Definitions/Rigorous

What is a rigorous definition?

A sequence of Riemann sums over an irregular partition of an interval. The number on top is the total area of the rectangles, which converges to the integral of the function. The approximation works as long as the width of each subdivision tends to zero. Credit: Kieff.

By collecting definitions from primary sources where it is claimed and hopefully defended that a particular definition is rigorous, it should be possible to provide relatively simple tests that demonstrate rigor (or rigour).

Hopefully, the rigorous definition will be a particular type of definition.

As the figure at the upper right suggests, a rigorous definition may be the limit of a series of definitions that are closer and closer approximations to the concept to be defined. This figure is a series of more accurate approximations to the area under the curve.

Definitions

Main resource: Definitions

Def. a "statement of the meaning of a word or word group or a sign or symbol" is called a definition.

Rigorous

Def. "allowing no abatement or mitigation [unyielding or inflexible]" is called rigorous.

Theory of rigorous definitions

“[D]efinitions are always of symbols, for only symbols have meanings for definitions to explain.” “A definition can be expressed in either of two ways,”

1. writing “about the symbol to be defined, or”
2. writing “about its referent.”

In the theory of definition, “the symbol being defined is called the definiendum, and the symbol or set of symbols used to explain the meaning of the definiendum is called the definiens.” “The definiens is not the meaning of the definiendum, but another symbol or group of symbols which, according to the definition, has the same meaning as the definiendum.”

“Anyone who introduces a new symbol has complete freedom to stipulate what meaning is to be given it.”

One reason for the introduction of a new symbol is that “[t]he emotive suggestions of familiar words are often disturbing to one interested only in their literal or informative meanings.”

Def. "[m]anifesting, exercising, or favoring rigour; allowing no abatement or mitigation; scrupulously accurate; exact; strict; severe; relentless" is called rigorous.

"[A]s, ... a rigorous definition or demonstration".

Axiomatic definitions

Main resources: Definitions/Axiomatics and Axiomatic definitions

It has been stated that "the rigorous definition of distance" fulfills "the three axioms that define an Euclidean metric" so that a "generalized metric can be defined using as distance an appropriate function ... that fulfills the three axioms of an Euclidean metric". Having met these three axioms as a criteria of an Euclidean metric, the definition of the generalized metric is said to be a "rigorous definition of distance".

An axiomatic definition is a rigorous definition: "the definition must clearly state the rules that are considered as binding, and on the other hand give the implementor enough freedom to achieve efficiency by leaving certain less important aspects undefined." This rigorous definition is for "an axiomatic definition of the programming language PASCAL".

Constipation

Def. “a bowel movement only every three days or less frequently” is called constipation. This is a rigorous definition of constipation.

Formal definition of limit

Def. (Formal definition of a limit)

Let $f(x)$ be a function defined on an open interval $D$ that contains $c$, except possibly at $x=c$.

Let $L$ be a number.

If, for every $\varepsilon >0$, there exists a $\delta >0$ such that for all $x\in D$ with

$0<\left|x-c\right|<\delta ,$

we have

$\left|f(x)-L\right|<\varepsilon$, then

$\lim _{x\to c}f(x)=L$ is called a limit.

This precise definition of limit is a rigorous definition, the formal definition, from Wikibooks Calculus/Formal Definition of the Limit.

Infinitesimals

Main resource: Infinitesimals

The Greek mathematician Archimedes (c.287 BC–c.212 BC), in The Method of Mechanical Theorems, was the first to propose a logically rigorous definition of infinitesimals. His Archimedean property defines a number x as infinite if it satisfies the conditions |x|>1, |x|>1+1, |x|>1+1+1, ..., and infinitesimal if x≠0 and a similar set of conditions holds for 1/x and the reciprocals of the positive integers. A number system is said to be Archimedean if it contains no infinite or infinitesimal members.

Religion

Main resource: Religions

Def. a "conjunction of ... a dogma, a moral law, and a cult or form of worship" is called a religion.

René Guénon proposed a rigorous definition of the term "religion".

Riemann integral

Main resource: Integrals

This image shows a region $S$ under a curve $f(x)$. Credit: 4C.

The Riemann integral [is] the first rigorous definition of the integral of a function on an interval.

"For Riemann's definition of his integral, see section 4, "Über der Begriff eines bestimmten Integrals und den Umfang seiner Gültigkeit", pages 101-103.

Let $f$ be a non-negative real-valued function of the interval $[a,b]$.

Let $S=\{(x,y)|0<y<f(x)\}$ be the region of the plane under the graph of the function $f$ and above the interval $[a,b]$ [on the x-axis]. We are interested in measuring the area of $S.$

Def. a finite sequence of numbers of the form

$\,a=x_{0}<x_{1}<x_{2}<\cdots <x_{n}=b\,$

is called a partition of an interval $[a,b]$.

