# George Pólya

# George Pólya

George Pólya | |
---|---|

Born | (1887-12-13)December 13, 1887 Budapest, Austria-Hungary |

Died | September 7, 1985(1985-09-07)(aged 97) Palo Alto, California |

Nationality | Hungarian (−1918) Swiss (1918–1947) American (1947–his death) ^{[1]} |

Alma mater | Eötvös Loránd University |

Known for | *Pólya–Szegő inequalityHow to Solve ItMultivariate Pólya distribution Pólya conjecture Pólya enumeration theorem Landau–Kolmogorov inequality Pólya–Vinogradov inequality Pólya inequality Pólya–Aeppli distribution Pólya urn model Fueter–Pólya theorem Hilbert–Pólya conjecture* |

Scientific career | |

Fields | Mathematics |

Institutions | ETH Zürich Stanford University |

Doctoral advisor | Lipót Fejér |

Doctoral students | Albert Edrei Hans Einstein Fritz Gassmann Albert Pfluger James J. Stoker Alice Roth |

Influenced | Imre Lakatos |

**George Pólya** (/ˈpoʊljə/; Hungarian: *Pólya György* [ˈpoːjɒ ˈɟørɟ]) (December 13, 1887 – September 7, 1985) was a Hungarian 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.^{[2]} He has been described as one of The Martians.^{[3]}

George Pólya | |
---|---|

Born | (1887-12-13)December 13, 1887 Budapest, Austria-Hungary |

Died | September 7, 1985(1985-09-07)(aged 97) Palo Alto, California |

Nationality | Hungarian (−1918) Swiss (1918–1947) American (1947–his death) ^{[1]} |

Alma mater | Eötvös Loránd University |

Known for | *Pólya–Szegő inequalityHow to Solve ItMultivariate Pólya distribution Pólya conjecture Pólya enumeration theorem Landau–Kolmogorov inequality Pólya–Vinogradov inequality Pólya inequality Pólya–Aeppli distribution Pólya urn model Fueter–Pólya theorem Hilbert–Pólya conjecture* |

Scientific career | |

Fields | Mathematics |

Institutions | ETH Zürich Stanford University |

Doctoral advisor | Lipót Fejér |

Doctoral students | Albert Edrei Hans Einstein Fritz Gassmann Albert Pfluger James J. Stoker Alice Roth |

Influenced | Imre Lakatos |

Life and works

Pólya was born in Budapest, Austria-Hungary to Anna Deutsch and Jakab Pólya, Hungarian Jews who had converted to the Roman Catholic faith in 1886.^{[4]} Although his parents were religious and he was baptized into the Roman Catholic Church, George Pólya grew up to be an agnostic.^{[5]} 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 Stanford Professor Emeritus for the rest of his life and career. He worked on a range of mathematical topics, including series, number theory, mathematical analysis, geometry, algebra, combinatorics, and probability.^{[6]} He was an Invited Speaker of the ICM in 1928 at Bologna,^{[7]} in 1936 at Oslo, and in 1950 at Cambridge, Massachusetts.

He died in Palo Alto, California, United States.

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.^{[8]} He wrote five books on the subject: *How to Solve It*, *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. Russian physicist Zhores I. Alfyorov (Nobel laureate in 2000) praised it, noting that he was a fan. The Australian-American mathematician Terence Tao used the book to prepare for the International Mathematical Olympiad. 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 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."^{[9]} In 1976 the Mathematical Association of America (MAA) established the George Pólya Award "for articles of expository excellence" published in the *College Mathematics Journal*.^{[10]} In 1987 the London Mathematical Society (LMS) established the Pólya Prize for "outstanding creativity in, imaginative exposition of, or distinguished contribution to, mathematics within the United Kingdom."^{[11]}

A mathematics center has been named in Pólya's honor at the University of Idaho in Moscow, Idaho. The mathematics center focuses mainly on tutoring students in the subjects of algebra and calculus.^{[12]}

Selected publications

Books

*Aufgaben und Lehrsätze aus der Analysis*, 1st edn. 1925.^{[13]}("Problems and theorems in analysis“). Springer, Berlin 1975 (with Gábor Szegő).

