LEIBNITZ 1691

Description

Leibnitz, Quadratura Arithmetica, Acta eruditorum, Lipsiae, Grosse & Gleditsch, 1691

 Gottfried Wilhelm Leibniz(1646 –1716) was a German polymath active as a mathematician, philosopher, scientist and diplomat. He is a prominent figure in both the history of philosophy and the history of mathematics. He wrote works on philosophy, theology, ethics, politics, law, history and philology. Leibniz also made major contributions to physics and technology, and anticipated notions that surfaced much later in probability theory, biology, medicine, geology, psychology, linguistics and computer science. In addition, he contributed to the field of library science by devising a cataloguing system whilst working at Wolfenbüttel library in Germany that would have served as a guide for many of Europe’s largest libraries.Leibniz’s contributions to a wide range of subjects were scattered in various learned journals, in tens of thousands of letters and in unpublished manuscripts. He wrote in several languages, primarily in Latin, French and German

As a philosopher, he was a leading representative of 17th-century rationalism and idealism. As a mathematician, his major achievement was the development of the main ideas of differential and integral calculus, independently of Isaac Newton’s contemporaneous developments. Mathematicians have consistently favored Leibniz’s notation as the conventional and more exact expression of calculus.

In the 20th century, Leibniz’s notions of the law of continuity and transcendental law of homogeneity found a consistent mathematical formulation by means of non-standard analysis. He was also a pioneer 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 the foundation of nearly all digital (electronic, solid-state, discrete logic) computers. This includes the Von Neumann architecture, which represents the standard “computer architecture” through from the second half of the 20th century to the present. Leibniz has been called the “founder of computer science”.

In philosophy and theology, Leibniz is most noted for his optimism, i.e. his conclusion that our world is, in a qualified sense, the best possible world that God could have created, a view sometimes lampooned by other thinkers, such as Voltaire in his satirical novella Candide. Leibniz, along with René Descartes and Baruch Spinoza, was one of the three influential early modern rationalists. His philosophy also assimilates elements of the scholastic tradition, notably the assumption that some substantive knowledge of reality can be achieved by reasoning from first principles or prior definitions. The work of Leibniz anticipated modern logic and still influences contemporary analytic philosophy, such as its adopted use of the term “possible world” to define modal notions.

In 1682, Leibniz, together with a fellow German philosopher and scientist, Otto Mencke (1644-1703), founded a scholarly journal, Acta Eruditorum [Reports of Scholars], in Leipzig. The journal was intended for the German-speaking regions of Europe, despite being written almost entirely in Latin. Acta Eruditorum, a monthly journal, would become the vehicle for much of the mathematical publication of Leibniz and the Bernoullis and would eventually be the forum through which Leibniz defended his priority in the development of calculus.

According to J. J. O’Connor’s and E. F. Robertson’s biography in the MacTutor History of Mathematics Archive, Leibniz “developed the basic features of his version of the calculus” while living in Paris during the 1670s:

In 1673 he was still struggling to develop a good notation for his calculus and his first calculations were clumsy. On 21 November 1675 he wrote a manuscript using the ∫f(x)dx notation for the first time. In the same manuscript the product rule for differentiation is given. By autumn 1676 Leibniz discovered the familiar d(xⁿ)=nxⁿ⁻¹dx for both integral and fractional n.

Leibniz began publishing his calculus results during the 1680s. He published three famous articles on the Calculus in Acta Eruditorum in 1684, 1686, and 1693:

• Nova Methodus pro Maximis et Minimis…, 1684
• De geometria recondite et analysi indivisibilium atque infinito-rum…, 1686
• Supplementum Geometrie Dimensorie…, 1693

Bibliography: J. J. O’Connor and E. F. Robertson, “Gottfried Wilhelm von Leibniz,” MacTutor History of Mathematics Archive.