Non-standard calculus: Wikis

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In mathematics, non-standard calculus is the name for the modern application of infinitesimals, in the sense of non-standard analysis, to differential and integral calculus. It provides a rigorous justification of what were previously believed to be purely formal calculations of infinitesimal calculus. Calculations with infinitesimals were widely used before the theory of limits replaced them in the 19th century. See history of calculus. To quote H. Jerome Keisler:

"In [1960], 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" (see Elementary calculus).

After the introduction of limits and prior to Robinson, mathematicians thought of infinitesimals in terms of "naive befogging" and "vague mystical ideas". Thus, Richard Courant wrote on page 81 of Differential and Integral Calculus, Vol I as follows:

We must, however, guard ourselves against thinking of dx as an "infinitely small quantity" or "infinitesimal", or of the integral as the "sum of an infinite number of infinitesimally small quantities". Such a conception would be devoid of any clear meaning; it is only a naive befogging of what we have previously carried out with precision."

and again on page 101:

We have no right to suppose that first Δx goes through something like a limiting process and reaches a value which is infinitesimally small but still not 0, so that Δx and Δy are replaced by "infinitely small quantities" or "infinitesimals" dx and dy, and that the quotient of these quantities is then formed. Such a conception of the derivative is incompatible with the clarity of ideas demanded in mathematics; in fact, it is entirely meaningless. For a great many simple-minded people it undoubtedly has a certain charm, the charm of mystery which is always associated with the word "infinite"; and in the early days of the differential calculus even Leibnitz [sic] himself was capable of combining these vague mystical ideas with a thoroughly clear understanding of the limiting process. It is true that this fog which hung round the foundations of the new science did not prevent Leibnitz [sic] or his great successors from finding the right path. But this does not release us from the duty of avoiding every such hazy idea in our building-up of the differential and integral calculus.

Contents

Motivation

To calculate the derivative of the function f(t)=t2 at the value x, both approaches agree on the algebraic manipulations:

 \frac{(x + \Delta x)^2 - x^2}{\Delta x} = 2 x + \Delta x \approx 2 x

This calculation becomes an example of a computation in non-standard calculus, once we interpret Δx as an infinitesimal, and \approx as the relation of being infinitely close. In more detail, note that the final term Δx must be discarded to define the derivative. In the standard approach, this is done by taking the limit as Δx tends to zero. In the non-standard approach, the quantity Δx is considered an infinitesimal number, namely a nonzero number which is closer to 0 than any nonzero standard real.

The derivative of f at x is then 2x, because it only differs from 2x by an infinitesimal quantity. Stripping away the error term, accomplished by an application of the standard part function, was historically considered paradoxical by some writers, most notably George Berkeley. In his famous criticism of calculus, he named the discarded error terms, the ghosts of departed quantities.

Definition of derivative

The hyperreals are 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 is surrounded by an infinitesimal "cloud" of hyperreal numbers infinitely close to it. To define the derivative of f in this approach, one no longer needs an infinite limiting process as in standard calculus. Instead, one sets

 f'(x) = \mathrm{st} \left( \frac{f(x+\epsilon)-f(x)}{\epsilon} \right),

where st is the standard part function, yielding the standard real number infinitely close to the hyperreal argument of st. The addition of st to the formula resolves the centuries-old paradox already severely criticized by George Berkeley (see Ghosts of departed quantities), and provides a rigorous basis to the approach of Newton, Leibniz, and Bonaventura Cavalieri.

Continuity

A real function f is continuous at x if for every hyperreal x' infinitely close to x, the value f(x') is also infinitely close to f(x).

Here to be precise, f would have to be replaced by its natural hyperreal extension usually denoted f* (see discussion of extension principle in main article at non-standard analysis).

Using the notation \approx for the relation of being infinitely close as above, the definition can be rewritten in an even shorter form as follows:

A function f is continuous at x if whenever x'\approx x, one has f^*(x')\approx f^*(x).

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 whenever |x-x' | < δ, one has |ƒ(x) − ƒ(x' )| <  ε.

