George Pólya > Quotes

“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.”
― Polya George

“If you can't solve a problem, then there is an easier problem you can solve: find it.”
― George Polya, Mathematical Discovery on Understanding, Learning, and Teaching Problem Solving, Volume I

“The first rule of style is to have something to say. The second rule of style is to control yourself when, by chance, you have two things to say; say first one, then the other, not both at the same time.”
― George Pólya, How to Solve It: A New Aspect of Mathematical Method

“Beauty in mathematics is seeing the truth without effort.”
― George Polya

“It is better to solve one problem five different ways, than to solve five problems one way.”
― 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

“Nothing is more important than to see the sources of invention which are, in my opinion, more interesting than the inventions themselves.”
― George Pólya, How to Solve It: A New Aspect of Mathematical Method

“Mathematics, you see, is not a spectator sport. To understand mathematics means to be able to do mathematics. And what does it mean [to be] doing mathematics? In the first place, it means to be able to solve mathematical problems.”
― George Pólya, How to Solve It: A New Aspect of Mathematical Method

“A great discovery solves a great problem, but there is a grain of discovery in the solution of any problem. Your problem may be modest, but if it challenges your curiosity and brings into play your inventive faculties, and if you solve it by your own means, you may experience the tension and enjoy the triumph of discovery.”
― George Pólya, How to Solve It: A New Aspect of Mathematical Method

“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, How to Solve It: A New Aspect of Mathematical Method

“The elegance of a mathematical theorem is directly proportional to the number of independent ideas one can see in the theorem and inversely proportional to the effort it takes to see them.”
― George Pólya, Mathematical Discovery on Understanding, Learning, and Teaching Problem Solving, Volume I

“I started studying law, but this I could stand just for one semester. I couldn't stand more. Then I studied languages and literature for two years. After two years I passed an examination with the result I have a teaching certificate for Latin and Hungarian for the lower classes of the gymnasium, for kids from 10 to 14. I never made use of this teaching certificate. And then I came to philosophy, physics, and mathematics. In fact, I came to mathematics indirectly. I was really more interested in physics and philosophy and thought about those. It is a little shortened but not quite wrong to say: I thought I am not good enough for physics and I am too good for philosophy. Mathematics is in between.”
― George Pólya

“It is generally useless to carry out details without having seen the main connection, or having made a sort of plan.”
― George Pólya, How to Solve It: A New Aspect of Mathematical Method

“If you cannot solve the proposed problem...try to solve first some related problem.”
― George Pólya, How to Solve It: A New Aspect of Mathematical Method

“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.”
― George Pólya, How to Solve It: A New Aspect of Mathematical Method

“Good problems and mushrooms of certain kinds have something in common; they grow in clusters.”
― George Pólya, How to Solve It: A New Aspect of Mathematical Method

“Analogy pervades all our thinking,”
― George Pólya, How to Solve It: A New Aspect of Mathematical Method

“One of the first and foremost duties of the teacher is not to give his students the impression that mathematical problems have little connection with each other, and no connection at all with anything else. We have a natural opportunity to investigate the connections of a problem when looking back at its solution.”
― George Pólya, How to Solve It: A New Aspect of Mathematical Method

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

“Here is a typical story about Mr. John Jones. Mr. Jones works in an office. He had hoped for a little raise but his hope, as hopes often are, was disappointed. The salaries of some of his colleagues were raised but not his. Mr. Jones could not take it calmly. He worried and worried and finally suspected that Director Brown was responsible for his failure in getting a raise. We cannot blame Mr. Jones for having conceived such a suspicion. There were indeed some signs pointing to Director Brown. The real mistake was that, after having conceived that suspicion, Mr. Jones became blind to all signs pointing in the opposite direction. He worried himself into firmly believing that Director Brown was his personal enemy and behaved so stupidly that he almost succeeded in making a real enemy of the director. The trouble with Mr. John Jones is that he behaves like most of us. He never changes his major opinions. He changes his minor opinions not infrequently and quite suddenly; but he never doubts any of his opinions, major or minor, as long as he has them. He never doubts them, or questions them, or examines them critically—he would especially hate critical examination, if he understood what that meant. Let us concede that Mr. John Jones is right to a certain extent. He is a busy man; he has his duties at the office and at home. He has little time for doubt or examination. At best, he could examine only a few of his convictions and why should he doubt one if he has no time to examine that doubt? Still, don’t do as Mr. John Jones does. Don’t let your suspicion, or guess, or conjecture, grow without examination till it becomes ineradicable. At any rate, in theoretical matters, the best of ideas is hurt by uncritical acceptance and thrives on critical examination. 2. A mathematical example. Of all quadrilaterals with”
― George Pólya, How to Solve It: A New Aspect of Mathematical Method

“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.”
― George Pólya, Mathematical Discovery on Understanding, Learning and Teaching Problem Solving, Volumes I and II

“It is foolish to answer a question that you do not understand. It is sad to work for an end that you do not desire.”
― George Pólya, How to Solve It: A New Aspect of Mathematical Method

“Solving problems is a practical art, like swimming, or skiing, or playing the piano: you can learn it only by imitation and practice”
― George Pólya

“Pedantry and mastery are opposite attitudes toward rules. 1. To apply a rule to the letter, rigidly, unquestioningly, in cases where it fits and in cases where it does not fit, is pedantry. Some pedants are poor fools; they never did understand the rule which they apply so conscientiously and so indiscriminately. Some pedants are quite successful; they understood their rule, at least in the beginning (before they became pedants), and chose a good one that fits in many cases and fails only occasionally. 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, How to Solve It: A New Aspect of Mathematical Method

“No idea is really bad, unless we are uncritical. What is really bad is to have no idea at all.”
― George Pólya, How to Solve It: A New Aspect of Mathematical Method

“Problems “in letters” are susceptible of more, and more interesting, tests than “problems in numbers”
― George Pólya, How to Solve It: A New Aspect of Mathematical Method

“you should be grateful for all new ideas, also for the lesser ones, also for the hazy ones, also for the supplementary ideas adding some precision to a hazy one, or attempting the correction of a less fortunate one.”
― George Pólya, How to Solve It: A New Aspect of Mathematical Method

“The teacher should help, but not too much and not too little, so that the student shall have a reasonable share of the work.”
― George Pólya, How to Solve It: A New Aspect of Mathematical Method

“Look around when you have got your first mushroom or
made your first discovery: they grow in clusters.”
― George Pólya

“Mathematics consists in proving the most obvious thing in the least obvious way.”
― George Pólya