Quotations

Nessuna umana investigazione si può dimandare vera scienza, se essa non passa per le matematiche dimostrazioni. Leonardo da Vinci (Trattato della pittura)

Mathematical reasoning may be regarded rather schematically as the exercise of a combination of two faculties, which we may call intuition and ingenuity. The activity of the intuition consists in making spontaneous judgments which are not the result of conscious trains of reasoning. . . The exercise of ingenuity in mathematics consists in aiding the intuition through suitable arrangements of propositions, and perhaps geometrical figures or drawings. It is intended that when these are really well arranged validity of the intuitive steps which are required cannot seriously be doubted. Alan Turing (PhD thesis 'System of logic based on ordinals')