The Non-standard analysis reference article from the English Wikipedia on 24-Apr-2004
(provided by Fixed Reference: snapshots of Wikipedia from wikipedia.org)

Non-standard analysis

Connect with a children's charity on your social network
In the most restricted sense, Nonstandard analysis or Non-standard analysis is that branch of mathematics that formulates analysis using a rigorous notion of infinitesimal number, where an element of an ordered field is infinitesimal if and only its absolute value is smaller than any element of the set

Ordered fields which have infinitesimal elements are also called Non archimedean.

Non-standard analysis was introduced in the early 1960's by the mathematician Abraham Robinson. Robinson' original approach was based on so-called non-standard models of the the field of real numbers. His classic foundational book on the subject Non-standard Analysis was published in 1966. The book has been reissued in paperback by Princeton University Press (see reference below) and is widely available in popular bookstores.

There are a number of technical issues that must be addressed by a theory of analysis sufficiently powerful to allow development of infinitesimal calculus. For example, not every ordered field with infinitesimals is sufficiently rich to allow such a development. See the article Hyperreals for a discussion of some of the relevant ideas.

Table of contents
1 Motivation
2 Approaches to Non-standard Analysis
3 Applications
4 Criticisms
5 References
6 External links

Motivation

\r\r\r\r\r\r\r\r \r\r\r\r\r\r\r\r \r\r\r\r\r\r\r\r \r\r\r\r\r\r\r\r \r\r\r\r\r\r\r\r \r\r\r\r\r\r\r\r \r\r\r\r\r\r\r\r \r\r\r\r\r\r\r\r \r\r\r\r\r\r\r\r \r\r\r\r\r\r\r\r \r\r\r\r\r\r\r\r \r\r\r\r\r\r\r\r \r\r\r\r\r\r\r\r \r\r\r\r\r\r\r\r \r\r\r\r\r\r\r\r \r\r\r\r\r\r\r\r \r\r\r\r\r\r\r\r \r\r\r\r\r\r\r\r \r\r\r\r\r\r\r\r \r\r\r\r\r\r\r\r \r\r\r\r\r\r\r\r \r\r\r\r\r\r\r\r \r\r\r\r\r\r\r\r \r\r\r\r\r\r\r\r \r\r\r\r\r\r\r\r \r\r\r\r\r\r\r\r \r\r\r\r\r\r\r\r \r\r\r\r\r\r\r\r \r\r\r\r\r\r\r\r \r\r\r\r\r\r\r\r \r\r\r\r\r\r\r\r \r\r\r\r\r\r\r\r \r\r\r\r\r\r\r\r \r\r\r\r\r\r\r\r \r\r\r\r\r\r\r\r \r\r\r\r\r\r\r\r \r\r\r\r\r\r\r\r \r\r\r\r\r\r\r\r \r\r\r\r\r\r\r\r \r\r\r\r\r\r\r\r \r\r\r\r\r\r\r\r \r\r\r\r\r\r\r\r \r\r\r\r\r\r\r\r \r\r\r\r\r\r\r\r \r\r\r\r\r\r\r\r \r\r\r\r\r\r\r\r \r\r\r\r\r\r\r\r \r\r\r\r\r\r\r\r \r\r\r\r\r\r\r\r \r\r\r\r\r\r\r\r \r\r\r\r\r\r\r\r \r\r\r\r\r\r\r\r \r\r\r\r\r\r\r\r \r\r\r\r\r\r\r\r \r\r\r\r\r\r\r\r \r\r\r\r\r\r\r\r \r\r\r\r\r\r\r\r There are at least three reasons to consider non-standard analysis: \r\r\r\r\r\r\r\r \r\r\r\r\r\r\r\r \r\r\r\r\r\r\r\r \r\r\r\r\r\r\r\r \r\r\r\r\r\r\r\r \r\r\r\r\r\r\r\r \r\r\r\r\r\r\r\r \r\r\r\r\r\r\r\r \r\r\r\r\r\r\r\r \r\r\r\r\r\r\r\r \r\r\r\r\r\r\r\r \r\r\r\r\r\r\r\r \r\r\r\r\r\r\r\r \r\r\r\r\r\r\r\r \r\r\r\r\r\r\r\r \r\r\r\r\r\r\r\r \r\r\r\r\r\r\r\r \r\r\r\r\r\r\r\r \r\r\r\r\r\r\r\r \r\r\r\r\r\r\r\r \r\r\r\r\r\r\r\r \r\r\r\r\r\r\r\r \r\r\r\r\r\r\r\r \r\r\r\r\r\r\r\r \r\r\r\r\r\r\r\r \r\r\r\r\r\r\r\r \r\r\r\r\r\r\r\r \r\r\r\r\r\r\r\r \r\r\r\r\r\r\r\r \r\r\r\r\r\r\r\r \r\r\r\r\r\r\r\r \r\r\r\r\r\r\r\r \r\r\r\r\r\r\r\r \r\r\r\r\r\r\r\r \r\r\r\r\r\r\r\r \r\r\r\r\r\r\r\r \r\r\r\r\r\r\r\r \r\r\r\r\r\r\r\r \r\r\r\r\r\r\r\r \r\r\r\r\r\r\r\r \r\r\r\r\r\r\r\r \r\r\r\r\r\r\r\r \r\r\r\r\r\r\r\r \r\r\r\r\r\r\r\r \r\r\r\r\r\r\r\r \r\r\r\r\r\r\r\r \r\r\r\r\r\r\r\r \r\r\r\r\r\r\r\r \r\r\r\r\r\r\r\r \r\r\r\r\r\r\r\r \r\r\r\r\r\r\r\r \r\r\r\r\r\r\r\r \r\r\r\r\r\r\r\r \r\r\r\r\r\r\r\r \r\r\r\r\r\r\r\r \r\r\r\r\r\r\r\r \r\r\r\r\r\r\r\r \r\r\r\r\r\r\r\r \r\r\r\r\r\r\r\r \r\r\r\r\r\r\r\r \r\r\r\r\r\r\r\r \r\r\r\r\r\r\r\r \r\r\r\r\r\r\r\r \r\r\r\r\r\r\r\r \r\r\r\r\r\r\r\r \r\r\r\r\r\r\r\r

