George Pólya - How to Solve It (1945)

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It is hard to have a good idea if we have little knowledge of the subject, and impossible to have it if we have no knowledge. Good ideas are based on past experience and formerly acquired knowledge.

George Pólya
My method to overcome a difficulty is to go round it.

George Pólya
Geometry is the art of correct reasoning on incorrect figures.

George Pólya
The first rule of discovery is to have brains and good luck. The second rule of discovery is to sit tight and wait till you get a bright idea.

George Pólya
The traditional mathematics professor of the popular legend is absentminded. He usually appears in public with a lost umbrella in each hand. He prefers to face a blackboard and to turn his back on the class. He writes a, he says b, he means c, but it should be d.

George Pólya
This principle is so perfectly general that no particular application is possible.

George Pólya
In order to solve a differential equation you look at it till a solution occurs to you.

George Pólya
A great discovery solves a great problem but there is a grain of discovery in the solution of any problem.

George Pólya
Mathematics presented with rigor is a systematic deductive science but mathematics in the making is an experimental inductive science.

George Pólya
There was a seminar for advanced students in Zürich that I was teaching and von Neumann was in the class. I came to a certain theorem, and I said it is not proved and it may be difficult. Von Neumann didn't say anything but after five minutes he raised his hand. When I called on him he went to the blackboard and proceeded to write down the proof. After that I was afraid of von Neumann.

George Pólya
What is the difference between a method and device? A method is a device which you use twice.

George Pólya
The cookbook gives a detailed description of ingredients and procedures but no proofs for its prescriptions or reasons for its recipes; the proof of the pudding is in the eating. … Mathematics cannot be tested in exactly the same manner as a pudding; if all sorts of reasoning are debarred, a course of calculus may easily become an incoherent inventory of indigestible information.

George Pólya
We need heuristic reasoning when we construct a strict proof as we need scaffolding when we erect a building.

George Pólya
Analogy pervades all our thinking, our everyday speech and our trivial conclusions as well as artistic ways of expression and the highest scientific achievements.

George Pólya
Euclid's manner of exposition, progressing relentlessly from the data to the unknown and from the hypothesis to the conclusion, is perfect for checking the argument in detail but far from being perfect for making understandable the main line of the argument.

George Pólya
To write and speak correctly is certainly necessary; but it is not sufficient. A derivation correctly presented in the book or on the blackboard may be inaccessible and uninstructive, if the purpose of the successive steps is incomprehensible, if the reader or listener cannot understand how it was humanly possible to find such an argument....

George Pólya
The best of ideas is hurt by uncritical acceptance and thrives on critical examination.

George Pólya
Pedantry and mastery are opposite attitudes toward rules. To apply a rule to the letter, rigidly, unquestioningly, in cases where it fits and in cases where it does not fit, is pedantry. … To apply a rule with natural ease, with judgment, noticing the cases where it fits, and without ever letting the words of the rule obscure the purpose of the action or the opportunities of the situation, is mastery.

George Pólya