Editing Mathematics (section)

=== Proposed definitions ===
{{Main|Definitions of mathematics}}

There is no general consensus about the definition of mathematics or its [[epistemology|epistemological status]]{{emdash}}that is, its place inside knowledge<!-- please, do not link "knowledge", since it is linked in the first paragraph of the preceding link. -->.<!-- <ref name="Mura" /><ref name="Runge" /> --> A great many professional mathematicians take no interest in a definition of mathematics, or consider it undefinable.<!-- <ref name="Mura" /> --> There is not even consensus on whether mathematics is an art or a science.<!-- <ref name="Runge" /> --> Some just say, "mathematics is what mathematicians do".<ref name="Mura">{{cite journal
 | title=Images of Mathematics Held by University Teachers of Mathematical Sciences
 | last=Mura | first=Roberta | date=Dec 1993
 | journal=Educational Studies in Mathematics
 | volume=25 | issue=4 | pages=375–85
 | doi=10.1007/BF01273907 | jstor=3482762 | s2cid=122351146
}}</ref><ref name="Runge">{{cite book
 | title=Iris Runge: A Life at the Crossroads of Mathematics, Science, and Industry
 | last1=Tobies
 | first1=Renate
 | author1-link=Renate Tobies
 | first2=Helmut
 | last2=Neunzert
 | publisher=Springer
 | year=2012
 | isbn=978-3-0348-0229-1
 | page=9
 | url={{GBurl|id=EDm0eQqFUQ4C|p=9}}
 | quote=[I]t is first necessary to ask what is meant by ''mathematics'' in general. Illustrious scholars have debated this matter until they were blue in the face, and yet no consensus has been reached about whether mathematics is a natural science, a branch of the humanities, or an art form.
 | access-date=June 20, 2015
}}</ref> A common approach is to define mathematics by its object of study.<ref>{{cite conference
 | title="What is Mathematics?" and why we should ask, where one should experience and learn that, and how to teach it
 | first1=Günter M. | last1=Ziegler | author1-link=Günter M. Ziegler
 | first2=Andreas | last2=Loos | editor-last=Kaiser | editor-first=G.
 | conference=Proceedings of the 13th International Congress on Mathematical Education
 | series=ICME-13 Monographs
 | date=November 2, 2017 | pages=63–77 | publisher=Springer
 | doi=10.1007/978-3-319-62597-3_5
| isbn=978-3-319-62596-6 }} (Sections "What is Mathematics?" and "What is Mathematics, Really?")</ref>{{sfn|Mura|1993|pp=379, 381}}{{sfn|Brown|Porter|1995|p=326}}<ref>{{cite journal
 | last=Strauss | first=Danie | year=2011
 | title=Defining mathematics
 | journal=Acta Academica
 | volume=43 | issue=4 | pages=1–28
 | url=https://www.researchgate.net/publication/290955899
 | access-date=November 25, 2022
}}</ref>

Aristotle defined mathematics as "the science of quantity" and this definition prevailed until the 18th century. However, Aristotle also noted a focus on quantity alone may not distinguish mathematics from sciences like physics; in his view, abstraction and studying quantity as a property "separable in thought" from real instances set mathematics apart.<ref name="Franklin">{{Cite book | last=Franklin | first=James | author-link=James Franklin (philosopher) | title=Philosophy of Mathematics | date= 2009 | isbn=978-0-08-093058-9 | pages=104–106 | publisher=Elsevier | url={{GBurl|id=mbn35b2ghgkC|p=104}} | access-date=June 20, 2015 }}</ref> In the 19th century, when mathematicians began to address topics{{mdash}}such as infinite sets{{mdash}}which have no clear-cut relation to physical reality, a variety of new definitions were given.<ref name="Cajori">{{cite book
 | title=A History of Mathematics
 | last=Cajori
 | first=Florian
 | author-link=Florian Cajori
 | publisher=American Mathematical Society (1991 reprint)
 | year=1893
 | isbn=978-0-8218-2102-2
 | pages=285–286
 | url={{GBurl|id=mGJRjIC9fZgC|p=285}}
 | access-date=June 20, 2015
}}</ref> With the large number of new areas of mathematics that have appeared since the beginning of the 20th century, defining mathematics by its object of study has become increasingly difficult.{{sfn|Devlin|2018|p=[https://books.google.com/books?id=gUb7CAAAQBAJ&pg=PA3 3]}} For example, in lieu of a definition, [[Saunders Mac Lane]] in ''[[Mathematics, form and function]]'' summarizes the basics of several areas of mathematics, emphasizing their inter-connectedness, and observes:<ref>{{cite book|author=Saunders Maclane|year=1986|title=Mathematics, form and function|publisher=Springer}}, page 409</ref>
{{blockquote|the development of Mathematics provides a tightly connected network of formal rules, concepts, and systems.  Nodes of this network are closely bound to procedures useful in human activities and to questions arising in science. The transition from activities to the formal Mathematical systems is guided by a variety of general insights and ideas.}}

Another approach for defining mathematics is to use its methods. For example, an area of study is often qualified as mathematics as soon as one can prove theorems{{emdash}}assertions whose validity relies on a proof, that is, a purely-logical deduction.{{efn|For example, logic belongs to philosophy since [[Aristotle]]. Circa the end of the 19th century, the [[foundational crisis of mathematics]] implied developments of logic that are specific to mathematics. This allowed eventually the proof of theorems such as [[Gödel's theorems]]. Since then, [[mathematical logic]] is commonly considered as an area of mathematics.}}<ref>{{cite journal | title=The Methodology of Mathematics | first1=Ronald | last1=Brown | author1-link=Ronald Brown (mathematician) | first2=Timothy | last2=Porter | journal=The Mathematical Gazette | volume=79 | issue=485 | pages=321–334 |year=1995 | doi=10.2307/3618304 | jstor=3618304 | s2cid=178923299 | url=https://cds.cern.ch/record/280311 | access-date=November 25, 2022 | archive-date=March 23, 2023 | archive-url=https://web.archive.org/web/20230323164159/https://cds.cern.ch/record/280311 | url-status=live }}</ref>{{verification failed|date=October 2024}}