In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, when a mathematical phenomenon runs counter to some intuition, then the phenomenon is sometimes called pathological. On the other hand, if a phenomenon does not run counter to intuition,
it is sometimes called well-behaved. These terms are sometimes useful in mathematical research and teaching, but there is no strict mathematical definition of pathological or well-behaved.
In analysis
A classic example of a pathology is the
Weierstrass function
In mathematics, the Weierstrass function is an example of a real-valued function (mathematics), function that is continuous function, continuous everywhere but Differentiable function, differentiable nowhere. It is an example of a fractal curve ...
, a function that is
continuous
Continuity or continuous may refer to:
Mathematics
* Continuity (mathematics), the opposing concept to discreteness; common examples include
** Continuous probability distribution or random variable in probability and statistics
** Continuous ...
everywhere but
differentiable
In mathematics, a differentiable function of one real variable is a function whose derivative exists at each point in its domain. In other words, the graph of a differentiable function has a non-vertical tangent line at each interior point in its ...
nowhere.
The sum of a differentiable
function
Function or functionality may refer to:
Computing
* Function key, a type of key on computer keyboards
* Function model, a structured representation of processes in a system
* Function object or functor or functionoid, a concept of object-oriente ...
and the Weierstrass function is again continuous but nowhere differentiable; so there are at least as many such functions as differentiable functions. In fact, using the
Baire category theorem, one can show that continuous functions are
generically nowhere differentiable.
Such examples were deemed pathological when they were first discovered:
To quote
Henri Poincaré
Jules Henri Poincaré ( S: stress final syllable ; 29 April 1854 – 17 July 1912) was a French mathematician, theoretical physicist, engineer, and philosopher of science. He is often described as a polymath, and in mathematics as "The ...
:
Since Poincaré, nowhere differentiable functions have been shown to appear in basic physical and biological processes such as
Brownian motion
Brownian motion, or pedesis (from grc, πήδησις "leaping"), is the random motion of particles suspended in a medium (a liquid or a gas).
This pattern of motion typically consists of random fluctuations in a particle's position insi ...
and in applications such as the
Black-Scholes model in finance.
In topology
One of the most notorious pathologies in topology is the
Alexander horned sphere
The Alexander horned sphere is a pathological object in topology discovered by .
Construction
The Alexander horned sphere is the particular embedding of a sphere in 3-dimensional Euclidean space obtained by the following construction, starting ...
, a counterexample showing that topologically embedding the sphere ''S''
2 in R
3 may fail to separate the space cleanly. As a counter-example, it motivated the extra condition of ''tameness'', which suppresses the kind of ''wild'' behavior the horned sphere exhibits.
Like many other pathologies, the horned sphere in a sense plays on infinitely fine, recursively generated structure, which in the limit violates ordinary intuition. In this case, the topology of an ever-descending chain of interlocking loops of continuous pieces of the sphere in the limit fully reflects that of the common sphere, and one would expect the outside of it, after an embedding, to work the same. Yet it does not: it fails to be
simply connected.
For the underlying theory, see
Jordan–Schönflies theorem
In mathematics, the Schoenflies problem or Schoenflies theorem, of geometric topology is a sharpening of the Jordan curve theorem by Arthur Schoenflies. For Jordan curves in the plane it is often referred to as the Jordan–Schoenflies theorem.
...
.
Well-behaved
Mathematician
A mathematician is someone who uses an extensive knowledge of mathematics in their work, typically to solve mathematical problems.
Mathematicians are concerned with numbers, data, quantity, structure, space, models, and change.
History
On ...
s (and those in related sciences) very frequently speak of whether a
mathematical
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
object—a
function
Function or functionality may refer to:
Computing
* Function key, a type of key on computer keyboards
* Function model, a structured representation of processes in a system
* Function object or functor or functionoid, a concept of object-oriente ...
, a
set
Set, The Set, SET or SETS may refer to:
Science, technology, and mathematics Mathematics
*Set (mathematics), a collection of elements
*Category of sets, the category whose objects and morphisms are sets and total functions, respectively
Electro ...
