George Pólya quotes

George Pólya was a Hungarian mathematician renowned for his work in problem solving, probability theory, and mathematical education. His influential books, especially "How to Solve It," have guided generations of students and teachers in the art of mathematical thinking. He is remembered as one of the most important educators and thinkers in mathematics of the twentieth century. Here are 37 of his quotes:

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)
Mathematics is the cheapest science. Unlike physics or chemistry, it does not require any expensive equipment. All one needs for mathematics is a pencil and paper. View quote
George Pólya
The best of ideas is hurt by uncritical acceptance and thrives on critical examination. View quote
George Pólya
If you cannot solve the proposed problem, try to solve first a simpler related problem. View quote
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. View quote
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. View quote
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. View quote
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.... View quote
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. View quote
George Pólya
In plausible reasoning the principal thing is to distinguish... a more reasonable guess from a less reasonable guess. View quote
George Pólya
We need heuristic reasoning when we construct a strict proof as we need scaffolding when we erect a building. (George Pólya)
I shall often discuss mathematical discoveries... I shall try to make up a likely story how the discovery could have happened. I shall try to emphasize the motives underlying the discovery, the plausible inferences that led to it... everything that deserves imitation. View quote
George Pólya
Good approximations often lead to better ones. View quote
George Pólya
For many of the stories told the final form resulted from a sort of informal psychological experiment. I discussed the subject with several different classes... Several passages... have been suggested by answers of my students, or... modified... by the reaction of my audience. View quote
George Pólya
Why should the typical student be interested in those wretched triangles? ...He is to be brought to see that without the knowledge of triangles there is not trigonometry; that without trigonometry we put back the clock millennia to Standard Darkness Time and antedate the Greeks. View quote
George Pólya
People tell you that wishful thinking is bad. Do not believe it, this is just one of those generally accepted errors. View quote
George Pólya
If we deal with our problem not knowing, or pretending not to know the general theory encompassing the concrete case before us, if we tackle the problem "with bare hands", we have a better chance to understand the scientist's attitude in general, and especially the task of the applied mathematician. View quote
George Pólya
Mathematics succeeds in dealing with tangible reality by being conceptual. We cannot cope with the full physical complexity; we must idealize. View quote
George Pólya
Facing any part of the observable reality, we are never in possession of complete knowledge, nor in a state of complete ignorance, although usually much closer to the latter state. View quote
George Pólya
Simplicity is worth buying if we do not have to pay too great a loss of precision for it. View quote
George Pólya
The result of the mathematician's creative work is demonstrative reasoning, a proof; but the proof is discovered by plausible reasoning, by guessing. If the learning of mathematics reflects to any degree the invention of mathematics, it must have a place for guessing, for plausible inference. (George Pólya)
Everyone knows that mathematics offers an excellent opportunity to learn demonstrative reasoning, but I contend also that there is no other subject in the usual curricula of the schools that affords a comparable opportunity to learn plausible reasoning. ...let us learn proving, but also let us learn guessing. View quote
George Pólya
The general or amateur student should also get a taste of demonstrative reasoning... he should acquire a standard with which he can compare alleged evidence of all sorts aimed at him in modern life. View quote
George Pólya
If we could be any mathematician in the history of the world (besides ourselves), who would we rather be? ...we narrowed the choice down to Euler and Pólya, and finally settled on George Pólya because of the sheer enjoyment of mathematics that he has conveyed by so many examples. View quote
George Pólya
Even if without the Scott's proverbial thrift, the difficulty of solving differential equations is an incentive to using them parsimoniously. View quote
George Pólya