Def. each x_i, x_i+1 is called a subinterval of the partition.

Notation: let $\Delta x$ represent $(x_{i+1}-x_{i}).$

Notation: let the symbol $s$ represent the area of the region $S$.

Def. the length of the longest subinterval,

$\max(x_{i+1}-x_{i}),$

where 0 ≤ i ≤ n-1" is called "[t]he mesh or norm of a partition.

Def. a partition together with a finite sequence of numbers t₀, t_n−1 subject to the conditions that for each i, x_i ≤ t_i ≤ x_i+1" is called a tagged partition $P(x,t)$ of an interval [a,b].

Let the two partitions $P(x,t)$ and $Q(y,s)$ be partitions of the interval [a,b].

Def. for each integer i with 0 ≤ i ≤ n there exists an integer r(i) such that x_i = y_r(i) and such that t_i = s_j for some j with r(i) ≤ j < r(i+1)" $Q(y,s)$ is called "a refinement of $P(x,t)$.

Def. one tagged partition is greater or equal to another when the former is a refinement of the latter is called a partial order on the set of all tagged partitions.

The Riemann integral of ƒ equals s if the following condition holds:

For all $\epsilon >0$, there exists $\delta >0$ "such that for any tagged partition $x_{0},\ldots ,x_{n}$ and $t_{0},\ldots ,t_{n-1}$ whose mesh is less than $\delta$, we have

$\left|\sum _{i=0}^{n-1}f(t_{i})(x_{i+1}-x_{i})-s\right|<\varepsilon .$

$\lim _{\Delta x\to 0}\sum _{i=0}^{n-1}f(t_{i})(x_{i+1}-x_{i})=s$ is denoted by:

$\int _{a}^{b}f(x)\,dx=s.$

Stem cells

Main resources: Cells/Stems and Stem cells

A rigorous definition of the term “stem cell” is as follows.

A cell is a stem cell if and only if it has the properties:

1. unlimited self-renewal and
2. within-tissue multipotentiality.

Def. a cell that has only "the properties [of] unlimited self-renewal and within-tissue multipotentiality" is called a stem cell.

This definition has limited flexibility in that it “does not necessarily exclude cross-tissue plasticity.”

Increasing rigor

The beam width is the single most important characteristic of a laser beam profile.

For astigmatic beams a more rigorous definition of the beam width is

$d_{\sigma x}=2{\sqrt {2}}\left(\langle x^{2}\rangle +\langle y^{2}\rangle +\gamma \left(\left(\langle x^{2}\rangle -\langle y^{2}\rangle \right)^{2}+4\langle xy\rangle ^{2}\right)^{1/2}\right)^{1/2}$

and

$d_{\sigma y}=2{\sqrt {2}}\left(\langle x^{2}\rangle +\langle y^{2}\rangle -\gamma \left(\left(\langle x^{2}\rangle -\langle y^{2}\rangle \right)^{2}+4\langle xy\rangle ^{2}\right)^{1/2}\right)^{1/2}.$

This definition also incorporates information about x-y-correlation $\langle xy\rangle$, but for circular symmetric beams, both definitions are the same.

σ is the standard deviation of the horizontal or vertical marginal distribution.

"$\langle x^{2}\rangle ={\frac {1}{P}}\int {I(x,y)(x-\langle x\rangle )^{2}dxdy},$

$\langle xy\rangle ={\frac {1}{P}}\int {I(x,y)(x-\langle x\rangle )(y-\langle y\rangle )dxdy},$

$\langle y^{2}\rangle ={\frac {1}{P}}\int {I(x,y)(y-\langle y\rangle )^{2}dxdy}$

$\langle x\rangle ={\frac {1}{P}}\int {I(x,y)xdxdy},$

$\langle y\rangle ={\frac {1}{P}}\int {I(x,y)ydxdy}$

The beam power is

$P=\int {I(x,y)dxdy}$

and

$\gamma =\operatorname {sgn} \left(\langle x^{2}\rangle -\langle y^{2}\rangle \right)={\frac {\langle x^{2}\rangle -\langle y^{2}\rangle }{|\langle x^{2}\rangle -\langle y^{2}\rangle |}}.$

The beam profile is the intensity (I) of the beam as measured at specific rectilinear coordinates (x,y): $I(x,y).$

Hypotheses

Main resource: Hypotheses

1. A rigorous definition should be possible for each word or symbol.

See also

• Dominant group
• Dominant group/Proof of concept
• Introduction to Limits
• Nonlinear finite elements/Sets
• Theory of definition

Further reading

• Irving M. Copi (1955). Introduction to Logic. New York: The MacMillan Company. pp. 472. 

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