Mathematics

==Philosophy==
{{Main|Philosophy of mathematics}}

===Reality===
The connection between mathematics and material reality has led to philosophical debates since at least the time of [[Pythagoras]]. The ancient philosopher [[Plato]] argued that abstractions that reflect material reality have themselves a reality that exists outside space and time. As a result, the philosophical view that mathematical objects somehow exist on their own in abstraction is often referred to as [[Mathematical Platonism|Platonism]]. Independently of their possible philosophical opinions, modern mathematicians may be generally considered as Platonists, since they think of and talk of their objects of study as real objects.<ref name=SEP-Platonism>{{cite encyclopedia |title=Platonism in Metaphysics |encyclopedia=The Stanford Encyclopedia of Philosophy |last=Balaguer |first=Mark |editor-last=Zalta |editor-first=Edward N. |year=2016 |edition=Spring 2016 |publisher=Metaphysics Research Lab, Stanford University |url=https://plato.stanford.edu/archives/spr2016/entries/platonism |access-date=April 2, 2022 |archive-date=January 30, 2022 |archive-url=https://web.archive.org/web/20220130174043/https://plato.stanford.edu/archives/spr2016/entries/platonism/ |url-status=live }}</ref>

[[Armand Borel]] summarized this view of mathematics reality as follows, and provided quotations of [[G. H. Hardy]], [[Charles Hermite]], [[Henri Poincaré]] and Albert Einstein that support his views.<ref name=borel />
{{blockquote| Something becomes objective (as opposed to "subjective") as soon as we are convinced that it exists in the minds of others in the same form as it does in ours and that we can think about it and discuss it together.<ref>See {{cite journal
 | first=L. | last=White | year=1947
 | title=The locus of mathematical reality: An anthropological footnote
 | journal=[[Philosophy of Science (journal)|Philosophy of Science]]
 | volume=14|issue=4 | pages=289–303
 | doi=10.1086/286957 | s2cid=119887253
 | id=189303 | postscript=;
}} also in {{cite book
 | first=J. R. | last=Newman | year=1956
 | title=The World of Mathematics
 | publisher=Simon and Schuster | location=New York
 | volume=4 | pages=2348–2364
}}</ref> Because the language of mathematics is so precise, it is ideally suited to defining concepts for which such a consensus exists. In my opinion, that is sufficient to provide us with a {{em|feeling}} of an objective existence, of a reality of mathematics ...}}

Nevertheless, Platonism and the concurrent views on abstraction do not explain the [[#Unreasonable effectiveness|unreasonable effectiveness]] of mathematics.<ref>{{cite book
 | title=The Software of the Universe, An Introduction to the History and Philosophy of Laws of Nature
 | first=Mauro
 | last=Dorato
 | year=2005
 | chapter=Why are laws mathematical?
 | pages=31–66
 | isbn=978-0-7546-3994-7
 | publisher=Ashgate
 | chapter-url=https://www.academia.edu/download/52076815/2ch.pdf
 | access-date=December 5, 2022
 | archive-url=https://web.archive.org/web/20230817111932/https://d1wqtxts1xzle7.cloudfront.net/52076815/2ch-libre.pdf?1488997736=&response-content-disposition=inline%3B+filename%3DChapter_2_of_the_book_the_software_of_th.pdf&Expires=1692274771&Signature=PXpNLBsmWMkz9YUs6~LUOfXNkmkCAmDfxQUoWOkGJKP4YqPGQUFMuP1I0xFycLZkL0dyfGwdGQ7mPk44nvmpM3YpKBSeVCZRXtDMiwgqs1JhEWrJovAhrchPLM1mGn3pw5P6LPo0sDZsl7uaPoZHMyCyJpayHvFtpyj1oUMIdmGuYM5P3euy1R87g6xlKyNAp~~BR5I4gVpopzLoeZn7d3oEnOOua0GjsqsZ6H9mEgcZMpH-qF8w9iFa9aSXFpqxagQwcVVkg7DXkOjVV5jyzctBUKQtOQQ~-9EN1y-c9pFV-Xu-NNuoN3Ij6K4SwvjYv0a8DMs8T5SVj1Kz9i4CEQ__&Key-Pair-Id=APKAJLOHF5GGSLRBV4ZA
 | archive-date=August 17, 2023
 | url-status=live
 }}</ref>

