Analogy pervades all our thinking, our everyday speech and our trivial conclusions as well as artistic ways… — George Pólya

We need heuristic reasoning when we construct a strict proof as we need scaffolding when we erect a building.

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.

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....