Mathematical reasoning may be regarded rather schematically as the exercise of a combination of two faculties, which we may call intuition and ingenu… - Alan Turing

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Mathematical reasoning may be regarded rather schematically as the exercise of a combination of two faculties, which we may call intuition and ingenuity.
... In pre-Gödel times it was thought by some that all the intuitive judgements of mathematics could be replaced by a finite number of rules. The necessity for intuition would then be entirely eliminated.
In our discussions, however, we have gone to the opposite extreme and eliminated not intuition but ingenuity, and this in spite of the fact that our aim has been in much the same direction.