Editing Pathological (mathematics) (section)

== 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.