Gottfried Leibniz

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edit classify history index Gottfried Leibniz

Gottfried Wilhelm von Leibniz (1 Jul 1646 - 14 Nov 1716) was a German philosopher and mathematician, writing primarily in Latin and French, who, independently of Newton, invented Calculus, invented the Binary_Number_System, and was a contributor to a vast array of subjects, including Philosophy, Physics, Technology, Politics, Law, Ethics, Theology, History and Philology. He is grouped along with René Descartes and Baruch Spinoza as one of the three great 17th Century “rationalists”. Leibniz’s works are scattered in journals and in tens of thousands of letters and unpublished manuscripts.

Life and Works

Gottfried was born in Leipzig to Friedrich Leibniz and Catherina Schmuck. His Father died when he was six, and his Mother exerted a profound influence on his philosophical thought later in life. However, his Father had been a Professor of Moral Philosophy at University of Leipzig, and Gottfried was left with his personal library. While schoolwork focused on a small canon of authorities, his Father’s library enabled him to study a wide variety of advanced works and accelerated his mastery of Latin, and he entered University at age 14, completed a Bachelor’s in Philosophy, and defended his Disputatio Metaphysica de Principio Individui, which addressed the Principle of Individuation, in 1663. A Master’s in Philosophy followed by 1664, and he published and defended a dissertation, Specimen Quaestionum Philosophicarum ex Jure collectarum, arguing for both a theoretical and a pedagogical relationship between Philosophy and the Law. After two years of legal studies, he was awarded a Bachelor’s in Law as well.

In 1666, at age 20, Leibniz published his first book, On the Art of Combinations, and enrolled in University of Altdorf, submitting a thesis, Disputatio de Casibus perplexis in Jure. Leibniz obtained a License and Doctorate in Law that year, something which normally took three years. Declining an offer of academic appointment at Altdorf, he worked as an alchemist in Nuremberg, and met Johann Christian von Boineburg, who hired him as an assistant. Leibniz was later appointed Assessor in the Court of Appeal, later lived for several years in Paris, and met Dutch physicist and mathematician Christiaan Huygens, who pushed Leibniz to further study, eventually including the invention of his version of the differential and integral Calculus. He met the leading French philosophers of the day, and studied the writings of René Descartes and Blaise Pascal, and quarrelled with Isaac Newton over who had invented Calculus.

In diplomatic endeavors, Leibniz at times verged on the unscrupulous, as was all too often the case with professional diplomats of his day. However, on several occasions he had backdated and altered personal manuscripts, and these actions put him in a bad light during the controversy over Calculus. Nevertheless, he was charming, well-mannered, and not without humor or imagination. Leibniz never married, and although he often worried about money, he left everything to his sole heir, his Sister’s stepson. Leibniz died in Hanover in 1716. At the time, he was so out of favor that neither George I (who happened to be near Hanover at the time) nor any fellow courtier other than his personal secretary attended the funeral. Even though Leibniz was a life member of the Royal Society and the Prussian Academy of Sciences, neither organization saw fit to honor his passing. His grave went unmarked for more than 50 years.

Leibnizian Philosophy

The philosophical thinking in Leibniz’s works appears fragmented because his philosophical writings consist mainly of a multitude of short pieces, journal articles, manuscripts published 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. Yet, Leibniz dated his beginning as a philosopher to his Discourse on Metaphysics, composed in 1686 as a commentary on a running dispute between Malebranche and Antoine Arnauld. 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.

Unlike Descartes and Spinoza, Leibniz had a thorough University education in Philosophy, and his lifelong scholastic and Aristotelian turn of mind, 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 own methods and concerns often anticipate the Logic, the “Analytic” and “Linguistic Philosophy of the 20th_Century.

