Editing George Pólya

{{Short description|Hungarian mathematician (1887–1985)}}
{{Hungarian name|Pólya György}}
{{use mdy dates|date=January 2022}}
{{Infobox scientist
| name              = George Pólya
| image             = ETH-BIB-Polya, Georg (1887-1985)-Portrait-Portr 13887-19-AL.tif
| caption           = Pólya, before 1935
| birth_date        = {{Birth date|1887|12|13}}
| birth_name        = Pólya György
| birth_place       = [[Budapest]], [[Austria-Hungary]]
| death_date        = {{Death date and age|1985|09|07|1887|12|13}}
| death_place       = [[Palo Alto, California]], U.S.
| citizenship       = Hungarian<br />Swiss (1918–1947)<br />American (1947–)<ref>[http://www.hls-dhs-dss.ch/textes/f/F31613.php George Polya in the Swiss historic lexicon.]</ref>
| fields            = [[Mathematics]]
| workplaces        = [[ETH Zürich]]<br />[[Stanford University]]
| alma_mater        = [[Eötvös Loránd University]]
| doctoral_advisor  = [[Lipót Fejér]]
| doctoral_students = {{ill|Albert Edrei|de|Albert Edrei}}<br />[[Hans Albert Einstein|Hans Einstein]]<br />[[Fritz Gassmann]]<br />[[Albert Pfluger]]<br />{{ill|Walter Saxer|de|Walter Saxer}}<br />[[James J. Stoker]]<br />[[Alice Roth]]
| known_for         = [[Pólya–Szegő inequality]]<br>''[[How to Solve It]]''<br />[[Multivariate Pólya distribution]]<br />[[Pólya conjecture]]<br />[[Pólya enumeration theorem]]<br />[[Landau–Kolmogorov inequality]]<br />[[Quadratic residue#The Pólya–Vinogradov inequality|Pólya–Vinogradov inequality]]<br />[[Pólya inequality]]<br />[[Pólya–Aeppli distribution]]<br />[[Pólya urn model]]<br />[[Fueter–Pólya theorem]]<br />[[Hilbert–Pólya conjecture]]<br />[[Jordan–Pólya number]]s
| awards            = 
}}

'''George Pólya''' ({{IPAc-en|ˈ|p|oʊ|l|j|ə}}; {{langx|hu|Pólya György}} {{IPA|hu|ˈpoːjɒ ˈɟørɟ|}}; December 13, 1887 – September 7, 1985) was a Hungarian-American [[mathematician]]. He was a professor of mathematics from 1914 to 1940 at [[ETH Zürich]] and from 1940 to 1953 at [[Stanford University]]. He made fundamental contributions to [[combinatorics]], [[number theory]], [[numerical analysis]] and [[probability theory]]. He is also noted for his work in [[heuristics]] and [[mathematics education]].<ref>{{cite book|author=Alexanderson, Gerald L.|author-link=Gerald L. Alexanderson|title=The random walks of George Pólya|year=2000|location=Washington, DC|publisher=Mathematical Association of America|isbn=9780883855287|url=https://books.google.com/books?id=OuHrR_6WEKsC}}</ref> He has been described as one of [[The Martians (scientists)|The Martians]],<ref>{{cite web|url=http://fizikaiszemle.hu/archivum/fsz9703/marsl.html|title=A marslakók legendája|last1=Marx|first1=György|authorlink1=György Marx|date=March 1997|access-date=September 11, 2022|archive-date=April 9, 2022|archive-url=https://web.archive.org/web/20220409175959/http://fizikaiszemle.hu/archivum/fsz9703/marsl.html|url-status=dead}}</ref> an informal category which included one of his most famous students at ETH Zurich, [[John von Neumann]].