*Mathematik und plausibles Schliessen*. Birkhäuser, Basel 1988,

– 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, Combinatoricswith 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. 145–254)with Godfrey Harold Hardy: John Edensor Littlewood

*Inequalities*, Cambridge University Press 1934*Mathematical methods in Science*^{[51]}, MAA, Washington D. C. 1977 (ed. Leon Bowden)with Gordon Latta:

*Complex Variables*, Wiley 1974with Robert E. Tarjan, Donald R. Woods:

*Notes on introductory combinatorics*, Birkhäuser 1983with Jeremy Kilpatrick:

*The Stanford mathematics problem book: with hints and solutions*, New York: Teachers College Press 1974with several co-authors:

*Applied combinatorical mathematics*, Wiley 1964 (ed. Edwin F. Beckenbach)with Gábor Szegő:

*Isoperimetric inequalities in mathematical physics*^{[52]}, Princeton, Annals of Mathematical Studies 27, 1951

Articles

"On the mean-value theorem corresponding to a given linear homogeneous differential equation".

*Trans. Amer. Math. Soc*.**24**(4): 312–324. 1922. doi:10.1090/s0002-9947-1922-1501228-5^{[53]}. MR 1501228^{[54]}."On Functions Whose Derivatives Do Not Vanish in a Given Interval"

^{[55]}.*Proc Natl Acad Sci U S A*.**27**(4): 216–218. 1941. doi:10.1073/pnas.27.4.216^{[56]}. PMC 1078308^{[57]}. PMID 16578010^{[58]}."Sur l'existence de fonctions entières satisfaisant à certaines conditions linéaires".

*Trans. Amer. Math. Soc*.**50**(1): 129–139. 1941. doi:10.2307/1989913^{[59]}. MR 0004304^{[60]}.with Ralph P. Boas, Jr.: "Generalizations of Completely Convex Functions"

^{[61]}.*Proc Natl Acad Sci U S A*.**27**(6): 323–325. 1941. doi:10.1073/pnas.27.6.323^{[62]}. PMC 1078330^{[63]}. PMID 16588467^{[64]}."On converse gap theorems".

*Trans. Amer. Math. Soc*.**52**(1): 65–71. 1942. doi:10.1090/s0002-9947-1942-0006577-0^{[65]}. MR 0006577^{[66]}.with Norbert Wiener: "On the oscillation of the derivatives of a periodic function".

*Trans. Amer. Math. Soc*.**52**(2): 249–256. 1942. doi:10.1090/s0002-9947-1942-0007169-x^{[67]}. MR 0007169^{[68]}."On the zeros of a derivative of a function and its analytic character".

*Bull. Amer. Math. Soc*. 49, Part 1 (3): 178–191. 1943. doi:10.1090/s0002-9904-1943-07853-6^{[69]}. MR 0007781^{[70]}."A Minimum Problem About the Motion of a Solid Through a Fluid"

^{[71]}.*Proc Natl Acad Sci U S A*.**33**(7): 218–221. 1947. doi:10.1073/pnas.33.7.218^{[72]}. PMC 1079030^{[73]}. PMID 16588747^{[74]}."Remark on Weyl's Note "Inequalities Between the Two Kinds of Eigenvalues of a Linear Transformation"

^{[75]}.*Proc Natl Acad Sci U S A*.**36**(1): 49–51. 1950. doi:10.1073/pnas.36.1.49^{[76]}. PMC 1063130^{[77]}. PMID 16588947^{[78]}.

See also

Landau–Kolmogorov inequality

Multivariate Pólya distribution

Pólya conjecture

Polya distribution

Pólya enumeration theorem

Pólya–Vinogradov inequality

Pólya inequality

Pólya urn model

Pólya's proof that there is no "horse of a different color"

The Martians (scientists)

## References

*The random walks of George Pólya*. Washington, DC: Mathematical Association of America.

*George Pólya: master of discovery 1887–1985*. Dale Seymour Publications. p. 50. ISBN 978-0-86651-611-2. 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.

*Faces of Mathematics, Third Edition*. New York, NY USA: HarperCollins College Publishers. p. 479. ISBN 0-06-501069-8.

*Atti del Congresso Internazionale dei Matematici: Bologna del 3 al 10 de settembre di 1928*. vol. 6. pp. 63–64.

*Mathematics Magazine*. Mathematics Magazine, Vol. 60, No. 5.

**60**(5): 283–291. doi:10.2307/2690409. JSTOR 2690409.

*Aufgaben und Lehrsätze aus der Analysis*, vols. 1 & 2, by George Pólya and Gábor Szegő" (PDF).

*Bull. Amer. Math. Soc*.

**34**(2): 233–234. doi:10.1090/s0002-9904-1928-04522-6.

*Isoperimetric inequalities in mathematical physics*