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Mathematics

Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

, non-standard calculus is the modern application of infinitesimals, in the sense of non-standard analysis

Non-standard analysis

Non-standard analysis is a branch of mathematics that formulates analysis using a rigorous notion of an infinitesimal number.Non-standard analysis was introduced in the early 1960s by the mathematician Abraham Robinson. He wrote:...

, to differential and integral calculus. It provides a rigorous justification for some arguments in calculus

Infinitesimal calculus

Infinitesimal calculus is the part of mathematics concerned with finding slope of curves, areas under curves, minima and maxima, and other geometric and analytic problems. It was independently developed by Gottfried Leibniz and Isaac Newton starting in the 1660s...

that were previously considered merely heuristic.

Calculations with infinitesimals were widely used before Karl Weierstrass

Karl Weierstrass

Karl Theodor Wilhelm Weierstrass was a German mathematician who is often cited as the "father of modern analysis".- Biography :Weierstrass was born in Ostenfelde, part of Ennigerloh, Province of Westphalia....

sought to replace them with the (ε, δ)-definition of limit

Limit of a function

In mathematics, the limit of a function is a fundamental concept in calculus and analysis concerning the behavior of that function near a particular input....

starting in the 1870s. (See history of calculus

History of calculus

Calculus, historically known as infinitesimal calculus, is a mathematical discipline focused on limits, functions, derivatives, integrals, and infinite series. Ideas leading up to the notions of function, derivative, and integral were developed throughout the 17th century, but the decisive step was...

.) For almost one hundred years thereafter, many mathematicians viewed infinitesimals as being naive and vague or meaningless.

Contrary to such views, Abraham Robinson

Abraham Robinson

Abraham Robinson was a mathematician who is most widely known for development of non-standard analysis, a mathematically rigorous system whereby infinitesimal and infinite numbers were incorporated into mathematics....

showed in 1960 that infinitesimals are precise, clear, and meaningful, building upon work by Edwin Hewitt

Edwin Hewitt

Edwin Hewitt was an American mathematician known for his work in abstract harmonic analysis and for his discovery, in collaboration with Leonard Jimmie Savage, of the Hewitt–Savage zero-one law.He received his Ph.D...

and Jerzy Łoś. According to Jerome Keisler

Howard Jerome Keisler

H. Jerome Keisler is an American mathematician, currently professor emeritus at University of Wisconsin–Madison. His research has included model theory and non-standard analysis.His Ph.D...

, "Robinson solved a three hundred year old problem by giving a precise treatment of infinitesimals. Robinson's achievement will probably rank as one of the major mathematical advances of the twentieth century."

Motivation

To calculate the derivative

of the function

at x, both approaches agree on the algebraic manipulations:

This is a non-standard computation using the hyperreals if we interpret Δx as an infinitesimal and let the symbol "

" be the relation being infinitely close.

In order to make f a real-valued function, we must dispense with the final term

. In the standard approach using only real numbers, that is done by taking the limit as

tends to zero. In the non-standard approach using hyperreal number

Hyperreal number

The system of hyperreal numbers represents a rigorous method of treating the infinite and infinitesimal quantities. The hyperreals, or nonstandard reals, *R, are an extension of the real numbers R that contains numbers greater than anything of the form1 + 1 + \cdots + 1. \, Such a number is...

s, the quantity

is taken to be an infinitesimal, a nonzero number that is closer to 0 than to any nonzero real. The manipulations displayed above then show that

is infinitely close to 2x, so the derivative of f at x is then 2x.

Discarding the "error term" is accomplished by an application of the standard part function

Standard part function

In non-standard analysis, the standard part function is a function from the limited hyperreals to the reals, which associates to every hyperreal, the unique real infinitely close to it. As such, it is a mathematical implementation of the historical concept of adequality introduced by Pierre de...

. Dispensing with infinitesimal error terms was historically considered paradoxical by some writers, most notably George Berkeley

George Berkeley

George Berkeley , also known as Bishop Berkeley , was an Irish philosopher whose primary achievement was the advancement of a theory he called "immaterialism"...