== Life and works ==
Pólya was born in [[Budapest]], [[Austria-Hungary]], to Anna Deutsch and Jakab Pólya, [[History of the Jews in Hungary|Hungarian Jews]] who had converted to [[Christianity]] in 1886.<ref>{{Cite web |url=http://www.gap-system.org/~history/Biographies/Polya.html |title=Polya biography |access-date=2009-07-04 |archive-url=https://web.archive.org/web/20120302204448/http://www.gap-system.org/~history/Biographies/Polya.html |archive-date=2012-03-02 |url-status=dead }}</ref> Although his parents were religious and he was baptized into the Catholic Church upon birth, George eventually grew up to be an [[agnostic]].<ref>{{cite book|title=George Pólya: master of discovery 1887–1985|year=1993|publisher=Dale Seymour Publications|isbn=978-0-86651-611-2|author=Harold D. Taylor, Loretta Taylor|page=50|quote=Plancherel was a military man, a colonel in the Swiss army, and a devout Catholic; Pólya did not like military ceremonies or activities, and he was an agnostic who objected to hierarchical religions.}}</ref> He received a PhD under [[Lipót Fejér]] in 1912, at [[Eötvös Loránd University]].<ref>{{cite web |title=George Pólya |url=https://mathgenealogy.org/id.php?id=13648 |website=Mathematics Genealogy Project |access-date=10 January 2023}}</ref> He was a professor of mathematics from 1914 to 1940 at [[ETH Zürich]] in Switzerland and from 1940 to 1953 at [[Stanford University]]. He remained a professor emeritus at Stanford for the rest of his career, working on a range of mathematical topics, including [[series (mathematics)|series]], [[number theory]], [[mathematical analysis]], [[geometry]], [[algebra]], [[combinatorics]], and [[probability]].<ref>{{cite book|last = Roberts|first= A. Wayne|year= 1995| title = Faces of Mathematics, Third Edition|publisher = HarperCollins College Publishers| location=New York, NY USA|isbn = 0-06-501069-8|pages=479}}</ref> He was invited to speak at the [[International Congress of Mathematicians|ICM]] at Bologna in 1928,<ref>{{cite book|author=Pólya, G.|chapter=Ueber eine Eigenschaft des Gaussschen Fehlergesetzes|title=''In:'' Atti del Congresso Internazionale dei Matematici: Bologna del 3 al 10 de settembre di 1928|volume=6|pages=63–64}}</ref> at Oslo in 1936 and at Cambridge, Massachusetts, in 1950.

On September 7, 1985, Pólya died in [[Palo Alto, California]]<ref>{{Cite web|last=Dembart|first=Lee|date=1985-09-08|title=George Polya, 97, Dean of Mathematicians, Dies|url=https://www.latimes.com/archives/la-xpm-1985-09-08-mn-2892-story.html|access-date=2022-01-20|website=Los Angeles Times|language=en-US}}</ref> due to complications of a stroke he suffered during that summer.

== Heuristics ==
Early in his career, Pólya wrote with [[Gábor Szegő]] two influential problem books, ''[[Problems and Theorems in Analysis]]'' (''I: Series, Integral Calculus, Theory of Functions'' and ''II: Theory of Functions. Zeros. Polynomials. Determinants. Number Theory. Geometry''). Later in his career, he spent considerable effort to identify systematic methods of problem-solving to further discovery and invention in mathematics for students, teachers, and researchers.<ref>{{cite journal|doi = 10.2307/2690409|last = Schoenfeld|first =Alan H.|title = Pólya, Problem Solving, and Education|journal = Mathematics Magazine|volume = 60|issue = 5|date=December 1987|pages = 283–291|publisher = Mathematics Magazine, Vol. 60, No. 5|jstor = 2690409|hdl = 10983/26321|hdl-access = free}}</ref> He wrote five books on the subject: ''[[How to Solve It]]'', ''[[Mathematics and plausible reasoning|Mathematics and Plausible Reasoning]]'' (''Volume I: Induction and Analogy in Mathematics'', and ''Volume II: Patterns of Plausible Inference''), and ''Mathematical Discovery: On Understanding, Learning, and Teaching Problem Solving'' (volumes 1 and 2).

In ''[[How to Solve It]]'', Pólya provides general [[heuristics]] for solving a gamut of problems, including both mathematical and non-mathematical problems. The book includes advice for teaching students of mathematics and a mini-encyclopedia of heuristic terms. It was translated into several languages and has sold over a million copies. The book is still used in [[mathematical education]]. [[Douglas Lenat]]'s [[Automated Mathematician]] and [[Eurisko]] artificial intelligence programs were inspired by Pólya's work.

In addition to his works directly addressing problem solving, Pólya wrote another short book called ''Mathematical Methods in Science'', based on a 1963 work supported by the National Science Foundation edited by [[Leon Bowden]] and published by the Mathematical Association of America (MAA) in 1977. As Pólya notes in the preface, Bowden carefully followed a tape recording of a course Pólya gave several times at Stanford in order to put the book together. Pólya notes in the preface "that the following pages will be useful, yet they should not be regarded as a finished expression."

