A short life of the author
George Pólya (13 December 1887 – 7 September 1985) was a Hungarian-American mathematician whose contributions spanned combinatorics, number theory, probability, and analysis, but whose greatest influence came through his writings on mathematical thinking and problem-solving. How to Solve It (1945) — a deceptively simple book about how to approach mathematical problems — has sold over a million copies, been translated into more than twenty languages, and become one of the most widely read books on mathematics ever written. Its influence extends far beyond mathematics into computer science, engineering education, and cognitive psychology.
Life
Pólya was born in Budapest into an assimilated Jewish family. He studied at the University of Budapest, then in Vienna and Göttingen, and received his doctorate from Budapest in 1912. He spent the years 1914–1940 at ETH Zürich, where he became one of the most productive mathematicians in Europe, publishing over 250 papers.
In 1940, fleeing the threat of Nazi persecution, he emigrated to the United States, where he joined the faculty at Stanford University. He remained at Stanford for the rest of his career, becoming increasingly focused on mathematics education and the psychology of mathematical discovery. He was still actively lecturing and writing well into his nineties.
How to Solve It (1945)
The book presents Pólya’s four-step method for solving problems: (1) Understand the problem — what is the unknown? what are the data? what are the conditions? (2) Devise a plan — find the connection between the data and the unknown; consider analogous problems, special cases, generalisations. (3) Carry out the plan — check each step. (4) Look back — can you check the result? can you derive it differently? can you use it for another problem?
The method is presented not as a rigid algorithm but as a set of heuristics — mental habits and questions that guide the problem-solver toward insight. Pólya filled the book with examples drawn from elementary mathematics (geometry, algebra, arithmetic) that demonstrate the heuristics in action.
The book’s genius lies in its attention to the process of thinking rather than the content of mathematics. Pólya treated mathematical problem-solving as a skill that could be taught — not through memorisation of techniques but through the cultivation of what he called “plausible reasoning”: the art of making educated guesses, drawing analogies, and working backward from the desired conclusion.
How to Solve It influenced the development of artificial intelligence (Allen Newell and Herbert Simon cited Pólya’s heuristics as a model for machine problem-solving), cognitive psychology (the study of expert thinking), and engineering education (the “design thinking” movement owes much to Pólya’s emphasis on problem understanding before solution).
Mathematics and Plausible Reasoning (1954)
Pólya’s two-volume masterwork — Induction and Analogy in Mathematics and Patterns of Plausible Inference — is a deeper, more technically demanding exploration of the ideas introduced in How to Solve It. The central argument is that mathematics advances not through formal deduction alone but through patterns of plausible reasoning — analogy, generalisation, specialisation, and the testing of conjectures — that are rational but not logically certain.
The books are filled with examples from the history of mathematics, showing how Euler, Laplace, Bernoulli, and others actually discovered their results — not through the polished proofs they published but through the messy, intuitive, analogical reasoning that preceded those proofs.
Mathematical Discovery (1962–1965)
A two-volume work aimed at a broader audience than Mathematics and Plausible Reasoning, Mathematical Discovery explores the psychology of invention in mathematics through problems drawn from geometry and combinatorics. It is the most accessible of Pólya’s pedagogical works after How to Solve It.
Research Mathematics
Pólya’s contributions to pure mathematics were substantial. The Pólya enumeration theorem (in combinatorics) is fundamental to the counting of symmetrical structures. His work on random walks (“the drunkard’s walk”) established foundational results in probability theory. His collaboration with Gábor Szegő on Problems and Theorems in Analysis (1925) produced one of the great problem books in mathematics — a work that trained generations of analysts.
Critical Standing
Pólya occupies a unique position in mathematics: revered as both a research mathematician and a pedagogue. How to Solve It is one of those rare books that has changed how an entire discipline thinks about itself — it made the process of mathematical thinking a legitimate subject of study and teaching. His influence on computer science is particularly notable: Donald Knuth, who was Pólya’s colleague at Stanford, credited him as a formative influence, and the heuristic problem-solving paradigm that Pólya articulated is visible in everything from algorithm design to the structure of programming interviews at major technology companies. The “understand, plan, execute, review” framework has been adopted — often without attribution — by software engineering, design thinking, and business strategy curricula worldwide.
Pólya’s pedagogy also anticipated the “growth mindset” movement by decades. His insistence that mathematical ability is not an innate gift but a skill developed through practice, reflection, and the cultivation of good mental habits was radical in an era that treated mathematical talent as a mysterious endowment. He was a gifted lecturer who continued teaching into his nineties, and former students consistently describe his classes as transformative — not because he made mathematics easy, but because he made the struggle of mathematical thinking visible, respectable, and ultimately productive.
Collecting Pólya
How to Solve It (1945, Princeton University Press) in first edition brings $200–$600 — a rare collectible in the popular mathematics category. Mathematics and Plausible Reasoning (1954, Princeton, 2 volumes) firsts are $100–$300 for the set. Problems and Theorems in Analysis (with Szegő, 1925, Springer) is a scholarly collectible. Later Princeton editions of How to Solve It are abundant and inexpensive.