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Gottfried Wilhelm Leibniz (1646 – 1716)

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Leibniz occupies a prominent place in the history of mathematics and the history of philosophy. He developed the infinitesimal calculus 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 calculators. While working on adding automatic multiplication and division to Pascal's calculator, he was the first to describe a pinwheel calculator in 1685 and invented the Leibniz wheel, used in the arithmometer, 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 is, in a restricted sense, the best possible one that God could have created. Leibniz, along with René Descartes and Baruch Spinoza, was one of the three great 17th century advocates of rationalism. The work of Leibniz anticipated modern logic and analytic philosophy, but his philosophy also looks back to the scholastic 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 and technology, and anticipated notions that surfaced much later in biology, medicine, geology, probability theory, psychology, linguistics, and information science. He wrote works on politics, law, ethics, theology, history, philosophy, and philology. 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to this vast array of subjects were scattered in various learned journals, in tens of thousands of letters, and in unpublished manuscripts.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.

Leibniz was the first to see that the coefficients of a system of linear equations could be arranged into an array, now called a matrix, which can be manipulated to find the solution of the system, if any. This method was later called Gaussian elimination. Leibniz's discoveries of Boolean algebra and of symbolic logic, also relevant to mathematics, are discussed in the preceding section. The best overview of Leibniz's writings on the calculus may be found in Bos (1974).

Leibniz is credited, along with Sir Isaac Newton, with the invention of infinitesimal calculus (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 = ∫ f(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 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 argument 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 also in De Motu, criticized these.

From 1711 until his death, Leibniz was engaged in a dispute with John Keill, Newton and others, over whether Leibniz had invented the calculus independently of Newton. This subject is treated at length in the article Leibniz-Newton controversy.

Infinitesimals were officially banned from mathematics by the followers of Karl Weierstrass, but 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, in the context of a field of hyperreal numbers. The resulting non-standard analysis can be seen as a belated vindication of Leibniz's mathematical reasoning. Robinson's transfer principle is a mathematical implementation of Leibniz's heuristic law of continuity.

Leibniz may have been the first computer scientist and information theorist. Early in life, he documented the binary numeral system (base 2), 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.

In 1671, Leibniz began to invent a machine that could execute all four arithmetical operations, gradually improving it over a number of years. This "Stepped Reckoner" attracted fair attention and was the basis of his election to the Royal Society in 1673. A number of such machines were made during his years in Hanover, by a craftsman working under Leibniz's supervision. It was not an unambiguous success because it did not fully mechanize the operation of carrying. Couturat 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 and Ada Lovelace. In 1679, while mulling over his binary arithmetic, Leibniz imagined a machine in which binary numbers were represented by marbles, governed by a rudimentary sort of punched cards. 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.

Selected works (The year given is usually that in which the work was completed, not of its eventual publication.):

1666. De Arte Combinatoria (On the Art of Combination); partially translated in Loemker ß1 and Parkinson (1966).
1671. Hypothesis Physica Nova (New Physical Hypothesis); Loemker ß8.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 ß35, 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; Farrer, A.M., and Huggard, E.M., trans., 1985 (1952). Wiener III.11 (part). An online translation is available at Project Gutenberg.
1714. Monadologie; translated by Nicholas Rescher, 1991. The Monadology: An Edition for Students. University of Pittsburg Press. Ariew and Garber 213, Loemker ß67, 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.

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