Editing Pathological (mathematics)

{{Short description|Counterintuitive mathematical object}}
{{redirect-distinguish|Well behaved|good behaviour (disambiguation){{!}}good behaviour}}

{{more citations needed|date=May 2013}}

[[File:WeierstrassFunction.svg|right|thumb|300px|The [[Weierstrass function]] is [[Continuous function|continuous]] everywhere but [[Differentiable function|differentiable]] nowhere.]]
In [[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''' or '''nice'''. These terms are sometimes useful in mathematical research and teaching, but there is no strict mathematical definition of pathological or well-behaved.<ref name=":1">{{Cite web|url=http://mathworld.wolfram.com/Pathological.html|title=Pathological|last=Weisstein|first=Eric W.|website=mathworld.wolfram.com|language=en|access-date=2019-11-29}}</ref>

==In analysis==
A classic example of a pathology is the [[Weierstrass function]], a function that is [[Continuous function|continuous]] everywhere but [[Differentiable function|differentiable]] nowhere.<ref name=":1" /> The sum of a differentiable [[Function (mathematics)|function]] 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 [[Generic property|generically]] nowhere differentiable.<ref>{{Cite web|url=https://www.math3ma.com/blog/baire-category-nowhere-differentiable-functions-part-one|title=Baire Category & Nowhere Differentiable Functions (Part One)|website=www.math3ma.com|access-date=2019-11-29}}</ref>

Such examples were deemed pathological when they were first discovered. To quote [[Henri Poincaré]]:<ref>{{Cite book |author=Kline, Morris |url=http://worldcat.org/oclc/1243569759 |title=Mathematical thought from ancient to modern times. |date=1990 |publisher=Oxford University Press |pages=973 |oclc=1243569759}}</ref>

{{blockquote|Logic sometimes breeds monsters. For half a century there has been springing up a host of weird functions, which seem to strive to have as little resemblance as possible to honest functions that are of some use. No more continuity, or else continuity but no derivatives, etc. More than this, from the point of view of logic, it is these strange functions that are the most general; those that are met without being looked for no longer appear as more than a particular case, and they have only quite a little corner left them.

Formerly, when a new function was invented, it was in view of some practical end. To-day they are invented on purpose to show our ancestors' reasonings at fault, and we shall never get anything more than that out of them.

If logic were the teacher's only guide, he would have to begin with the most general, that is to say, with the most weird, functions. He would have to set the beginner to wrestle with this collection of monstrosities. If you don't do so, the logicians might say, you will only reach exactness by stages.|[[Henri Poincaré]]|Science and Method (1899)|source=(1914 translation), page 125}}

Since Poincaré, nowhere differentiable functions have been shown to appear in basic physical and biological processes such as [[Brownian motion]] and in applications such as the [[Black-Scholes]] model in finance.

''Counterexamples in Analysis'' is a whole book of such counterexamples.<ref>{{Cite book |last=Gelbaum |first=Bernard R. |url=https://www.worldcat.org/oclc/527671 |title=Counterexamples in analysis |date=1964 |publisher=Holden-Day |others=John M. H. Olmsted |isbn=0-486-42875-3 |location=San Francisco |oclc=527671}}</ref>

Another example of pathological function is [[Paul du Bois-Reymond|Du-Bois Reymond]] [[continuous function]], that can't be represented as a [[Fourier series]].<ref>{{Cite book |last=Jahnke |first=Hans Niels |title=A history of analysis |date=2003 |publisher=American mathematical society |isbn=978-0-8218-2623-2 |series=History of mathematics |location=Providence (R.I.) |pages=187}}</ref>

==In topology==
One famous counterexample in topology is the [[Alexander horned sphere]], showing that topologically embedding the sphere ''S''<sup>2</sup> in '''R'''<sup>3</sup> may fail to separate the space cleanly. As a counterexample, it motivated mathematicians to define the ''tameness'' property, which suppresses the kind of ''wild'' behavior exhibited by the horned sphere, [[wild knot]], and other similar examples.<ref>{{Cite web|url=http://mathworld.wolfram.com/AlexandersHornedSphere.html|title=Alexander's Horned Sphere|last=Weisstein|first=Eric W.|website=mathworld.wolfram.com|language=en|access-date=2019-11-29}}</ref>

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]].