Historical

\r \r Much of the earliest development of the infinitesimal calculus by Newton and Leibniz was formulated using expressions such as infinitesimal number and vanishing quantity. As noted in the article on
Hyperreals, these formulations were widely criticed by Bishop Berkeley and others. It was a challenge to develop a consistent theory of analysis using infinitesimals and it is arguable that the first person to solve this in a satisfactory way was Abraham Robinson, see reference below.\r\r\r\r\r\r\r\r \r\r\r\r\r\r\r\r \r\r\r\r\r\r\r\r \r\r\r\r\r\r\r\r \r\r\r\r\r\r\r\r \r\r\r\r\r\r\r\r \r\r\r\r\r\r\r\r \r\r\r\r\r\r\r\r \r\r\r\r\r\r\r\r \r\r\r\r\r\r\r\r \r\r\r\r\r\r\r\r \r\r\r\r\r\r\r\r \r\r\r\r\r\r\r\r \r\r\r\r\r\r\r\r \r\r\r\r\r\r\r\r \r\r\r\r\r\r\r\r \r\r\r\r\r\r\r\r \r\r\r\r\r\r\r\r \r\r\r\r\r\r\r\r \r\r\r\r\r\r\r\r \r\r\r\r\r\r\r\r \r\r\r\r\r\r\r\r \r\r\r\r\r\r\r\r \r\r\r\r\r\r\r\r \r\r\r\r\r\r\r\r \r\r\r\r\r\r\r\r \r\r\r\r\r\r\r\r \r\r\r\r\r\r\r\r \r\r\r\r\r\r\r\r \r\r\r\r\r\r\r\r \r\r\r\r\r\r\r\r \r\r\r\r\r\r\r\r \r\r\r\r\r\r\r\r \r\r\r\r\r\r\r\r \r\r\r\r\r\r\r\r \r\r\r\r\r\r\r\r \r\r\r\r\r\r\r\r \r\r\r\r\r\r\r\r \r\r\r\r\r\r\r\r \r\r\r\r\r\r\r\r \r\r\r\r\r\r\r\r \r\r\r\r\r\r\r\r \r\r\r\r\r\r\r\r \r\r\r\r\r\r\r\r \r\r\r\r\r\r\r\r \r\r\r\r\r\r\r\r \r\r\r\r\r\r\r\r \r\r\r\r\r\r\r\r \r\r\r\r\r\r\r\r \r\r\r\r\r\r\r\r \r\r\r\r\r\r\r\r \r\r\r\r\r\r\r\r \r\r\r\r\r\r\r\r \r\r\r\r\r\r\r\r \r\r\r\r\r\r\r\r \r\r\r\r\r\r\r\r \r\r\r\r\r\r\r\r \r\r\r\r\r\r\r\r \r\r\r\r\r\r\r\r \r\r\r\r\r\r\r\r \r\r\r\r\r\r\r\r \r\r\r\r\r\r\r\r \r\r\r\r\r\r\r\r \r\r\r\r\r\r\r\r \r\r\r\r\r\r\r\r \r\r\r\r\r\r\r\r

