jury is still out

Gottfried Leibniz

[lahyb-nits; Ger. lahyp-nits]

Gottfried Wilhelm Leibniz (also Leibnitz or von Leibniz; 1 July 1646 – 14 November 1716) was a German polymath who wrote primarily in Latin and French.

He occupies an equally grand place in both the history of philosophy and the history of mathematics. He invented calculus independently of Newton, and his notation is the one in general use since then. He also discovered the binary system, foundation of virtually all modern computer architectures. In philosophy, he is mostly remembered for optimism, i.e. his conclusion that our universe is, in a restricted sense, the best possible one God could have made. He was, along with René Descartes and Baruch Spinoza, one of the three greatest 17th-century rationalists, but his philosophy also looks back to the scholastic tradition and anticipates modern logic and analysis. Leibniz also made major contributions to physics and technology, and anticipated notions that surfaced much later in biology, medicine, geology, probability theory, psychology, linguistics, and information science. He also wrote on politics, law, ethics, theology, history, and philology, even occasional verse. His contributions to this vast array of subjects are scattered in journals and in tens of thousands of letters and unpublished manuscripts. As of 2008, there is no complete edition of Leibniz's writings.


Early life

Gottfried Leibniz was born on 1 July 1646 in Leipzig to Friedrich Leibniz and Catherina Schmuck. His father passed away when he was six, so he learned his religious and moral values from his mother. These would exert a profound influence on his philosophical thought in later life. As an adult, he often styled himself "von Leibniz", and many posthumous editions of his works gave his name on the title page as "Freiherr [Baron] G. W. von Leibniz." However, no document has been found confirming that he was ever granted a patent of nobility.

When Leibniz was six years old, his father, a Professor of Moral Philosophy at the University of Leipzig, died, leaving a personal library to which Leibniz was granted free access from age seven onwards. By 12, he had taught himself Latin, which he used freely all his life, and had begun studying Greek.

He entered his father's university at age 14, and completed university studies by 20, specializing in law and mastering the standard university courses in classics, logic, and scholastic philosophy. However, his education in mathematics was not up to the French and British standards. In 1666 (age 20), he published his first book, also his habilitation thesis in philosophy, On the Art of Combinations. When Leipzig declined to assure him a position teaching law upon graduation, Leibniz submitted the thesis he had intended to submit at Leipzig to the University of Altdorf instead, and obtained his doctorate in law in five months. He then declined an offer of academic appointment at Altdorf, and spent the rest of his life in the service of two major German noble families.


Leibniz's first position was as a salaried alchemist in Nuremberg, even though he knew nothing about the subject. He soon met Johann Christian von Boineburg (1622–1672), the dismissed chief minister of the Elector of Mainz, Johann Philipp von Schönborn. Von Boineburg hired Leibniz as an assistant, and shortly thereafter reconciled with the Elector and introduced Leibniz to him. Leibniz then dedicated an essay on law to the Elector in the hope of obtaining employment. The stratagem worked; the Elector asked Leibniz to assist with the redrafting of the legal code for his Electorate. In 1669, Leibniz was appointed Assessor in the Court of Appeal. Although von Boineburg died late in 1672, Leibniz remained under the employment of his widow until she dismissed him in 1674.


Thus Leibniz began several years in Paris. Soon after arriving, he met Dutch physicist and mathematician Christiaan Huygens and realised that his own knowledge of mathematics and physics was spotty. 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. He met Malebranche and Antoine Arnauld, the leading French philosophers of the day, and studied the writings of Descartes and Pascal, unpublished as well as published. He befriended a German mathematician, Ehrenfried Walther von Tschirnhaus; they corresponded for the rest of their lives.

When it became clear that France would not implement its part of Leibniz's Egyptian plan, the Elector sent his nephew, escorted by Leibniz, on a related mission to the English government in London, early in 1673. There Leibniz came into acquaintance of Henry Oldenburg and John Collins. After demonstrating a calculating machine to the Royal Society he had been designing and building since 1670, the first such machine that could execute all four basic arithmetical operations, the Society made him an external member. The mission ended abruptly when news reached it of the Elector's death, whereupon Leibniz promptly returned to Paris and not, as had been planned, to Mainz.


