with prescient ideas. By proposing that the earth has a molten core, he anticipated modern geology. From his study of comparative anatomy and fossils, he worked out a primal organismic theory. In psychology, he anticipated the distinction between conscious and unconscious states, and in sociology, he laid the ground for communication theory. In public health, he advocated establishing a medical administrative authority, with powers over epidemiology and veterinary medicine. In economics, he proposed tax reforms and a national insurance scheme, and discussed the balance of trade.

He was also an inventor and engineer, and has been claimed as the "father of applied science", insisting that theory be combined with practical application. He designed wind-driven propellers and water pumps, mining machines to extract ore, hydraulic presses, water desalination systems, lamps, submarines, clocks, etc, and (with Denis Papin) even an early steam engine.

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