Intuition and formalism

April 15, 2011

« Intuitive insight and formal proof are two different ways of perceiving the truth, comparable to the perception of a material object through two different senses, sight and touch. Intuitive insight may rush far ahead of formal proof. […] Formal manipulation of logical rules and algebraic formulas may get far ahead of intution. Almost everybody can see at once that 3 straight lines, taken at random, divide the plane into 7 parts […] Scarcely anybody is able to see, even straining his attention to the utmost, that 5 planes, taken at random, divide space into 26 parts. Yet it can be rigidly proved that the right number is actually 26, and the proof is not even long or difficult. »
~ “How to solve it”. G. Polya.

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