, a
space
Space is the boundless three-dimensional extent in which objects and events have relative position and direction. In classical physics, physical space is often conceived in three linear dimensions, although modern physicists usually consider ...
of one sort or another—is "well-behaved". While the term has no fixed formal definition, it generally refers to the quality of satisfying a list of prevailing conditions, which might be dependent on context, mathematical interests, fashion, and taste. To ensure that an object is "well-behaved", mathematicians introduce further axioms to narrow down the domain of study. This has the benefit of making analysis easier, but produces a
loss of generality
''Without loss of generality'' (often abbreviated to WOLOG, WLOG or w.l.o.g.; less commonly stated as ''without any loss of generality'' or ''with no loss of generality'') is a frequently used expression in mathematics. The term is used to indicat ...
of any conclusions reached.
In both pure and applied mathematics (e.g.,
optimization
Mathematical optimization (alternatively spelled ''optimisation'') or mathematical programming is the selection of a best element, with regard to some criterion, from some set of available alternatives. It is generally divided into two subfi ...
,
numerical integration
In analysis, numerical integration comprises a broad family of algorithms for calculating the numerical value of a definite integral, and by extension, the term is also sometimes used to describe the numerical solution of differential equations ...
,
mathematical physics
Mathematical physics refers to the development of mathematics, mathematical methods for application to problems in physics. The ''Journal of Mathematical Physics'' defines the field as "the application of mathematics to problems in physics and t ...
), ''well-behaved'' also means not violating any assumptions needed to successfully apply whatever analysis is being discussed.
The opposite case is usually labeled "pathological". It is not unusual to have situations in which most cases (in terms of
cardinality
In mathematics, the cardinality of a set is a measure of the number of elements of the set. For example, the set A = \ contains 3 elements, and therefore A has a cardinality of 3. Beginning in the late 19th century, this concept was generalized ...
or
measure
Measure may refer to:
* Measurement, the assignment of a number to a characteristic of an object or event
Law
* Ballot measure, proposed legislation in the United States
* Church of England Measure, legislation of the Church of England
* Mea ...
) are pathological, but the pathological cases will not arise in practice—unless constructed deliberately.
The term "well-behaved" is generally applied in an absolute sense—either something is well-behaved or it is not. For example:
*In
algorithmic inference Algorithmic inference gathers new developments in the statistical inference methods made feasible by the powerful computing devices widely available to any data analyst. Cornerstones in this field are computational learning theory, granular computi ...
, a
well-behaved statistic
Although the term well-behaved statistic often seems to be used in the scientific literature in somewhat the same way as is well-behaved in mathematics (that is, to mean "non-pathological") it can also be assigned precise mathematical meaning, and ...
is monotonic, well-defined, and
sufficient
In logic and mathematics, necessity and sufficiency are terms used to describe a conditional or implicational relationship between two statements. For example, in the conditional statement: "If then ", is necessary for , because the truth of ...
.
*In
Bézout's theorem
Bézout's theorem is a statement in algebraic geometry concerning the number of common zeros of polynomials in indeterminates. In its original form the theorem states that ''in general'' the number of common zeros equals the product of the deg ...
, two
polynomial
In mathematics, a polynomial is an expression consisting of indeterminates (also called variables) and coefficients, that involves only the operations of addition, subtraction, multiplication, and positive-integer powers of variables. An exa ...
s are well-behaved, and thus the formula given by the theorem for the number of their intersections is valid, if their polynomial greatest common divisor is a constant.
*A
meromorphic function
In the mathematical field of complex analysis, a meromorphic function on an open subset ''D'' of the complex plane is a function that is holomorphic on all of ''D'' ''except'' for a set of isolated points, which are pole (complex analysis), pole ...
is a ratio of two well-behaved functions, in the sense of those two functions being
holomorphic
In mathematics, a holomorphic function is a complex-valued function of one or more complex variables that is complex differentiable in a neighbourhood of each point in a domain in complex coordinate space . The existence of a complex derivati ...
.
