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You should not put too much trust in any unproved conjecture, even if it has been propounded by a great authority, even if it has been propounded by yourself. You should try to prove it or disprove it.

George Polya

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George Polya
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More quotes by George Polya

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The best of ideas is hurt by uncritical acceptance and thrives on critical examination.
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The open secret of real success is to throw your whole personality into your problem.
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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.
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The world is anxious to admire that apex and culmination of modern mathematics: a theorem so perfectly general that no particular application of it is feasible.
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The first and foremost duty of the high school in teaching mathematics is to emphasize methodical work in problem solving...The teacher who wishes to serve equally all his students, future users and nonusers of mathematics, should teach problem solving so that it is about one-third mathematics and two-thirds common sense.
George Polya
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Beauty in mathematics is seeing the truth without effort.
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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
George Polya
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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 Polya
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If you cannot solve the proposed problem try to solve first some related problem.
George Polya
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The result of the mathematician's creative work is demonstrative reasoning, a proof but the proof is discovered by plausible reasoning, by guessing.
George Polya
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Geometry is the science of correct reasoning on incorrect figures.
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The principle is so perfectly general that no particular application of it is possible.
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My method to overcome a difficulty is to go round it.
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Look around when you have got your first mushroom or made your first discovery: they grow in clusters.
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Mathematics consists in proving the most obvious thing in the least obvious way.
George Polya
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Solving problems is a practical art, like swimming, or skiing, or playing the piano: you can learn it only by imitation and practice.
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There exist a lot of questions that the fools can ask, and the intelligent cannot answer.
George Polya
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When introduced at the wrong time or place, good logic may be the worst enemy of good teaching.
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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.
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A mathematics teacher is a midwife to ideas.
George Polya