George Pólya 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.

Mathematics is being lazy. Mathematics is letting the principles do the work for you so that you do not have to do the work for yourself.

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.

there is nothing to learn about reasoning and invention if the motive and purpose of the most conspicuous step remain incomprehensible.

Perseverance kills the game.

If you cannot solve the proposed problem...try to solve first some related problem.

Even fairly good students, when they have obtained the solution of the problem and written down neatly the argument, shut their books and look for something else. Doing so, they miss an important and instructive phase of the work. ... A good teacher should understand and impress on his students the view that no problem whatever is completely exhausted.

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

No idea is really bad, unless we are uncritical. What is really bad is to have no idea at all.

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.

If you can't solve a problem, then there is an easier problem you can solve: find it.

Quite often, when an idea that could be helpful presents itself, we do not appreciate it, for it is so inconspicuous. The expert has, perhaps, no more ideas than the inexperienced, but appreciates more what he has and uses it better.

Beauty in mathematics is seeing the truth without effort.

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.

"Now and then, teaching may approach poetry, and now and then it may approach profanity. May I tell you a little story about the great Einstein? I listened once to Einstein as he talked to a group of physicists in a party. "Why have all the electrons the same charge?" said he. "Well, why are all the little balls in the goat dung of the same size?" Why did Einstein say such things? Just to make some snobs to raise their eyebrows? He was not disinclined to do so, I think. Yet, probably, it went deeper. I do not think that the overheard remark of Einstein was quite casual. At any rate, I learnt something from it: Abstractions are important; use all means to make them more tangible. Nothing is too good or too bad, too poetical or too trivial to clarify your abstractions. As Montaigne put it: The truth is such a great thing that we should not disdain any means that could lead to it. Therefore, if the spirit moves you to be a little poetical, or a little profane, in your class, do not have the wrong kind of inhibition." - George Polya's Mathematical Discovery, Volume 11, pp 102, 1962.