*The
Karush–Kuhn–Tucker conditions
In mathematical optimization, the Karush–Kuhn–Tucker (KKT) conditions, also known as the Kuhn–Tucker conditions, are first derivative tests (sometimes called first-order necessary conditions) for a solution in nonlinear programming to be o ...
are first-order necessary conditions for a solution in a well-behaved
nonlinear programming
In mathematics, nonlinear programming (NLP) is the process of solving an optimization problem where some of the constraints or the objective function are nonlinear. An optimization problem is one of calculation of the extrema (maxima, minima or s ...
problem to be optimal; a problem is referred to as well-behaved if some regularity conditions are satisfied.
*In
probability
Probability is the branch of mathematics concerning numerical descriptions of how likely an Event (probability theory), event is to occur, or how likely it is that a proposition is true. The probability of an event is a number between 0 and ...
, events contained in the
probability space
In probability theory, a probability space or a probability triple (\Omega, \mathcal, P) is a mathematical construct that provides a formal model of a random process or "experiment". For example, one can define a probability space which models t ...
's corresponding
sigma-algebra are well-behaved, as are
measurable
In mathematics, the concept of a measure is a generalization and formalization of Geometry#Length, area, and volume, geometrical measures (length, area, volume) and other common notions, such as mass and probability of events. These seemingly ...
functions.
Unusually, the term could also be applied in a comparative sense:
*In
calculus
Calculus, originally called infinitesimal calculus or "the calculus of infinitesimals", is the mathematical study of continuous change, in the same way that geometry is the study of shape, and algebra is the study of generalizations of arithm ...
:
**
Analytic function
In mathematics, an analytic function is a function that is locally given by a convergent power series. There exist both real analytic functions and complex analytic functions. Functions of each type are infinitely differentiable, but complex an ...
s are better-behaved than general
smooth function
In mathematical analysis, the smoothness of a function (mathematics), function is a property measured by the number of Continuous function, continuous Derivative (mathematics), derivatives it has over some domain, called ''differentiability cl ...
s.
**Smooth functions are better-behaved than general differentiable functions.
**Continuous
differentiable
In mathematics, a differentiable function of one real variable is a function whose derivative exists at each point in its domain. In other words, the graph of a differentiable function has a non-vertical tangent line at each interior point in its ...
functions are better-behaved than general continuous functions. The larger the number of times the function can be differentiated, the more well-behaved it is.
**
Continuous function
In mathematics, a continuous function is a function such that a continuous variation (that is a change without jump) of the argument induces a continuous variation of the value of the function. This means that there are no abrupt changes in value ...
s are better-behaved than
Riemann-integrable functions on compact sets.
**Riemann-integrable functions are better-behaved than
Lebesgue-integrable functions.
**Lebesgue-integrable functions are better-behaved than general functions.
*In
topology
In mathematics, topology (from the Greek language, Greek words , and ) is concerned with the properties of a mathematical object, geometric object that are preserved under Continuous function, continuous Deformation theory, deformations, such ...
,
continuous
Continuity or continuous may refer to:
Mathematics
* Continuity (mathematics), the opposing concept to discreteness; common examples include
** Continuous probability distribution or random variable in probability and statistics
** Continuous ...
functions are better-behaved than discontinuous ones.
**
Euclidean space
Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's Elements, Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics ther ...
is better-behaved than
non-Euclidean geometry
In mathematics, non-Euclidean geometry consists of two geometries based on axioms closely related to those that specify Euclidean geometry. As Euclidean geometry lies at the intersection of metric geometry and affine geometry, non-Euclidean geo ...
.
**Attractive
fixed points are better-behaved than repulsive fixed points.
**
Hausdorff topologies are better-behaved than those in arbitrary
general topology
In mathematics, general topology is the branch of topology that deals with the basic set-theoretic definitions and constructions used in topology. It is the foundation of most other branches of topology, including differential topology, geomet ...
.
**
Borel set
In mathematics, a Borel set is any set in a topological space that can be formed from open sets (or, equivalently, from closed sets) through the operations of countable union, countable intersection, and relative complement. Borel sets are named ...
s are better-behaved than arbitrary
sets of
real number
In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every real ...
s.
