Mathematics

=== 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 name=wigner1960>{{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 is 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 /><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>