“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.

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Alan Turing 33

British mathematician, logician, cryptanalyst, and computer… 1912–1954

Harold Chestnut (1917–2001) American engineer

Source: Systems Engineering Tools, (1965), p. 108; As cited in: Alberto Ortiz (1992) Systems engineering concept demonstration, process model http://www.dtic.mil/dtic/tr/fulltext/u2/a265469.pdf. p. 12

“The very same reason which one man may regard as a motive for taking care to prolong his life may be regarded by another man as a motive for shooting himself.”

Miguel de Unamuno (1864–1936) 19th-20th century Spanish writer and philosopher

The Tragic Sense of Life (1913), XI : The Practical Problem

“Moral judgments may conflict with one another. Two conflicting moral judgments cannot both be right. This does not mean that the moral intuition itself is subject to error, but merely that we may fail to disentangle the intuition itself from irrelevances, or may unconsciously pretend to have an intuition when we actually have it not. The intuition itself is infallible; but we can never be sure that we have it, or that we have not confused it, or expressed it falsely in words. In the same way sense-experience is infallible, but we may unconsciously pretend to have it when we have, it not, and we may misdescribe it, and so on.”

Olaf Stapledon (1886–1950) British novelist and philosopher

Philosophy and Living (1939)

“But how can we avoid the use of human language? The… symbol. Only by using a symbolic language not yet usurped by those vague ideas of space, time, continuity which have their origin in intuition and tend to obscure pure reason—only thus may we hope to build mathematics on the solid foundation of logic.”

Tobias Dantzig (1884–1956) American mathematician

p, 125
Number: The Language of Science (1930)

“The more advanced the sciences have become, the more they have tended to enter the domain of mathematics, which is a sort of center towards which they converge. We can judge of the perfection to which a science has come by the facility, more or less great, with which it may be approached by calculation.”

Adolphe Quetelet (1796–1874) Belgian astronomer, mathematician, statistician and sociologist

Edward Mailly, Essai sur la vie et les ouv rages de Quetelet in the Annuaire de Vacadimie royale des sciences des lettres et des beaux-arts de Belgique (1875) Vol. xli pp. 109-297 found also in "Conclusions" of Instructions populaires sur le calcul des probabilités p. 230

“The last thirty years [1925 to 1955] have seen an enormous improvement in the position of geometry as a branch of mathematics, or, rather, have seen the re-integration of geometry into the main fabric of mathematics. Indeed, one can go further and say that with the restoration of geometry to its rightful place in the mathematical scheme the process of fragmentation which had been doing so much harm to mathematics has been reversed, and we may look forward to the day in which there are no longer analysts, algebraists, geometers and so on, but simply mathematicians. Mathematical research has two aspects, motivation and technique, and when the latter gains control the result is apt to be excessive specialisation. The revolution of geometrical thought, and the reinstatement of geometry as one of the major mathematical disciplines, have helped to bring about a unification of mathematics which we may justly regard as one of the major contributions of the last quarter century to the subject.”

W. V. D. Hodge (1903–1975) British mathematician

W. V. D. Hodge, Changing Views of Geometry. Presidential Address to the Mathematical Association, 14th April, 1955, The Mathematical Gazette 39 (329) (1955), 177-183.

“A model may be pictorial, descriptive, qualitative, or generally approximate in nature, or it may be mathematical and quantitative in nature and reasonably precise.”

Harold Chestnut (1917–2001) American engineer

Source: Systems Engineering Tools, (1965), p. 108

“There are a number of theorems in ordinary algebra, which, though apparently proved to be true only for symbols representing numbers, admit of a much more extended application. Such theorems depend only on the laws of combination to which the symbols are subject, and are therefore true for all symbols, whatever their nature may be, which are subject to the same laws of combination. The laws with which we have here concern are few in number, and may be stated in the following manner. Let a, b represent two operations, u, v two subjects on which they operate, then the laws are
(1) ab(u) = ba (u),
(2) a(u + v) = a (u) + a (v),
(3) am. an. u = am + n. u.
The first of these laws is called the commutative law, and symbols which are subject to it are called commutative symbols. The second law is called distributive, and the symbols subject to it distributive symbols. The third law is not so much a law of combination of the operation denoted by a, but rather of the operation performed on a, which is indicated by the index affixed to a. It may be conveniently called the law of repetition, since the most obvious and important case of it is that in which m and n are integers, and am therefore indicates the repetition m times of the operation a.”

Duncan Gregory (1813–1844) British mathematician

That these are the laws employed in the demonstration of the principal theorems in Algebra, a slight examination of the processes will easily shew ; but they are not confined to symbols of numbers ; they apply also to the symbol used to denote differentiation.
p. 237 http://books.google.com/books?id=8lQ7AQAAIAAJ&pg=PA237; Highlighted section cited in: George Boole " Mr Boole on a General Method in Analysis http://books.google.com/books?pg=PA225-IA15&id=aGwOAAAAIAAJ&hl," Philosophical Transactions, Vol. 134 (1844), p. 225; Other section (partly) cited in: James Gasser (2000) A Boole Anthology: Recent and Classical Studies in the Logic of George Boole,, p. 52
Examples of the processes of the differential and integral calculus, (1841)

“Knowledge which is divorced from justice may be called cunning rather than wisdom.”

Marcus Tullius Cicero (-106–-43 BC) Roman philosopher and statesman

“The arithmetization of mathematics… which began with Weierstrass… had for its object the separation of purely mathematical concepts, such as number and correspondence and aggregate, from intuitional ideas, which mathematics had acquired from long association with geometry and mechanics.
These latter, in the opinion of the formalists, are so firmly entrenched in mathematical thought that in spite of the most careful circumspection in the choice of words, the meaning concealed behind these words, may influence our reasoning. For the trouble with human words is that they possess content, whereas the purpose of mathematics is to construct pure thought.”

Tobias Dantzig (1884–1956) American mathematician

p, 125
Number: The Language of Science (1930)

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

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