Editing Abstract nonsense

{{short description|Tongue-in-cheek description of category theory and abstract mathematics}}In [[mathematics]], '''abstract nonsense''', '''general abstract nonsense''', '''generalized abstract nonsense''', and '''general nonsense''' are nonderogatory terms used by [[mathematician]]s to describe long, theoretical parts of a proof they skip over when readers are expected to be familiar with them.<ref name="mathworld" /> These terms are mainly used for abstract methods related to [[category theory]] and [[homological algebra]]. More generally, "abstract nonsense" may refer to a proof that relies on category-theoretic methods, or even to the study of category theory itself.

==Background==

Roughly speaking, category theory is the study of the general form, that is, categories of mathematical theories, without regard to their content.  As a result, [[mathematical proof]]s that rely on category-theoretic ideas often seem out-of-context, somewhat akin to a [[non sequitur (literary device)|non sequitur]]. Authors sometimes dub these proofs "abstract nonsense" as a light-hearted way of alerting readers to their abstract nature. Labeling an argument "abstract nonsense" is usually ''not'' intended to be derogatory,<ref name="monastyrsky">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.
:"''In algebra, the term “abstract nonsense” has a definite meaning without any pejorative connotation.''"</ref><ref name="mathworld" /> and is instead used jokingly,<ref name="maclane"/> in a [[self-deprecating]] way,<ref name="rotman"/> affectionately,<ref name="lang"/> or even as a compliment to the generality of the argument. [[Alexander Grothendieck]] was critical of this notion, and stated that:
{{Citation bloc|The introduction of the [[zero|cipher 0]] or the [[Group (mathematics)|group]] concept was general nonsense too, and mathematics was more or less stagnating for thousands of years because nobody was around to take such childish steps... <ref>{{cite web |title=Correspondance Alexander Grothendieck - Ronald Brown |url=https://webusers.imj-prg.fr/~leila.schneps/grothendieckcircle/Letters/LettersGrothendieckRBrown.pdf |publisher=Société Mathématique de France}}</ref>}}

Certain ideas and constructions in mathematics share a uniformity throughout many domains, unified by category theory. Typical methods include the use of [[classifying space]]s and [[universal property|universal properties]], use of the [[Yoneda lemma]], [[natural transformation]]s between [[functor]]s, and [[diagram chasing]].<ref>{{Citation|last=Marquis|first=Jean-Pierre|title=Category Theory|date=2019|url=https://plato.stanford.edu/archives/fall2019/entries/category-theory/|encyclopedia=The Stanford Encyclopedia of Philosophy|editor-last=Zalta|editor-first=Edward N.|edition=Fall 2019|publisher=Metaphysics Research Lab, Stanford University|access-date=2019-10-27}}</ref>

When an audience can be assumed to be familiar with the general form of such arguments, mathematicians will use the expression "Such and such is true by abstract nonsense" rather than provide an elaborate explanation of particulars.<ref name="mathworld">{{MathWorld| urlname=AbstractNonsense | title=Abstract Nonsense|author=Macura, Wiktor K.}}</ref> For example, one might say that "By abstract nonsense, [[product (category theory)|products]] are unique up to isomorphism when they exist", instead of arguing about how these isomorphisms can be derived from the [[universal property]] that defines the product. This allows one to skip proof details that can be considered trivial or not providing much insight, focusing instead on genuinely innovative parts of a larger proof.

==History==

The term predates the foundation of category theory as a subject itself. Referring to a joint paper with [[Samuel Eilenberg]] that introduced the notion of a "[[category (mathematics)|category]]" in 1942, [[Saunders Mac Lane]] wrote the subject was 'then called "general abstract nonsense"'.<ref name="maclane">Saunders Mac Lane. "[http://www.pnas.org/content/94/12/5983.full The PNAS way back then]". ''Proc. Natl. Acad. Sci. USA'' Vol. 94, pp.&nbsp;5983–5985, June 1997.
:"''The first of these papers is a more striking case; it introduced the very abstract idea of a "category"—a subject then called "general abstract nonsense"!''"</ref> The term is often used to describe the application of category theory and its techniques to less abstract domains.<ref>[https://web.archive.org/web/20040726025100/http://www.csupomona.edu/~jis/1999/lord.pdf An Application of Abstract Nonsense to Surface Area], Harriet Lord</ref><ref>[http://www.edsko.net/tcd/talks/cattheory.pdf Abstract Nonsense for Functional Programmers] {{Webarchive|url=https://web.archive.org/web/20150206192631/http://www.edsko.net/tcd/talks/cattheory.pdf |date=2015-02-06 }}, Edsko de Vries</ref>

The 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'', Br. J. Philos. Sci., 41 (1990) p 355.
: "''Steenrod jokingly tagged category theory 'abstract nonsense' and made it central to his axiomatics for homology''"</ref><ref name="rotman">Joseph Rotman, "''An Introduction to Homological Algebra'', by Charles A. Weibel" (book review), Bull. Am. Math. Soc., 33:4 (Oct. 1996) 473–476.
:"''The self-deprecating phrase ''general abstract nonsense'' (due to Steenrod) was promulgated by Eilenberg and Mac Lane, two of the major innovators of homological algebra, to highlight this aspect of the subject.''"</ref><ref name="lang">[[Serge Lang]], "Algebra" Second Edition, Addison Wesley, 1984, p 175</ref> himself one of the developers of the categorical point of view.

==Notes and references==
{{Reflist|30em}}

==External links==
{{wiktionary|abstract nonsense}}
* [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]

[[Category:Mathematical terminology]]
[[Category:Category theory]]