Philosophical Essays and Texts of Leibniz

Mathematician

Although the mathematical notion of function was implicit in trigonometric and logarithmic tables, which existed in his day, Leibniz was the first, in 1692 and 1694, to employ it explicitly, to denote any of several geometric concepts derived from a curve, such as abscissa, ordinate, tangent, chord, and the perpendicular.[54] In the 18th century, "function" lost these geometrical associations.

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).[55]

Calculus

Leibniz is credited, along with Sir Isaac Newton, with the discovery of calculus (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 = ƒ(x).[56] 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.[57] 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 exploited infinitesimals in developing the calculus, 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. A recent study argues that Leibnizian calculus was free of contradictions, and was better grounded than Berkeley's empiricist criticisms.[58]

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.

The use of infinitesimals in mathematics was frowned upon by 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, while the standard part function implements the Leibnizian transcendental law of homogeneity.

Topology

Leibniz was the first to use the term analysis situs,[59] 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:

Although for Leibniz the situs of a sequence of points is completely determined by the distance between them and is altered if those distances are altered, his admirer Euler, in the famous 1736 paper solving the Königsberg Bridge Problem and its generalizations, used the term geometria situs in such a sense that the situs remains unchanged under topological deformations. He mistakenly credits Leibniz with originating this concept. ...it is sometimes not realized that Leibniz used the term in an entirely different sense and hence can hardly be considered the founder of that part of mathematics.[60]

But Hideaki Hirano argues differently, quoting Mandelbrot:[61]

To sample Leibniz' scientific works is a sobering experience. Next to calculus, and to other thoughts that have been carried out to completion, the number and variety of premonitory thrusts is overwhelming. We saw examples in 'packing,'... My Leibniz mania is further reinforced by finding that for one moment its hero attached importance to geometric scaling. In "Euclidis Prota"..., which is an attempt to tighten Euclid's axioms, he states,...: 'I have diverse definitions for the straight line. The straight line is a curve, any part of which is similar to the whole, and it alone has this property, not only among curves but among sets.' This claim can be proved today.[62]

Thus the fractal geometry promoted by Mandelbrot drew on Leibniz's notions of self-similarity and the principle of continuity: natura non saltum facit. We also see that when Leibniz wrote, in a metaphysical vein, that "the straight line is a curve, any part of which is similar to the whole", he was anticipating topology by more than two centuries. As for "packing", Leibniz told to his friend and correspondent Des Bosses to imagine a circle, then to inscribe within it three congruent circles with maximum radius; the latter smaller circles could be filled with three even smaller circles by the same procedure. This process can be continued infinitely, from which arises a good idea of self-similarity. Leibniz's improvement of Euclid's axiom contains the same concept.


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