House of Hanover, 1676–1716

Leibniz managed to delay his arrival in Hanover until the end of 1676, after making one more short journey to London, where he possibly was shown some of Newton's unpublished work on the calculus. This fact was deemed evidence supporting the accusation, made decades later, that he had stolen the calculus from Newton. On the journey from London to Hanover, Leibniz stopped in The Hague where he met Leeuwenhoek, the discoverer of microorganisms. He also spent several days in intense discussion with Spinoza, who had just completed his masterwork, the Ethics. Leibniz respected Spinoza's powerful intellect, but was dismayed by his conclusions that contradicted both Christian and Jewish orthodoxy.


Among the few people in north Germany to warm to Leibniz were the Electress Sophia of Hanover (1630–1714), her daughter Sophia Charlotte of Hanover (1668–1705), the Queen of Prussia and her avowed disciple, and Caroline of Ansbach, the consort of her grandson, the future George II. To each of these women he was correspondent, adviser, and friend. In turn, they all warmed to him more than did their spouses and the future king George I of Great Britain.





In 1711, while traveling in northern Europe, the Russian Tsar Peter the Great stopped in Hanover and met Leibniz, who then took some interest in matters Russian over the rest of his life. In 1712, Leibniz began a two year residence in Vienna, where he was appointed Imperial Court Councillor to the Habsburgs. On the death of Queen Anne in 1714, Elector Georg Ludwig became King George I of Great Britain, under the terms of the 1701 Act of Settlement. Even though Leibniz had done much to bring about this happy event, it was not to be his hour of glory. Despite the intercession of the Princess of Wales, Caroline of Ansbach, George I forbade Leibniz to join him in London until he completed at least one volume of the history of the Brunswick family his father had commissioned nearly 30 years earlier. Moreover, for George I to include Leibniz in his London court would have been deemed insulting to Newton, who was seen as having won the calculus priority dispute and whose standing in British official circles could not have been higher. Finally, his dear friend and defender, the dowager Electress Sophia, died in 1714.


Leibniz never married. He complained on occasion about money, but the fair sum he left to his sole heir, his sister's stepson, proved that the Brunswicks had, by and large, paid him well. In his diplomatic endeavors, he at times verged on the unscrupulous, as was all too often the case with professional diplomats of his day. On several occasions, Leibniz backdated and altered personal manuscripts, actions which cannot be excused or defended and which put him in a bad light during the calculus controversy. On the other hand, he was charming, well-mannered, and not without humor and imagination; he had many friends and admirers all over Europe.


Leibniz's philosophical thinking appears fragmented, because his philosophical writings consist mainly of a multitude of short pieces: journal articles, manuscripts published long after his death, and many letters to many correspondents. He wrote only two philosophical treatises, and the one he published in his lifetime, the Théodicée of 1710, is as much theological as philosophical.

Leibniz dated his beginning as a philosopher to his Discourse on Metaphysics, which he composed in 1686 as a commentary on a running dispute between Malebranche and Antoine Arnauld. This led to an extensive and valuable correspondence with Arnauld; it and the Discourse were not published until the 19th century. In 1695, Leibniz made his public entrée into European philosophy with a journal article titled "New System of the Nature and Communication of Substances". Between 1695 and 1705, he composed his New Essays on Human Understanding, a lengthy commentary on John Locke's 1690 An Essay Concerning Human Understanding, but upon learning of Locke's 1704 death, lost the desire to publish it, so that the New Essays were not published until 1765. The Monadologie, composed in 1714 and published posthumously, consists of 90 aphorisms.

Leibniz met Spinoza in 1676, read some of his unpublished writings, and has since been suspected of appropriating some of Spinoza's ideas. While Leibniz admired Spinoza's powerful intellect, he was also forthrightly dismayed by Spinoza's conclusions, especially when these were inconsistent with Christian orthodoxy.

Unlike Descartes and Spinoza, Leibniz had a thorough university education in philosophy. His lifelong scholastic and Aristotelian turn of mind betrayed the strong influence of one of his Leipzig professors, Jakob Thomasius, who also supervised his BA thesis in philosophy. Leibniz also eagerly read Francisco Suárez, a Spanish Jesuit respected even in Lutheran universities. Leibniz was deeply interested in the new methods and conclusions of Descartes, Huygens, Newton, and Boyle, but viewed their work through a lens heavily tinted by scholastic notions. Yet it remains the case that Leibniz's methods and concerns often anticipate the logic, and analytic and linguistic philosophy of the 20th century.

