Mathematics

=== Mathematical logic and set theory ===
{{Main|Mathematical logic|Set theory}}
[[File:Venn A intersect B.svg|thumb|The [[Venn diagram]] is a commonly used method to illustrate the relations between sets.]]

The two subjects of mathematical logic and set theory have belonged to mathematics since the end of the 19th century.<ref name=Ewald_2018>{{cite web
 | first=William
 | last=Ewald
 | date=November 17, 2018
| title=The Emergence of First-Order Logic
 | website=Stanford Encyclopedia of Philosophy
 | url=https://plato.stanford.edu/entries/settheory-early/
 | access-date=November 2, 2022
 | archive-date=May 12, 2021
| archive-url=https://web.archive.org/web/20210512135148/https://plato.stanford.edu/entries/settheory-early/
 | url-status=live
 }}</ref><ref name="Ferreirós_2020">{{cite web
 | first=José
 | last=Ferreirós
 | date=June 18, 2020
| title=The Early Development of Set Theory
 | website=Stanford Encyclopedia of Philosophy
 | url=https://plato.stanford.edu/entries/settheory-early/
 | access-date=November 2, 2022
 | archive-date=May 12, 2021
| archive-url=https://web.archive.org/web/20210512135148/https://plato.stanford.edu/entries/settheory-early/
 | url-status=live
 }}</ref> Before this period, sets were not considered to be mathematical objects, and [[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>

Before [[Georg Cantor|Cantor]]'s study of [[infinite set]]s, mathematicians were reluctant to consider [[actual infinite|actually infinite]] collections, and considered [[infinity]] to be the result of endless [[enumeration]]. Cantor's work offended many mathematicians not only by considering actually infinite sets<ref>{{cite web | first=Natalie | last=Wolchover | author-link=Natalie Wolchover | date=December 3, 2013 | title=Dispute over Infinity Divides Mathematicians | website=[[Scientific American]] | url=https://www.scientificamerican.com/article/infinity-logic-law/ | access-date=November 1, 2022 | archive-date=November 2, 2022 | archive-url=https://web.archive.org/web/20221102011848/https://www.scientificamerican.com/article/infinity-logic-law/ | url-status=live }}</ref> but by showing that this implies different sizes of infinity, per [[Cantor's diagonal argument]]. This led to the [[controversy over Cantor's theory|controversy over Cantor's set theory]].<ref>{{cite web
 | title=Wittgenstein's analysis on Cantor's diagonal argument
 | last=Zhuang | first=C. | website=[[PhilArchive]]
 | url=https://philarchive.org/archive/ZHUWAO
 | access-date=November 18, 2022 }}</ref>

In the same period, various areas of mathematics concluded the former intuitive definitions of the basic mathematical objects were insufficient for ensuring [[mathematical rigour]]. Examples of such intuitive definitions are "a set is a collection of objects", "natural number is what is used for counting", "a point is a shape with a zero length in every direction", "a curve is a trace left by a moving point", etc.

This became the foundational crisis of mathematics.<ref>{{cite web
 | title="Clarifying the nature of the infinite": the development of metamathematics and proof theory
 | first1=Jeremy
 | last1=Avigad
 | author1-link=Jeremy Avigad
 | first2=Erich H.
 | last2=Reck
 | date=December 11, 2001
| work=Carnegie Mellon Technical Report CMU-PHIL-120
 | url=https://www.andrew.cmu.edu/user/avigad/Papers/infinite.pdf
 | access-date=November 12, 2022
 | archive-date=October 9, 2022
| archive-url=https://web.archive.org/web/20221009074025/https://www.andrew.cmu.edu/user/avigad/Papers/infinite.pdf
 | url-status=live
 }}</ref> It was eventually solved in mainstream mathematics by systematizing the axiomatic method inside a [[Zermelo–Fraenkel set theory|formalized set theory]]. Roughly speaking, each mathematical object is defined by the set of all similar objects and the properties that these objects must have.<ref name=Warner_2013/> For example, in [[Peano arithmetic]], the natural numbers are defined by "zero is a number", "each number has a unique successor", "each number but zero has a unique predecessor", and some rules of reasoning.<ref>{{cite book
 | title=Numbers, Sets and Axioms: The Apparatus of Mathematics
 | first=Alan G.
 | last=Hamilton
 | pages=3–4
 | year=1982
 | isbn=978-0-521-28761-6
 | publisher=Cambridge University Press
 | url={{GBurl|id=OXfmTHXvRXMC|p=3}}
 | access-date=November 12, 2022
}}</ref> This [[abstraction (mathematics)|mathematical abstraction]] from reality is embodied in the modern philosophy of [[Formalism (philosophy of mathematics)|formalism]], as founded by [[David Hilbert]] around 1910.<ref name="Snapper">{{Cite journal |doi=10.2307/2689412 |title=The Three Crises in Mathematics: Logicism, Intuitionism, and Formalism |journal=Mathematics Magazine |date=September 1979 |first=Ernst |last=Snapper |author-link=Ernst Snapper |volume=52 |issue=4 |pages=207–216 |jstor=2689412 }}</ref>