Uniform continuity

A function f on an interval I is uniformly continuous if its natural extension f* in I* has the following property (see Keisler, Foundations of Infinitesimal Calculus ('07), p. 45) depending only on the pointwise cluster:

for every pair of hyperreals x and y in I*, if x\approx y then f^*(x)\approx f^*(y).

This definition has a reduced quantifier complexity when compared with the standard (ε, δ)-definition. 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 of the real numbers.

Furthermore, the hyperreal definition as stated above is local in the sense that it only depends on the monad 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 continuous (in the sense of the formula above) at a 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 continuous at every positive infinite hyperreal point.

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

x^2\,

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

Concerning quantifier complexity, the following remarks were made by Kevin Houston:[1]

The number of quantifiers in a mathematical statement gives a rough measure of the statement’s complexity. Statements involving three or more quantifiers can be difficult to understand. This is the main reason why it is hard to understand the rigorous definitions of limit, convergence, continuity and differentiability in analysis as they have many quantifiers.

In fact, it is the alternation of the \forall and \exists that causes the complexity.

Heine-Cantor theorem

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

c = \operatorname{st}(x) = \operatorname{st}(y).\,

Since the function is assumed continuous at c, we have

f(x)\approx f(c)\approx f(y),\,

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

Why is the squaring function not uniformly continuous?

Let f(x) = x2 defined on \mathbb{R}. Let N\in \mathbb{R}^* be an infinite hyperreal. The hyperreal number N + \tfrac{1}{N} is infinitely close to N. Meanwhile, the difference

 f(N+\tfrac{1}{N}) - f(N) = N^2 + 2 + \tfrac{1}{N^2} - N^2 = 2 + \tfrac{1}{N^2}

is not infinitesimal. Therefore 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 (Fitzpatrick 2006, Example 3.15).

Dirichlet function

Consider the Dirichlet function

I_Q(x)=\begin{cases}1\, \mbox{ if }\, x\, \mbox{ is rational} \\0\, \mbox{ if }\, x\, \mbox{ is irrational} \end{cases}.

It is well-known that the function is discontinuous at every point. Let us check this in terms of the non-standard definition of continuity above, for instance let us show that the Dirichlet function is not continuous at π. Consider the continued fraction approximation an of π. Now let the index n be an infinite hyperinteger. By the transfer principle, the natural extension of the Dirichlet function takes the value 1 at an. Note that the hyperrational point an is infinitely close to π. Thus the natural extension of the Dirichlet function takes different values 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 st, namely

\lim_{x\to a} f(x) = L\,

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 \{x_n|n\in \mathbb{N}\}\;, if L\in \mathbb{R}\; we say L is the limit of the sequence and write

 L = \lim_{n \to \infty} x_n

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

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

L = \lim_{n \to \infty} x_n\Longleftrightarrow \forall \epsilon>0\;, \exists N \in \mathbb{N}\;, \forall n \in \mathbb{N} : n >N \rightarrow d(x_n,L)<\epsilon.\;

Intermediate value theorem

As another illustration of the power of Robinson's approach, we present a short proof of the intermediate value theorem (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. Consider a partition of [a,b] into N intervals of equal length, with partition points xi as i runs from 0 to N. Consider the collection I of indices such that f(xi)>0. Let i0 be the least element in I (such an element exists by the transfer principle, as I is an internal set; see non-standard analysis). Then the real number

c=\mathrm{st}(x_{i_0})

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

Robinson's Argument

In an axiomatic formal treatment of numbers, there are symbols for numbers such as 1,2,3,4, rational numbers like 4/5, 7/17, and certain irrational quantities which have a precise definition: \scriptstyle \pi for example can be defined as the smallest positive value where sin(x) is zero. But there are infinitely many numbers which cannot be given an explicit name, because the set of all names is countable, while the real numbers are not.