The Principles

Leibniz variously invoked one or another of seven fundamental philosophical Principles:

  • Identity of indiscernibles. Two things are identical if and only if they share 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.
  • Continuity. Natura non saltum facit. A mathematical analog to this principle would proceed as follows. If a function describes a transformation of something to which continuity applies, then its domain and range are both dense sets.
  • Optimism. "God assuredly always chooses the best.

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 is his theory of monads, as exposited in Monadologie. Monads are to the metaphysical realm what atoms are to the physical/phenomenal. Monads are the ultimate elements of the universe. The monads are "substantial forms of being" with the following properties: they are eternal, indecomposable, individual, subject to their own laws, un-interacting, and each reflecting the entire universe in a pre-established harmony (a historically important example of panpsychism). Monads are centers of force; substance is force, while space, matter, and motion are merely phenomenal.


Monads are purported to having gotten rid of the problematic:

The monadology was thought arbitrary, even eccentric, in Leibniz's day and since.

Theodicy and optimism


The statement that "we live in the best of all possible worlds" drew scorn, most notably from Voltaire, who lampooned it in his comic novella Candide by having the character Dr. Pangloss (a parody of Leibniz) repeat it like a mantra. Thus the adjective "panglossian", which describes one who believes that the world about us is the best possible one.

The mathematician Paul du Bois-Reymond, in his "Leibnizian Thoughts in Modern Science", wrote that Leibniz thought of God as a mathematician:

As is well known, the theory of the maxima and minima of functions was indebted to him for the greatest progress through the discovery of the method of tangents. Well, he conceives God in the creation of the world like a mathematician who is solving a minimum problem, or rather, in our modern phraseology, a problem in the calculus of variations the question being to determine among an infinite number of possible worlds, that for which the sum of necessary evil is a minimum.

A cautious defense of Leibnizian optimism would invoke certain scientific principles that emerged in the two centuries since his death and that are now thoroughly established: the principle of least action, the conservation of mass, and the conservation of energy. In addition, the modern observations that lead to the Fine-tuned Universe arguments seem to support his view:

  1. The 3+1 dimensional structure of spacetime may be ideal. In order to sustain complexity such as life, a universe probably requires three spatial and one temporal dimension. Most universes deviating from 3+1 either violate some fundamental physical laws, or are impossible. The mathematically richest number of spatial dimensions is also 3 (in the sense of topological nontriviality).
  2. The universe, solar system, and Earth are the "best possible" in that they enable intelligent life to exist. Such life exists on Earth only because the Earth, solar system, and Milky Way possess a number of unusual characteristics.
  3. The most sweeping form of optimism derives from the Anthropic Principle. Physical reality can be seen as grounded in the numerical values of a handful of dimensionless constants, the best known of which are the fine structure constant and the ratio of the rest mass of the proton to the electron. Were the numerical values of these constants to differ by a few percent from their observed values, it is unlikely that the resulting universe would contain complex structures.

Our physical laws, universe, solar system, and home planet are all "best" in the sense that they enable complex structures such as galaxies, stars, and, ultimately, intelligent life. On the other hand, it is also reasonable to believe that life might be more intelligent given some other set of circumstances.

Symbolic thought



Leibniz's calculus ratiocinator, which resembles symbolic logic, 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 Begriffsschrift and in writings by Charles Peirce and his students in the 1880s, and hence well after Boole and De Morgan began that logic in 1847.

Leibniz thought symbols were important for human understanding. He attached so much importance to the invention of good notations that he attributed all his discoveries in mathematics to this. His notation for the infinitesimal calculus is an example of his skill in this regard. Charles Peirce, a 19th-century pioneer of semiotics, shared Leibniz's passion for symbols and notation, and his belief that these are essential to a 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 characters, and the symbols of astronomy and chemistry, he deemed not real. Instead, he proposed the creation of a characteristica universalis or "universal characteristic", built on an alphabet of human thought in which each fundamental concept would be represented by a unique "real" character:





Formal logic



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 variables ranging over some universe of discourse.






