In
nonstandard analysis
The history of calculus is fraught with philosophical debates about the meaning and logical validity of fluxions or infinitesimal numbers. The standard way to resolve these debates is to define the operations of calculus using (ε, δ)-definitio ...
, the standard part function is a function from the limited (finite)
hyperreal number
In mathematics, the system of hyperreal numbers is a way of treating infinite and infinitesimal (infinitely small but non-zero) quantities. The hyperreals, or nonstandard reals, *R, are an extension of the real numbers R that contains numbers ...
s to the real numbers. Briefly, the standard part function "rounds off" a finite hyperreal to the nearest real. It associates to every such hyperreal
, the unique real
infinitely close to it, i.e.
is
infinitesimal
In mathematics, an infinitesimal number is a quantity that is closer to zero than any standard real number, but that is not zero. The word ''infinitesimal'' comes from a 17th-century Modern Latin coinage ''infinitesimus'', which originally referr ...
. As such, it is a mathematical implementation of the historical concept of
adequality Adequality is a technique developed by Pierre de Fermat in his treatise ''Methodus ad disquirendam maximam et minimam''[Pierre de Fermat
Pierre de Fermat (; between 31 October and 6 December 1607 – 12 January 1665) was a French mathematician who is given credit for early developments that led to infinitesimal calculus, including his technique of adequality. In particular, he ...](_blank)
,
[Karin Usadi Katz and Mikhail G. Katz (2011) A Burgessian Critique of Nominalistic Tendencies in Contemporary Mathematics and its Historiography. ]Foundations of Science
''Foundations of Science'' is a peer-reviewed interdisciplinary academic journal focussing on methodological and philosophical topics concerning the structure and the growth of science. It is the official journal of the Association for Foundati ...
.
Se
arxiv
The authors refer to the Fermat-Robinson standard part. as well as
Leibniz
Gottfried Wilhelm (von) Leibniz . ( – 14 November 1716) was a German polymath active as a mathematician, philosopher, scientist and diplomat. He is one of the most prominent figures in both the history of philosophy and the history of ma ...
's
Transcendental law of homogeneity
In mathematics, the transcendental law of homogeneity (TLH) is a heuristic principle enunciated by Gottfried Wilhelm Leibniz most clearly in a 1710 text entitled ''Symbolismus memorabilis calculi algebraici et infinitesimalis in comparatione potent ...
.
The standard part function was first defined by
Abraham Robinson
Abraham Robinson (born Robinsohn; October 6, 1918 – April 11, 1974) was a mathematician who is most widely known for development of nonstandard analysis, a mathematically rigorous system whereby infinitesimal and infinite numbers were reincorp ...
who used the notation
for the standard part of a hyperreal
(see Robinson 1974). This concept plays a key role in defining the concepts of the calculus, such as continuity, the derivative, and the integral, in
nonstandard analysis
The history of calculus is fraught with philosophical debates about the meaning and logical validity of fluxions or infinitesimal numbers. The standard way to resolve these debates is to define the operations of calculus using (ε, δ)-definitio ...
. The latter theory is a rigorous formalization of calculations with
infinitesimal
In mathematics, an infinitesimal number is a quantity that is closer to zero than any standard real number, but that is not zero. The word ''infinitesimal'' comes from a 17th-century Modern Latin coinage ''infinitesimus'', which originally referr ...
s. The standard part of ''x'' is sometimes referred to as its shadow.
Definition
![Standard part function with two continua](https://upload.wikimedia.org/wikipedia/commons/8/8f/Standard_part_function_with_two_continua.svg)
Nonstandard analysis deals primarily with the pair
, where the
hyperreal
Hyperreal may refer to:
* Hyperreal numbers, an extension of the real numbers in mathematics that are used in non-standard analysis
* Hyperreal.org, a rave culture website based in San Francisco, US
* Hyperreality, a term used in semiotics and po ...
s
are an
ordered field
In mathematics, an ordered field is a field together with a total ordering of its elements that is compatible with the field operations. The basic example of an ordered field is the field of real numbers, and every Dedekind-complete ordered fiel ...
extension of the reals
, and contain infinitesimals, in addition to the reals. In the hyperreal line every real number has a collection of numbers (called a
monad
Monad may refer to:
Philosophy
* Monad (philosophy), a term meaning "unit"
**Monism, the concept of "one essence" in the metaphysical and theological theory
** Monad (Gnosticism), the most primal aspect of God in Gnosticism
* ''Great Monad'', a ...
, or halo) of hyperreals infinitely close to it. The standard part function associates to a
finite
Finite is the opposite of infinite. It may refer to:
* Finite number (disambiguation)
* Finite set, a set whose cardinality (number of elements) is some natural number
* Finite verb, a verb form that has a subject, usually being inflected or marke ...
hyperreal
Hyperreal may refer to:
* Hyperreal numbers, an extension of the real numbers in mathematics that are used in non-standard analysis
* Hyperreal.org, a rave culture website based in San Francisco, US
* Hyperreality, a term used in semiotics and po ...
''x'', the unique standard real number ''x''
0 that is infinitely close to it. The relationship is expressed symbolically by writing
:
The standard part of any
infinitesimal
In mathematics, an infinitesimal number is a quantity that is closer to zero than any standard real number, but that is not zero. The word ''infinitesimal'' comes from a 17th-century Modern Latin coinage ''infinitesimus'', which originally referr ...
is 0. Thus if ''N'' is an infinite
hypernatural
In nonstandard analysis, a hyperinteger ''n'' is a hyperreal number that is equal to its own integer part. A hyperinteger may be either finite or infinite. A finite hyperinteger is an ordinary integer. An example of an infinite hyperinteger is ...
, then 1/''N'' is infinitesimal, and
If a hyperreal
is represented by a Cauchy sequence
in the
ultrapower
The ultraproduct is a mathematical construction that appears mainly in abstract algebra and mathematical logic, in particular in model theory and set theory. An ultraproduct is a quotient of the direct product of a family of structures. All factor ...
construction, then
:
More generally, each finite
defines a
Dedekind cut
In mathematics, Dedekind cuts, named after German mathematician Richard Dedekind but previously considered by Joseph Bertrand, are а method of construction of the real numbers from the rational numbers. A Dedekind cut is a partition of the r ...
on the subset
(via the total order on
) and the corresponding real number is the standard part of ''u''.
Not internal
The standard part function "st" is not defined by an
internal set
In mathematical logic, in particular in model theory and nonstandard analysis, an internal set is a set that is a member of a model.
The concept of internal sets is a tool in formulating the transfer principle, which concerns the logical relation ...
. There are several ways of explaining this. Perhaps the simplest is that its domain L, which is the collection of limited (i.e. finite) hyperreals, is not an internal set. Namely, since L is bounded (by any infinite hypernatural, for instance), L would have to have a least upper bound if L were internal, but L doesn't have a least upper bound. Alternatively, the range of "st" is
, which is not internal; in fact every internal set in
that is a subset of
is necessarily ''finite'', see (Goldblatt, 1998).
Applications
All the traditional notions of calculus can be expressed in terms of the standard part function, as follows.
Derivative
The standard part function is used to define the derivative of a function ''f''. If ''f'' is a real function, and ''h'' is infinitesimal, and if ''f''′(''x'') exists, then
:
Alternatively, if
, one takes an infinitesimal increment
, and computes the corresponding
. One forms the ratio
. The derivative is then defined as the standard part of the ratio:
:
Integral
Given a function
on