To any formal axiomatic description of the real numbers it is possible to add axioms that describe numbers which otherwise would be inaccessible. For Robinson's argument, the new axioms are of the form:

  1. There exists a number \scriptstyle\epsilon>0
  2. \scriptstyle\epsilon is smaller than 1/2
  3. \scriptstyle\epsilon is smaller than 1/3
  4. \scriptstyle\epsilon is smaller than 1/4

and so on, making an infinite list of axioms. Any finite number of these axioms is consistent, meaning that it cannot lead to a logical contradiction, because there actually is an explicitly nameable \scriptstyle \epsilon which is smaller than 1/N for any N. But when the list is allowed to be infinite, the symbol ε becomes a quantity which cannot be named--- since it is positive, but smaller than the reciprocal of any rational number. It has become infinitesimal.

In logic, deductions are not allowed to use infinitely many statements--- the deductions must end after finitely many steps. So any deduction from this infinite list of axioms can never reach a contradiction. The reason is that such a deduction only uses finitely many of the axioms, and would therefore also be valid if only finitely many of the axioms are kept. But any finite truncation of this list of axioms is consistent.

So the theory of the real numbers including infinitesimals is equally consistent with the theory of real numbers without them, although it requires an infinite list of new axioms to make sure that the infinitesimals are there.

Along with the infinitesimal quantity \scriptstyle\epsilon come all the quantities which can be formed by multiplication and addition. These are the polynomials in positive and negative powers of \scriptstyle\epsilon, the nonstandard numbers. A nonstandard real number is finite if it involves no negative powers of \scriptstyle\epsilon, and it is infinitesimal if it additionally doesn't have a constant term. Since ε is a quantity in a theory of the real numbers, any function which is defined for all positive real numbers sufficiently close to zero is consistently defined for ε too, although its value is just as impossible to name explicitly as \scriptstyle\epsilon itself.

The standard part A of a nonstandard number B is the unique standard number such that A-B is infinitesimal.

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

 \Delta_h f := \operatorname{st} \frac{[*f](x+h)-[*f](x)}{h}

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

This fact follows from the transfer principle of non-standard analysis and overspill.

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

 \sum_{k=0}^{n-1} f(\xi_k) (x_{k+1} - x_k)

where

a = x_0 \leq \xi_0 \leq x_1 \leq \ldots x_{n-1} \leq \xi_{n-1} \leq x_n = b.

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

 \sup_k (x_{k+1} - x_k)

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

 S_M = \operatorname{st} \sum_{k=0}^{n-1} [*f](\xi_k) (x_{k+1} - x_k)

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 functions on intervals of hyperreal numbers.

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

 \Delta_h f = \operatorname{st} \frac{f(x+h)-f(x)}{h}

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 ba is a bounded hyperreal. Suppose furthermore that

 |f'(x)| \leq M \quad a \leq x \leq b.

Then for some infinitesimal ε

 |f(b) - f(a)| \leq M (b-a) + \epsilon.

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:

a = x_0 < x_1< \cdots < x_{N-1} < x_N = b

Then

 |f(b) - f(a)| \leq \sum_{k=1}^{N-1} |f(x_{k+1}) - f(x_{k})| \leq \sum_{k=1}^{N-1} \left\{|f'(x_k)| + \epsilon_k\right\}|x_{k+1} - x_{k}|.

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

 |f(b) - f(a)| \leq \sum_{k=1}^{N-1} (M + \epsilon)(x_{k+1} - x_{k}) = M(b-a) + \epsilon (b-a)

from which the result follows.

External links

See also

Notes

  1. ^ Kevin Houston, How to Think Like a Mathematician, ISBN 9780521719780

References

  • Fitzpatrick, Patrick (2006), Advanced Calculus, Brooks/Cole  
  • H. Jerome Keisler: Elementary Calculus: An Approach Using Infinitesimals. First edition 1976; 2nd edition 1986. (This book is now out of print. The publisher has reverted the copyright to the author, who has made available the 2nd edition in .pdf format available for downloading at http://www.math.wisc.edu/~keisler/calc.html.)
  • H. Jerome Keisler: Foundations of Infinitesimal Calculus, available for downloading at http://www.math.wisc.edu/~keisler/foundations.html (10 jan '07)
  • The Origins of Cauchy's Rigorous Calculus, Judith Grabiner, Dover Publications (February 4, 2005), ISBN 0486438155
  • Bishop, Errett (1975), "The crisis in contemporary mathematics", Historia Math. (4): 507–517  
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