“[M]athematics is not a popular subject... The reason for this is to be found in the common superstition that [it] is but a continuation... of the fine art of arithmetic, of juggling with numbers. [We] combat that superstition, by offering, instead of formulas, figures that may be looked at and that may easily be supplemented by models which the reader may construct. This book... bring[s] about a greater enjoyment of mathematics, by making it easier... to penetrate the essence of mathematics without... a laborious course of studies.”

— David Hilbert , Geometry and the Imagination

Preface
Geometry and the Imagination (1952)

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David Hilbert 30

German prominent mathematician 1862–1943

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“The final truth about a phenomenon resides in the mathematical description of it; so long as there is no imperfection in this, our knowledge of the phenomenon is complete. We go beyond the mathematical formula at our own risk; we may find a model or a picture which helps us understand it, but we have no right to expect this, and our failure to find such a model or picture need not indicate that either our reasoning or our knowledge is at fault. The making of models or pictures to explain mathematical formulas and the phenomena they describe is not a step towards, but a step away from reality; it is like making a graven image of a spirit.”

James Jeans book The Mysterious Universe

The Mysterious Universe (1930)

“Superstition may be defined as constructive religion which has grown incongruous with intelligence.”

John Tyndall (1820–1893) British scientist

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“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

“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

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“Simulation is the use of models and/or the actual conditions of either the thing being modeled or the environment in which it operates, with the models or conditions in physical, mathematical, or some other form. The purpose of simulation is to explore the various results which might be obtained from the real system by subjecting the model to representative environments which are equivalent to, or in some way representative of, the situations it is desired to understand or investigate. Simulation may involve system hardware and the actual physical environment, or it may involve mathematical models subjected to mathematical forcing or disturbance functions representative of the systems conditions to be studied.”

Harold Chestnut (1917–2001) American engineer

Source: Systems Engineering Tools, (1965), p. 111 as cited in

“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.

“We are told … statements that are not mathematical or logical formulae may look as if they were necessarily or certainly true, but they only look like that. They cannot really be either necessary or certain.”

Robert Maynard Hutchins (1899–1977) philosopher and university president

Great Books: The Foundation of a Liberal Education (1954)

“Thus mathematics may be defined as the subject in which we never know what we are talking about, nor whether what we are saying is true.”

Bertrand Russell (1872–1970) logician, one of the first analytic philosophers and political activist

Source: 1910s, Mysticism and Logic and Other Essays http://archive.org/stream/mysticism00russuoft/mysticism00russuoft_djvu.txt (1918), Ch. 5: Mathematics and the Metaphysicians

“Arithmetic and geometry, according to Plato, are the two wings of the mathematician. The object indeed of all mathematical questions, is to determine the ratios of numbers, or of magnitudes; and it may even be said, to continue the comparison of the ancient philosopher, that arithmetic is the mathematician's right wing; for it is an incontestable truth, that geometrical determinations would, for the most part, present nothing satisfactory to the mind, if the ratios thus determined could not be reduced to numerical ratios. This justifies the common practice, which we shall here follow, of beginning with arithmetic.”

Jacques Ozanam (1640–1718) French mathematician

Source: Recreations in Mathematics and Natural Philosophy, (1803), p. 2

“If the system exhibits a structure which can be represented by a mathematical equivalent, called a mathematical model, and if the objective can be also so quantified, then some computational method may be evolved for choosing the best schedule of actions among alternatives. Such use of mathematical models is termed mathematical programming.”

George Dantzig (1914–2005) American mathematician

Source: Linear programming and extensions (1963), p. 2

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