George Pólya Quotes

44 Sourced Quotes

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
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.

George Pólya
If you wish to learn swimming you have to go into the water and if you wish to become a problem solver you have to solve problems.

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.

George Pólya
If you have to prove a theorem, do not rush. First of all, understand fully what the theorem says, try to see clearly what it means. Then check the theorem; it could be false. Examine the consequences, verify as many particular instances as are needed to convince yourself of the truth. When you have satisfied yourself that the theorem is true, you can start proving it.

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.

George Pólya
If the proof starts from axioms, distinguishes several cases, and takes thirteen lines in the text book... it may give the youngsters the impression that mathematics consists in proving the most obvious things in the least obvious way.

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

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
In plausible reasoning the principal thing is to distinguish... a more reasonable guess from a less reasonable guess.

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.

George Pólya
Simplicity is worth buying if we do not have to pay too great a loss of precision for it.

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
Mathematics succeeds in dealing with tangible reality by being conceptual. We cannot cope with the full physical complexity; we must idealize.

George Pólya