Abstract nonsense


Abstract nonsense

Abstract nonsense, or general abstract nonsense, alternatively general nonsense, is a popular term used by mathematicians to describe certain kinds of arguments and concepts in category theory or applications.

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" in 1942, Saunders Mac Lane wrote the subject was 'then called "general abstract nonsense"'. [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"!"] The term is often used to describe the application of category theory and its techniques to less abstract domains. [ [http://www.csupomona.edu/~jis/1999/lord.pdf An Application of Abstract Nonsense to Surface Area] , Harriet Lord] [ [https://www.cs.tcd.ie/~devriese/talks/cattheory.pdf Abstract Nonsense for Functional Programmers] , Edsko de Vries]

The term is believed to have been coined by the mathematician Norman Steenrod, [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."] himself one of the developers of the categorical point of view. This term is used by practitioners as an indication of mathematical sophistication (or possession of a deeper perspective) rather than as a derogatory designation. [ 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."]

Certain ideas and constructions in mathematics display a uniformity throughout many domains. The unifying theme is category theory. When their 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. [MathWorld| urlname=AbstractNonsense | title=Abstract Nonsense]

Examples

Typical instances are arguments involving diagram chasing, application of the definition of universal property, definition of natural transformations between functors, use of the Yoneda lemma, arguments exploiting classifying spaces, and so on.

To spell out a concrete example, consider a 3-manifold M with positive Betti number. One would like to show that M admits a map to the 2-sphere which is "non-trivial", i.e. non-homotopic to the constant map. By a "general nonsense argument", there is a
f: M o K(mathbb{Z},2) to the Eilenberg-MacLane space, corresponding to a non-trivial element in H2(M). Since K(Z,2) is complex projective space and the latter admits a skeleton structure with no cells in odd dimensions, we can apply the cellular approximation theorem to conclude that the map "f" can be pushed down to the 2-skeleton, which happens to be the 2-sphere. Now surely there are more concrete, "hands-on" constructions of such a map, but the conceptual approach above would probably appeal to the elite.

Notes and references

External links

* [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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