Editing Experimental mathematics

{{for|the mathematical journal of the same name|Experimental Mathematics (journal)}}
'''Experimental mathematics''' is an approach to [[mathematics]] in which computation is used to investigate mathematical objects and identify properties and patterns.<ref>{{Mathworld|urlname=ExperimentalMathematics|title=Experimental Mathematics}}</ref> It has been defined as "that branch of mathematics that concerns itself ultimately with the codification and transmission of insights within the mathematical community through the use of experimental (in either the Galilean, Baconian, Aristotelian or Kantian sense) exploration of [[conjecture]]s and more informal beliefs and a careful analysis of the data acquired in this pursuit."<ref>[http://oldweb.cecm.sfu.ca/organics/vault/expmath/expmath/html/node16.html Experimental Mathematics: A Discussion] {{Webarchive|url=https://web.archive.org/web/20080121081424/http://oldweb.cecm.sfu.ca/organics/vault/expmath/expmath/html/node16.html |date=2008-01-21 }} by J. Borwein, P. Borwein, R. Girgensohn and S. Parnes</ref>

As expressed by [[Paul Halmos]]: "Mathematics is not a [[deductive science]]—that's a cliché. When you try to prove a theorem, you don't just list the [[Premise|hypotheses]], and then start to reason. What you do is [[trial and error]], experimentation, guesswork. You want to find out what the facts are, and what you do is in that respect similar to what a laboratory technician does."<ref>''I Want to be a Mathematician: An Automathography'' (1985), p. 321 (in 2013 reprint)</ref>

==History==
Mathematicians have always practiced experimental mathematics. Existing records of early mathematics, such as [[Babylonian mathematics]], typically consist of lists of numerical examples illustrating algebraic identities. However, modern mathematics, beginning in the 17th century, developed a tradition of publishing results in a final, formal and abstract presentation. The numerical examples that may have led a mathematician to originally formulate a general theorem were not published, and were generally forgotten.

Experimental mathematics as a separate area of study re-emerged in the twentieth century, when the invention of the electronic computer vastly increased the range of feasible calculations, with a speed and precision far greater than anything available to previous generations of mathematicians. A significant milestone and achievement of experimental mathematics was the discovery in 1995 of the [[Bailey–Borwein–Plouffe formula]] for the binary digits of [[Pi|π]]. This formula was discovered not by formal reasoning, but instead
by numerical searches on a computer; only afterwards was a rigorous [[Mathematical proof|proof]] found.<ref>[http://crd.lbl.gov/~dhbailey/dhbpapers/pi-quest.pdf The Quest for Pi] {{Webarchive|url=https://web.archive.org/web/20110927044106/http://crd.lbl.gov/~dhbailey/dhbpapers/pi-quest.pdf |date=2011-09-27 }} by [[David H. Bailey (mathematician)|David H. Bailey]], [[Jonathan Borwein|Jonathan M. Borwein]], [[Peter Borwein|Peter B. Borwein]] and [[Simon Plouffe]].</ref>

==Objectives and uses==
The objectives of experimental mathematics are "to generate understanding and insight; to generate and confirm or confront conjectures; and generally to make mathematics more tangible, lively and fun for both the professional researcher and the novice".<ref>{{cite book |title= Mathematics by Experiment: Plausible Reasoning in the 21st Century|last= Borwein|first= Jonathan |author2=Bailey, David|year= 2004|publisher= A.K. Peters|isbn= 978-1-56881-211-3|pages=vii}}</ref>

The uses of experimental mathematics have been defined as follows:<ref>{{cite book |title= Mathematics by Experiment: Plausible Reasoning in the 21st Century|last= Borwein|first= Jonathan |author2=Bailey, David|year= 2004|publisher= A.K. Peters|isbn= 978-1-56881-211-3|pages=2}}</ref>

#Gaining insight and intuition.
#Discovering new patterns and relationships.
#Using graphical displays to suggest underlying mathematical principles.
#Testing and especially falsifying conjectures.
#Exploring a possible result to see if it is worth formal proof.
#Suggesting approaches for formal proof.
#Replacing lengthy hand derivations with computer-based derivations.
#Confirming analytically derived results.

