The result of the mathematician's creative work is demonstrative reasoning, a proof; but the proof is discovered by plausible reasoning, by guessing.

Analogy pervades all our thinking, our everyday speech and our trivial conclusions as well as artistic ways of expression and the highest scientific achievements.

Hilbert once had a student in mathematics who stopped coming to his lectures, and he was finally told the young man had gone off to become a poet. Hilbert is reported to have remarked: 'I never thought he had enough imagination to be a mathematician.'

Even fairly good students, when they have obtained the solution of the problem and written down neatly the argument, shut their books and look for something else. Doing so, they miss an important and instructive phase of the work. ... A good teacher should understand and impress on his students the view that no problem whatever is completely exhausted.

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.

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.

The best of ideas is hurt by uncritical acceptance and thrives on critical examination.