Pathological (mathematics)

In mathematics, the term pathological is used to refer to a specific example, to express the attitude that its 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. In that case, the Baire category theorem was later used to show, quite to the contrary, that such behaviour was typical and even generic.

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 embedding topologically a 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) those searching for the 'pathological' are like experimentalists, interested in knocking down potential theorems proposed (by 'theorists'); though this should all take place within mathematics. What is created especially can have some undesirable, unusual, or other properties that make it difficult to contain or explain within a theory. But that point of view is probably biased, by preconceptions.

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