George Polya Quotes - Page 3

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• 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 Polya

  Teaching, Writing, Science

  "How to Solve It" by George Polya, 2nd edition, (p. xv), 1957.

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• Epitaph on Newton: Nature and Nature's law lay hid in night: God said, "Let Newton be!," and all was light. [added by Sir John Collings Squire: It did not last: the Devil shouting "Ho. Let Einstein be," restored the status quo] [Aaron Hill's version: O'er Nature's laws God cast the veil of night, Out blaz'd a Newton's soul and all was light.

  George Polya

  Night, Law, Light

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• 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 Polya

  Book, Writing, Purpose

  "How to Solve It" by George Polya, Expanded Princeton Science Library Edition, (p. 207), 2004.

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• The teacher can seldom afford to miss the questions: What is the unknown? What are the data? What is the condition? The student should consider the principal parts of the problem attentively, repeatedly, and from from various sides.

  George Polya

  Teacher, Science, Data

  George Pólya (1957). “How to solve it: a new aspect of mathematical method”

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• 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 Polya

  Book, Obvious Things, Giving

  George Pólya (1968). “Mathematical discovery: on understanding, learning, and teaching problem solving”

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• The cookbook gives a detailed description of ingredients and procedures but no proofs for its prescriptions or reasons for its recipes; the proof of the pudding is in the eating. ... Mathematics cannot be tested in exactly the same manner as a pudding; if all sorts of reasoning are debarred, a course of calculus may easily become an incoherent inventory of indigestible information.

  George Polya

  Giving, Pudding, Inventory

  "How to Solve It" by George Polya, Expanded Princeton Science Library Edition, (p. 219), 2004.

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• In the "commentatio" (note presented to the Russian Academy) in which his theorem on polyhedra (on the number of faces, edges and vertices) was first published Euler gives no proof. In place of a proof, he offers an inductive argument: he verifies the relation in a variety of special cases. There is little doubt that he also discovered the theorem, as many of his other results, inductively.

  George Polya

  Numbers, Discovery, Giving

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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

  Data, Perfect, Progress

  George Pólya (1945). “How to Solve it: A New Aspect of Mathematical Method”

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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.

  George Polya

  Math, Discovery, Way

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