== Legacy ==
There are three prizes named after Pólya, causing occasional confusion of one for another. In 1969 the [[Society for Industrial and Applied Mathematics]] (SIAM) established the [[Pólya Prize (SIAM)|George Pólya Prize]], given alternately in two categories for "a notable application of combinatorial theory" and for "a notable contribution in another area of interest to George Pólya."<ref>[http://www.siam.org/prizes/sponsored/polya.php Society for Industrial and Applied Mathematics George Pólya Prize]</ref>

In 1976 the [[Mathematical Association of America]] (MAA) established the [[George Pólya Award]] "for articles of expository excellence" published in the ''[[College Mathematics Journal]]''.<ref>[http://www.maa.org/Awards/polya.html Mathematical Association of America George Pólya Award]</ref> In 1987 the [[London Mathematical Society]] (LMS) established the [[Pólya Prize (LMS)|Pólya Prize]] for "outstanding creativity in, imaginative exposition of, or distinguished contribution to, mathematics within the United Kingdom."<ref>{{Cite web |url=http://www.lms.ac.uk/activities/prizes_com/index.html |title=London Mathematical Society Polya Prize |access-date=2009-10-09 |archive-url=https://web.archive.org/web/20100510090317/http://www.lms.ac.uk/activities/prizes_com/index.html |archive-date=2010-05-10 |url-status=dead }}</ref> In 1991, the MAA established the George Pólya Lectureship series.<ref name="MAA-Pólya Lectures">{{cite web | title=Pólya Lectures | website=Mathematical Association of America | url=https://www.maa.org/programs-and-communities/member-communities/maa-awards/lecture-awards/p%C3%B3lya-lecturers | access-date=2022-11-15}}</ref>

[[Stanford University]] has a Polya Hall named in his honor.<ref>{{Cite web |url=http://maps.stanford.edu/ada/building-ada.cfm?FACIL_ID=14-160 |title=POLYA HALL, 14-160 |access-date=2020-04-03}}</ref>

== Selected publications ==
[[File:George Polya and Gabor Szego in Berlin.jpg|thumb|Pólya and Gábor Szegő in Berlin, delivering the manuscript of ''[[Aufgaben und Lehrsätze aus der Analysis]]'' to Springer.]]

=== Books ===
* ''[[Aufgaben und Lehrsätze aus der Analysis]]'', 1st edn. 1925.<ref>{{cite journal|author=Tamarkin, J. D.|author-link=Jacob Tamarkin|title=Review: ''Aufgaben und Lehrsätze aus der Analysis'', vols. 1 & 2, by George Pólya and Gábor Szegő|journal=Bull. Amer. Math. Soc.|year=1928|volume=34|issue=2|pages=233–234|url=https://www.ams.org/journals/bull/1928-34-02/S0002-9904-1928-04522-6/S0002-9904-1928-04522-6.pdf |archive-url=https://web.archive.org/web/20150402094843/http://www.ams.org/journals/bull/1928-34-02/S0002-9904-1928-04522-6/S0002-9904-1928-04522-6.pdf |archive-date=2015-04-02 |url-status=live|doi=10.1090/s0002-9904-1928-04522-6|doi-access=free}}</ref> ("Problems and theorems in analysis“). Springer, Berlin 1975 (with [[Gábor Szegő]]).
# ''Reihen''. 1975, 4th edn., {{ISBN|3-540-04874-X}}.
# ''Funktionentheorie, Nullstellen, Polynome, Determinanten, Zahlentheorie''. 1975, 4th edn., {{ISBN|3-540-05456-1}}.
* ''Mathematik und plausibles Schliessen<!-- sic -->''.  Birkhäuser, Basel 1988,
# ''Induktion und Analogie in der Mathematik'', 3rd edn., {{ISBN|3-7643-1986-0}} (Wissenschaft und Kultur; 14).
# ''Typen und Strukturen plausibler Folgerung'', 2nd edn., {{ISBN|3-7643-0715-3}} (Wissenschaft und Kultur; 15).
*  –  English translation: ''[[Mathematics and Plausible Reasoning]]'', Princeton University Press 1954, 2 volumes (Vol. 1: ''Induction and Analogy in Mathematics'', Vol. 2: ''Patterns of Plausible Inference'')
* ''Schule des Denkens. Vom Lösen mathematischer Probleme'' ("How to solve it"). 4th edn. Francke Verlag, Tübingen 1995, {{ISBN|3-7720-0608-6}} (Sammlung Dalp).
*  –  English translation: ''[[How to Solve It]]'', Princeton University Press 2004 (with foreword by [[John Horton Conway]] and added exercises)
* ''Vom Lösen mathematischer Aufgaben''. 2nd edn. Birkhäuser, Basel 1983, {{ISBN|3-7643-0298-4}} (Wissenschaft und Kultur; 21).
*  –  English translation: ''Mathematical Discovery: On Understanding, Learning and Teaching Problem Solving'', 2 volumes, Wiley 1962 (published in one vol. 1981)
* ''Collected Papers'', 4 volumes, MIT Press 1974 (ed. Ralph P. Boas). Vol. 1: Singularities of Analytic Functions, Vol. 2: Location of Zeros, Vol. 3: Analysis, Vol. 4: Probability, Combinatorics
* with R. C. Read: ''Combinatorial enumeration of groups, graphs, and chemical compounds'', Springer Verlag 1987 (English translation of ''Kombinatorische Anzahlbestimmungen für Gruppen, Graphen und chemische Verbindungen'', Acta Mathematica, vol. 68, 1937, pp.&nbsp;145–254) {{ISBN|978-0387964133}}
* with [[Godfrey Harold Hardy]]: [[John Edensor Littlewood]] ''Inequalities'', Cambridge University Press 1934
* [https://archive.org/details/MATHMETODSSCIENCE ''Mathematical Methods in Science''], MAA, Washington D. C. 1977 (ed. Leon Bowden) {{ISBN|0-88385-626-3}}
* with Gordon Latta: ''Complex Variables'', Wiley 1974
* with [[Robert E. Tarjan]], Donald R. Woods: ''Notes on introductory combinatorics'', Birkhäuser 1983 {{ISBN|978-3-7643-3170-2}}
* with Jeremy Kilpatrick: ''The Stanford mathematics problem book: with hints and solutions'', New York: Teachers College Press 1974
* with several co-authors: ''Applied combinatorial mathematics'', Wiley 1964 (ed. [[Edwin F. Beckenbach]])
* with Gábor Szegő: [https://books.google.com/books?id=HWc8gFW2XXgC ''Isoperimetric inequalities in mathematical physics''], Princeton, Annals of Mathematical Studies 27, 1951

