Editing Pure mathematics

{{Short description|Mathematics independent of applications}}
{{Essay-like|date=December 2023}}
{{Math topics TOC}}
[[File:E8Petrie.svg|thumb|251x251px|Pure mathematics studies the properties and structure of abstract objects,<ref>{{cite web|url=https://www.liverpool.ac.uk/mathematical-sciences/research/pure-mathematics/|title=Pure Mathematics|publisher=[[University of Liverpool]]|access-date=2022-03-24}}</ref> such as the [[E8 (mathematics)|E8 group]], in [[group theory]]. This may be done without focusing on concrete applications of concepts in the physical world.]]
'''Pure mathematics''' is the study of mathematical concepts independently of any application outside [[mathematics]]. These concepts may originate in real-world concerns, and the results obtained may later turn out to be useful for practical applications, but pure mathematicians are not primarily motivated by such applications. Instead, the appeal is attributed to the intellectual challenge and [[Mathematical beauty|aesthetic beauty]] of working out the logical consequences of basic principles.

While pure mathematics has existed as an activity since at least [[ancient Greece]], the concept was elaborated upon around the year 1900,<ref>{{MacTutor|id=Sadleirian_Professors|title=Sadleirian Professors|last=Piaggio|first=H. T. H.|class=Extras}}</ref> after the introduction of theories with counter-intuitive properties (such as [[non-Euclidean geometries]] and [[Georg Cantor|Cantor's]] theory of infinite sets), and the discovery of apparent paradoxes (such as [[continuous function]]s that are nowhere [[differentiable function|differentiable]], and [[Russell's paradox]]). This introduced the need to renew the concept of [[mathematical rigor]] and rewrite all mathematics accordingly, with a systematic use of [[axiomatic method]]s. This led many mathematicians to focus on mathematics for its own sake, that is, pure mathematics.

Nevertheless, almost all mathematical theories remained motivated by problems coming from the real world or from less abstract mathematical theories. Also, many mathematical theories, which had seemed to be totally pure mathematics, were eventually used in applied areas, mainly [[physics]] and [[computer science]]. A famous early example is [[Isaac Newton]]'s demonstration that his [[law of universal gravitation]] implied that [[planet]]s move in orbits that are [[conic section]]s, geometrical curves that had been studied in antiquity by [[Apollonius of Perga|Apollonius]]. Another example is the problem of [[factorization|factoring]] large [[integer]]s, which is the basis of the [[RSA cryptosystem]], widely used to secure [[internet]] communications.<ref>{{cite journal |url=https://www.msri.org/people/members/sara/articles/rsa.pdf |journal=SIAM News |volume=36 |issue=5 |date=June 2003 |title=Still Guarding Secrets after Years of Attacks, RSA Earns Accolades for its Founders |first=Sara |last=Robinson }}</ref>

It follows that, currently, the distinction between pure and [[applied mathematics]] is more a philosophical point of view or a mathematician's preference rather than a rigid subdivision of mathematics.<ref>{{Cite journal |last1=Koperski |first1=Jeffrey |title=Mathematics |journal=European Journal for Philosophy of Science |volume=12 |issue=1 |pages=Article 12 |year=2022 |doi=10.1007/s13194-021-00435-9 |url=https://link.springer.com/content/pdf/10.1007/s13194-021-00435-9.pdf |access-date=October 16, 2024}}</ref>

==History==

===Ancient Greece===
Ancient Greek mathematicians were amongst the earliest to make a distinction between pure and applied mathematics. [[Plato]] helped to create the gap between "arithmetic", now called [[number theory]], and "logistic", now called [[arithmetic]]. Plato regarded logistic (arithmetic) as appropriate for businessmen and men of war who "must learn the art of numbers or [they] will not know how to array [their] troops" and arithmetic (number theory) as appropriate for philosophers "because [they have] to arise out of the sea of change and lay hold of true being."<ref>{{cite book |first=Carl B. |last=Boyer |author-link=Carl Benjamin Boyer |title=A History of Mathematics |edition=Second |publisher=John Wiley & Sons, Inc. |year=1991 |isbn=0-471-54397-7 |chapter=The age of Plato and Aristotle |pages=[https://archive.org/details/historyofmathema00boye/page/86 86] |chapter-url=https://archive.org/details/historyofmathema00boye/page/86 }}</ref> In this wise [[Euclid of Alexandria]], when asked by one of his students of what use was the study of geometry, asked his slave to give the student threepence, "since he must make gain of what he learns."<ref>{{cite book |first=Carl B. |last=Boyer |author-link=Carl Benjamin Boyer |title=A History of Mathematics |edition=Second |publisher=John Wiley & Sons, Inc. |year=1991 |isbn=0-471-54397-7 |chapter=Euclid of Alexandria |pages=[https://archive.org/details/historyofmathema00boye/page/101 101] |chapter-url=https://archive.org/details/historyofmathema00boye/page/101 }}</ref> The Greek mathematician [[Apollonius of Perga]], asked about the usefulness of some of his theorems in Book IV of ''Conics'',   asserted that<ref name="Apollonius">{{cite book |first=Carl B. |last=Boyer |author-link=Carl Benjamin Boyer |title=A History of Mathematics |edition=Second |publisher=John Wiley & Sons, Inc. |year=1991 |isbn=0-471-54397-7 |chapter=Apollonius of Perga |pages=[https://archive.org/details/historyofmathema00boye/page/152 152] |chapter-url=https://archive.org/details/historyofmathema00boye/page/152 }}</ref>
<blockquote>They are worthy of acceptance for the sake of the demonstrations themselves, in the same way as we accept many other things in mathematics for this and for no other reason.</blockquote>
And since many of his results were not applicable to the science or engineering of his day, Apollonius further argued in the preface of the fifth book of ''Conics'' that the subject is one of those that "...seem worthy of study for their own sake."<ref name="Apollonius" />