''[[Counterexamples in Topology]]'' is a whole book of such counterexamples.<ref>{{Cite book |last=Steen |first=Lynn Arthur |url=https://www.worldcat.org/oclc/32311847 |title=Counterexamples in topology |date=1995 |publisher=Dover Publications |others=J. Arthur Seebach |isbn=0-486-68735-X |location=New York |oclc=32311847}}</ref>

== Well-behaved ==

[[Mathematician]]s (and those in related sciences) very frequently speak of whether a [[mathematics|mathematical]] object&mdash;a [[Function (mathematics)|function]], a [[Set (mathematics)|set]], a [[Space (mathematics)|space]] of one sort or another&mdash;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]] of any conclusions reached.

In both pure and applied mathematics (e.g., [[Optimization (mathematics)|optimization]], [[numerical integration]], [[mathematical physics]]), ''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]] or [[measure (mathematics)|measure]]) 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&mdash;either something is well-behaved or it is not. For example:

*In [[algorithmic inference]], a [[well-behaved statistic]] is monotonic, well-defined, and [[Sufficient statistic|sufficient]].
*In [[Bézout's theorem]], two [[polynomial]]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]] is a ratio of two well-behaved functions, in the sense of those two functions being [[Holomorphic function|holomorphic]].
*The [[Karush–Kuhn–Tucker conditions]] are first-order necessary conditions for a solution in a well-behaved [[nonlinear programming]] problem to be optimal; a problem is referred to as well-behaved if some regularity conditions are satisfied.
*In [[probability]], events contained in the [[probability space]]'s corresponding [[sigma-algebra]] are well-behaved, as are [[measurable]] functions.

Unusually, the term could also be applied in a comparative sense:

*In [[calculus]]:
**[[Analytic function]]s are better-behaved than general [[smooth function]]s.
**Smooth functions are better-behaved than general differentiable functions.
**Continuous [[Differentiable function|differentiable]] 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]]s are better-behaved than [[Riemann integration|Riemann-integrable]] functions on compact sets.
**Riemann-integrable functions are better-behaved than [[Lebesgue integration|Lebesgue-integrable]] functions.
**Lebesgue-integrable functions are better-behaved than general functions.
*In [[topology]]:
**[[Continuous function (topology)|Continuous]] functions are better-behaved than discontinuous ones.
**[[Euclidean space]] is better-behaved than [[non-Euclidean geometry]].
**Attractive [[fixed point (mathematics)|fixed points]] are better-behaved than repulsive fixed points.
**[[Hausdorff topology|Hausdorff topologies]] are better-behaved than those in arbitrary [[general topology]].
**[[Borel set]]s are better-behaved than arbitrary [[set (mathematics)|sets]] of [[real number]]s.
**Spaces with [[integer]] dimension are better-behaved than spaces with [[fractal dimension]].
*In [[abstract algebra]]:
**[[Group (mathematics)|Groups]] are better-behaved than [[Magma (algebra)|magmas]] and [[semigroup]]s.
**[[Abelian group]]s are better-behaved than non-Abelian groups.
**[[Finitely-generated abelian group|Finitely-generated Abelian group]]s are better-behaved than non-finitely-generated Abelian groups.
**[[wiktionary:finite|Finite]]-[[dimension (vector space)|dimensional]] [[vector space]]s are better-behaved than [[Infinity|infinite]]-dimensional ones.
**[[Field (mathematics)|Fields]] are better-behaved than [[skew field]]s or general [[ring (mathematics)|rings]].
**Separable [[field extension]]s are better-behaved than non-separable ones.
**[[Normed division algebra]]s are better-behaved than general composition algebras.

