Mathematics

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

There is no general consensus about a definition of mathematics or its [[epistemology|epistemological status]]{{emdash}}that is, its place among other human activities.<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 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" /> This makes sense, as there is a strong consensus among them about what is mathematics and what is not. Most proposed definitions try 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 }}</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 appeared since the beginning of the 20th century and continue to appear, defining mathematics by this object of study becomes an impossible task.

Another approach for defining mathematics is to use its methods. So, an area of study can be qualified as mathematics as soon as one can prove theorems{{emdash}}assertions whose validity relies on a proof, that is, a purely-logical deduction.<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 | date=January 2000 | 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> Others take the perspective that mathematics is an investigation of axiomatic set theory, as this study is now a foundational discipline for much of modern mathematics.<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>