**Spaces with
integer
An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign (−1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the language ...
dimension are better-behaved than spaces with
fractal dimension
In mathematics, more specifically in fractal geometry, a fractal dimension is a ratio providing a statistical index of complexity comparing how detail in a pattern (strictly speaking, a fractal pattern) changes with the scale at which it is me ...
.
*In
abstract algebra
In mathematics, more specifically algebra, abstract algebra or modern algebra is the study of algebraic structures. Algebraic structures include groups, rings, fields, modules, vector spaces, lattices, and algebras over a field. The term ''a ...
:
**
Groups
A group is a number of persons or things that are located, gathered, or classed together.
Groups of people
* Cultural group, a group whose members share the same cultural identity
* Ethnic group, a group whose members share the same ethnic ide ...
are better-behaved than
magmas
Magma () is the molten or semi-molten natural material from which all igneous rocks are formed. Magma is found beneath the surface of the Earth, and evidence of magmatism has also been discovered on other terrestrial planets and some natural sa ...
and
semigroup
In mathematics, a semigroup is an algebraic structure consisting of a set together with an associative internal binary operation on it.
The binary operation of a semigroup is most often denoted multiplicatively: ''x''·''y'', or simply ''xy'', ...
s.
**
Abelian group
In mathematics, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on the order in which they are written. That is, the group operation is commut ...
s are better-behaved than non-Abelian groups.
**
Finitely-generated Abelian group
In abstract algebra, an abelian group (G,+) is called finitely generated if there exist finitely many elements x_1,\dots,x_s in G such that every x in G can be written in the form x = n_1x_1 + n_2x_2 + \cdots + n_sx_s for some integers n_1,\dots, n ...
s are better-behaved than non-finitely-generated Abelian groups.
**
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 ...
-
dimensional
In physics and mathematics, the dimension of a mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any point within it. Thus, a line has a dimension of one (1D) because only one coordi ...
vector space
In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called ''vectors'', may be added together and multiplied ("scaled") by numbers called '' scalars''. Scalars are often real numbers, but can ...
s are better-behaved than
infinite
Infinite may refer to:
Mathematics
* Infinite set, a set that is not a finite set
*Infinity, an abstract concept describing something without any limit
Music
*Infinite (group), a South Korean boy band
*''Infinite'' (EP), debut EP of American m ...
-dimensional ones.
**
Fields
Fields may refer to:
Music
* Fields (band), an indie rock band formed in 2006
* Fields (progressive rock band), a progressive rock band formed in 1971
* ''Fields'' (album), an LP by Swedish-based indie rock band Junip (2010)
* "Fields", a song b ...
are better-behaved than
skew field
Skew may refer to:
In mathematics
* Skew lines, neither parallel nor intersecting.
* Skew normal distribution, a probability distribution
* Skew field or division ring
* Skew-Hermitian matrix
* Skew lattice
* Skew polygon, whose vertices do not l ...
s or general
rings
Ring may refer to:
* Ring (jewellery), a round band, usually made of metal, worn as ornamental jewelry
* To make a sound with a bell, and the sound made by a bell
:(hence) to initiate a telephone connection
Arts, entertainment and media Film and ...
.
**Separable
field extension
In mathematics, particularly in algebra, a field extension is a pair of fields E\subseteq F, such that the operations of ''E'' are those of ''F'' restricted to ''E''. In this case, ''F'' is an extension field of ''E'' and ''E'' is a subfield of ...
s are better-behaved than non-separable ones.
**
Normed division algebra
In mathematics, Hurwitz's theorem is a theorem of Adolf Hurwitz (1859–1919), published posthumously in 1923, solving the Hurwitz problem for finite-dimensional unital real non-associative algebras endowed with a positive-definite quadratic f ...
s are better-behaved than general composition algebras.
