Editing Philosophy of mathematics (section)

==Major themes==
===Reality===
{{excerpt|Mathematics#Reality}}

=== 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>

=== Relationship with sciences ===
{{further|Relationship between mathematics and physics}}

Mathematics is used in most [[science]]s for [[Mathematical model|modeling]] phenomena, which then allows predictions to be made from experimental laws.<ref>{{cite book | title=Modelling Mathematical Methods and Scientific Computation | first1=Nicola | last1=Bellomo | first2=Luigi | last2=Preziosi | publisher=CRC Press | date=December 22, 1994 | page=1 | isbn=978-0-8493-8331-1 | series=Mathematical Modeling | volume=1 | url={{GBurl|id=pJAvWaRYo3UC}} | access-date=November 16, 2022 }}</ref> The independence of mathematical truth from any experimentation implies that the accuracy of such predictions depends only on the adequacy of the model.<ref>{{cite journal
 | title=Mathematical Models and Reality: A Constructivist Perspective
 | first=Christian | last=Hennig
 | journal=Foundations of Science
 | volume=15 | pages=29–48 | year=2010
 | doi=10.1007/s10699-009-9167-x
 | s2cid=6229200 | url=https://www.researchgate.net/publication/225691477
 | access-date=November 17, 2022
}}</ref> Inaccurate predictions, rather than being caused by invalid mathematical concepts, imply the need to change the mathematical model used.<ref>{{cite journal | title=Models in Science | date=February 4, 2020 | first1=Roman | last1=Frigg | author-link=Roman Frigg | first2=Stephan | last2=Hartmann | author2-link=Stephan Hartmann | website=Stanford Encyclopedia of Philosophy | url=https://seop.illc.uva.nl/entries/models-science/ | access-date=November 17, 2022 | archive-date=November 17, 2022 | archive-url=https://web.archive.org/web/20221117162412/https://seop.illc.uva.nl/entries/models-science/ | url-status=live }}</ref> For example, the [[perihelion precession of Mercury]] could only be explained after the emergence of [[Einstein]]'s [[general relativity]], which replaced [[Newton's law of gravitation]] as a better mathematical model.<ref>{{cite book | last=Stewart | first=Ian | author-link=Ian Stewart (mathematician) | chapter=Mathematics, Maps, and Models | title=The Map and the Territory: Exploring the Foundations of Science, Thought and Reality | pages=345–356 | publisher=Springer | year=2018 | editor1-first=Shyam | editor1-last=Wuppuluri | editor2-first=Francisco Antonio | editor2-last=Doria | isbn=978-3-319-72478-2 | series=The Frontiers Collection | chapter-url={{GBurl|id=mRBMDwAAQBAJ|p=345}} | doi=10.1007/978-3-319-72478-2_18 | access-date=November 17, 2022 }}</ref>