Pedagogical

\r \r Some authors maintain that use of infinitesimals is more intuitive and more easily grasped by students than the so-called "epsilon-delta" approach to analytic concepts. See Jerome Keisler's book referenced below. This approach can sometimes provide easier proofs of results which are somewhat tedious in epsilon-delta formulation of analysis. For example, proving the
chain rule for differentiation is easier in a non-standard setting. Much of the simplification comes\rfrom applying very easy rules of nonstandard arithmetic, viz:\r
infinitesimal x bounded = infiniteimal\r\r\r\r\r\r
infinitesimal + infinitesimal = infinitesimal \r\r\r
together with the transfer principle mentioned below. Critics of non-standard analysis maintain that these simplifications are really illusory since they merely mask use of elementary epsilon-delta arguments. \r\r\r\r\r\r One stunning pedagogical application of non-standard analysis is Edward Nelson's treatment of the theory of stochastic processes, presented in his monograph Radically Elementary Probability Theory.\r\r\r\r\r\r \r\r\r\r\r\r Though there is no scientific evidence either way on the pedagogical claim, the view that non-standard analysis simplifies teaching of Calculus and is easier for students to grasp is still a minority view.\r\r\r\r\r\r\r\r \r\r\r\r\r\r\r\r \r\r\r\r\r\r\r\r \r\r\r\r\r\r\r\r \r\r\r\r\r\r\r\r \r\r\r\r\r\r\r\r \r\r\r\r\r\r\r\r \r\r\r\r\r\r\r\r \r\r\r\r\r\r\r\r \r\r\r\r\r\r\r\r \r\r\r\r\r\r\r\r \r\r\r\r\r\r\r\r \r\r\r\r\r\r\r\r \r\r\r\r\r\r\r\r \r\r\r\r\r\r\r\r \r\r\r\r\r\r\r\r \r\r\r\r\r\r\r\r \r\r\r\r\r\r\r\r \r\r\r\r\r\r\r\r \r\r\r\r\r\r\r\r \r\r\r\r\r\r\r\r \r\r\r\r\r\r\r\r \r\r\r\r\r\r\r\r \r\r\r\r\r\r\r\r \r\r\r\r\r\r\r\r \r\r\r\r\r\r\r\r \r\r\r\r\r\r\r\r \r\r\r\r\r\r\r\r \r\r\r\r\r\r\r\r \r\r\r\r\r\r\r\r \r\r\r\r\r\r\r\r \r\r\r\r\r\r\r\r \r\r\r\r\r\r\r\r \r\r\r\r\r\r\r\r \r\r\r\r\r\r\r\r \r\r\r\r\r\r\r\r \r\r\r\r\r\r\r\r \r\r\r\r\r\r\r\r \r\r\r\r\r\r\r\r \r\r\r\r\r\r\r\r \r\r\r\r\r\r\r\r \r\r\r\r\r\r\r\r \r\r\r\r\r\r\r\r \r\r\r\r\r\r\r\r \r\r\r\r\r\r\r\r \r\r\r\r\r\r\r\r \r\r\r\r\r\r\r\r \r\r\r\r\r\r\r\r \r\r\r\r\r\r\r\r \r\r\r\r\r\r\r\r \r\r\r\r\r\r\r\r \r\r\r\r\r\r\r\r \r\r\r\r\r\r\r\r \r\r\r\r\r\r\r\r \r\r\r\r\r\r\r\r \r\r\r\r\r\r\r\r \r\r\r\r\r\r\r\r \r\r\r\r\r\r\r\r \r\r\r\r\r\r\r\r \r\r\r\r\r\r\r\r \r\r\r\r\r\r\r\r \r\r\r\r\r\r\r\r \r\r\r\r\r\r\r\r \r\r\r\r\r\r\r\r \r\r\r\r\r\r\r\r \r\r\r\r\r\r\r\r

Technical

Some recent work has been done in analysis using concepts from non-standard analysis particularly in investigating limiting processes of statistics and mathematical physics.\r\r\r\r\r\r\r\r \r\r\r\r\r\r\r\r \r\r\r\r\r\r\r\r \r\r\r\r\r\r\r\r \r\r\r\r\r\r\r\r \r\r\r\r\r\r\r\r \r\r\r\r\r\r\r\r \r\r\r\r\r\r\r\r \r\r\r\r\r\r\r\r The Albeverio et-al reference below discusses some of these applications.

Approaches to Non-standard Analysis

There are two very different approaches to non-standard analysis. The semantic or model-theoretic approach and the syntactic approach.

