calculus
The infinitesimal calculus is a technique developed independently by Gottfried Wilhelm Leibniz and Isaac Newton to operate differential and integral calculus . It provides a method to describe a function consistently on arbitrarily small (i.e. infinitesimal ) sections . Early attempts to quantify infinitely small intervals had failed because of contradictions and paradoxes of division .
For today's analysis , which works with limit values and not with infinitesimal numbers , the term is usually not used - however, with the so-called non - standard analysis, a consistent infinitesimal calculation has existed since the 1960s .
history
René Descartes and Bonaventura Cavalieri were important pioneers of the infinitesimal calculus . Descartes first developed methods to use algebra or arithmetic operations to solve geometric problems. Cavalieri recognized that geometrical figures are ultimately composed of infinitesimal elements.
Gottfried Wilhelm Leibniz developed the method of differences in the seventies of the 17th century. He understood a curve as an infinite corner, so that a tangent ultimately had to intersect the curve in an infinitely small distance. Below this infinitely small tangent section there is an infinitesimal slope triangle, in which the differences in the function values determine the slope of the tangent.
Leibniz also recognized that calculating the area under a curve is the inverse operation for the formation of differences - in other words: the integral calculus is the inverse (like minus and plus) of the differential calculus, or the problem of calculating the area is the inverse tangent problem. Here Leibniz determined the area under a curve as the sum of infinitely narrow rectangles.
At about the same time as Leibniz, the English natural scientist Sir Isaac Newton also developed a principle of calculus. However, he did not regard curves and lines as a sequence of an infinite number of points in the Cavalieri sense, but rather as the result of constant movement. He named an enlarged or flowing quantity as fluent , the speed of enlargement or movement as a fluxion and thus as an infinitely small time interval. This enabled him to determine the speed of the movement from the length of a path traveled (i.e. calculate the derivative) and, conversely, calculate the length of the path from a given speed (i.e. create the antiderivative).
With Newton, areas were not determined as the sum of infinitesimal partial areas, but rather the concept of deriving was placed in the center. This enabled him to derive very clear rules for everyday use. However, compared to Leibniz, his concept had some conceptual inaccuracies.
Leibniz looked at a curve by creating the slope triangle and thus arriving at the tangent. Newton, on the other hand, looked at the movement of a point in time, made the time interval infinitely small, so that the increase in movement also disappeared and thus had the opportunity to calculate the derivative, i.e. the slope, at a point.
Leibniz published his calculus in 1684, followed by Newton in 1687, but Leibniz's system of symbols prevailed because of its elegant spelling and simpler calculations. Leibniz was later attacked by Newton's followers for stealing Newton's ideas from a correspondence between the two of them in 1676. This led to a plagiarism lawsuit that was investigated by a commission from the Royal Society of London in 1712 . The commission, influenced by Newton, found Leibniz falsely guilty. This dispute then strained the relationship between English and continental mathematicians for decades. Today, both Newton's and Leibniz's method are considered to have been developed independently of one another.
With his philosophical-mathematical investigations into mathematical infinity, Nikolaus von Kues is considered to be the pioneer of infinitesimal calculus.
Calculus today
Inspired by Gödel's theorem of completeness and a resulting “non-standard model of natural numbers” that knows infinitely large “natural” numbers, Abraham Robinson developed a consistent infinitesimal calculus in the early 1960s, which is now mostly referred to as non-standard analysis and which is based on Leibniz 'Builds up ideas.
Today, infinitesimal analysis is used in parts of applied mathematics, stochastics, physics and economics, for example to construct mathematical models that can work with extreme differences in size. An example of an (often intuitive) use in atomic physics is the agreement that particles are “infinitely far” apart and therefore “almost not” influence each other. Another intuitively correct example from stochastics is the statement made again and again by pupils and students that some events should be assigned an "infinitely small" but really positive probability. Corresponding event spaces can be modeled with the help of infinitesimals.
See also
literature
- S. Albeverio, JE Fenstad, R. Hoegh-Krohn, T. Lindstrom: Non standard methods in stochastic analysis and mathematical physics . Academic Press, 1986.
- CB Boyer: The history of the calculus and its conceptual development . Dover, New York 1949.
- O. Deiser: Real numbers. The classical continuum and the natural consequences . Springer, Berlin 2007.
- W. Dunham: The calculus gallery. Masterpieces from Newton to Lebesgue . Princeton University Press, Princeton, New Jersey 2005.
- CH Edwards Jr .: The historical development of the calculus . Springer, New York 1979.
- Heinz-Jürgen Heß: Invention of the infinitesimal calculus . In: Erwin Stein , Albert Heinekamp (ed.): Gottfried Wilhelm Leibniz - The work of the great philosopher and universal scholar as mathematician, physicist, technician . Gottfried Wilhelm Leibniz Society, Hanover 1990, pp. 24–31. ISBN 3-9800978-4-6 .
- H.-N. Jahnke (ed.): History of Analysis . Spectrum, Heidelberg 1999.
- H. Kaiser / W. Nöbauer: History of Mathematics . Oldenbourg, 2003, pp. 202-263.
- H.-H. Körle: The Fantastic History of Analysis. Your problems and methods since Democritus and Archimedes. In addition, the basic terms of today . Oldenbourg, Munich 2009.
- M. Kordos: Forays through the history of mathematics . 1999.
- D. Laugwitz: Numbers and Continuum . BI, Mannheim 1986.
- A. Robinson: Non-Standard Analysis . 1966.
- K. Volkert: History of Analysis . BI, Mannheim 1988.
- W. Walter: Analysis 1 . Springer, Berlin 1997.