Definition of derivative

The hyperreals

Hyperreal number

can be constructed in the framework of ZFC, the standard axiomatisation of set theory used elsewhere in mathematics. To give an intuitive idea for the hyperreal approach, note that, naively speaking, non-standard analysis postulates the existence of positive numbers ε which are infinitely small, meaning that ε is smaller than any standard positive real, yet greater than zero. Every real number x is surrounded by an infinitesimal "cloud" of hyperreal numbers infinitely close to it. To define the derivative of f at a standard real number x in this approach, one no longer needs an infinite limiting process as in standard calculus. Instead, one sets

where st is the standard part function

Standard part function

, yielding the real number infinitely close to the hyperreal argument of st, and

is the natural extension of

to the hyperreals.

Continuity

A real function f is continuous at a standard real number x if for every hyperreal x' infinitely close to x, the value f(x' ) is also infinitely close to f(x). This captures Cauchy's definition of continuity.

Here to be precise, f would have to be replaced by its natural hyperreal extension usually denoted f^* (see discussion of Transfer principle

Transfer principle

In model theory, a transfer principle states that all statements of some language that are true for some structure are true for another structure...

in main article at non-standard analysis).

Using the notation

for the relation of being infinitely close as above,
the definition can be extended to arbitrary (standard or non-standard) points as follows:

A function f is microcontinuous at x if whenever

, one has

Here the point x' is assumed to be in the domain of (the natural extension of) f.

The above requires fewer quantifiers than the (ε, δ)-definition familiar from standard elementary calculus:

f is continuous at x if for every ε > 0, there exists a δ > 0 such that for every x' , whenever |x − x' | < δ, one has |ƒ(x) − ƒ(x' )| < ε.

Uniform continuity

A function f on an interval I is uniformly continuous

Uniform continuity

In mathematics, a function f is uniformly continuous if, roughly speaking, it is possible to guarantee that f and f be as close to each other as we please by requiring only that x and y are sufficiently close to each other; unlike ordinary continuity, the maximum distance between x and y cannot...

if its natural extension f* in I* has the following property (see Keisler, Foundations of Infinitesimal Calculus ('07), p. 45):

for every pair of hyperreals x and y in I*, if

then

.

In terms of microcontinuity defined in the previous section, this can be stated as follows: a real function is uniformly continuous if its natural extension f* is microcontinuous at every point of the domain of f*.

This definition has a reduced quantifier complexity when compared with the standard (ε, δ)-definition. Namely, the epsilontic definition of uniform continuity requires four quantifiers, while the infinitesimal definition requires only two quantifiers. It has the same quantifier complexity as the definition of uniform continuity in terms of sequences in standard calculus, which however is not expressible in the first-order language

First-order logic

First-order logic is a formal logical system used in mathematics, philosophy, linguistics, and computer science. It goes by many names, including: first-order predicate calculus, the lower predicate calculus, quantification theory, and predicate logic...

of the real numbers.

Furthermore, the hyperreal definition as stated above is local in the sense that it only depends on the monad

Monad (non-standard analysis)

In non-standard analysis, a monad is the set of points infinitely close to a given point.Given a hyperreal number x in R*, the monad of x is the set...

of each point in I*. Meanwhile, the standard (ε,δ)-definition is global in the sense that it is formulated in terms of pairs of points.

The localness of the hyperreal definition can be illustrated by the following three examples.

Example 1: a function f is uniformly continuous on the semi-open interval (0,1], if and only if its natural extension f* is microcontinuous (in the sense of the formula above) at every positive infinitesimal, in addition to continuity at the standard points of the interval.

Example 2: a function f is uniformly continuous on the semi-open interval [0,∞) if and only if it is continuous at the standard points of the interval, and in addition, the natural extension f* is microcontinuous at every positive infinite hyperreal point.

Example 3: similarly, the failure of uniform continuity for the squaring function

is due to the absence of microcontinuity at a single infinite hyperreal point, see below.

Concerning quantifier complexity, the following remarks were made by Kevin Houston:

Heine–Cantor theorem

The fact that a continuous function on a compact interval I is necessarily uniformly continuous (the Heine–Cantor theorem

Heine–Cantor theorem

In mathematics, the Heine–Cantor theorem, named after Eduard Heine and Georg Cantor, states that if M and N are metric spaces and M is compact then every continuous functionis uniformly continuous....