==Tools and techniques==
Experimental mathematics makes use of [[numerical methods]] to calculate approximate values for [[integral]]s and [[infinite series]]. [[Arbitrary precision arithmetic]] is often used to establish these values to a high degree of precision – typically 100 significant figures or more. [[Integer relation algorithm]]s are then used to search for relations between these values and [[mathematical constant]]s. Working with high precision values reduces the possibility of mistaking a [[mathematical coincidence]] for a true relation. A formal proof of a conjectured relation will then be sought – it is often easier to find a formal proof once the form of a conjectured relation is known.

If a [[counterexample]] is being sought or a large-scale [[proof by exhaustion]] is being attempted, [[distributed computing]] techniques may be used to divide the calculations between multiple computers.

Frequent use is made of general [[mathematical software]] or domain-specific software written for attacks on problems that require high efficiency. Experimental mathematics software usually includes [[error detection and correction]] mechanisms, integrity checks and redundant calculations designed to minimise the possibility of results being invalidated by a hardware or software error.

==Applications and examples==
Applications and examples of experimental mathematics include:

*Searching for a counterexample to a conjecture
**Roger Frye used experimental mathematics techniques to find the smallest counterexample to [[Euler's sum of powers conjecture]].
**The [[ZetaGrid]] project was set up to search for a counterexample to the [[Riemann hypothesis]].
**Tomás Oliveira e Silva<ref>{{cite web|first=Tomás|last=Silva|website=Institute of Electronics and Informatics Engineering of Aveiro|title=Computational verification of the 3x+1 conjecture|url=http://sweet.ua.pt/tos/3x+1.html|archive-url=https://web.archive.org/web/20130318045112/http://www.ieeta.pt/~tos/3x+1.html|archive-date=18 March 2013|date=28 December 2015|url-status=live}}</ref> searched for a counterexample to the [[Collatz conjecture]].
*Finding new examples of numbers or objects with particular properties
**The [[Great Internet Mersenne Prime Search]] is searching for new [[Mersenne prime]]s.
**The Great Periodic Path Hunt is searching for new periodic paths.
**[[distributed.net]]'s OGR project searched for optimal [[Golomb ruler]]s.
**The [[PrimeGrid]] project is searching for the smallest [[Riesel number|Riesel]] and [[Sierpiński number|Sierpiński]] numbers.
*Finding serendipitous numerical patterns
**[[Edward Lorenz]] found the [[Lorenz attractor]], an early example of a chaotic [[dynamical system]], by investigating anomalous behaviours in a numerical weather model.
**The [[Ulam spiral]] was discovered by accident.
**The pattern in the [[Ulam number]]s was discovered by accident.
**[[Mitchell Feigenbaum]]'s discovery of the [[Feigenbaum constant]] was based initially on numerical observations, followed by a rigorous proof.
*Use of computer programs to check a large but finite number of cases to complete a [[computer-assisted proof|computer-assisted]] [[proof by exhaustion]]
**[[Thomas Callister Hales|Thomas Hales]]'s proof of the [[Kepler conjecture]].
**Various proofs of the [[four colour theorem]].
**[[Clement Lam]]'s proof of the non-existence of a [[projective plane|finite projective plane]] of order 10.<ref>{{cite journal |author=Clement W. H. Lam |title=The Search for a Finite Projective Plane of Order 10 |journal=[[American Mathematical Monthly]] |volume=98 |issue=4 |year=1991 |pages=305–318 |url=http://www.cecm.sfu.ca/organics/papers/lam/ |doi=10.2307/2323798|jstor=2323798 |url-access=subscription }}</ref>
**Gary McGuire proved a minimum uniquely solvable [[Sudoku]] requires 17 clues.<ref>{{cite news|last1=arXiv|first1=Emerging Technology from the|title=Mathematicians Solve Minimum Sudoku Problem|url=https://www.technologyreview.com/s/426554/mathematicians-solve-minimum-sudoku-problem/|access-date=27 November 2017|work=MIT Technology Review|language=en}}</ref>
*Symbolic validation (via [[computer algebra]]) of conjectures to motivate the search for an analytical proof
**Solutions to a special case of the quantum [[three-body problem]] known as the [[hydrogen molecule-ion]] were found standard quantum chemistry basis sets before realizing they all lead to the same unique analytical solution in terms of a ''generalization'' of the [[Lambert W function]].  Related to this work is the isolation of a previously unknown link between gravity theory and quantum mechanics in lower dimensions (see [[Quantum gravity#The dilaton|quantum gravity]] and references therein).
**In the realm of relativistic [[N-body problem|many-bodied mechanics]], namely the [[t-symmetry|time-symmetric]] [[Wheeler–Feynman absorber theory]]: the equivalence between an advanced [[Liénard–Wiechert potential]] of particle ''j'' acting on particle ''i'' and the corresponding potential for particle ''i'' acting on particle ''j'' was demonstrated exhaustively to order <math> 1/c^{10} </math> before being proved mathematically.  The Wheeler-Feynman theory has regained interest because of [[quantum nonlocality]].
**In the realm of linear optics, verification of the series expansion of the [[Slowly varying envelope approximation|envelope]] of the electric field for [[Ultrashort pulse#Wave packet propagation in nonisotropic media|ultrashort light pulses travelling in non isotropic media]].  Previous expansions had been incomplete: the outcome revealed an extra term vindicated by experiment.
*Evaluation of [[series (mathematics)|infinite series]], [[infinite product]]s and [[integral]]s (also see [[symbolic integration]]), typically by carrying out a high precision numerical calculation, and then using an [[integer relation algorithm]] (such as the [[Inverse Symbolic Calculator]]) to find a linear combination of mathematical constants that matches this value. For example, the following identity was rediscovered by Enrico Au-Yeung, a student of [[Jonathan Borwein]] using computer search and [[PSLQ algorithm]] in 1993:<ref>{{cite journal |author=Bailey, David |title=New Math Formulas Discovered With Supercomputers |journal=NAS News |year=1997 |volume=2 |issue=24 |url=https://www.nas.nasa.gov/About/Gridpoints/PDF/nasnews_V02_N24_1997.pdf}}</ref><ref>H. F. Sandham and Martin Kneser, The American mathematical monthly, Advanced problem 4305, Vol. 57, No. 4 (Apr., 1950), pp. 267-268</ref>

