Exceptions
A similar but distinct phenomenon is that of exceptional objects (and exceptional isomorphisms), which occurs when there are a "small" number of exceptions to a general pattern – quantitatively, 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, as discussed in prevalence, above – e.g., almost all real numbers are irrational.
Subjectively, exceptional objects (such as the icosahedron or sporadic simple groups) 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 algebras are included in the theory of semisimple Lie algebras: 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. 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 will also in general study the narrower theory from which the original examples were drawn.
Read more about this topic: Pathological (mathematics)
Famous quotes containing the word exceptions:
“... people were so ridiculous with their illusions, carrying their fools’ caps unawares, thinking their own lies opaque while everybody else’s were transparent, making themselves exceptions to everything, as if when all the world looked yellow under a lamp they alone were rosy.”
—George Eliot [Mary Ann (or Marian)
“For true poetry, complete poetry, consists in the harmony of contraries. Hence, it is time to say aloud—and it is here above all that exceptions prove the rule—that everything that exists in nature exists in art.”
—Victor Hugo (1802–1885)