George Polya, one of the most remarkable mathematicians of the 20th Century, who made fundamental contributions to a wide range of topics and to the theory of problem solving, died Saturday in Palo Alto. He was 97 years old and had suffered a stroke earlier this summer.
In addition to a prodigious lifetime output of more than 250 papers, Polya in 1945 wrote “How to Solve It,” which explains in non-technical terms how to think about invention, discovery, creativity and analysis. The book has been translated into 15 languages and has sold more than 1 million copies, making it one of the most widely circulated mathematics books in history.
A native of Hungary, Polya came to this country and to Stanford University in 1940. He retired from the university in 1953 but continued doing innovative mathematics well into his 90s, an extraordinary contribution. Conventional wisdom holds that important discoveries in mathematics are made by young people.
In addition, Polya devoted himself after retirement to mathematics education, pioneering the problem-solving approach to teaching math. In 1963, the Mathematical Assn. of America gave him its distinguished service award “for his constructive influence on mathematical education in the widest sense.”
“He has given a new dimension to problem-solving by emphasizing the organic building up of elementary steps into a complex proof, and conversely, the decomposition of mathematical invention into smaller steps,” the citation read. “Problem solving a la Polya serves not only to develop mathematical skill but also teaches constructive reasoning in general.”
Polya’s work on the strategy of innovation and discovery was based on the assumption that the ability to create can be taught using rules of thumb and know-how. He summarized his global approach to problem-solving as follows:
“First: you have to understand the problem.
“Second: find the connection between the data and the unknown. You may be obliged to consider auxiliary problems if an immediate connection cannot be found. You should obtain eventually a plan of the solution.
“Third: Carry out your plan.
“Fourth: Examine the solution obtained.”
Then he broke down the advice into individual strategies:
“If you cannot solve the proposed problem, look around for an appropriate related problem.
“Narrow the condition.
“Widen the condition.
“Seek a counterexample.
“Guess and test.
“Divide and conquer.
“Change the conceptual mode.”
In pure mathematics, Polya made important discoveries in such diverse fields as probability, real and complex analysis, combinatorics, geometry, number theory and mathematical physics. The single most important work of his life was “Problems and Theorems in Analysis,” co-authored with Gabor Szego and published in 1925, when Polya was at the Swiss Federal Institute of Technology, where he taught for 26 years.
But the breadth of Polya’s contributions is evidenced by the fact that his name appears on significant concepts in a variety of fields. Probability theory contains a “Polya criterion;" complex function theory contains “Polya peaks,” “the Polya representation” and the “Polya gap theorem;" combinatorics contains the “Polya enumeration theorem” and the Polya Prize in Combinatorial Theory and Its Applications given by the Society for Industrial and Applied Mathematics.
In a paper written in 1921, Polya introduced the term “random walk,” which became a branch of probability theory that can be used to describe a coin toss or the stock market.
Polya was born in Budapest on Dec. 13, 1887. He attended the University of Budapest, intending to study law, but he found it boring and dropped it after a semester. He turned to language and literature and then to philosophy, where a professor told him that physics and mathematics would help him understand the discipline. That advice determined his life’s work. Polya got his Ph.D at Budapest in 1912.
In an interview shortly after his 90th birthday published in “Mathematical People: Profiles and Interviews” (Birkhauser Boston: 1985), Polya explained:
“I came to mathematics indirectly. I was really more interested in physics and philosophy and thought about those. It is a little shortened but not quite wrong to say: I thought I am not good enough for physics and I am too good for philosophy. Mathematics is in between.”
Praise From Colleague
In a special issue of the Journal of Graph Theory marking Polya’s 90th birthday, the mathematician Frank Harary wrote: “George Polya is not only a distinguished gentleman but a most kind and gentle man: his ebullient enthusiasm, the twinkle in his eye, his tremendous curiosity, his generosity with his time, his spry energetic walk, his warm genuine friendliness, his welcoming visitors into his home and showing them his pictures of great mathematicians he has known--these are all components of his happy personality.”
Polya was a member of the American Academy of Arts and Sciences, the National Academy of Sciences of the United States, the Academie des Sciences in Paris, the Hungarian Academy and the Academie Internationale de Philosophie des Sciences in Brussels. His collected papers were published in four volumes by the MIT Press in 1984.
He is survived by his wife of 67 years, the former Stella Vera Weber.
Memorial services are pending. The family requests that donations be made to the George Polya Memorial Book Fund, Stanford University Department of Mathematics, Stanford, Calif. 94305.