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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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George Polya
Mathematics
Result
Demonstrative
Results
Guessing
Creative
Plausible
Science
Mathematician
Work
Reasoning
Discovered
Proof

More quotes by George Polya

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The open secret of real success is to throw your whole personality into your problem.
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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.
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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 principle is so perfectly general that no particular application of it is possible.
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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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My method to overcome a difficulty is to go round it.
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Mathematics consists in proving the most obvious thing in the least obvious way.
George Polya
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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.
George Polya
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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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It may be more important in the mathematics class how you teach than what you teach.
George Polya
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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.
George Polya
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Geometry is the science of correct reasoning on incorrect figures.
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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
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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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I am intentionally avoiding the standard term which, by the way, did not exist in Euler's time. One of the ugliest outgrowths of the new math was the premature introduction of technical terms.
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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.
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A mathematics teacher is a midwife to ideas.
George Polya
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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 Polya
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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 Polya