Difference between revisions of "Abstract nonsense"

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'''''Abstract nonsense''''' is a popular term used by [[mathematician]]s to describe certain kinds of arguments and concepts in [[category theory]]. This term is believed to have been coined by the mathematician [[Norman Steenrod]],<ref>Colin McLarty, ''The Uses and Abuses of the History of Topos Theory'', Brit. J. Phil. Sci, 41 (1990) p 355.
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"Abstract nonsense" is a humorous term applied by mathematicians to criticize what they consider to be unnecessary abstractions used by [[category theory]], particularly when used to solve a problem that could be solved more directly.
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: "''[[Norman Steenrod|Steenrod]] jokingly tagged category theory 'abstract nonsense' and made it central to his axiomatics for homology''" </ref><ref>Joseph Rotman, ''Review of "An Introduction to Homological Algebra"'', Bull. Amer. Math. Soc., 33, (1995), p 475</ref> himself one of the developers of the categorical point of view. This term is used by practitioners as an indication of mathematical sophistication or [[Cool (aesthetic)|coolness]] rather than as a derogatory designation.<ref> Michael Monastyrsky, ''Some Trends in Modern Mathematics and the Fields Medal.'' Can. Math. Soc. Notes, March and April 2001, Volume 33, nos. 2 and 3.  Online version available at http://www.fields.utoronto.ca/aboutus/FieldsMedal_Monastyrsky.pdf.
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[[category:mathematics]]
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:"''In algebra, the term “abstract nonsense” has a definite meaning without any pejorative connotation.''"</ref>
 
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Certain ideas and constructions in mathematics display a uniformity throughout many domains.  The unifying theme is category theory. Rather than enter an elaborate discussion on particulars of arguments, mathematicians will use the expression ''Such and such is true by abstract nonsense''. Typical instances are arguments involving [[diagram chasing]], application of the definition of [[universal property]], definition of [[natural transformation]]s between [[functor]]s, use of the [[Yoneda lemma]] and so on.
 
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Other, possibly less flattering, characterizations of abstract reasoning have been recorded, although they have not acquired the status of [[mathematical jargon]]. For example,  [[Paul Albert Gordan]] described a proof of [[David Hilbert]]'s in [[invariant theory]], saying "This is not [[mathematics]]; this is [[theology]]."<ref>Hermann Weyl, ''David Hilbert. 1862-1943'', Obituary Notices of Fellows of the Royal Society (1944).
 
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The proof in question was the (non-constructive) existence of a finite basis for invariants. This proof seemed to counter the sensibilities of Gordan, whom Weyl described in his Hilbert obituary as a great "algorithmician."</ref>
 
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==Notes and references==
 
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<references/>
 
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==External links==
 
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* [http://www.math.harvard.edu/~elkies/M55a.05/nonsense.html Usage in mathematical exposition] from [http://www.math.harvard.edu/~elkies/ Noam Elkies' class notes]
 
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[[Category:Mathematical terminology]][[Category:Category theory]]
 
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[[fr:Abstract nonsense]]
 
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[[ja:アブストラクト・ナンセンス]]
 

Revision as of 17:54, August 19, 2007

"Abstract nonsense" is a humorous term applied by mathematicians to criticize what they consider to be unnecessary abstractions used by category theory, particularly when used to solve a problem that could be solved more directly.

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