) admits a succinct non-standard proof. Let x, y be hyperreals in (the natural extension of) I. Since I is bounded, both x and y admit standard parts

Standard part function

. Since I is closed, st(x) and st(y) belong to I. If x and y are infinitely close, then by the triangle inequality, they have the same standard part

Since the function is assumed continuous at c, we have

and therefore f(x) and f(y) are infinitely close.

Why is the squaring function not uniformly continuous?

Let f(x) = x² defined on

. Let

be an infinite hyperreal. The hyperreal number

is infinitely close to N. Meanwhile, the difference

is not infinitesimal. Therefore f* fails to be microcontinuous at N. Thus, the squaring function is not uniformly continuous, according to the definition in uniform continuity above.

A similar proof may be given in the standard setting .

Example: Dirichlet function

Consider the Dirichlet function

It is well-known that the function is discontinuous at every point. Let us check this in terms of the hyperreal definition of continuity above, for instance let us show that the Dirichlet function is not continuous at π. Consider the continued fraction approximation a_n of π. Now let the index n be an infinite hypernatural number. By the transfer principle

Transfer principle

In model theory, a transfer principle states that all statements of some language that are true for some structure are true for another structure...

, the natural extension of the Dirichlet function takes the value 1 at a_n. Note that the hyperrational point a_n is infinitely close to π. Thus the natural extension of the Dirichlet function takes different values (0 and 1) at these two infinitely close points, and therefore the Dirichlet function is not continuous at π.

Limit

While the thrust of Robinson's approach is that one can dispense with the limit-theoretic approach using multiple quantifiers, the notion of limit can be easily recaptured in terms of the standard part function

Standard part function

st, namely

if and only if whenever the difference x − a is infinitesimal, the difference ƒ(x) − L is infinitesimal, as well, or in formulas:

if st(x) = a then st(ƒ(x)) = L,

cf. (ε, δ)-definition of limit.

Limit of sequence

Given a sequence of real numbers

, if

we say L is the limit of the sequence and write

if for every infinite hypernatural n, we have st(x_n)=L (here the extension principle is used to define x_n for every hyperinteger n).

This definition has no quantifier alternations.The standard (ε, δ)-style definition on the other hand does have quantifier alternations:

Extreme value theorem

To show that a real continuous function f on [0,1] has a maximum, let N be an infinite hyperinteger

Hyperinteger

In non-standard analysis, a hyperinteger N is a hyperreal number equal to its own integer part. A hyperinteger may be either finite or infinite. A finite hyperinteger is an ordinary integer...

. The interval [0, 1] has a natural hyperreal extension. The function ƒ is also naturally extended to hyperreals between 0 and 1. Consider the partition of the hyperreal interval [0,1] into N subintervals of equal infinitesimal

Infinitesimal

Infinitesimals have been used to express the idea of objects so small that there is no way to see them or to measure them. The word infinitesimal comes from a 17th century Modern Latin coinage infinitesimus, which originally referred to the "infinite-th" item in a series.In common speech, an...

length 1/N, with partition points x_i = i /N as i "runs" from 0 to N. In the standard setting (when N is finite), a point with the maximal value of ƒ can always be chosen among the N+1 points x_i, by induction. Hence, by the transfer principle

Transfer principle

In model theory, a transfer principle states that all statements of some language that are true for some structure are true for another structure...

, there is a hyperinteger i₀ such that 0 ≤ i₀ ≤ N and

for all i = 0, …, N (an alternative explanation is that every hyperfinite set admits a maximum). Consider the real point

where st is the standard part function

Standard part function

. An arbitrary real point x lies in a suitable sub-interval of the partition, namely

, so that st(x_i) = x. Applying st to the inequality

, we obtain

. By continuity of ƒ we have

.
Hence ƒ(c) ≥ ƒ(x), for all x, proving c to be a maximum of the real function ƒ. See .

Intermediate value theorem

As another illustration of the power of Robinson

Abraham Robinson

's approach, we present a short proof of the intermediate value theorem

Intermediate value theorem

In mathematical analysis, the intermediate value theorem states that for each value between the least upper bound and greatest lower bound of the image of a continuous function there is at least one point in its domain that the function maps to that value....