Pathological examples
Pathological examples often have some undesirable or unusual properties that make it difficult to contain or explain within a theory. Such pathological behaviors often prompt new investigation and research, which leads to new theory and more general results. Some important historical examples of this are:
*The discovery of
irrational number
In mathematics, the irrational numbers (from in- prefix assimilated to ir- (negative prefix, privative) + rational) are all the real numbers that are not rational numbers. That is, irrational numbers cannot be expressed as the ratio of two integ ...
s by the school of
Pythagoras
Pythagoras of Samos ( grc, Πυθαγόρας ὁ Σάμιος, Pythagóras ho Sámios, Pythagoras the Samos, Samian, or simply ; in Ionian Greek; ) was an ancient Ionians, Ionian Ancient Greek philosophy, Greek philosopher and the eponymou ...
in ancient Greece; for example, the length of the diagonal of a
unit square
In mathematics, a unit square is a square whose sides have length . Often, ''the'' unit square refers specifically to the square in the Cartesian plane with corners at the four points ), , , and .
Cartesian coordinates
In a Cartesian coordinate ...
, that is
.
*The discovery of
complex number
In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the form ...
s in the 16th century in order to find the roots of
cubic and
quartic polynomial functions.
*The rationals are
countable
In mathematics, a set is countable if either it is finite or it can be made in one to one correspondence with the set of natural numbers. Equivalently, a set is ''countable'' if there exists an injective function from it into the natural numbers; ...
. That is, every
rational number
In mathematics, a rational number is a number that can be expressed as the quotient or fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (e.g. ). The set of all ration ...
can be mapped to a unique
integer
An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign (−1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the language ...
.
*Some
number field
In mathematics, an algebraic number field (or simply number field) is an extension field K of the field of rational numbers such that the field extension K / \mathbb has finite degree (and hence is an algebraic field extension).
Thus K is a f ...
s have
rings of integers
In mathematics, the ring of integers of an algebraic number field K is the ring (mathematics), ring of all algebraic integers contained in K. An algebraic integer is a root of a polynomial, root of a monic polynomial with integer coefficients: x^n+ ...
that do not form a
unique factorization domain
In mathematics, a unique factorization domain (UFD) (also sometimes called a factorial ring following the terminology of Bourbaki) is a ring in which a statement analogous to the fundamental theorem of arithmetic holds. Specifically, a UFD is an ...
, for example the field
.
*The discovery of
fractal
In mathematics, a fractal is a geometric shape containing detailed structure at arbitrarily small scales, usually having a fractal dimension strictly exceeding the topological dimension. Many fractals appear similar at various scales, as illu ...
s and other "rough" geometric objects (see
Hausdorff dimension
In mathematics, Hausdorff dimension is a measure of ''roughness'', or more specifically, fractal dimension, that was first introduced in 1918 by mathematician Felix Hausdorff. For instance, the Hausdorff dimension of a single point is zero, of ...
).
*
Weierstrass function
In mathematics, the Weierstrass function is an example of a real-valued function (mathematics), function that is continuous function, continuous everywhere but Differentiable function, differentiable nowhere. It is an example of a fractal curve ...
, a
real
Real may refer to:
Currencies
* Brazilian real (R$)
* Central American Republic real
* Mexican real
* Portuguese real
* Spanish real
* Spanish colonial real
Music Albums
* ''Real'' (L'Arc-en-Ciel album) (2000)
* ''Real'' (Bright album) (2010) ...
-valued function on the
real line
In elementary mathematics, a number line is a picture of a graduated straight line (geometry), line that serves as visual representation of the real numbers. Every point of a number line is assumed to correspond to a real number, and every real ...
, that is
continuous
Continuity or continuous may refer to:
Mathematics
* Continuity (mathematics), the opposing concept to discreteness; common examples include
** Continuous probability distribution or random variable in probability and statistics
** Continuous ...
everywhere but
differentiable
In mathematics, a differentiable function of one real variable is a function whose derivative exists at each point in its domain. In other words, the graph of a differentiable function has a non-vertical tangent line at each interior point in its ...
nowhere.
*
Test functions
Distributions, also known as Schwartz distributions or generalized functions, are objects that generalize the classical notion of functions in mathematical analysis. Distributions make it possible to differentiate functions whose derivatives d ...
in real analysis and distribution theory, which are
infinitely differentiable function
In mathematical analysis, the smoothness of a function is a property measured by the number of continuous derivatives it has over some domain, called ''differentiability class''. At the very minimum, a function could be considered smooth if ...
s on the real line that are 0 everywhere outside of a given limited
interval. An example of such a function is the test function,