Polya 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.

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

Beauty in mathematics is seeing the truth without effort.

If you can’t solve a problem, then there is an easier problem you can’t solve: find it. —George Polya When

Here he became acquainted with an eighteen-year-old schoolgirl named Pelageia Onufrieva, the fiancée of one of his fellow exiles, Petr Chizhikov. The future dictator flirted openly with the girl and gave her a book with the inscription, “To clever, nasty Polya from the oddball Iosif.” When Pelageia left Vologda, Jughashvili sent her facetious cards, such as: “I claim a kiss from you conveyed via Petka [Chizhikov]. I kiss you back, and I don’t just kiss you, but passionately (simple kissing isn’t worth it). Iosif.”7 In his personal files, Stalin kept a photograph of Chizhikov and Onufrieva dating to his time in Vologda: a serious, pretty, round-faced girl in glasses and a serious young man with regular features and a moustache and beard. The jocular cards, presents, and photograph attest to the thirty-three-year-old Jughashvili’s interest in the young woman but do not prove that he was romantically involved with her. We

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.

Si no puedes resolver un problema, entonces existe un problema más fácil que no puedes resolver: Encuéntralo”. - George Polya

The pragmatic approach entails “being concerned primarily with developing creativity” (Sternberg, 2000, p. 5), as opposed to understanding it. Polya’s (1954) emphasis on the use of a variety of heuristics for solving mathematical problems of varying complexity is an example of a pragmatic approach. Thus, heuristics can be viewed as a decision-making mechanism which leads the mathematician down a certain path, the outcome of which may or may not be fruitful. The popular technique of brainstorming, often used in corporate or other business settings, is another example of inducing creativity by seeking as many ideas or solutions as possible in a non-critical setting.

The psychodynamic approach to studying creativity is based on the idea that creativity arises from the tension between conscious reality and unconscious drives (Hadamard, 1945; Poincaré, 1948, Sternberg, 2000, Wallas, 1926; Wertheimer, 1945). The four-step Gestalt model (preparation-incubationillumination-verification) is an example of the use of a psychodynamic approach to studying creativity. It should be noted that the gestalt model has served as kindling for many contemporary problem-solving models (Polya, 1945; Schoenfeld, 1985; Lester, 1985). Early psychodynamic approaches to creativity were used to construct case studies of eminent creators such as Albert Einstein, but the behaviorists criticized this approach because of the difficulty in measuring proposed theoretical constructs.

The preceding excerpt indicates that mathematicians tend to work on more than one problem at a given time. Do mathematicians switch back and forth between problems in a completely random manner, or do they employ and exhaust a systematic train of thought about a problem before switching to a different problem? Many of the mathematicians reported using heuristic reasoning, trying to prove something one day and disprove it the next day, looking for both examples and counterexamples, the use of "manipulations" (Polya, 1954) to gain an insight into the problem. This indicates that mathematicians do employ some of the heuristics made explicit by Polya. It was unclear whether the mathematicians made use of computers to gain an experimental or computational insight into the problem.

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