Editing Philosophy of mathematics (section)

=== Logic and rigor ===
{{see also|Logic|Mathematics#Rigor|Mathematics#Mathematical logic and set theory}}

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 [[syllogism]]s or [[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>

The rules of rigorous reasoning have been established by the [[ancient Greek philosophers]] under the name of ''logic''. Logic is not specific to mathematics, but, in mathematics, the standard of rigor is much higher than elsewhere.

For many centuries, logic, although used for mathematical proofs, belonged to philosophy and was not specifically studied by mathematicians.<ref>{{Cite journal
 | title=The Road to Modern Logic—An Interpretation
 | last=Ferreirós
 | first=José
 | journal=Bulletin of Symbolic Logic
 | volume=7
 | issue=4
 | pages=441–484
 | date=2001
 | doi=10.2307/2687794
 | jstor=2687794
 | hdl=11441/38373
 | s2cid=43258676
 | url=https://idus.us.es/xmlui/bitstream/11441/38373/1/The%20road%20to%20modern%20logic.pdf
 | access-date=November 11, 2022
 | archive-url=https://web.archive.org/web/20230202133703/https://idus.us.es/bitstream/handle/11441/38373/The%20road%20to%20modern%20logic.pdf?sequence=1
 | archive-date=February 2, 2023
 | url-status=live
 }}</ref> Circa the end of the 19th century, several [[paradox]]es made questionable the logical foundation of mathematics, and consequently the validity of the whole of mathematics. This has been called the [[foundational crisis of mathematics]]. Some of these paradoxes consist of results that seem to contradict the common intuition, such as the possibility to construct valid [[non-Euclidean geometries]] in which the [[parallel postulate]] is wrong, the [[Weierstrass function]] that is [[continuous function|continuous]] but nowhere  [[differentiable function|differentiable]], and the study by [[Georg Cantor]] of [[infinite sets]], which led to consider several sizes of infinity (infinite [[cardinality|cardinals]]). Even more striking, [[Russell's paradox]] shows that the phrase "the set of all sets" is self contradictory.

Several methods have been proposed to solve the problem by changing of logical framework, such as  [[constructive mathematics]] and [[intuitionistic logic]]. Roughly speaking, the first one consists of requiring that every existence theorem must provide an explicit example, and the second one excludes from mathematical reasoning the [[law of excluded middle]] and [[double negation elimination]]. 

These logics have less inference rules than classical logic. On the other hand classical logic was a [[first-order logic]], which means roughly that [[quantifier (logic)|quantifier]]s cannot be applied to infinite sets. This means, for example that the sentence "every set of [[natural number]]s has a least element" is nonsensical in any formalization of classical logic. This led to the introduction of [[higher-order logic]]s, which are presently used commonly in mathematics.

The problems of [[foundation of mathematics]] has been eventually resolved with the rise of [[mathematical logic]] as a new area of mathematics. In this framework, a mathematical or [[logical theory]] consists of a [[formal language]] that defines the [[well-formed formula|well-formed of assertions]], a set of basic assertions called [[axioms]] and a set of [[inference rule]]s that allow producing new assertions from one or several known assertions. A ''theorem'' of such a theory is either an axiom or an assertion that can be obtained from previously known theorems by the application of an inference rule. The [[Zermelo–Fraenkel set theory]] with the [[axiom of choice]], generally called ''ZFC'', is a higher-order logic in which all mathematics have been restated; it is used implicitely in all mathematics texts that do not specify explicitly on which foundations they are based. Moreover, the other proposed foundations can be modeled and studied inside ZFC.

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. In particular, proofs are rarely written in full details, and some steps of a proof are generally considered as ''trivial'', ''easy'', or ''straightforward'', and therefore left to the reader. As most proof errors occur in these skipped steps, a new proof requires to be verified by other specialists of the subject, and can be considered as reliable only after having been accepted by the community of the specialists, which may need several years.<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>

Also, 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 | author2-link = Amy Roth 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>