Leibniz is credited, along with Isaac Newton, with the discovery of infinitesimal calculus. According to Leibniz's notebooks, a critical breakthrough occurred on 11 November 1675, when he employed integral calculus for the first time to find the area under the function y = x. He introduced several notations used to this day, for instance the integral sign ∫ representing an elongated S, from the Latin word summa and the d used for differentials, from the Latin word differentia. This ingenious and suggestive notation for the calculus is probably his most enduring mathematical legacy. Leibniz did not publish anything about his calculus until 1684. The product rule of differential calculus is still called "Leibniz's law". In addition, the theorem that tells how and when to differentiate under the integral sign is called the Leibniz integral rule.

Leibniz's approach to the calculus fell well short of later standards of rigor (the same can be said of Newton's). We now see a Leibniz "proof" as being in truth mostly a heuristic hodgepodge mainly grounded in geometric intuition. Leibniz also freely invoked mathematical entities he called infinitesimals, manipulating them in ways suggesting that they had paradoxical algebraic properties. George Berkeley, in a tract called The Analyst and elsewhere , ridiculed this and other aspects of the early calculus, pointing out that natural science grounded in the calculus required just as big of a leap of faith as theology grounded in Christian revelation.

From 1711 until his death, Leibniz's life was envenomed by a long dispute with John Keill, Newton, and others, over whether Leibniz had invented the calculus independently of Newton, or whether he had merely invented another notation for ideas that were fundamentally Newton's.

Modern, rigorous calculus emerged in the 19th century, thanks to the efforts of Augustin Louis Cauchy, Bernhard Riemann, Karl Weierstrass, and others, who based their work on the definition of a limit and on a precise understanding of real numbers. Their work discredited the use of infinitesimals to justify calculus. Yet, infinitesimals survived in science and engineering, and even in rigorous mathematics, via the fundamental computational device known as the differential. Beginning in 1960, Abraham Robinson worked out a rigorous foundation for Leibniz's infinitesimals, using model theory. The resulting nonstandard analysis can be seen as a belated vindication of Leibniz's mathematical reasoning.


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. There are two takes on this situation. On the one hand, Mates, citing a 1954 paper in German by Jacob Freudenthal, argues:


But Hirano argues differently, quoting Mandelbrot:



Scientist and engineer



Leibniz contributed a fair amount to the statics and dynamics emerging about him, often disagreeing with Descartes and Newton. He devised a new theory of motion (dynamics) based on kinetic energy and potential energy, which posited space as relative, whereas Newton felt strongly space was absolute. An important example of Leibniz's mature physical thinking is his Specimen Dynamicum of 1695.

Until the discovery of subatomic particles and the quantum mechanics governing them, many of Leibniz's speculative ideas about aspects of nature not reducible to statics and dynamics made little sense. For instance, he anticipated Albert Einstein by arguing, against Newton, that space, time and motion are relative, not absolute. Leibniz's rule in interacting theories plays a role in supersymmetry and in the lattices of quantum mechanics. The principle of sufficient reason has been invoked in recent cosmology, and his identity of indiscernibles in quantum mechanics, a field some even credit him with having anticipated in some sense. Those who advocate digital philosophy, a recent direction in cosmology, claim Leibniz as a precursor.

The vis viva

Leibniz's vis viva (Latin for living force) is mv2, twice the modern kinetic energy. He realized that the total energy would be conserved in certain mechanical systems, so he considered it an innate motive characteristic of matter. Here too his thinking gave rise to another regrettable nationalistic dispute. His vis viva was seen as rivaling the conservation of momentum championed by Newton in England and by Descartes in France; hence academics in those countries tended to neglect Leibniz's idea. Engineers eventually found vis viva useful, so that the two approaches eventually were seen as complementary.

Other natural science


Social science




Information technology

Leibniz may have been the first computer scientist and information theorist. Early in life, he discovered the binary number system (base 2), which is used on computers, then revisited that system throughout his career. He anticipated Lagrangian interpolation and algorithmic information theory. His calculus ratiocinator anticipated aspects of the universal Turing machine. In 1934, Norbert Wiener claimed to have found in Leibniz's writings a mention of the concept of feedback, central to Wiener's later cybernetic theory.






Advocate of scientific societies


Lawyer, moralist

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



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. He believed that Europe would adopt a uniform religion. He reiterated these proposals in 1715.






Leibniz was perhaps the first major European intellect to take a close interest in Chinese civilization, which he knew by corresponding with, and reading other work by, European Christian missionaries posted in China. He concluded that Europeans could learn much from the Confucian ethical tradition. He mulled over the possibility that the Chinese characters were an unwitting form of his universal characteristic. He noted with fascination how the I Ching hexagrams correspond to the binary numbers from 0 to 111111, and concluded that this mapping was evidence of major Chinese accomplishments in the sort of philosophical mathematics he admired.

