George Pólya (; hu, Pólya György, ; December 13, 1887 – September 7, 1985) was a Hungarian

mathematician A mathematician is someone who uses an extensive knowledge of mathematics in their work, typically to solve mathematical problems. Mathematicians are concerned with numbers, data, quantity, structure, space, models, and change. History On ...

. He was a professor of mathematics from 1914 to 1940 at

ETH Zürich (colloquially) , former_name = eidgenössische polytechnische Schule , image = ETHZ.JPG , image_size = , established = , type = Public , budget = CHF 1.896 billion (2021) , rector = Günther Dissertori , president = Joël Mesot , ac ...

and from 1940 to 1953 at

Stanford University Stanford University, officially Leland Stanford Junior University, is a private research university in Stanford, California. The campus occupies , among the largest in the United States, and enrolls over 17,000 students. Stanford is consider ...

. He made fundamental contributions to

combinatorics Combinatorics is an area of mathematics primarily concerned with counting, both as a means and an end in obtaining results, and certain properties of finite structures. It is closely related to many other areas of mathematics and has many appl ...

number theory Number theory (or arithmetic or higher arithmetic in older usage) is a branch of pure mathematics devoted primarily to the study of the integers and arithmetic function, integer-valued functions. German mathematician Carl Friedrich Gauss (1777� ...

numerical analysis Numerical analysis is the study of algorithms that use numerical approximation (as opposed to symbolic computation, symbolic manipulations) for the problems of mathematical analysis (as distinguished from discrete mathematics). It is the study of ...

and

probability theory Probability theory is the branch of mathematics concerned with probability. Although there are several different probability interpretations, probability theory treats the concept in a rigorous mathematical manner by expressing it through a set o ...

. He is also noted for his work in

heuristics A heuristic (; ), or heuristic technique, is any approach to problem solving or self-discovery that employs a practical method that is not guaranteed to be optimal, perfect, or rational, but is nevertheless sufficient for reaching an immediate, ...

and

mathematics education In contemporary education, mathematics education, known in Europe as the didactics or pedagogy of mathematics – is the practice of teaching, learning and carrying out scholarly research into the transfer of mathematical knowledge. Although re ...

. He has been described as one of The Martians, an informal category which included one of his most famous students at ETH Zurich,

John Von Neumann John von Neumann (; hu, Neumann János Lajos, ; December 28, 1903 – February 8, 1957) was a Hungarian-American mathematician, physicist, computer scientist, engineer and polymath. He was regarded as having perhaps the widest cove ...

Life and works

Pólya was born in

Budapest Budapest (, ; ) is the capital and most populous city of Hungary. It is the ninth-largest city in the European Union by population within city limits and the second-largest city on the Danube river; the city has an estimated population ...

Austria-Hungary Austria-Hungary, often referred to as the Austro-Hungarian Empire,, the Dual Monarchy, or Austria, was a constitutional monarchy and great power in Central Europe between 1867 and 1918. It was formed with the Austro-Hungarian Compromise of ...

, to Anna Deutsch and Jakab Pólya, Hungarian Jews who had converted to

Christianity Christianity is an Abrahamic monotheistic religion based on the life and teachings of Jesus of Nazareth. It is the world's largest and most widespread religion with roughly 2.38 billion followers representing one-third of the global pop ...

in 1886. Although his parents were religious and he was baptized into the Catholic Church upon birth, George eventually grew up to be an agnostic. He was a professor of mathematics from 1914 to 1940 at

in Switzerland and from 1940 to 1953 at

. He remained a Professor Emeritus at Stanford for the rest of his career, working on a range of mathematical topics, including

series Series may refer to: People with the name * Caroline Series (born 1951), English mathematician, daughter of George Series * George Series (1920–1995), English physicist Arts, entertainment, and media Music * Series, the ordered sets used in ...

mathematical analysis Analysis is the branch of mathematics dealing with continuous functions, limit (mathematics), limits, and related theories, such as Derivative, differentiation, Integral, integration, measure (mathematics), measure, infinite sequences, series (m ...

geometry Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is c ...

algebra Algebra () is one of the broad areas of mathematics. Roughly speaking, algebra is the study of mathematical symbols and the rules for manipulating these symbols in formulas; it is a unifying thread of almost all of mathematics. Elementary a ...

, and

probability Probability is the branch of mathematics concerning numerical descriptions of how likely an Event (probability theory), event is to occur, or how likely it is that a proposition is true. The probability of an event is a number between 0 and ...

. He was invited to speak at the ICM at Bologna in 1928, at Oslo in 1936 and at Cambridge, Massachusetts, in 1950. He died on September 7, 1985, in

Palo Alto Palo Alto (; Spanish for "tall stick") is a charter city in the northwestern corner of Santa Clara County, California, United States, in the San Francisco Bay Area, named after a coastal redwood tree known as El Palo Alto. The city was estab ...

