"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.
“The processes that may be employed, to give to portions of inert matter, precise movements resembling those of organized beings, are innumerable, as they consist of an indefinite number and variety of cords pulleys, toothed-wheels, nails, screws, levers, inclined-planes, as well as agencies of air, water, fire, light, &c., combined in endless modes to produce a desired effect Ingenuity has been long exercised on such combinations, chiefly for public amusement or mystification, without any object of utility.”
Source: The Philosophy of Manufactures, 1835, p. 9
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Andrew Ure 14
Scottish doctor and chemist 1778–1857Related quotes
“Falsehood has an infinity of combinations, but truth has only one mode of being.”
Source: Small Houses: Their Economic Design and Construction (1922), Ch. II
Source: The Functions of the Executive (1938), p. 141
Small Houses: Their Economic Design and Construction (1922)
Context: Economy in building consists in the aggregate of a great number of savings, which when considered separately may seem trivial, but when combined are important. The list of those here provided for... may be divided into classes as follows:<!-- Introduction
For every one of those Sumptions, are Aliquot Parts of a b c d e, except the last, (which is the whole,) and instead thereof, 1 is also an Aliquot Part; which makes the number of Aliquot Parts, the same with the Number of Sumptions. Only here is to be understood, (which the Rule should have intimated;) that, all the Numbers proposed, are to be Prime Numbers, and each distinct from the other. For if any of them be Compound Numbers, or any Two of them be the same, the Rule for Aliquot Parts will not hold.
Source: A Discourse of Combinations, Alterations, and Aliquot Parts (1685), Ch.I Of the variety of Elections, or Choice, in taking or leaving One or more, out of a certain Number of things proposed.
Source: An Introduction to Medical Literature, Including a System of Practical Nosology (1823), p. 5