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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.
Often the usefulness of a theorem is justified by saying examples which don't meet the assumptions (counterexamples) are pathological. 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.
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.
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:
- The discovery of irrational numbers by the ancient Greeks.
- The discovery of number fields whose integers do not admit unique factorisation.
- 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.
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.
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.
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.
See also: well-behaved.
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.
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