Monads and Theodicy

Leibniz’s best known contribution to Metaphysics is his Monadologie. Monads are to the metaphysical realm what atoms are to the physical and phenomenal. Monads are the ultimate elements of the Universe, “substantial forms of being”, eternal, indecomposable, individual, subject to their own laws, un-interacting, and each reflecting the entire universe in a pre-established harmony. Monads are centers of force, and substance is force, while Space, Matter, and Motion are merely phenomenal. The ontological essence of a monad is its irreducible simplicity. Each monad is like a little mirror of the Universe. Thus, for Leibniz, monads remove the problematic interaction between Mind and Matter (Descartes), and the lack of Individuation (Spinoza).

Leibniz’s Principles:
  • Identity/Contradiction: If a proposition is true, then its negation is false and vice versa.
  • Identity of Indiscernibles: Two things are identical if and only if they share the same and only the same properties.
  • Principle of Sufficient Reason: The reason for anything to exist, any event to occur, any truth to obtain.
  • Pre-Established Harmony: 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.
  • Continuum Mechanics: 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.
  • Plenitude Principle: The best of all possible worlds will contain all possibilities, with our finite experience of eternity giving no reason to dispute nature’s perfection.

Logic and Mathematics

Between Aristotle’s time and the day of George Boole and Augustus De Morgan, Leibniz was an avid student of Language and Logic, and enunciated the properties of what we now call Logical Conjunction, Disjunction, Negation, Identity, Sets and even the Empty Set. Yet, Leibniz published nothing on Formal Logic in his lifetime, and what he did write on the subject consists of working drafts. Bertrand Russell went so far as to claim Leibniz had already developed Logic to a level reached 200 years later.

Although the mathematical notion of a Function was implicit in trigonometric and logarithmic tables extant in his day, Leibniz was the first, in 1692 and 1694, to employ it to denote any of several geometric concepts derived from a curve, such as the Abscissa, Ordinate, Tangent, Chord, and the Perpendicular. Leibniz was the first to see that the co-efficients of a system of linear equations could be arranged into an Array, now called a Matrix, and manipulated to find the solution of the system, if any. This method was later called Gaussian Elimination.

Famously, Leibniz is credited, along with Sir Isaac Newton, with the development of the 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 Graph of a Function. He also introduced several notations used today, such as 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, although the Product Rule of Differential Calculus is still called “Leibniz’s Law”, and the theorem showing how and when to differentiate under the integral sign is called the Leibniz Integral Rule. However, both Leibniz’s and Newtwon’s approaches to Calculus fall well short of current standards of rigour, relying as much on heuristics and intuition.

Dynamism and Technology

Leibniz contributed to the notions of Stasis and Dynamics emerging at the time, and often disagreed with Descartes and Newton. He devised a theory of motion, called Dynamics, or Dynamism, based on Kinetic and Potential Energies. His Dynamism posited Space as Relative, whereas Newton felt strongly space was Absolute. Until the discovery of subatomic particles and Quantum Mechanics governing them, many of Leibniz’s speculative ideas about aspects of Nature made little sense to contemporaries. Yet, he anticipated Albert Einstein by arguing, against Newton, that Space, Time and Motion are Relative, not Absolute, an idea often and erroneously credited to Einstein.

In proposing the Earth has a molten core, Leibniz anticipated the discoveries of Modern Geology. In Embryology, he was a pre-formationist, but also proposed organisms are the outcome of a combination of an infinite number of possible microstructures and of their powers. In the Life_Sciences and Paleontology, he revealed his study of comparative Anatomy and Fossils. One of his principal works, Protogaea, has recently been published in English for the first time, wherein he worked out an early Organismic Theory, anticipating current Biochemistry and Ecology.

In 1906, Garland published a volume of Leibniz’s writings bearing on his many practical inventions and engineering work. Few of these writings have been translated into English, yet it is understood that Leibniz was a designer or wind-driven propellers and water pumps, mining machines to extract ore, hydraulic presses, lamps, submarines, and clocks. With Denis Papin, Leibniz invented a steam engine, and even proposed a method for desalinating water. From 1680 to 1685, he had struggled to overcome the chronic flooding afflicting the ducal silver mines in the Harz Mountains, but did not succeed.

