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:
“Every declaration of love contains an unstated list of exceptions and demands.”
—Mason Cooley (b. 1927)
“Skepticism is unbelief in cause and effect. A man does not see, that, as he eats, so he thinks: as he deals, so he is, and so he appears; he does not see that his son is the son of his thoughts and of his actions; that fortunes are not exceptions but fruits; that relation and connection are not somewhere and sometimes, but everywhere and always; no miscellany, no exemption, no anomaly,—but method, and an even web; and what comes out, that was put in.”
—Ralph Waldo Emerson (1803–1882)
“... 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)