There is still a philosophical debate whether mathematics is a science. However, in practice, mathematicians are typically grouped with scientists, and mathematics shares much in common with the physical sciences. Like them, it is [[falsifiable]], which means in mathematics that if a result or a theory is wrong, this can be proved by providing a [[counterexample]]. Similarly as in science, [[mathematical theory|theories]] and results (theorems) are often obtained from [[experimentation]].<ref>{{Cite web|url=https://undsci.berkeley.edu/article/mathematics|title=The science checklist applied: Mathematics|website=Understanding Science |publisher=University of California, Berkeley |access-date=October 27, 2019|archive-url=https://web.archive.org/web/20191027021023/https://undsci.berkeley.edu/article/mathematics|archive-date=October 27, 2019|url-status=live}}</ref> In mathematics, the experimentation may consist of computation on selected examples or of the study of figures or other representations of mathematical objects (often mind representations without physical support). For example, when asked how he came about his theorems, Gauss once replied "durch planmässiges Tattonieren" (through systematic experimentation).<ref>{{cite book | last=Mackay | first=A. L. | year=1991 | title=Dictionary of Scientific Quotations | location=London | page=100 | isbn=978-0-7503-0106-0 | publisher=Taylor & Francis | url={{GBurl|id=KwESE88CGa8C|q=durch planmässiges Tattonieren}} | access-date=March 19, 2023 }}</ref> However, some authors emphasize that mathematics differs from the modern notion of science by not {{em|relying}} on empirical evidence.<ref>{{cite book | last1 = Bishop | first1 = Alan | year = 1991 | chapter = Environmental activities and mathematical culture | title = Mathematical Enculturation: A Cultural Perspective on Mathematics Education | chapter-url = {{GBurl|id=9AgrBgAAQBAJ|p=54}} | pages = 20–59 | location = Norwell, Massachusetts | publisher = Kluwer Academic Publishers | isbn = 978-0-7923-1270-3 | access-date = April 5, 2020 }}</ref><ref>{{cite book | title=Out of Their Minds: The Lives and Discoveries of 15 Great Computer Scientists | last1=Shasha | first1=Dennis Elliot | author1-link=Dennis Elliot Shasha | last2=Lazere | first2=Cathy A. | publisher=Springer | year=1998 | page=228 | isbn=978-0-387-98269-4 }}</ref><ref>{{cite book | last=Nickles | first=Thomas | year=2013 | chapter=The Problem of Demarcation | title=Philosophy of Pseudoscience: Reconsidering the Demarcation Problem | page=104 | location=Chicago | publisher=The University of Chicago Press | isbn=978-0-226-05182-6 }}</ref><ref>{{Cite magazine | year=2014| last=Pigliucci| first=Massimo | author-link=Massimo Pigliucci | title=Are There 'Other' Ways of Knowing? | magazine=[[Philosophy Now]]| url=https://philosophynow.org/issues/102/Are_There_Other_Ways_of_Knowing | access-date=April 6, 2020| archive-date=May 13, 2020 | archive-url=https://web.archive.org/web/20200513190522/https://philosophynow.org/issues/102/Are_There_Other_Ways_of_Knowing | url-status=live}}</ref>
<!-- What precedes is only one aspect of the relationship between mathematics and other sciences. Other aspects are considered in the next subsections. -->

==== Unreasonable effectiveness ====

The [[The Unreasonable Effectiveness of Mathematics in the Natural Sciences|unreasonable effectiveness of mathematics]] is a phenomenon that was named and first made explicit by physicist [[Eugene Wigner]].<ref>{{cite journal
 | title=The Unreasonable Effectiveness of Mathematics in the Natural Sciences
 | last=Wigner | first=Eugene | author-link=Eugene Wigner
 | journal=[[Communications on Pure and Applied Mathematics]]
 | volume=13 | issue=1 | pages=1–14 | year=1960
 | doi=10.1002/cpa.3160130102 | bibcode=1960CPAM...13....1W
 | s2cid=6112252 | url=https://math.dartmouth.edu/~matc/MathDrama/reading/Wigner.html
 | url-status=live | archive-url=https://web.archive.org/web/20110228152633/http://www.dartmouth.edu/~matc/MathDrama/reading/Wigner.html
 | archive-date=February 28, 2011 | df=mdy-all
}}</ref> It is the fact that many mathematical theories (even the "purest") have applications outside their initial object. These applications may be completely outside their initial area of mathematics, and may concern physical phenomena that were completely unknown when the mathematical theory was introduced.<ref>{{cite journal
 | title=Revisiting the 'unreasonable effectiveness' of mathematics
 | first=Sundar | last=Sarukkai
 | journal=Current Science
 | volume=88 | issue=3 | date=February 10, 2005 | pages=415–423
 | jstor=24110208
}}</ref> Examples of unexpected applications of mathematical theories can be found in many areas of mathematics.