===19th century===
The term itself is enshrined in the full title of the [[Sadleirian Professor of Pure Mathematics|Sadleirian Chair]], "Sadleirian Professor of Pure Mathematics", founded (as a professorship) in the mid-nineteenth century. The idea of a separate discipline of ''pure'' mathematics may have emerged at that time. The generation of [[Carl Friedrich Gauss|Gauss]] made no sweeping distinction of the kind between ''pure'' and ''applied''. In the following years, specialisation and professionalisation (particularly in the [[Weierstrass]] approach to [[mathematical analysis]]) started to make a rift more apparent.

===20th century===
At the start of the twentieth century mathematicians took up the [[axiomatic method]], strongly influenced by [[David Hilbert]]'s example. The logical formulation of pure mathematics suggested by [[Bertrand Russell]] in terms of a [[Quantifier (logic)|quantifier]] structure of [[Proposition (mathematics)|proposition]]s seemed more and more plausible, as large parts of mathematics became axiomatised and thus subject to the simple criteria of ''[[rigorous proof]]''.

Pure mathematics, according to a view that can be ascribed to the [[Bourbaki group]], is what is proved. "Pure mathematician" became a recognized vocation, achievable through training. That said, the case has been made pure mathematics is useful in [[engineering education]]:<ref>[[A. S. Hathaway]] (1901) [https://www.ams.org/journals/bull/1901-07-06/S0002-9904-1901-00797-5/S0002-9904-1901-00797-5.pdf "Pure mathematics for engineering students"], [[Bulletin of the American Mathematical Society]] 7(6):266–71.</ref>

:There is a training in habits of thought, points of view, and intellectual comprehension of ordinary engineering problems, which only the study of higher mathematics can give.

==Generality and abstraction==
[[File:Banach-Tarski Paradox.svg|thumbnail|right|350px|An illustration of the [[Banach–Tarski paradox]], a famous result in pure mathematics. Although it is proven that it is possible to convert one sphere into two using nothing but cuts and rotations, the transformation involves objects that cannot exist in the physical world.]]
One central concept in pure mathematics is the idea of generality; pure mathematics often exhibits a trend towards increased generality. Uses and advantages of generality include the following:

* Generalizing theorems or mathematical structures can lead to deeper understanding of the original theorems or structures
* Generality can simplify the presentation of material, resulting in shorter proofs or arguments that are easier to follow.
* One can use generality to avoid duplication of effort, proving a general result instead of having to prove separate cases independently, or using results from other areas of mathematics.
* Generality can facilitate connections between different branches of mathematics. [[Category theory]] is one area of mathematics dedicated to exploring this commonality of structure as it plays out in some areas of math.

Generality's impact on [[intuition (knowledge)|intuition]] is both dependent on the subject and a matter of personal preference or learning style.  Often generality is seen as a hindrance to intuition, although it can certainly function as an aid to it, especially when it provides analogies to material for which one already has good intuition.

As a prime example of generality, the [[Erlangen program]] involved an expansion of [[geometry]] to accommodate [[non-Euclidean geometries]] as well as the field of [[topology]], and other forms of geometry, by viewing geometry as the study of a space together with a [[Group (mathematics)|group]] of transformations.  The study of [[number]]s, called [[algebra]] at the beginning undergraduate level, extends to [[abstract algebra]] at a more advanced level; and the study of [[function (mathematics)|function]]s, called [[calculus]] at the college freshman level becomes [[mathematical analysis]] and [[functional analysis]] at a more advanced level.  Each of these branches of more ''abstract'' mathematics have many sub-specialties, and there are in fact many connections between pure mathematics and applied mathematics disciplines. A steep rise in [[abstraction]] was seen mid 20th century.

In practice, however, these developments led to a sharp divergence from [[physics]], particularly from 1950 to 1983. Later this was criticised, for example by [[Vladimir Arnold]], as too much [[David Hilbert|Hilbert]], not enough [[Henri Poincaré|Poincaré]]. The point does not yet seem to be settled: [[string theory]] pulls one way towards abstraction, while [[discrete mathematics]] pulls back towards proof as central.

==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}}

==See also==
{{portal|Mathematics}}
*[[Applied mathematics]]
*[[Logic]]
*[[Metalogic]]
*[[Metamathematics]]

==References==
{{reflist|2}}

==External links==
{{Wikiquote}}
*[https://uwaterloo.ca/pure-mathematics/about-pure-math/what-is-pure-math ''What is Pure Mathematics?''] – Department of Pure Mathematics, [[University of Waterloo Faculty of Mathematics|University of Waterloo]]
*[https://fair-use.org/bertrand-russell/the-principles-of-mathematics/ ''The Principles of Mathematics''] by [[Bertrand Russell]]

{{Areas of mathematics | state=collapsed}}

{{DEFAULTSORT:Pure Mathematics}}
[[Category:Fields of mathematics]]
[[Category:Abstraction]]