::<math>
\begin{align}
\sum_{k=1}^\infty \frac{1}{k^2}\left(1+\frac{1}{2}+\frac{1}{3}+\cdots+\frac{1}{k}\right)^2 = \frac{17\pi^4}{360}.
\end{align}</math>

*Visual investigations
**In [[Indra's Pearls (book)|Indra's Pearls]], [[David Mumford]] and others investigated various properties of [[Möbius transformation]] and the [[Schottky group]] using computer generated images of the [[Group (mathematics)|groups]] which: ''furnished convincing evidence for many conjectures and lures to further exploration''.<ref>{{cite book | last = Mumford | first = David |author2=Series, Caroline |author3=Wright, David |title = Indra's Pearls: The Vision of Felix Klein | publisher = Cambridge | date = 2002 | isbn = 978-0-521-35253-6 |pages=viii}}</ref>

== Plausible but false examples ==
{{main| mathematical coincidence}}

Some plausible relations hold to a high degree of accuracy, but are still not true. One example is:

:<math>
\int_{0}^{\infty}\cos(2x)\prod_{n=1}^{\infty}\cos\left(\frac{x}{n}\right)\mathrm{d}x = \frac{\pi}{8}.</math>

The two sides of this expression actually differ after the 42nd decimal place.<ref name=bailey>David H. Bailey and Jonathan M. Borwein, [http://crd.lbl.gov/~dhbailey/dhbpapers/math-future.pdf Future Prospects for Computer-Assisted Mathematics] {{Webarchive|url=https://web.archive.org/web/20110720013038/http://crd.lbl.gov/~dhbailey/dhbpapers/math-future.pdf |date=2011-07-20 }}, December 2005</ref>

