Quotes from work
Systems of Logic Based on Ordinals

Alan Turing

Systems of Logic Based on Ordinals was the PhD dissertation of the mathematician Alan Turing.Turing’s thesis is not about a new type of formal logic, nor was he interested in so-called ‘ranked logic’ systems derived from ordinal or relative numbering, in which comparisons can be made between truth-states on the basis of relative veracity. Instead, Turing investigated the possibility of resolving the Godelian incompleteness condition using Cantor’s method of infinites. This condition can be stated thus- in all systems with finite sets of axioms, an exclusive-or condition applies to expressive power and provability; ie one can have power and no proof, or proof and no power, but not both.

“Mathematical reasoning may be regarded rather schematically as the exercise of a combination of two facilities, which we may call intuition and ingenuity.”

Alan Turing Systems of Logic Based on Ordinals

"Systems of Logic Based on Ordinals," section 11: The purpose of ordinal logics (1938), published in Proceedings of the London Mathematical Society, series 2, vol. 45 (1939)
In a footnote to the first sentence, Turing added: "We are leaving out of account that most important faculty which distinguishes topics of interest from others; in fact, we are regarding the function of the mathematician as simply to determine the truth or falsity of propositions."
Context: Mathematical reasoning may be regarded rather schematically as the exercise of a combination of two facilities, which we may call intuition and ingenuity. The activity of the intuition consists in making spontaneous judgements 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.