In mathematics, abstract nonsense, general abstract nonsense, and general nonsense are terms used by mathematicians to describe abstract methods related to category theory. More generally, “abstract nonsense” may refer to a proof that relies on category theoretic methods, or even to the study of category theory itself.
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 proofs that rely on category theoretic ideas often seem out of context, somewhat akin to a non sequitur. Authors sometimes dub these proofs “abstract nonsense” as a light-hearted way of alerting readers to their abstract nature. Generality is lauded within the field of mathematics, so labeling an argument "abstract nonsense" is not intended[according to whom?] to be derogatory, and is often in fact, a compliment.[clarification needed]
Certain ideas and constructions in mathematics share a uniformity throughout many domains, unified by category theory. Typical methods include the use of classifying spaces and universal properties[clarification needed], use of the Yoneda lemma[clarification needed], natural transformations between functors[clarification needed], an diagram chasing[clarification needed].
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[clarification needed] rather than provide an elaborate explanation of particulars.
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" in 1942, Saunders Mac Lane wrote the subject was 'then called "general abstract nonsense"'. The term is often used to describe the application of category theory and its techniques to less abstract domains.
The term is believed[according to whom?] to have been coined by the mathematician Norman Steenrod[when?], himself one of the developers of the categorical point of view. This term is used by practitioners as an indication of mathematical abstraction rather than as a derogatory designation.
Consider the example of showing that a 3-manifold M with positive 2nd Betti number admits a map to the 2-sphere which is "non-trivial", i.e. non-homotopic to the constant map. Since K(Z,2) is a complex projective space and the latter admits a skeleton structure with no cells in odd dimensions, by the cellular approximation theorem, a general nonsense argument[clarification needed], the map f can be pushed down to the 2-skeleton, which happens to be the 2-sphere.
Thus, there is a map
to the Eilenberg-MacLane space, corresponding to a non-trivial element in H2(M).
Though this proof establishes the truth of the statement in question, the proof technique has little to do with the topology or geometry of the 2-sphere, let alone 3-manifolds, as it relies on more general categorical principles. Because of the reliance on these abstract principles, the result is independent of subtler geometric details, so offers little geometric insight into the nature of such a map. On the other hand, the proof is surprisingly short and clean, and a “hands-on” approach involving the physical construction of such a map would be potentially laborious. A reader expecting a long, difficult proof might be surprised—or even delighted by this.
Notes and references
- Macura, Wiktor K. "Abstract Nonsense". MathWorld.
- 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."
- Saunders Mac Lane. "The PNAS way back then". Proc. Natl. Acad. Sci. USA Vol. 94, pp. 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"!"
- An Application of Abstract Nonsense to Surface Area, Harriet Lord
- Abstract Nonsense for Functional Programmers, Edsko de Vries
- Colin McLarty, The Uses and Abuses of the History of Topos Theory, Brit. J. Phil. Sci, 41 (1990) p 355.
- "Steenrod jokingly tagged category theory 'abstract nonsense' and made it central to his axiomatics for homology"
- Joseph Rotman, "An Introduction to Homological Algebra, by Charles A. Weibel" (book review), Bull. Amer. 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."
- Serge Lang, "Algebra" Second Edition, Addison Wesley, 1984, p 175
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