=== Articles ===
*{{cite journal|title=On the mean-value theorem corresponding to a given linear homogeneous differential equation|journal=Trans. Amer. Math. Soc.|year=1922|volume=24|issue=4|pages=312–324|mr=1501228|doi=10.1090/s0002-9947-1922-1501228-5|doi-access=free|last1=Pólya|first1=G.}}
*{{cite journal|last1=Pólya|first1=G.|title=Untersuchungen Über Lücken Und Singularitäten Von Potenzreihen|journal=Ann. Math.|year=1933|volume=34|issue=4|pages=731–77|doi=10.2307/1968698|doi-access=free|jstor=1968698}}
*{{cite journal|title=On Functions Whose Derivatives Do Not Vanish in a Given Interval|journal=Proc Natl Acad Sci U S A|year=1941|volume=27|issue=4|pages=216–218|pmc=1078308|doi=10.1073/pnas.27.4.216|pmid=16578010|last1=Polya|first1=G.|bibcode=1941PNAS...27..216P|doi-access=free}}
*{{cite journal|title=Sur l'existence de fonctions entières satisfaisant à certaines conditions linéaires|journal=Trans. Amer. Math. Soc.|year=1941|volume=50|issue=1|pages=129–139|mr=0004304|doi=10.2307/1989913|jstor=1989913|last1=Polya|first1=G.}}
*with [[Ralph P. Boas, Jr.]]: {{cite journal|title=Generalizations of Completely Convex Functions|journal=Proc Natl Acad Sci U S A|year=1941|volume=27|issue=6|pages=323–325|pmc=1078330|pmid=16588467|doi=10.1073/pnas.27.6.323|last1=Boas|first1=R. P.|last2=Polya|first2=G.|bibcode=1941PNAS...27..323B|doi-access=free}}<!-- the pmc index has "Generalization" instead of "Generalizations" -->
*{{cite journal|title=On converse gap theorems|journal=Trans. Amer. Math. Soc.|year=1942|issue=1|volume=52|pages=65–71|mr=0006577|doi=10.1090/s0002-9947-1942-0006577-0|doi-access=free|last1=Pólya|first1=George}}
*with [[Norbert Wiener]]: {{cite journal|title=On the oscillation of the derivatives of a periodic function|journal=Trans. Amer. Math. Soc.|year=1942|volume=52|issue=2|pages=249–256|mr=0007169|doi=10.1090/s0002-9947-1942-0007169-x|doi-access=free|last1=Pólya|first1=George|last2=Wiener|first2=Norbert}}
*{{cite journal|title=On the zeros of a derivative of a function and its analytic character|journal=Bull. Amer. Math. Soc.|year=1943|volume=49, Part 1|issue=3|pages=178–191|mr=0007781|doi=10.1090/s0002-9904-1943-07853-6|doi-access=free|url=https://projecteuclid.org/journals/bulletin-of-the-american-mathematical-society/volume-49/issue-3.P1/On-the-zeros-of-the-derivatives-of-a-function-and/bams/1183505036.pdf}}
*{{cite journal|title=A Minimum Problem About the Motion of a Solid Through a Fluid|journal=Proc Natl Acad Sci U S A|year=1947|volume=33|issue=7|pages=218–221|pmc=1079030|pmid=16588747|doi=10.1073/pnas.33.7.218|last1=Polya|first1=G.|bibcode=1947PNAS...33..218P|doi-access=free}}
*{{cite journal|title=Remark on Weyl's Note "Inequalities Between the Two Kinds of Eigenvalues of a Linear Transformation|journal=Proc Natl Acad Sci U S A|year=1950|volume=36|issue=1|pages=49–51|pmc=1063130|doi=10.1073/pnas.36.1.49|pmid=16588947|last1=Polya|first1=G.|bibcode=1950PNAS...36...49P|doi-access=free}}