The "nature" of the objects defined this way is a philosophical problem that mathematicians leave to philosophers, even if many mathematicians have opinions on this nature, and use their opinion{{emdash}}sometimes called "intuition"{{emdash}}to guide their study and proofs. The approach allows considering "logics" (that is, sets of allowed deducing rules), theorems, proofs, etc. as mathematical objects, and to prove theorems about them. For example, [[Gödel's incompleteness theorems]] assert, roughly speaking that, in every [[Consistency|consistent]] [[formal system]] that contains the natural numbers, there are theorems that are true (that is provable in a stronger system), but not provable inside the system.<ref name=Raatikainen_2005>{{cite journal | title=On the Philosophical Relevance of Gödel's Incompleteness Theorems | first=Panu | last=Raatikainen | journal=Revue Internationale de Philosophie | volume=59 | issue=4 | date=October 2005 | pages=513–534 | doi=10.3917/rip.234.0513 | url=https://www.cairn.info/revue-internationale-de-philosophie-2005-4-page-513.htm | jstor=23955909 | s2cid=52083793 | access-date=November 12, 2022 | archive-date=November 12, 2022 | archive-url=https://web.archive.org/web/20221112212555/https://www.cairn.info/revue-internationale-de-philosophie-2005-4-page-513.htm | url-status=live }}</ref> This approach to the foundations of mathematics was challenged during the first half of the 20th century by mathematicians led by [[L. E. J. Brouwer|Brouwer]], who promoted [[intuitionistic logic]], which explicitly lacks the [[law of excluded middle]].<ref>{{cite web
 | title=Intuitionistic Logic
 | date=September 4, 2018
| first=Joan
 | last=Moschovakis
 | author-link=Joan Moschovakis
 | website=Stanford Encyclopedia of Philosophy
 | url=https://plato.stanford.edu/entries/logic-intuitionistic/
 | access-date=November 12, 2022
 | archive-date=December 16, 2022
| archive-url=https://web.archive.org/web/20221216154821/https://plato.stanford.edu/entries/logic-intuitionistic/
 | url-status=live
 }}</ref><ref>{{cite journal
 | title=At the Heart of Analysis: Intuitionism and Philosophy
 | first=Charles | last=McCarty
 | journal=Philosophia Scientiæ, Cahier spécial 6
 | year=2006 | pages=81–94 | doi=10.4000/philosophiascientiae.411
| doi-access=free}}</ref>

These problems and debates led to a wide expansion of mathematical logic, with subareas such as [[model theory]] (modeling some logical theories inside other theories), [[proof theory]], [[type theory]], [[computability theory]] and [[computational complexity theory]].<ref name=MSC/> Although these aspects of mathematical logic were introduced before the rise of [[computer]]s, their use in [[compiler]] design, [[computer program|program certification]], [[proof assistant]]s and other aspects of [[computer science]], contributed in turn to the expansion of these logical theories.<ref>{{cite web
 | last1=Halpern | first1=Joseph | author1-link=Joseph Halpern
 | last2=Harper | first2=Robert | author2-link=Robert Harper (computer scientist)
 | last3=Immerman | first3=Neil | author3-link=Neil Immerman
 | last4=Kolaitis | first4=Phokion | author4-link=Phokion Kolaitis
 | last5=Vardi | first5=Moshe | author5-link=Moshe Vardi
 | last6=Vianu | first6=Victor | author6-link=Victor Vianu
 | title=On the Unusual Effectiveness of Logic in Computer Science
 | url=https://www.cs.cmu.edu/~rwh/papers/unreasonable/basl.pdf
 | access-date=January 15, 2021 | date=2001 | archive-date=March 3, 2021
| archive-url=https://web.archive.org/web/20210303115643/https://www.cs.cmu.edu/~rwh/papers/unreasonable/basl.pdf
 | url-status=live }}</ref>