Mathematics

=== 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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