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.

Alan Turing— "Systems of Logic Based on Ordinals," section 11: The purpose of ordinal logics (1938), published in

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."

Instruction tables will have to be made up by mathematicians with computing experience and perhaps a certain puzzle-solving ability. There need be no real danger of it ever becoming a drudge, for any processes that are quite mechanical may be turned over to the machine itself.

Alan Turing— "Proposed Electronic Calculator" (1946), a report for National Physical Laboratory, Teddington; published in

A man provided with paper, pencil, and rubber, and subject to strict discipline, is in effect a universal machine.

Alan Turing— "Intelligent Machinery: A Report by A. M. Turing," (Summer 1948), submitted to the National Physical Laboratory (1948) and published in

Science is a differential equation. Religion is a boundary condition.

Alan Turing— Epigram to Robin Gandy (1954); reprinted in Andrew Hodges,

The idea behind digital computers may be explained by saying that these machines are intended to carry out any operations which could be done by a human computer.

Alan Turing— p. 436 (Computing Machinery and Intelligence (1950))

We can only see a short distance ahead, but we can see plenty there that needs to be done.

Alan Turing— p. 460 (Computing Machinery and Intelligence (1950))

His high-pitched voice already stood out above the general murmur of well-behaved junior executives grooming themselves for promotion within the Bell corporation. Then he was suddenly heard to say: "No, I'm not interested in developing a powerful brain. All I'm after is just a mediocre brain, something like the President of the American Telephone and Telegraph Company."

Alan Turing— Andrew Hodges,

Describing an incident which occurred in the New York AT & T lab cafeteria in 1943

Turing had a strong predeliction for working things out from first principles, usually in the first instance without consulting any previous work on the subject, and no doubt it was this habit which gave his work that characteristically original flavor. I was reminded of a remark which Beethoven is reputed to have made when he was asked if he had heard a certain work of Mozart which was attracting much attention. He replied that he had not, and added "neither shall I do so, lest I forfeit some of my own originality."

Alan Turing— James H. Wilkinson, "Some Comments from a Numerical Analyst", 1970 Turing Award lecture,

He was particularly fond of little programming tricks (some people would say that he was too fond of them to be a "good" programmer) and would chuckle with boyish good humor at any little tricks I may have used.

Alan Turing— James H. Wilkinson, "Some Comments from a Numerical Analyst", 1970 Turing Award lecture,

Although a mathematician, Turing took quite an interest in the engineering side of computer design. There was some discussion in 1947 as to whether a cheaper substance than mercury could not be found for use as an ultrasonic delay medium. Turing's contribution to this discussion was to advocate the use of gin, which he said contained alcohol and water in just the right proportions to give a zero temperature coefficient of propagation velocity at room temperature.

Alan Turing— Maurice V. Wilkes, "Computers Then and Now",