Note that both these approaches apply to other areas of mathematics beyond number theory, including algebra and topology.

The semantic approach is by far the most popular approach to non-standard analysis. The original formulation of non-standard analysis falls into this category of approaches. As developed by Robinson in his papers, it is based on studying saturated models of a theory. Since Robinson's first work appeared, a simpler semantic approach has been developed using purely set-theoretic objects called superstructures. In this approach a model of a theory is replaced by an object called a superstructure V(S) over a set S. Starting from a superstructure V(S) one constructs another object *V(S) using the ultrapower construction together with a mapping

which satisfies the transfer principle. The map * relates formal properties of of V and *V. Moreover it is possible to consider a simpler form of saturation called countable saturation. This simplified approach is also more suitable for use by mathematicians not specialists in model theory or logic.

The syntactic approach requires much less logic and model theory to understand and use. This approach was developed in the mid 1970's by the mathematician Edward Nelson. Nelson introduced an entirely axiomatic formulation of non-standard analysis that he called Internal Set Theory or IST. IST is an extension of Zermelo-Fraenkel set theory in that alongside the basic binary membership relation , it introduces a new unary predicate standard which can be applied to elements of the mathematical universe together with some axioms for reasoning with this new predicate.

Despite its elegance and simplicity, syntactic non-standard analysis requires a great deal of care in applying the principle of set formation (formally known as the axiom of comprehension) which mathematicians usually take for granted. As Nelson points out a common fallacy in reasoning in IST is that of illegal set formation. For instance, there is no set in IST whose elements are precisely the standard integers.

Applications

Despite some initial hope in the mathematical community that non-standard analysis would alter the way mathematicians thought about and reasoned with real numbers, this expectation never materialized. Moreover the list of new applications is still very small. Indeed there are not many results that have been proved first using non-standard analysis. One of these results is the theorem proven by Abraham Robinson and Allen Bernstein that every polynomially compact linear operator on a Hilbert space has an invariant subspace, proven before classic functional analysis techniques were developed that deal with such problems.

Other results are more along the line of reinterpreting or reproving previously known results. Of particular interest is Kamae's proof of the individual ergodic theorem or van den Dries and Wilkie's treatment of Gromov's Theorem on Groups of Polynomial Growth. There are also applications of non-standard analysis to the theory of processes, particularly constructions of brownian motion as random walks. The Albeverio et-al reference below has an excellent introduction to this area of research.


As an application to mathematical education, H. Jerome Keisler has written a practical elementary text that develops differential and integral calculus using infintesimals. Thus for instance, a function f is differentiable if and only if for every infinitesimal h


exists and is independent of h.

Criticisms

Despite the elegance and appeal of some aspects of non-standard analysis, there is a great deal of skepticism in the mathematical community about whether this machinery really adds anything that can not just as easily be achieved by standard methods. One particularly noted critic of non-standard analysis is the Fields Medalist Alain Connes. In his now famous book Non Commutative Geometry he offers an alternative approach to infinitesimals using ideals of compact operators on a Hilbert space.

Despite these criticisms, however, there is absolutely no controversy about the mathematical validity of the approach and the results of non-standard analysis.

References

\r\r \r\r \r\r \r\r \r\r \r\r \r\r \r\r \r\r \r\r \r\r \r\r \r\r \r\r \r\r \r\r \r\r \r\r \r\r \r\r \r\r \r\r \r\r \r\r \r\r \r\r \r\r \r\r \r\r \r\r \r\r \r\r \r\r \r\r \r\r \r\r \r\r \r\r \r\r \r\r \r\r \r\r \r\r \r\r \r\r \r\r \r\r \r\r \r\r \r\r \r\r \r\r \r\r \r\r \r\r \r\r \r\r \r\r \r\r \r\r \r\r \r\r \r\r \r\r \r\r \r\r \r\r \r\r \r\r \r\r \r\r \r\r \r\r \r\r \r\r \r\r \r\r \r\r \r\r \r\r \r\r \r\r \r\r \r\r \r\r \r\r \r\r \r\r \r\r \r\r \r\r \r\r \r\r \r\r \r\r \r\r \r\r \r\r \r\r \r\r \r\r \r\r \r\r \r\r \r\r \r\r \r\r \r \r \r \r\r \r\r \r\r \r\r \r\r \r\r \r\r \r\r \r\r \r\r \r\r \r\r \r\r \r\r \r\r \r\r \r\r \r\r \r\r \r\r \r\r \r\r \r\r \r\r \r\r \r\r \r\r \r\r \r\r \r\r

External links