Another example is that the maximum [[Height of a polynomial|height]] (maximum absolute value of coefficients) of all the factors of ''x''<sup>''n''</sup> − 1 appears to be the same as the height of the ''n''th [[cyclotomic polynomial]]. This was shown by computer to be true for ''n'' < 10000 and was expected to be true for all ''n''. However, a larger computer search showed that this equality fails to hold for ''n'' = 14235, when the height of the ''n''th cyclotomic polynomial is 2, but maximum height of the factors is 3.<ref>The height of Φ<sub>4745</sub> is 3 and 14235 = 3 x 4745. See Sloane sequences {{OEIS2C|id=A137979}} and {{OEIS2C|id=A160338}}.</ref>

==Practitioners==
The following [[mathematician]]s and [[computer scientist]]s have made significant contributions to the field of experimental mathematics:
{{div col|colwidth=20em}}
*[[Fabrice Bellard]]
*[[David H. Bailey (mathematician)|David H. Bailey]]
*[[Jonathan Borwein]]
*[[David Epstein (mathematician)|David Epstein]]
*[[Helaman Ferguson]]
*[[Ronald Graham]]
*[[Thomas Callister Hales]]
*[[Donald Knuth]]
*[[Clement Lam]]
*[[Oren Patashnik]]
*[[Simon Plouffe]]
*[[Eric Weisstein]]
*[[Stephen Wolfram]]
*[[Doron Zeilberger]]
*[[A.J. Han Vinck]]
{{div col end}}

== See also ==
* [[Borwein integral]]
* [[Computer-aided proof]]
* ''[[Proofs and Refutations]]''
* [[Experimental Mathematics (journal)|''Experimental Mathematics'' (journal)]]
* Institute for Experimental Mathematics

==References==
{{reflist|2}}

== External links ==
* [http://www.tandfonline.com/loi/uexm20 Experimental Mathematics] (Journal)
* [http://www.cecm.sfu.ca/ Centre for Experimental and Constructive Mathematics (CECM)] at [[Simon Fraser University]]
* [https://web.archive.org/web/20071229010931/http://www.crme.soton.ac.uk/ Collaborative Group for Research in Mathematics Education] at [[University of Southampton]]
* [https://web.archive.org/web/20080422084455/http://oldweb.cecm.sfu.ca/organics/papers/bailey/paper/html/paper.html Recognizing Numerical Constants] by [[David H. Bailey (mathematician)|David H. Bailey]] and [[Simon Plouffe]]
* [https://archive.today/20121223084303/http://www.soton.ac.uk/~crime/research/expmath/ Psychology of Experimental Mathematics]
* [http://www.experimentalmath.info/ Experimental Mathematics Website] (Links and resources)
* [https://gwtokarsky.github.io/ The Great Periodic Path Hunt Website] (Links and resources)
* [https://www.nersc.gov/news-publications/news/science-news/2000/an-algorithm-for-the-ages/ An Algorithm for the Ages: PSLQ, A Better Way to Find Integer Relations] (Alternative [http://www.lbl.gov/Science-Articles/Archive/pi-algorithm.html link] {{Webarchive|url=https://web.archive.org/web/20210213222154/http://www2.lbl.gov/Science-Articles/Archive/pi-algorithm.html |date=2021-02-13 }})
* [http://www.mathrix.org/experimentalAIT/ Experimental Algorithmic Information Theory]
* [http://www.experimentalmath.info/books/expmath-probs.pdf Sample Problems of Experimental Mathematics] by [[David H. Bailey (mathematician)|David H. Bailey]] and [[Jonathan Borwein|Jonathan M. Borwein]]
* [http://crd.lbl.gov/~dhbailey/dhbpapers/tenproblems.pdf Ten Problems in Experimental Mathematics] {{Webarchive|url=https://web.archive.org/web/20110610051846/http://crd.lbl.gov/~dhbailey/dhbpapers/tenproblems.pdf |date=2011-06-10 }} by [[David H. Bailey (mathematician)|David H. Bailey]], [[Jonathan Borwein|Jonathan M. Borwein]], Vishaal Kapoor, and [[Eric W. Weisstein]]
* [http://www.iem.uni-due.de/ Institute for Experimental Mathematics] {{Webarchive|url=https://web.archive.org/web/20150210195137/http://www.iem.uni-due.de/ |date=2015-02-10 }} at [[University of Duisburg-Essen]]

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[[Category:Experimental mathematics| ]]