alternative formulation of calculus using a logically rigorous notion of infinitesimal numbers
Nonstandard analysis is a logically precise method for the use of infinitesimals in calculus and mathematical analysis. Abraham Robinson originated the mathematics of nonstandard analysis in the 1960s.
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- There are three main tools in nonstandard analysis. One is the transference principle, which roughly states that the same assertions of the formal language are true in the standard universe as in the nonstandard universe. It is typically used by proving a desired result in the nonstandard universe, and then, noting that the result is expressible in the language, concluding that it holds in the standard universe as well.
Another technique is concurrence. This is a logical technique that guarantees that the extended structure contains all possible completions, compactifications, and so forth.
The third technique is internality. A set s of elements of the nonstandard universe is internal if s itself is an element of the nonstandard universe; otherwiise, s is external. A surprislingly useful method of proof is one by reductio ad absurdum in which the contradiction is that some set one knows to be external would in fact be internal under the assumption being refuted.
- Nonstandard analysis has proved to be a natural framework for studying the local properties of Banach spaces. The central construction in this approach is the nonstandard hull, introduced by Luxemburg ... . Not only is this a useful tool in studying the local theory of Banach space geometry, but also nonstandard hulls arise naturally in many other places within nonstandard analysis.
- We describe and analyze a parametrization of fractal ‘‘curves’’ (i.e., fractal of topological dimension 1). The nondifferentiability of fractals and their infinite length forbid a complete description based on usual real numbers. We show that using nonstandard analysis it is possible to solve this problem: A class of nonstandard curves (whose standard part is the usual fractal) is defined so that a curvilinear coordinate along the fractal can be built, this being the first step towards the possible definition and study of a fractal space. We mention fields of physics to which such a formalism could be applied in the future.
- It was in 1966 that A. Robinson's book ... on nonstandard analysis ... appeared. In it, a first rigorous foundation of the theory of infinitesmals was developed. In fact, A. Robinson had been using theorems in mathematical logic in the fifties to derive known mathematical results in a neoclassical way. His methods were based on the theory of models and in particular on the Löwenheim-Skolem theorem.
- ... in the author's opinion this is the most important advantage of nonstandard analysis over standard analysis: To have convenient (almost "explicit") representations of certain objects like Hahn-Banach limits for which by standard methods more or less only their mere existence can be proved with the axiom of choice.