Everything about Non-standard Analysis totally explained
Non-standard analysis is a branch of
mathematics that formulates
analysis using a rigorous notion of
infinitesimal, where a non-zero element of an
ordered field F is infinitesimal if and only if its
absolute value is smaller than any element of
F of the form 1/
n, for
n a natural number. Ordered fields that have infinitesimal elements are also called
non-Archimedean. More generally, non-standard analysis is any form of mathematics that relies on
non-standard models and the
transfer principle. A field which satisfies the transfer principle for real numbers is a
hyperreal field, and non-standard real analysis uses these fields as
non-standard models of the real numbers.
Non-standard analysis was introduced in the early 1960s by the mathematician
Abraham Robinson. Robinson's original approach was based on these non-standard models of the field of real numbers. His classic foundational book on the subject
Non-standard Analysis was published in
1966 and is still in print.
Several technical issues must be addressed to develop a calculus of infinitesimals. For example, it isn't enough to construct an ordered field with infinitesimals. See the article on
hyperreal numbers for a discussion of some of the relevant ideas.
Motivation
There are at least three reasons to consider non-standard analysis:
Historical
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
hyperreal numbers, these formulations were widely criticized by
Bishop Berkeley and others. It was a challenge to develop a consistent theory of analysis using infinitesimals and it's arguable that the first person to solve this in a satisfactory way was Abraham Robinson, see reference below.
In 1958 Curt Schmieden and Detlef Laugwitz published an Article "Eine Erweiterung der Infinitesimalrechnung" - "An Extension of Infinitesimal Calculus" (see reference below), which proposed a construction of a ring containing infinitesimals. The ring was constructed from sequences of real numbers. Two sequences were considered equivalent if they differed only in a finite number of elements. Arithmetic operations were defined elementwise. However, the ring constructed in this way contains zero divisors and thus can't be a field.
Pedagogical
Some authors maintain that the use of infinitesimals is more intuitive and more easily grasped by students than the so-called "epsilon-delta" approach to analytic concepts. See
H. 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 from applying very easy rules of nonstandard arithmetic, viz:
» :infinitesimal × bounded = infinitesimal
» :infinitesimal + infinitesimal = infinitesimal
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.
One 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.
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. 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. Both these approaches apply to other areas of mathematics beyond analysis, including number theory, algebra and topology.
The
semantic approach is by far the most popular approach to non-standard analysis. Robinson's original formulation of non-standard analysis falls into this category. As developed by him in his papers, it's based on studying models (in particular
saturated models) of a
theory. Since Robinson's work first appeared, a simpler semantic approach (due to
Elias Zakon) 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 V(
S) → *V(
S)
which satisfies the
transfer principle. The map * relates formal properties of V(
S) and *V(
S). Moreover it's possible to consider a simpler form of saturation called
countable saturation. This simplified approach is also more suitable for use by mathematicians who are 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-1970s 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's 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 in mathematics is still very small. 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. Upon reading a preprint of the Bernstein-Robinson paper, Paul Halmos reinterpreted their proof using standard techniques. Both papers appeared back-to-back in the same issue of the Pacific Journal of Mathematics. Some of the ideas used in Halmos' proof reappeared many years later in Halmos' own work on quasi-triangular operators.
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 stochastic processes, particularly constructions of Brownian motion as random walks. The Albeverio et-al reference below has an excellent introduction to this area of research.
Applications to calculus
As an application to
mathematical education, H. Jerome Keisler has written a practical elementary text that develops differential and integral calculus using the hyperreal numbers, which, as we've seen has infinitesimal elements. These applications of non-standard analysis depend on the existence of the
standard part of a limited hyperreal
r. The standard part of
r, denoted st(
r), is a standard real number infinitely close to
r.
The standard part may not always be defined. In the following illustrative examples we'll use the map
* mentioned above which applies to sets, functions etc. Moreover, as is commonly the case, we assume that for real numbers
r,
*r is identical to
r. This expresses the condition that
R is considered to be embedded in
*R. One of the expository devices Keisler uses is that of an imaginary infinite power microscope to distinguish points infinitely close together.
Criticisms
Despite the elegance and appeal of some aspects of non-standard analysis, there's a great deal of skepticism in the mathematical community about whether this machinery really adds anything that can't just as easily be achieved by
standard methods. One noted critic of non-standard analysis is the
Fields Medalist
Alain Connes, as evidenced by the following quote:
» The answer given by nonstandard analysis, a so-called nonstandard real, is equally deceiving. From every nonstandard real number one can construct canonically a subset of the interval [0,1], which isn't
Lebesgue measurable. No such set can be exhibited (Stern, 1985). This implies that not a single nonstandard real number can actually be exhibited.
A. Connes
Noncommutative Geometry and Space-Time, Page 55 in
The Geometric Universe, Huggett et al. The point of Connes' criticism is that nonstandard
hyperreals are as fictitious as
non-measurable sets. These sets can be shown to exist, assuming the
axiom of choice of
set theory, but are not constructible. Non-measurable sets are usually considered pathological, a sort of irritant that must be tolerated in order to have the axiom of choice available.
In his now famous book
Non Commutative Geometry, Connes offers an alternative approach to infinitesimals based on ideals of compact operators on a Hilbert space. In this treatment, the
Dixmier trace plays a central role, but its definition is itself dependent on the choice of a free
ultrafilter on the natural numbers, which is certainly nonconstructive. Moreover, Robinson notes on page 48 of the 1966 edition of his book that his theory doesn't require the axiom of choice but can also be based upon the
ultrafilter lemma. Robinson infinitesimals can also be obtained using a free ultrafilter over the natural numbers.
These criticisms notwithstanding, however, there's absolutely no controversy about the mathematical validity of the approach and the results of non-standard analysis.
Logical framework
Given any set
S, the
superstructure over a set
S is the set
V(
S)
defined by the conditions
»
exists and is independent of
h. In this case
f'(x) is a real number and is the derivative of
f at
x.
Further Information
Get more info on 'Non-standard Analysis'.
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