=={{anchor|Pathological example}}Pathological examples==
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{{original research|date=August 2019}}
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:

*{{anchor|In voting}}[[Instant-runoff voting|Ranked-choice voting]] is commonly described as a pathological [[social choice function]], because of its tendency to eliminate candidates for [[Monotonicity criterion|winning too many votes]].<ref name=":0">{{Cite journal |last1=Doron |first1=Gideon |last2=Kronick |first2=Richard |date=1977 |title=Single Transferrable Vote: An Example of a Perverse Social Choice Function |url=https://www.jstor.org/stable/2110496 |journal=American Journal of Political Science |volume=21 |issue=2 |pages=303–311 |doi=10.2307/2110496 |jstor=2110496 |issn=0092-5853}}</ref>
*The discovery of [[irrational number]]s by the school of [[Pythagoras]] in ancient Greece; for example, the length of the diagonal of a [[unit square]], that is <math>\sqrt{2}</math>.
*The discovery of [[complex number]]s in the 16th century in order to find the roots of [[Cubic function|cubic]] and [[Quartic function|quartic]] [[polynomial function]]s. 
*Some [[number field]]s have [[ring of integers|rings of integers]] that do not form a [[unique factorization domain]], for example the [[Field extension|extended field]] <math>\mathbb{Q}(\sqrt{-5})</math>.
*The discovery of [[fractal]]s and other "rough" geometric objects (see [[Hausdorff dimension]]).
*[[Weierstrass function]], a [[real number|real]]-valued function on the [[real line]], that is [[continuous function|continuous]] everywhere but [[Differentiable function|differentiable]] nowhere.<ref name=":1" />
*[[Test functions]] in real analysis and distribution theory, which are [[infinitely differentiable function]]s on the real line that are 0 everywhere outside of a given limited [[Interval (mathematics)|interval]]. An example of such a function is the test function, <math display="block">\varphi(t) = \begin{cases}
e^{-1/(1-t^2)}, & -1<t<1, \\
0, & \text{otherwise}.
\end{cases}</math>
*The [[Cantor set]] is a subset of the interval <math>[0,1]</math> that has [[measure (mathematics)|measure]] zero but is [[uncountable]].
*The [[fat Cantor set]] is [[nowhere dense]] but has positive [[measure (mathematics)|measure]].
*The [[Fabius function]] is everywhere [[smoothness|smooth]] but nowhere [[analytic function|analytic]].
*[[Volterra's function]] is [[differentiable function|differentiable]] with [[bounded function|bounded]] derivative everywhere, but the derivative is not [[Riemann-integrable]].
*The Peano [[space-filling curve]] is a continuous [[surjective]] function that maps the unit interval <math>[0,1]</math> onto <math>[0,1]\times[0,1]</math>.
*The [[Dirichlet function]], which is the [[indicator function]] for rationals, is a bounded function that is not [[Riemann integrable]]. 
*The [[Cantor function]] is a [[monotonic]] continuous surjective function that maps <math>[0,1]</math> onto <math>[0,1]</math>, but has zero derivative [[almost everywhere]].
*The [[Minkowski question-mark function]] is continuous and ''strictly'' increasing but has zero derivative almost everywhere.
*Satisfaction classes containing "intuitively false" arithmetical statements can be constructed for [[countable]], recursively saturated [[model theory|models]] of [[Peano arithmetic]]. {{Citation needed|date=April 2018}}
*The [[Osgood curve]] is a [[Jordan curve]] (unlike most [[space-filling curves]]) of positive [[area]].
*An [[exotic sphere]] is [[homeomorphic]] but not [[diffeomorphic]] to the standard Euclidean [[n-sphere]].