=== 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>

=== Rigor ===
{{See also|Logic}}
Mathematical reasoning requires [[Mathematical rigor|rigor]]. This means that the definitions must be absolutely unambiguous and the [[proof (mathematics)|proof]]s must be reducible to a succession of applications of [[inference rule]]s,{{efn|This does not mean to make explicit all inference rules that are used. On the contrary, this is generally impossible, without [[computer]]s and [[proof assistant]]s. Even with this modern technology, it may take years of human work for writing down a completely detailed proof.}} without any use of empirical evidence and [[intuition]].{{efn|This does not mean that empirical evidence and intuition are not needed for choosing the theorems to be proved and to prove them.}}<ref>{{cite journal | title=Mathematical Rigor and Proof | first=Yacin | last=Hamami | journal=The Review of Symbolic Logic | volume=15 | issue=2 | date=June 2022 | pages=409–449 | url=https://www.yacinhamami.com/wp-content/uploads/2019/12/Hamami-2019-Mathematical-Rigor-and-Proof.pdf | access-date=November 21, 2022 | doi=10.1017/S1755020319000443 | s2cid=209980693 | archive-date=December 5, 2022 | archive-url=https://web.archive.org/web/20221205114343/https://www.yacinhamami.com/wp-content/uploads/2019/12/Hamami-2019-Mathematical-Rigor-and-Proof.pdf | url-status=live }}</ref> Rigorous reasoning is not specific to mathematics, but, in mathematics, the standard of rigor is much higher than elsewhere. Despite mathematics' [[concision]], rigorous proofs can require hundreds of pages to express. The emergence of [[computer-assisted proof]]s has allowed proof lengths to further expand,{{efn|For considering as reliable a large computation occurring in a proof, one generally requires two computations using independent software}}<ref>{{harvnb|Peterson|1988|p=4}}: "A few complain that the computer program can't be verified properly." (in reference to the Haken–Apple proof of the [[Four color theorem|Four Color Theorem]])</ref> such as the 255-page [[Feit–Thompson theorem]].{{efn|The book containing the complete proof has more than 1,000 pages.}} The result of this trend is a philosophy of the [[Quasi-empiricism in mathematics|quasi-empiricist]] proof that can not be considered infallible, but has a probability attached to it.<ref name=Kleiner_1991/>

The concept of rigor in mathematics dates back to ancient Greece, where their society encouraged logical, deductive reasoning. However, this rigorous approach would tend to discourage exploration of new approaches, such as irrational numbers and concepts of infinity. The method of demonstrating rigorous proof was enhanced in the sixteenth century through the use of symbolic notation. In the 18th century, social transition led to mathematicians earning their keep through teaching, which led to more careful thinking about the underlying concepts of mathematics. This produced more rigorous approaches, while transitioning from geometric methods to algebraic and then arithmetic proofs.<ref name=Kleiner_1991/>

At the end of the 19th century, it appeared that the definitions of the basic concepts of mathematics were not accurate enough for avoiding paradoxes (non-Euclidean geometries and [[Weierstrass function]]) and contradictions (Russell's paradox). This was solved by the inclusion of axioms with the [[Apodicticity|apodictic]] inference rules of mathematical theories; the re-introduction of axiomatic method pioneered by the ancient Greeks.<ref name=Kleiner_1991/> It results that "rigor" is no more a relevant concept in mathematics, as a proof is either correct or erroneous, and a "rigorous proof" is simply a [[pleonasm]]. Where a special concept of rigor comes into play is in the socialized aspects of a proof, wherein it may be demonstrably refuted by other mathematicians. After a proof has been accepted for many years or even decades, it can then be considered as reliable.<ref>{{cite journal
 | title=On the Reliability of Mathematical Proofs
 | first=V. Ya. | last=Perminov
 | journal=Philosophy of Mathematics
 | volume=42 | issue=167 (4) | year=1988 | pages=500–508
 | publisher=Revue Internationale de Philosophie
}}</ref>

Nevertheless, the concept of "rigor" may remain useful for teaching to beginners what is a mathematical proof.<ref>{{cite journal
 | title=Teachers' perceptions of the official curriculum: Problem solving and rigor
 | first1=Jon D. | last1=Davis | first2=Amy Roth | last2=McDuffie
 | first3=Corey | last3=Drake | first4=Amanda L. | last4=Seiwell
 | journal=International Journal of Educational Research
 | volume=93 | year=2019 | pages=91–100
 | doi=10.1016/j.ijer.2018.10.002 | s2cid=149753721 }}</ref>
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