, California,

United States The United States of America (U.S.A. or USA), commonly known as the United States (U.S. or US) or America, is a country primarily located in North America. It consists of 50 states, a federal district, five major unincorporated territorie ...

Heuristics

Early in his career, Pólya wrote with

Gábor Szegő Gábor Szegő () (January 20, 1895 – August 7, 1985) was a Hungarian-American mathematician. He was one of the foremost mathematical analysts of his generation and made fundamental contributions to the theory of orthogonal polynomials and ...

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. He wrote five books on the subject: ''

How to Solve It ''How to Solve It'' (1945) is a small volume by mathematician George Pólya describing methods of problem solving. Four principles ''How to Solve It'' suggests the following steps when solving a mathematical problem: # First, you have to ''und ...

'', '' 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 ''

'', Pólya provides general

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 Douglas Bruce Lenat (born 1950) is the CEO of Cycorp, Inc. of Austin, Texas, and has been a prominent researcher in artificial intelligence; he was awarded the biannual IJCAI Computers and Thought Award in 1976 for creating the machine learning p ...

'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 The Society for Industrial and Applied Mathematics (SIAM) has three prizes named after George Pólya: the George Pólya Prize for Mathematical Exposition, established in 2013; the George Pólya Prize in Applied Combinatorics, established in 1969 ...

, 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." In 1976 the Mathematical Association of America (MAA) established the

George Pólya Award The George Pólya Award is presented annually by the Mathematical Association of America (MAA) for articles of expository excellence that have been published in The College Mathematics Journal. The award was established in 1976 and up to two aw ...

"for articles of expository excellence" published in the ''

College Mathematics Journal The ''College Mathematics Journal'' is an expository magazine aimed at teachers of college mathematics, particular those teaching the first two years. It is published by Taylor & Francis on behalf of the Mathematical Association of America and is ...

''. 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." In 1991, the MAA established the George Pólya Lectureship series.

has a Polya Hall named in his honor.

Selected publications

Books

* ''Aufgaben und Lehrsätze aus der Analysis'', 1st edn. 1925. ("Problems and theorems in analysis“). Springer, Berlin 1975 (with

). # ''Reihen''. 1975, 4th edn., . # ''Funktionentheorie, Nullstellen, Polynome, Determinanten, Zahlentheorie''. 1975, 4th edn., . * ''Mathematik und plausibles Schliessen''. Birkhäuser, Basel 1988, # ''Induktion und Analogie in der Mathematik'', 3rd edn., (Wissenschaft und Kultur; 14). # ''Typen und Strukturen plausibler Folgerung'', 2nd edn., (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, (Sammlung Dalp). * – English translation: ''

'', Princeton University Press 2004 (with foreword by

John Horton Conway John Horton Conway (26 December 1937 – 11 April 2020) was an English mathematician active in the theory of finite groups, knot theory, number theory, combinatorial game theory and coding theory. He also made contributions to many branches ...

and added exercises) * ''Vom Lösen mathematischer Aufgaben''. 2nd edn. Birkhäuser, Basel 1983, (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. 145–254) * with

Godfrey Harold Hardy Godfrey Harold Hardy (7 February 1877 – 1 December 1947) was an English mathematician, known for his achievements in number theory and mathematical analysis. In biology, he is known for the Hardy–Weinberg principle, a basic principle of pop ...

John Edensor Littlewood John Edensor Littlewood (9 June 1885 – 6 September 1977) was a British mathematician. He worked on topics relating to analysis, number theory, and differential equations, and had lengthy collaborations with G. H. Hardy, Srinivasa Ramanu ...

''Inequalities'', Cambridge University Press 1934
''Mathematical Methods in Science''
MAA, Washington D. C. 1977 (ed. Leon Bowden) * with Gordon Latta: ''Complex Variables'', Wiley 1974 * with

Robert E. Tarjan Robert Endre Tarjan (born April 30, 1948) is an American computer scientist and mathematician. He is the discoverer of several graph algorithms, including Tarjan's off-line lowest common ancestors algorithm, and co-inventor of both splay trees ...

, Donald R. Woods: ''Notes on introductory combinatorics'', Birkhäuser 1983 * 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
''Isoperimetric inequalities in mathematical physics''
Princeton, Annals of Mathematical Studies 27, 1951

Articles

* * * *with

Ralph P. Boas, Jr. Ralph Philip Boas Jr. (August 8, 1912 – July 25, 1992) was a mathematician, teacher, and journal editor. He wrote over 200 papers, mainly in the fields of real analysis, real and complex analysis.. Biography He was born in Walla Walla, Washi ...

: * *with Norbert Wiener: * * *

References

External links

The George Pólya Award
* *
George Pólya, Gábor Szegö, ''Problems and theorems in analysis'' (1998)
*
George Pólya on UIUC's WikEd

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