General nonsense
In mathematics , the term general nonsense ( English abstract nonsense , general nonsense ) summarizes proofs that use abstract, category-theoretical arguments. As a rule, the term should not be interpreted in a derogatory manner.
history
The General Theory of Natural Equivalences , published in 1942 by Samuel Eilenberg and Saunders MacLane , is generally considered to be the hour of birth of category theory. According to MacLane, the theory of categories was pejoratively called general abstract nonsense by some people at the time .
The popularization of the term with a non-pejorative meaning is now attributed to Norman Steenrod .
There is an exercise in the first two editions of Serge Lang's well-known algebra textbook :
"Take any book on homological algebra, and prove all the theorems without looking at the proofs given in that book."
- ("Take any book on homological algebra and prove all the theorems without looking at the proofs in that book.")
"Homological algebra was invented by Eilenberg-MacLane. General category theory (ie, the theory of arrow-theoretic results) is generally known as abstract nonsense (the terminology is due to Steenrod)."
- (Homological algebra was invented by Eilenberg-MacLane. General category theory, that is, the theory of abstract morphisms, is commonly known as abstract nonsense, the terminology traced back to Steenrod.)
In the third edition of 1993 this task is missing, but in the introduction to Part Four: Homological Algebra :
In the forties and fifties (mostly in the works of Cartan, Eilenberg, MacLane, and Steenrod, see [CaE 57]), it was realized that there was a systematic way of developing certain relationships of linear algebra, depending only on fairly general constructions which were mostly arrow-theoretical, and were affectionately called abstract nonsense by Steenrod.
- (In the 1940s and 50s (especially in the work of Cartan, Eilenberg, MacLane, and Steenrod) it was noticed that there was a systematic way of developing certain linear algebra relationships that only depends on fairly general constructions, of which the Most are morphism-theoretical and have been affectionately described by Steenrod as abstract nonsense.)
Examples
Typical examples are
- Diagram hunting evidence : a method of proof in homological algebra that exploits the commutativity of diagrams , the injectivity , surjectivity or bijectivity of morphisms, or the exactness of sequences . An example is the construction of the connecting homomorphism in the proof of the snake lemma .
- Use a universal property or adjoint functors .
- Applications of the lemma of Yoneda .
Individual evidence
- ↑ Michael Barr: Re: Who said: General Abstract Nonsense. May 20, 1998
- ^ Saunders Mac Lane: The PNAS way back then. What is meant is Eilenberg-MacLane: Natural isomorphisms in group theory. Proc. Natl. Acad. Sci. USA 1942, No. 28, pp. 537-543.
- ^ 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"
- ^ Charles A. Weibel: Book review on Joseph Rotman: An Introduction to Homological Algebra. Bull. Amer. Math. Soc., 33: 4 (October 1996), pp. 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. "
- ↑ 2nd edition from 1984 (Lang's book is considered a standard work. From the review in the Notices of the American Mathematical Society : "Lang's Algebra changed the way graduate algebra is taught, retaining classical topics but introducing language and ways of thinking from category theory and homological algebra. It has affected all subsequent graduate-level algebra books. ")