Mathematics

=== Psychology (aesthetic, creativity and intuition) ===
The validity of a mathematical theorem relies only on the rigor of its proof, which could theoretically be done automatically by a [[computer program]]. This does not mean that there is no place for creativity in a mathematical work. On the contrary, many important mathematical results (theorems) are solutions of problems that other mathematicians failed to solve, and the invention of a way for solving them may be a fundamental way of the solving process.<ref>{{cite journal
 | title=The Outlook of the Mathematicians' Creative Processes
 | first=Narges | last=Yaftian
 | journal=Procedia - Social and Behavioral Sciences
 | volume=191 | date=June 2, 2015 | pages=2519–2525
 | doi=10.1016/j.sbspro.2015.04.617
| doi-access=free}}</ref><ref>{{cite journal
 | title=The Frontage of Creativity and Mathematical Creativity
 | first1=Mehdi | last1=Nadjafikhah | first2=Narges | last2=Yaftian
 | journal=Procedia - Social and Behavioral Sciences
 | volume=90 | date=October 10, 2013 | pages=344–350
 | doi=10.1016/j.sbspro.2013.07.101
| doi-access=free}}</ref> An extreme example is [[Apery's theorem]]: [[Roger Apery]] provided only the ideas for a proof, and the formal proof was given only several months later by three other mathematicians.<ref>{{cite journal
 | title=A proof that Euler missed... Apéry's Proof of the irrationality of ζ(3)
 | first=A.
 | last=van der Poorten
 | journal=[[The Mathematical Intelligencer]]
 | volume=1
 | issue=4
 | year=1979
 | pages=195–203
 | doi=10.1007/BF03028234
 | s2cid=121589323
 | url=http://pracownicy.uksw.edu.pl/mwolf/Poorten_MI_195_0.pdf
 | access-date=November 22, 2022
 | archive-date=September 6, 2015
| archive-url=https://web.archive.org/web/20150906015716/http://pracownicy.uksw.edu.pl/mwolf/Poorten_MI_195_0.pdf
 | url-status=live
 }}</ref>

Creativity and rigor are not the only psychological aspects of the activity of mathematicians. Some mathematicians can see their activity as a game, more specifically as solving [[puzzle]]s.<ref>{{cite book
 | title=Famous Puzzles of Great Mathematicians
 | first=Miodrag
 | last=Petkovi
 | date=September 2, 2009
| publisher=American Mathematical Society
 | pages=xiii–xiv
 | isbn=978-0-8218-4814-2
 | url={{GBurl|id=AZlwAAAAQBAJ|pg=PR13}}
 | access-date=November 25, 2022
}}</ref> This aspect of mathematical activity is emphasized in [[recreational mathematics]].

Mathematicians can find an [[aesthetic]] value to mathematics. Like [[beauty]], it is hard to define, it is commonly related to ''elegance'', which involves qualities like [[simplicity]], [[symmetry]], completeness, and generality. G. H. Hardy in ''[[A Mathematician's Apology]]'' expressed the belief that the aesthetic considerations are, in themselves, sufficient to justify the study of pure mathematics. He also identified other criteria such as significance, unexpectedness, and inevitability, which contribute to mathematical aesthetic.<ref>{{cite book
 | title=A Mathematician's Apology
 | last=Hardy | first=G. H. | author-link=G. H. Hardy
 | publisher=Cambridge University Press | year=1940
 | url=https://archive.org/details/hardy_annotated/
 | isbn=978-0-521-42706-7 | access-date=November 22, 2022
}} See also ''[[A Mathematician's Apology]]''.</ref> [[Paul Erdős]] expressed this sentiment more ironically by speaking of "The Book", a supposed divine collection of the most beautiful proofs. The 1998 book ''[[Proofs from THE BOOK]]'', inspired by Erdős, is a collection of particularly succinct and revelatory mathematical arguments. Some examples of particularly elegant results included are Euclid's proof that there are infinitely many prime numbers and the [[fast Fourier transform]] for [[harmonic analysis]].<ref>{{cite journal
 | title=Reflections on Paul Erdős on His Birth Centenary, Part II
 | first1=Noga | last1=Alon | first2=Dan | last2=Goldston
 | first3=András | last3=Sárközy | first4=József | last4=Szabados
 | first5=Gérald | last5=Tenenbaum | first6=Stephan Ramon | last6=Garcia
 | first7=Amy L. | last7=Shoemaker
 | journal=Notices of the American Mathematical Society
 | date=March 2015 | volume=62 | issue=3 | pages=226–247
 | editor1-first=Krishnaswami | editor1-last=Alladi
 | editor2-first=Steven G. | editor2-last=Krantz
 | doi=10.1090/noti1223
| doi-access=free }}</ref>

Some feel that to consider mathematics a science is to downplay its artistry and history in the seven traditional [[liberal arts]].<ref>See, for example [[Bertrand Russell]]'s statement "Mathematics, rightly viewed, possesses not only truth, but supreme beauty ..." in his {{cite book | title=History of Western Philosophy | year=1919 | page=60 }}</ref> One way this difference of viewpoint plays out is in the philosophical debate as to whether mathematical results are ''created'' (as in art) or ''discovered'' (as in science).<ref name=borel>{{Cite journal
 | last=Borel | first=Armand | author-link=Armand Borel
 | title=Mathematics: Art and Science
 | journal=The Mathematical Intelligencer
 | volume=5 | issue=4 | pages=9–17 | year=1983
 | publisher=Springer | issn=1027-488X
 | doi=10.4171/news/103/8| doi-access=free
}}</ref> The popularity of recreational mathematics is another sign of the pleasure many find in solving mathematical questions.