In mathematics, a pathological example is one whose properties are (or should be considered) untypically bad. The classical case is probably that of some everywhere continuous functions that are in fact nowhere differentiable, such as the Weierstrass function. In that case, the Baire category theorem was later used to show, quite to the contrary, that such behaviour was typical and even generic. This highlights the fact that the term pathological is subjective, and its meaning in any particular case resides in the community of mathematicians, not within the subject matter of mathematics itself. Mathematics, often abbreviated maths in Commonwealth English and math in American English, is the study of abstraction. ... Mathematicians (and those in related sciences) very frequently speak of whether a mathematical object -- a number, a function, a set, a space of one sort or another -- is well-behaved or not. ... In mathematics, a continuous function is one in which arbitrarily small changes in the input produce arbitrarily small changes in the output. ... In mathematics, the derivative of a function is one of the two central concepts of calculus. ... In mathematics, the Weierstrass function was the first example found of a kind of function with the property that it is continuous everywhere but differentiable nowhere. ... In mathematics, the Baire category theorem is an important tool in the study of complete spaces, such as Banach spaces and Hilbert spaces, that arise in topology and functional analysis. ...

Often the usefulness of a theorem is justified by saying examples which don't meet the assumptions (counterexamples) are pathological. This process is sometimes referred to as "Reductio Ad Absurdum". A famous case is the Alexander horned sphere, a counterexample showing that topologically embedding the sphere S² in R³ may fail to "separate the space cleanly", unless an extra condition of tameness is used to suppress possible wild behaviour. In logic, and especially in its applications to mathematics and philosophy, a counterexample is an exception to a proposed general rule, i. ... A drawing of Alexanders horned sphere Alexanders Horned Sphere is one of the most famous pathological examples in mathematics. ...

One can therefore say that (particularly in mathematical analysis and set theory) those searching for the "pathological" are like experimentalists, interested in knocking down potential theorems, in contrast to finding general statements widely applicable. Each activity has its role within mathematics. Analysis is that branch of mathematics which deals with the real numbers, complex numbers, and their functions. ... Set theory is the mathematical theory of sets, which represent collections of abstract objects. ...

Pathological examples often have some undesirable or unusual properties that make it difficult to contain or explain within a theory. Such pathological behaviour often prompts new investigation which leads to new theory and "general" results. For example, three important historical examples of this are the following:

1. The discovery of irrational numbers by the ancient Greeks.
2. The discovery of number fields whose integers do not admit unique factorisation.
3. The discovery of the fractals and other "rough" geometric objects.

At the time of their discovery, each of these were considered highly pathological; today, each has been assimilated and explained by an extensive general theory. In mathematics, an irrational number is any real number that is not a rational number, i. ... In mathematics, an algebraic number field (or simply number field) is a finite field extension of the rational numbers Q. That is, it is a field which contains Q and has finite dimension when considered as a vector space over Q. The study of algebraic number fields, and these days... A fractal is a geometric object which can be divided into parts, each of which is similar to the original object. ...

Again, to reiterate, it should be pointed out that such judgments about what is or is not pathological are inherently subjective and depend on both training and experience — what is pathological to one researcher may very well be standard behaviour 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 are finite. For Wikipedia statistics, see m:Statistics Statistics is the science and practice of developing human knowledge through the use of empirical data expressed in quantitative form. ... Standard Cauchy_Lorentz probability distribution function The Cauchy_Lorentz distribution is a probability distribution with probability density function where x0 is the location parameter, specifying the location of the peak of the distribution, and γ is the scale parameter which specifies the half_width at half_maximum (HWHM). ... Central limit theorems are a set of weak-convergence results in probability theory. ...

The best-known paradoxes such as the Banach-Tarski paradox and Hausdorff paradox are based on the existence of non-measurable sets. Mathematicians, unless they take the minority position of denying the axiom of choice, are in general resigned to living with such sets. For other meanings of Paradox, see Paradox (disambiguation). ... The Banach-Tarski paradox: A sphere can be decomposed and reassembled into two spheres the same size as the original. ... In mathematics, the Hausdorff paradox, named after Felix Hausdorff, states that if you remove some countable subset of the sphere S2, the remainder can be divided into three subsets A, B and C such that A, B, C and B ∪C are all congruent. ... In mathematics, a non-measurable set is a set whose structure is so complicated it sheds light on the very notion of length, area or volume. ... In mathematics, the axiom of choice is an axiom of set theory. ...

Other examples include the Peano space-filling curve which maps the unit interval [0, 1] continuously onto [0, 1] × [0, 1], and the Cantor set which is a subset of the interval [0, 1] and has the pathological property that it is uncountable, yet its measure is zero. Intuitively, a continuous curve in the 2-dimensional plane or in the 3-dimensional space can be thought of as the path of a continuously moving point. To eliminate the inherent vagueness of this notion, Jordan in 1887 introduced the following rigorous definition, which has since been adopted as the... The Cantor set, introduced by German mathematician Georg Cantor, is a remarkable construction involving only the real numbers between zero and one. ...

See also: well-behaved. Mathematicians (and those in related sciences) very frequently speak of whether a mathematical object -- a number, a function, a set, a space of one sort or another -- is well-behaved or not. ...

Computer science uses this term in a slightly different sense with regard to the study of algorithms. 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 complexity, or even its correctness. For example, hash tables generally have pathological inputs: sets of keys that collide on hash values. 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 Byzantine. Computer science (academically, CS, CSC or compsci) encompasses a variety of topics that relates to computation, like abstract analysis of algorithms, formal grammars, and subjects such as programming languages, program design, software and computer hardware. ... Flowcharts are often used to represent algorithms. ... Complexity theory is part of the theory of computation dealing with the resources required during computation to solve a given problem. ... In computer science, a hash table is a data structure that speeds up searching for information by a particular aspect of that information, called a key. ... Hash collision is a term in computer programming for a situation that occurs when two distinct inputs into a hash function produce identical outputs. ... In fault-tolerant distributed computing, a Byzantine failure is an arbitrary fault that occurs during the execution of an algorithm by a distributed system. ...

External links

• Pathological Structures & Fractals (http://www.mountainman.com.au/fractal_00.htm) - Extract of an article by Freeman Dyson, "Characterising Irregularity", Science, May 1978

This article incorporates material from pathological  (http://planetmath.org/?op=getobj&from=objects&id=6310) on PlanetMath, which is licensed under the GFDL. Freeman Dyson at Harvard University in 2004 Freeman John Dyson (born December 15, 1923) is an English-born American physicist and mathematician. ... PlanetMath is a free, collaborative, online mathematics encyclopedia. ...