A notable example is the [[prime factorization]] of natural numbers that was discovered more than 2,000 years before its common use for secure [[internet]] communications through the [[RSA cryptosystem]].<ref>{{cite book
 | chapter=History of Integer Factoring
 | pages=41–77
 | first=Samuel S. Jr.
 | last=Wagstaff
 | title=Computational Cryptography, Algorithmic Aspects of Cryptography, A Tribute to AKL
 | editor1-first=Joppe W.
 | editor1-last=Bos
 | editor2-first=Martijn
 | editor2-last=Stam
 | series=London Mathematical Society Lecture Notes Series 469
 | publisher=Cambridge University Press
 | year=2021
 | chapter-url=https://www.cs.purdue.edu/homes/ssw/chapter3.pdf
 | access-date=November 20, 2022
 | archive-date=November 20, 2022
| archive-url=https://web.archive.org/web/20221120155733/https://www.cs.purdue.edu/homes/ssw/chapter3.pdf
 | url-status=live
 }}</ref> A second historical example is the theory of [[ellipse]]s. They were studied by the [[Greek mathematics|ancient Greek mathematicians]] as [[conic section]]s (that is, intersections of [[cone]]s with planes). It was almost 2,000 years later that [[Johannes Kepler]] discovered that the [[trajectories]] of the planets are ellipses.<ref>{{cite web
 | title=Curves: Ellipse
 | website=MacTutor
 | publisher=School of Mathematics and Statistics, University of St Andrews, Scotland
 | url=https://mathshistory.st-andrews.ac.uk/Curves/Ellipse/
 | access-date=November 20, 2022
 | archive-date=October 14, 2022
| archive-url=https://web.archive.org/web/20221014051943/https://mathshistory.st-andrews.ac.uk/Curves/Ellipse/
 | url-status=live
 }}</ref>

In the 19th century, the internal development of geometry (pure mathematics) led to definition and study of non-Euclidean geometries, spaces of dimension higher than three and [[manifold]]s. At this time, these concepts seemed totally disconnected from the physical reality, but at the beginning of the 20th century, [[Albert Einstein]] developed the [[theory of relativity]] that uses fundamentally these concepts. In particular, [[spacetime]] of [[special relativity]] is a non-Euclidean space of dimension four, and spacetime of [[general relativity]] is a (curved) manifold of dimension four.<ref>{{cite web
 | title=Beyond the Surface of Einstein's Relativity Lay a Chimerical Geometry
 | first=Vasudevan
 | last=Mukunth
 | website=The Wire
 | date=September 10, 2015
| url=https://thewire.in/science/beyond-the-surface-of-einsteins-relativity-lay-a-chimerical-geometry
 | access-date=November 20, 2022
 | archive-date=November 20, 2022
| archive-url=https://web.archive.org/web/20221120191206/https://thewire.in/science/beyond-the-surface-of-einsteins-relativity-lay-a-chimerical-geometry
 | url-status=live
 }}</ref><ref>{{cite journal
 | title=The Space-Time Manifold of Relativity. The Non-Euclidean Geometry of Mechanics and Electromagnetics
 | first1=Edwin B. | last1=Wilson | first2=Gilbert N. | last2=Lewis
 | journal=Proceedings of the American Academy of Arts and Sciences
 | volume=48 | issue=11 | date=November 1912 | pages=389–507
 | doi=10.2307/20022840 | jstor=20022840 }}</ref>

A striking aspect of the interaction between mathematics and physics is when mathematics drives research in physics. This is illustrated by the discoveries of the [[positron]] and the [[omega baryon|baryon]] <math>\Omega^{-}.</math> In both cases, the equations of the theories had unexplained solutions, which led to conjecture of the existence of an unknown [[particle]], and the search for these particles. In both cases, these particles were discovered a few years later by specific experiments.<ref name="Borel-1983">{{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><ref>{{cite journal
 | title=Discovering the Positron (I)
 | first=Norwood Russell | last=Hanson | author-link=Norwood Russell Hanson
 | journal=The British Journal for the Philosophy of Science
 | volume=12 | issue=47 | date=November 1961 | pages=194–214
 | publisher=The University of Chicago Press
 | jstor=685207 | doi=10.1093/bjps/xiii.49.54
}}</ref><ref>{{cite journal
 | title=Avoiding reification: Heuristic effectiveness of mathematics and the prediction of the Ω<sup>–</sup> particle
 | first=Michele | last=Ginammi
 | journal=Studies in History and Philosophy of Science Part B: Studies in History and Philosophy of Modern Physics
 | volume=53 | date=February 2016 | pages=20–27
 | doi=10.1016/j.shpsb.2015.12.001
| bibcode=2016SHPMP..53...20G }}</ref>