== See also ==
* [[Integer-valued polynomial]]
* [[Laguerre–Pólya class]]
* [[Landau–Kolmogorov inequality]]
* [[Multivariate Pólya distribution]]
* [[Characterization of probability distributions|Pólya's characterization theorem]]
* [[Hermite class|Pólya class]]
* [[Pólya conjecture]]
* [[Polya distribution]]
* [[Pólya enumeration theorem]]
* [[Quadratic residue#The Pólya–Vinogradov inequality|Pólya–Vinogradov inequality]]
* [[Pólya inequality]]
* [[Pólya urn model]]
* [[Analytic continuation#Pólya's theorem|Pólya's theorem]]
* [[All horses are the same color|Pólya's proof that there is no "horse of a different color"]]
* [[Wallpaper group]]
*[[The Martians (scientists)]]

== References ==
{{reflist}}

== External links ==
{{Wikiquote}}
* [http://www.maa.org/Awards/polya.html The George Pólya Award]
* {{MacTutor Biography|id=Polya}}
* {{MathGenealogy |id=13648}}
* [https://books.google.com/books?id=-BoU6EhyU84C&q=Problems+and+theorems+in+analysis George Pólya, Gábor Szegö, '''''Problems and theorems in analysis''''' (1998)]
* {{webarchive |url=https://web.archive.org/web/20090728190041/http://geocities.com/polyapower/ |date=July 28, 2009 |title=PolyaPower—an introduction to Polya's Heuristics }}
* [https://web.archive.org/web/20090214210940/http://wik.ed.uiuc.edu/index.php/P%C3%B3lya%2C_George George Pólya on UIUC's WikEd]
* [https://web.archive.org/web/20060916224323/http://histsoc.stanford.edu/pdfmem/PolyaG.pdf Memorial Resolution]
* {{Biographical Memoirs|polya-george}}
* {{Vimeo|id=48768091|title="Polya Guessing"}}

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{{DEFAULTSORT:Polya, George}}
[[Category:1887 births]]
[[Category:1985 deaths]]
[[Category:20th-century Hungarian mathematicians]]
[[Category:Mathematics popularizers]]
[[Category:American agnostics]]
[[Category:American people of Hungarian-Jewish descent]]
[[Category:Hungarian Jews]]
[[Category:American statisticians]]
[[Category:Hungarian emigrants to Switzerland]]
[[Category:Combinatorialists]]
[[Category:Academic staff of ETH Zurich]]
[[Category:Hungarian agnostics]]
[[Category:Hungarian statisticians]]
[[Category:Complex analysts]]
[[Category:Mathematical analysts]]
[[Category:Members of the United States National Academy of Sciences]]
[[Category:Mathematicians from Budapest]]
[[Category:Swiss emigrants to the United States]]
[[Category:Stanford University Department of Mathematics faculty]]