Lemma (mathematics)
In mathematics, informal logic and argument mapping, a lemma (Template:Plural form: lemmas or lemmata) is a generally minor, proven proposition which is used as a stepping stone to a larger result. For that reason, it is also known as a "helping theorem" or an "auxiliary theorem".<ref>Higham, Nicholas J. (1998). Handbook of Writing for the Mathematical Sciences. Society for Industrial and Applied Mathematics. pp. 16. ISBN 0-89871-420-6.</ref><ref name=":0">"Definition of lemma | Dictionary.com". www.dictionary.com. Retrieved 2019-11-28.</ref> In many cases, a lemma derives its importance from the theorem it aims to prove; however, a lemma can also turn out to be more important than originally thought.<ref name=":1">Richeson, Dave (2008-09-23). "What is the difference between a theorem, a lemma, and a corollary?". David Richeson: Division by Zero. Retrieved 2019-11-28.</ref>
It is also used generally in scholarship and philosophy.<ref>[1] "Lemma." Merriam-Webster.com Dictionary, Merriam-Webster.</ref><ref>Loewen, Nathan R. B. Beyond the Problem of Evil. Lexington Books. March 12, 2018. ISBN 9781498555739 p. 47</ref>
Etymology
From the Ancient Greek λῆμμα, (perfect passive εἴλημμαι) something received or taken. Thus something taken for granted in an argument. <ref name="OED">"Oxford English Dictionary". www.oed.com. Oxford University Press. Retrieved 26 April 2023.</ref>
Comparison with theorem
There is no formal distinction between a lemma and a theorem, only one of intention (see Theorem terminology). However, a lemma can be considered a minor result whose sole purpose is to help prove a more substantial theorem – a step in the direction of proof.<ref name=":1"/>
Well-known lemmas
Some powerful results in mathematics are known as lemmas, first named for their originally minor purpose. These include, among others:
- Bézout's lemma
- Burnside's lemma
- Dehn's lemma
- Euclid's lemma
- Farkas' lemma
- Fatou's lemma
- Gauss's lemma (any of several named after Carl Friedrich Gauss)
- Greendlinger's lemma
- Itô's lemma
- Jordan's lemma
- Nakayama's lemma
- Poincaré's lemma
- Riesz's lemma
- Schur's lemma
- Schwarz's lemma
- Sperner's lemma
- Urysohn's lemma
- Vitali covering lemma
- Yoneda's lemma
- Zorn's lemma
While these results originally seemed too simple or too technical to warrant independent interest, they have eventually turned out to be central to the theories in which they occur.
See also