As polymath



Posthumous reputation

When Leibniz died, his reputation was in decline. He was remembered for only one book, the Théodicée, whose supposed central argument Voltaire lampooned in his Candide. Voltaire's depiction of Leibniz's ideas was so influential that many believed it to be an accurate description (this misapprehension may still be the case among certain lay people). Thus Voltaire and his Candide bear some of the blame for the lingering failure to appreciate and understand Leibniz's ideas. Leibniz had an ardent disciple, Christian Wolff, whose dogmatic and facile outlook did Leibniz's reputation much harm. In any event, philosophical fashion was moving away from the rationalism and system building of the 17th century, of which Leibniz had been such an ardent exponent. His work on law, diplomacy, and history was seen as of ephemeral interest. The vastness and richness of his correspondence went unrecognized.



In 1900, Bertrand Russell published a critical study of Leibniz's metaphysics. Shortly thereafter, Louis Couturat published an important study of Leibniz, and edited a volume of Leibniz's heretofore unpublished writings, mainly on logic. While their conclusions, especially Russell's, were subsequently challenged and often dismissed, they made Leibniz somewhat respectable among 20th-century analytical and linguistic philosophers in the English-speaking world (Leibniz had already been of great influence to many Germans such as Bernhard Riemann). For example, Leibniz's phrase salva veritate, meaning interchangeability without loss of or compromising the truth, recurs in Willard Quine's writings. Nevertheless, the secondary 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).



Writings and edition



The extant parts of the critical edition of Leibniz's writings are organized as follows:

  • Series 5. Historical and Linguistic Writings. Inactive.

The systematic cataloguing of all of Leibniz's Nachlass began in 1901. It was hampered by two world wars, the NS dictatorship (with Jewish genocide, including an employee of the project, and other personal consequences), and decades of German division (two states with the cold war's "iron curtain" in between, separating scholars and also scattered portions of his literary estates). 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, Münster, Hannover and Berlin have jointly published 25 volumes of the critical edition, with an average of 870 pages, and prepared index and concordance works.

Selected works



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

See also


Works cited

  • Barrow, John D. and Frank J. Tipler, 1986. The Anthropic Cosmological Principle. Oxford Univ. Press.
  • Du Bois-Reymond, Paul, 18nn. "Leibnizian Thoughts in Modern Science".
  • Hall, A. R., 1980. Philosophers at War: The Quarrel between Newton and Leibniz. Cambridge Univ. Press.
  • Jolley, Nicholas, ed., 1995. The Cambridge Companion to Leibniz. Cambridge Univ. Press.
  • LeClerc, Ivor, ed., 1973. The Philosophy of Leibniz and the Modern World. Vanderbilt Univ. Press.
  • Lovejoy, Arthur O., 1957 (1936). "Plenitude and Sufficient Reason in Leibniz and Spinoza" in his The Great Chain of Being. Harvard Uni. Press: 144–82. Reprinted in Frankfurt, H. G., ed., 1972. Leibniz: A Collection of Critical Essays. Anchor Books.
  • Mates, Benson, 1986. The Philosophy of Leibniz: Metaphysics and Language. Oxford Univ. Press.
  • Mercer, Christia, 2001. Leibniz's metaphysics: Its Origins and Development. Cambridge Univ. Press.
  • Morris, Simon Conway, 2003. Life's Solution: Inevitable Humans in a Lonely Universe. Cambridge Uni. Press.
  • Perkins, Franklin, 2004. Leibniz and China: A Commerce of Light. Cambridge Univ. Press.
  • Riley, Patrick, 1996. Leibniz's Universal Jurisprudence: Justice as the Charity of the Wise. Harvard Univ. Press.
  • Rutherford, Donald, 1998. Leibniz and the Rational Order of Nature. Cambridge Univ. Press.
  • Struik, D. J., 1969. A Source Book in Mathematics, 1200–1800. Harvard Uni. Press.
  • Wilson, Catherine, 1989. 'Leibniz's Metaphysics''. Princeton Univ. Press.
  • Woolhouse, R.S., and Francks, R., 1998. Leibniz: Philosophical Texts. Oxford Uni. Press.

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