Early in life, Leibniz also documented a Binary Number System (base 2), which is currently used in computer programming (1s and Zeroes), and revisited the system throughout his career. From Lagrangian Interpolation and Agorithmic Information Theory, to a Calculus Ratiocinator, and aspects of the universal Turing Machine, we also find the concept of Feedback and a machine which could execute all four arithmetical operations. This was the basis of his election to the Royal Society in 1673, and a number of such machines were made by a craftsman working under Leibniz’s supervision. Reports suggest an unpublished note by Leibniz, dated 1674, describes a machine capable of performing some of the algebraic operations, all centuries before common computing. Modern electronic digital computers use, instead of Leibniz’s marbles moving by Gravity, shift registers, voltage gradients, and pulses of electrons, but otherwise operate much as Leibniz envisioned by 1679.

While serving as Librarian of the Ducal Libraries in Hanover and Wolfenbuettel, Leibniz became one founder of Library Science. The latter library was enormous for its day, containing 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, with no apparent knowlede the only other such system in use at the Bodleian Library at Oxford University. He helped called on publishers to distribute abstracts of their new titles produced each year, all in a standard form to help indexing. He had hoped this, along with an empirical Database would include everything printed from Gutenberg‘s Bible to Leibniz’s day. Neither proposal met with success at the time, but similar practices are standard now, including the Library of Congress and the British Library.

Further Reading

Works by Leibniz

  • 1666. De Arte Combinatoria (On the Art of Combination)
  • 1671. Hypothesis Physica Nova (New Physical Hypothesis)
  • 1673 Confessio Philosophi (A Philosopher’s Creed)
  • 1684. Nova methodus pro maximis et minimis (New method for maximums and minimums)
  • 1686. Discourse on Metaphysics
  • 1703. Explication de l’Arithmetique Binaire (Explanation of Binary Arithmetic)
  • 1710. Theodicee; Farrer, A.M., and Huggard, E.M., trans., 1985 (1952)
  • 1714. Monadologie
  • 1765. Nouveaux essais sur l’entendement humain


  • Aiton, Eric J., 1985. Leibniz: A Biography. Hilger (UK).
  • Alexander, H G (ed) The Leibniz-Clarke Correspondence. Manchester: Manchester University Press, 1956.
  • Ariew, R & D Garber, 1989. Leibniz: Philosophical Essays. Hackett.
  • arrow, John D. and Frank J. Tipler, 1986. The Anthropic Cosmological Principle. Oxford Univ. Press.
  • Cook, Daniel, and Rosemont, Henry Jr., 1994. Leibniz: Writings on China. Open Court.
  • Couturat, Louis, 1901. La Logique de Leibniz. Paris: Felix Alcan.
  • Davis, Martin, 2000. The Universal Computer: The Road from Leibniz to Turing. WW Norton.
  • Du Bois-Reymond, Paul, 18nn. “Leibnizian Thoughts in Modern Science”.
  • Grattan-Guinness, Ivor, 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 Univ. Press.
  • Hirano, Hideaki, 1997. “Cultural Pluralism And Natural Law.” Unpublished.
  • Hostler, J., 1975. Leibniz’s Moral Philosophy. UK: Duckworth.
  • 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.
  • Jolley, Nicholas, ed., 1995. The Cambridge Companion to Leibniz. Cambridge Univ. Press.
  • LeClerc, Ivor, ed., 1973. The Philosophy of Leibniz and the Modern World. Vanderbilt Univ. Press.
  • Loemker, Leroy, 1969 (1956). Leibniz: Philosophical Papers and Letters. Reidel.
  • 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.
  • Mandelbrot, Benoit, 1977. The Fractal Geometry of Nature. Freeman.
  • 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.
  • Ward, P. D., and Brownlee, D., 2000. Rare Earth: Why Complex Life is Uncommon in the Universe. Springer Verlag.
  • Struik, D. J., 1969. A Source Book in Mathematics, 1200 - 1800. Harvard Uni. Press.
  • Wiener, Philip, 1951. Leibniz: Selections. Scribner.
  • 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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