(Bolzano's theorem) using infinitesimals.

Let f be a continuous function on [a,b] such that f(a)<0 while f(b)>0. Then there exists a point c in [a,b] such that f(c)=0.

The proof proceeds as follows. Let N be an infinite hyperinteger

Hyperinteger

In non-standard analysis, a hyperinteger N is a hyperreal number equal to its own integer part. A hyperinteger may be either finite or infinite. A finite hyperinteger is an ordinary integer...

. Consider a partition of [a,b] into N intervals of equal length, with partition points x_i as i runs from 0 to N. Consider the collection I of indices such that f(x_i)>0. Let i₀ be the least element in I (such an element exists by the transfer principle

Transfer principle

In model theory, a transfer principle states that all statements of some language that are true for some structure are true for another structure...

, as I is a hyperfinite

Hyperfinite

Hyperfinite may refer to:*Hyperfinite set*von Neumann algebra...

set; see non-standard analysis

Non-standard analysis

). Then the real number

is the desired zero of f.
Such a proof reduces the quantifier

Predicate logic

In mathematical logic, predicate logic is the generic term for symbolic formal systems like first-order logic, second-order logic, many-sorted logic or infinitary logic. This formal system is distinguished from other systems in that its formulae contain variables which can be quantified...

complexity of a standard proof of the IVT.

Basic theorems

If f is a real valued function defined on an interval [a, b], then the transfer operator applied to f, denoted by *f, is an internal, hyperreal-valued function defined on the hyperreal interval [*a, *b].

Theorem. Let f be a real-valued function defined on an
interval [a, b]. Then f is differentiable at a < x < b if and only if
for every non-zero infinitesimal h, the value

is independent of h. In that case, the common value is the derivative of f at x.

This fact follows from the transfer principle

Transfer principle

In model theory, a transfer principle states that all statements of some language that are true for some structure are true for another structure...

of non-standard analysis and overspill

Overspill

In non-standard analysis, a branch of mathematics, overspill is a widely used proof technique...

.

Note that a similar result holds for differentiability at the endpoints a, b provided the sign of the infinitesimal h is suitably restricted.

For the second theorem, we consider the Riemann integral. This integral is defined as the limit, if it exists, of a directed family of Riemann sums; these are sums of the form

where

We will call such a sequence of values a partition or mesh and

the width of the mesh. In the definition of the Riemann integral, the limit of the Riemann sums is taken as the width of the mesh goes to 0.

Theorem. Let f be a real-valued function defined on an
interval [a, b]. Then f is Riemann-integrable on [a, b] if and only if
for every internal mesh of infinitesimal width, the quantity

is independent of the mesh. In this case, the common value is
the Riemann integral of f over [a, b].

Applications

One immediate application is an extension of the standard definitions of differentiation and integration to internal function

Internal set

In mathematical logic, in particular in model theory and non-standard analysis, an internal set is a set that is a member of a model.Internal set is the key tool in formulating the transfer principle, which concerns the logical relation between the properties of the real numbers R, and the...

s on intervals of hyperreal numbers.

An internal hyperreal-valued function f on [a, b] is S-differentiable at x, provided

exists and is independent of the infinitesimal h. The value is the S derivative at x.

Theorem. Suppose f is S-differentiable at every point of [a, b] where b − a is a bounded hyperreal. Suppose furthermore that

Then for some infinitesimal ε

To prove this, let N be a non-standard natural number. Divide the interval [a, b] into N subintervals by placing N − 1 equally spaced intermediate points:

Then

Now the maximum of any internal set of infinitesimals is infinitesimal. Thus all the ε_k's are dominated by an infinitesimal ε. Therefore,

from which the result follows.

External links

The source of this article is wikipedia, the free encyclopedia. The text of this article is licensed under the GFDL.

Motivation

Definition of derivative

Continuity

Uniform continuity

Heine–Cantor theorem

Why is the squaring function not uniformly continuous?

Example: Dirichlet function

Limit

Limit of sequence

Extreme value theorem

Intermediate value theorem

Basic theorems

Applications

See also

External links