Infinitesimal calculus
The calculus is an independently developed by Leibniz and Newton technique to operate the differential and integral calculus. It provides a method, a function on arbitrary (ie infinitesimal ) sections describe consistent. Early attempts to take infinitesimal intervals quantitatively had failed to contradictions and paradoxes division.
For today's analysis, which works with limit values and not with Infinitesimalzahlen, the term is not commonly used, but there exists the so-called non-standard analysis since the 1960s, a contradiction-free calculus.
History of calculus
Gottfried Wilhelm Leibniz
Key pioneers of the infinitesimal calculus were René Descartes and Bonaventura Cavalieri. Descartes developed the first methods to use algebra or arithmetic operations in solving geometrical problems. Cavalieri realized that geometric figures are ultimately composed of infinitesimal elements.
Gottfried Wilhelm Leibniz developed in the seventies of the 17th century, the method of differences. He understood a curve as a Unendlicheck, so that a tangent ultimately had to cut the curve into an infinitely small distance. Under this infinitesimal tangent section gives an infinitesimal slope triangle, in which the differences of function values determine the slope of the tangent.
Leibniz also realized that calculate the surface area under a curve the inverse operation to the formation of differences - in other words, the integral calculus is the reverse ( as plus and minus ) of the differential calculus or the problem of area calculation is the inverse tangent problem. Here Leibniz certain the area under a curve as the sum of infinitely narrow rectangles.
About the same time Leibniz also developed the English scientist Sir Isaac Newton, a principle of calculus. However, he considered curves and lines not in the sense of Cavalieri as a series of infinitely many points, but as a result of continuous movement. He named an enlarged or flowing size as Fluente, the rate of increase or movement as Fluxion and so as an infinitely small time interval. So he could from the length of a route traversed the speed of movement determine (ie calculate the derivative ) and vice versa from a given speed calculate the length of the path (ie the antiderivative create ).
With Newton being land the concept of deriving were not determined as the sum of infinitesimal part of space, but placed at the center. So he could derive quite intuitive rules for everyday use. His concept was compared to Leibniz, however, some conceptual inaccuracies.
Leibniz considered a curve by investing the slope triangle and so comes to the tangent. Newton, however, looking at the movement of a point in time, the time interval can be infinitely small, so that the motion gain disappears and thus had the ability to drain, thus, the slope to be calculated at a point.
Leibniz published his calculations in 1684, after which followed Newton in 1687, but sat down Leibniz's system of signs for its elegant style and the simpler computations by. Leibniz was later attacked by followers of Newton, he stole the ideas of Newton from a correspondence between the two in 1676. This led to a plagiarism lawsuit, which was examined in 1712 by a commission of the Royal Society of London. The Commission, influenced by Newton, Leibniz spoke falsely guilty. This dispute then loaded for decades the relationship between English and Continental mathematicians. Today, both Newton and Leibniz's method are developed as independent.
Nicholas of Cusa applies his philosophical and mathematical studies on mathematical infinity as a pioneer of the calculus.
Calculus today
Inspired by Gödel's completeness theorem and a consequent " non-standard model of natural numbers " that knows infinitely large "natural" numbers, Abraham Robinson developed in the early 1960s a consistent calculus, which is now usually referred to as non-standard analysis and the principle of Leibniz ' based ideas.
Today the Infinitesimal Analysis is used in parts of applied mathematics, stochastics, physics and economics, about to construct mathematical models that can work with extreme differences in size. An example of a (often intuitive ) use is in atomic physics the agreement that particle " infinitely far " from each other and are therefore " almost not " affect. Another intuitively correct example from the stochastics is that of students repeatedly made observation that some events an " infinitely small ", but strictly positive probability should be assigned. Corresponding event rooms can be modeled with the help of infinitesimals.