Editing Mathematics (section)

== Symbolic notation and terminology ==
{{Main|Mathematical notation|Language of mathematics|Glossary of mathematics}}
[[File:Sigma summation notation.svg|thumb|An explanation of the sigma (Σ) [[summation]] notation|class=skin-invert-image]]
Mathematical notation is widely used in science and [[engineering]] for representing complex [[concept]]s and [[property (philosophy)|properties]] in a concise, unambiguous, and accurate way. This notation consists of [[glossary of mathematical symbols|symbols]] used for representing [[operation (mathematics)|operation]]s, unspecified numbers, [[relation (mathematics)|relation]]s and any other mathematical objects, and then assembling them into [[expression (mathematics)|expression]]s and formulas.<ref>{{cite conference |last=Wolfram |first=Stephan |date=October 2000 |author-link=Stephen Wolfram |title=Mathematical Notation: Past and Future |conference=MathML and Math on the Web: MathML International Conference 2000, Urbana Champaign, USA |url=https://www.stephenwolfram.com/publications/mathematical-notation-past-future/ |url-status=live |archive-url=https://web.archive.org/web/20221116150905/https://www.stephenwolfram.com/publications/mathematical-notation-past-future/ |archive-date=November 16, 2022 |access-date=February 3, 2024}}</ref> More precisely, numbers and other mathematical objects are represented by symbols called variables, which are generally [[Latin alphabet|Latin]] or [[Greek alphabet|Greek]] letters, and often include [[subscript]]s. Operation and relations are generally represented by specific [[Glossary of mathematical symbols|symbols]] or [[glyph]]s,<ref>{{cite journal |last1=Douglas |first1=Heather |last2=Headley |first2=Marcia Gail |last3=Hadden |first3=Stephanie |last4=LeFevre |first4=Jo-Anne |author4-link=Jo-Anne LeFevre |date=December 3, 2020 |title=Knowledge of Mathematical Symbols Goes Beyond Numbers |journal=Journal of Numerical Cognition |volume=6 |issue=3 |pages=322–354 |doi=10.5964/jnc.v6i3.293 |doi-access=free |eissn=2363-8761 |s2cid=228085700}}</ref> such as {{math|+}} ([[plus sign|plus]]), {{math|×}} ([[multiplication sign|multiplication]]), <math display =inline>\int</math> ([[integral sign|integral]]), {{math|1==}} ([[equals sign|equal]]), and {{math|<}} ([[less-than sign|less than]]).<ref name=AMS>{{cite web |last1=Letourneau |first1=Mary |last2=Wright Sharp |first2=Jennifer |date=October 2017 |title=AMS Style Guide |page=75 |publisher=[[American Mathematical Society]] |url=https://www.ams.org/publications/authors/AMS-StyleGuide-online.pdf |url-status=live |archive-url=https://web.archive.org/web/20221208063650/https://www.ams.org//publications/authors/AMS-StyleGuide-online.pdf |archive-date=December 8, 2022 |access-date=February 3, 2024}}</ref> All these symbols are generally grouped according to specific rules to form expressions and formulas.<ref>{{cite journal |last1=Jansen |first1=Anthony R. |last2=Marriott |first2=Kim |last3=Yelland |first3=Greg W. |year=2000 |title=Constituent Structure in Mathematical Expressions |journal=Proceedings of the Annual Meeting of the Cognitive Science Society |volume=22 |publisher=[[University of California Merced]] |eissn=1069-7977 |oclc=68713073 |url=https://escholarship.org/content/qt35r988q9/qt35r988q9.pdf |url-status=live |archive-url=https://web.archive.org/web/20221116152222/https://escholarship.org/content/qt35r988q9/qt35r988q9.pdf |archive-date=November 16, 2022 |access-date=February 3, 2024}}</ref> Normally, expressions and formulas do not appear alone, but are included in sentences of the current language, where expressions play the role of [[noun phrase]]s and formulas play the role of [[clause]]s.

Mathematics has developed a rich terminology covering a broad range of fields that study the properties of various abstract, idealized objects and how they interact. It is based on rigorous [[Technical definition|definitions]] that provide a standard foundation for communication. An axiom or [[postulate]] is a mathematical statement that is taken to be true without need of proof. If a mathematical statement has yet to be proven (or disproven), it is termed a [[conjecture]]. Through a series of rigorous arguments employing [[deductive reasoning]], a statement that is [[formal proof|proven]] to be true becomes a theorem. A specialized theorem that is mainly used to prove another theorem is called a [[Lemma (mathematics)|lemma]]. A proven instance that forms part of a more general finding is termed a [[corollary]].<ref>{{cite book |last=Rossi |first=Richard J. |year=2006 |title=Theorems, Corollaries, Lemmas, and Methods of Proof |series=Pure and Applied Mathematics: A Wiley Series of Texts, Monographs and Tracts |publisher=[[John Wiley & Sons]] |pages=1–14, 47–48 |isbn=978-0-470-04295-3 |lccn=2006041609 |oclc=64085024}}</ref>

Numerous technical terms used in mathematics are [[neologism]]s, such as ''[[polynomial]]'' and ''[[homeomorphism]]''.<ref>{{cite web |url=https://mathshistory.st-andrews.ac.uk/Miller/mathword/ |title=Earliest Uses of Some Words of Mathematics |website=MacTutor |publisher=[[University of St. Andrews]] |publication-place=Scotland, UK |url-status=live |archive-url=https://web.archive.org/web/20220929032236/https://mathshistory.st-andrews.ac.uk/Miller/mathword/ |archive-date=September 29, 2022 |access-date=February 3, 2024}}</ref> Other technical terms are words of the common language that are used in an accurate meaning that may differ slightly from their common meaning. For example, in mathematics, "[[logical disjunction|or]]" means "one, the other or both", while, in common language, it is either ambiguous or means "one or the other but not both" (in mathematics, the latter is called "[[exclusive or]]"). Finally, many mathematical terms are common words that are used with a completely different meaning.<ref>{{cite journal |last=Silver |first=Daniel S. |date=November–December 2017 |title=The New Language of Mathematics |journal=The American Scientist |volume=105 |number=6 |pages=364–371 |publisher=[[Sigma Xi]] |doi=10.1511/2017.105.6.364 |doi-access=free |issn=0003-0996 |lccn=43020253 |oclc=1480717 |s2cid=125455764}}</ref> This may lead to sentences that are correct and true mathematical assertions, but appear to be nonsense to people who do not have the required background. For example, "every [[free module]] is [[flat module|flat]]" and "a [[field (mathematics)|field]] is always a [[ring (mathematics)|ring]]".