“The great advances in mathematics have not been made by logic but by creative imagination.”

— George Frederick James Temple

100 Years of Mathematics: a Personal Viewpoint (1981)
Context: Logical analysis is indispensable for an examination of the strength of a mathematical structure, but it is useless for its conception and design. The great advances in mathematics have not been made by logic but by creative imagination.

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George Frederick James Temple21

British mathematician 1901–1992

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“There cannot be a greater mistake than that of looking superciliously upon the practical applications of science. The life and soul of science is its practical application; and just as the great advances in mathematics have been made through the desire of discovering the solution of problems which were of a highly practical kind in mathematical science, so in physical science many of the greatest advances that have been made from the beginning of the world to the present time have been made in earnest desire to turn the knowledge of the properties of matter to some purpose useful to mankind.”

William Thomson (1824–1907) British physicist and engineer

Lecture on "Electrical Units of Measurement" (3 May 1883), published in Popular Lectures Vol. I, p. 73, as quoted in The Life of Lord Kelvin (1910) by Silvanus Phillips Thompson

“As in Mathematics in general, the really great advances, embodying new ideas of far-reaching fruitfullness, have been due to an exceedingly small number of great men… there are periods when for a long series of centuries no advance was made; when the results obtained in a more enlightened age have been forgotten. We observe the times of revival, when the older learning has been rediscovered, and when the results of the progress made in distinct countries have been made available as the starting points of new efforts and a fresh period of activity.”

E. W. Hobson (1856–1933) British mathematician

Source: Squaring the Circle (1913), p. 10

“Only an unashamed dogmatist would dare to assert that the issue has finally been resolved now, in favor of the view that, outside logic or mathematics, the method of modern science is the only method to employ in seeking knowledge. The dogmatist who made this assertion would have to be more than ashamed. He would have to blind himself to the fact that his own assertion was not established by the experimental method, nor made as an indisputable conclusion of mathematical reasoning or of purely logical analysis.”

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

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

“As a science mathematics has been adapted to the description of natural phenomena, and the great practitioners in this field… have never concerned themselves with the logical foundations of mathematics, but have boldly taken a pragmatic view of mathematics as an intellectual machine which works successfully. Description has been verified by further observation, still more strikingly be prediction, and sometimes, more ominously, by control of natural forces. Happily, unresolved problems… still remain as challenges.”

George Frederick James Temple (1901–1992) British mathematician

100 Years of Mathematics: a Personal Viewpoint (1981)

“There is a logic of language and a logic of mathematics.”

Thomas Merton (1915–1968) Priest and author

The Secular Journal of Thomas Merton (1959)
Context: There is a logic of language and a logic of mathematics. The former is supple and lifelike, it follows our experience. The latter is abstract and rigid, more ideal. The latter is perfectly necessary, perfectly reliable: the former is only sometimes reliable and hardly ever systematic. But the logic of mathematics achieves necessity at the expense of living truth, it is less real than the other, although more certain. It achieves certainty by a flight from the concrete into abstraction. Doubtless, to an idealist, this would seem to be a more perfect reality. I am not an idealist. The logic of the poet — that is, the logic of language or the experience itself — develops the way a living organism grows: it spreads out towards what it loves, and is heliotropic, like a plant.

“The mere formulation of a problem is far more often essential than its solution, which may be merely a matter of mathematical or experimental skill. To raise new questions, new possibilities, to regard old problems from a new angle requires creative imagination and makres real advances in science.”

Albert Einstein (1879–1955) German-born physicist and founder of the theory of relativity

http://umich.edu/~scps/html/01chap/html/summary.htm
Attributed in posthumous publications, Einstein and Religion (1999)

“The calculus was the first achievement of modern mathematics and it is difficult to overestimate its importance. I think it defines more unequivocally than anything else the inception of modern mathematics; and the system of mathematical analysis, which is its logical development, still constitutes the greatest technical advance in exact thinking.”

John Von Neumann (1903–1957) Hungarian-American mathematician and polymath

As quoted in Bigeometric Calculus: A System with a Scale-Free Derivative (1983) by Michael Grossman, and in Single Variable Calculus (1994) by James Stewart.

“Pixar is the most technically advanced creative company; Apple is the most creatively advanced technical company.”

Steve Jobs (1955–2011) American entrepreneur and co-founder of Apple Inc.

As quoted in Fortune (21 February 2005)
2000s

“Every great advance in science has issued from a new audacity of imagination.”

John Dewey (1859–1952) American philosopher, psychologist, and educational reformer

The Quest for Certainty (1929), Ch. XI
Misc. Quotes
Source: The Quest for Certainty: A Study of the Relation of Knowledge and Action

“Only mathematics and mathematical logic can say as little as the physicist means to say.”

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

The Scientific Outlook (1931)
1930s
Context: Ordinary language is totally unsuited for expressing what physics really asserts, since the words of everyday life are not sufficiently abstract. Only mathematics and mathematical logic can say as little as the physicist means to say.

“The great advances in mathematics have not been made by logic but by creative imagination.”

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George Frederick James Temple21

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