At the time of their discovery, each of these was considered highly pathological; today, each has been assimilated into modern mathematical theory. These examples prompt their observers to correct their beliefs or intuitions, and in some cases necessitate a reassessment of foundational definitions and concepts. Over the course of history, they have led to more correct, more precise, and more powerful mathematics. For example, the Dirichlet function is Lebesgue integrable, and convolution with test functions is used to approximate any locally integrable function by smooth functions.<ref group="Note">The approximations converge [[almost everywhere]] and in the [[space of locally integrable functions]].</ref>

Whether a behavior is pathological is by definition subject to personal intuition. Pathologies depend on context, training, and experience, and what is pathological to one researcher may very well be standard behavior to another.

Pathological examples can show the importance of the assumptions in a theorem. For example, in [[statistics]], the [[Cauchy distribution]] does not satisfy the [[central limit theorem]], even though its symmetric [[bell-shape]] appears similar to many distributions which do; it fails the requirement to have a mean and standard deviation which exist and that are finite.

Some of the best-known [[paradox]]es, such as [[Banach–Tarski paradox]] and [[Hausdorff paradox]], are based on the existence of [[non-measurable set]]s. Mathematicians, unless they take the minority position of denying the [[axiom of choice]], are in general resigned to living with such sets.{{Citation needed|date=November 2019}}

==Computer science==
In [[computer science]], ''pathological'' has a slightly different sense with regard to the study of [[algorithm]]s. Here, an input (or set of inputs) is said to be ''pathological'' if it causes atypical behavior from the algorithm, such as a violation of its average case [[Computational complexity theory|complexity]], or even its correctness. For example, [[hash table]]s generally have pathological inputs: sets of keys that [[Hash collision|collide]] on hash values. [[Quicksort]] normally has <math>O(n \log{n})</math> time complexity, but deteriorates to <math>O(n^2)</math> when it is given input that triggers suboptimal behavior.

The term is often used pejoratively, as a way of dismissing such inputs as being specially designed to break a routine that is otherwise sound in practice (compare with ''[[Byzantine failure|Byzantine]]''). On the other hand, awareness of pathological inputs is important, as they can be exploited to mount a [[denial-of-service attack]] on a computer system. Also, the term in this sense is a matter of subjective judgment as with its other senses. Given enough run time, a sufficiently large and diverse user community (or other factors), an input which may be dismissed as pathological could in fact occur (as seen in the [[Ariane 5 Flight 501|first test flight]] of the [[Ariane 5]]).

== Exceptions ==
{{main|Exceptional object}}
A similar but distinct phenomenon is that of [[exceptional object]]s (and [[exceptional isomorphism]]s), which occurs when there are a "small" number of exceptions to a general pattern (such as a finite set of exceptions to an otherwise infinite rule). By contrast, in cases of pathology, often most or almost all instances of a phenomenon are pathological (e.g., almost all real numbers are irrational).

Subjectively, exceptional objects (such as the [[icosahedron]] or [[sporadic simple group]]s) are generally considered "beautiful", unexpected examples of a theory, while pathological phenomena are often considered "ugly", as the name implies. Accordingly, theories are usually expanded to include exceptional objects. For example, the [[exceptional Lie algebra]]s are included in the theory of [[semisimple Lie algebra]]s: the axioms are seen as good, the exceptional objects as unexpected but valid.

By contrast, pathological examples are instead taken to point out a shortcoming in the axioms, requiring stronger axioms to rule them out. For example, requiring tameness of an embedding of a sphere in the [[Schönflies problem]]. In general, one may study the more general theory, including the pathologies, which may provide its own simplifications (the real numbers have properties very different from the rationals, and likewise continuous maps have very different properties from smooth ones), but also the narrower theory, from which the original examples were drawn.

== See also ==

* [[Fractal curve]]
* [[List of mathematical jargon]]
*[[Runge's phenomenon]]
*[[Gibbs phenomenon]]
*[[Paradoxical set]]

== References ==
<references />

== Notes ==
<references group="Note" />

==External links ==
*[http://www.mountainman.com.au/fractal_00.htm Pathological Structures & Fractals] – Extract of an article by [[Freeman Dyson]], "Characterising Irregularity", Science, May 1978

{{PlanetMath attribution|id=6310|title=pathological}}

[[Category:Mathematical terminology]]