Editing Pure mathematics (section)

==Pure vis- applied mathematics==
Mathematicians have always had differing opinions regarding the distinction between pure and applied mathematics. One of the most famous (but perhaps misunderstood) modern examples of this debate can be found in [[G.H. Hardy]]'s 1940 essay ''[[A Mathematician's Apology]]''.

It is widely believed that Hardy considered applied mathematics to be ugly and dull. Although it is true that Hardy preferred pure mathematics, which he often compared to [[painting]] and [[poetry]], Hardy saw the distinction between pure and applied mathematics to be simply that applied mathematics sought to express ''physical'' truth in a mathematical framework, whereas pure mathematics expressed truths that were independent of the physical world. Hardy made a separate distinction in mathematics between what he called "real" mathematics, "which has permanent aesthetic value", and "the dull and elementary parts of mathematics" that have practical use.<ref>{{Cite journal |last=Levinson |first=Norman |date=1970 |title=Coding Theory: A Counterexample to G. H. Hardy's Conception of Applied Mathematics |url=https://www.jstor.org/stable/2317708 |journal=The American Mathematical Monthly |volume=77 |issue=3 |pages=249–258 |doi=10.2307/2317708 |jstor=2317708 |issn=0002-9890|url-access=subscription }}</ref>

Hardy considered some physicists, such as [[Albert Einstein|Einstein]] and [[Paul Dirac|Dirac]], to be among the "real" mathematicians, but at the time that he was writing his ''Apology'', he considered [[general relativity]] and [[quantum mechanics]] to be "useless", which allowed him to hold the opinion that only "dull" mathematics was useful. Moreover, Hardy briefly admitted that—just as the application of [[Matrix (mathematics)|matrix theory]] and [[group theory]] to physics had come about unexpectedly—the time may come where some kinds of beautiful, "real" mathematics may be useful as well.

Another insightful view is offered by American mathematician [[Andy Magid]]:
{{quote|I've always thought that a good model here could be drawn from ring theory. In that subject, one has the subareas of [[commutative ring|commutative ring theory]] and [[non-commutative ring|non-commutative ring theory]]. An uninformed observer might think that these represent a dichotomy, but in fact the latter subsumes the former: a non-commutative ring is a not-necessarily-commutative ring. If we use similar conventions, then we could refer to applied mathematics and nonapplied mathematics, where by the latter we ''mean not-necessarily-applied mathematics''... [emphasis added]<ref name=Magid>[[Andy Magid]] (November 2005) [https://www.ams.org/notices/200510/commentary.pdf Letter from the Editor], [[Notices of the American Mathematical Society]], page 1173</ref>}}

[[Friedrich Engels]] argued in his 1878 book ''[[Anti-Dühring]]'' that "it is not at all true that in pure mathematics the mind deals only with its own creations and imaginations. The concepts of number and figure have not been invented from any source other than the world of reality".<ref name=engels>{{cite book |last1=Engels |first1=Frederick |title=Marx Engels Collected Works (Volume 25) |date=1987 |publisher=Progress Publishers |location=Moscow |isbn=0-7178-0525-5 |page=33-133 |edition=English }}</ref>{{rp|36}} He further argued that "Before one came upon the idea of deducing the form of a cylinder from the rotation of a rectangle about one of its sides, a number of real rectangles and cylinders, however imperfect in form, must have been examined. Like all other sciences, mathematics arose out of the needs of men...But, as in every department of thought, at a certain stage of development the laws, which were abstracted from the real world, become divorced from the real world, and are set up against it as something independent, as laws coming from outside, to which the world has to conform."<ref name=engels/>{{rp|37}}