"It seems clear that [set theory] violates against the essence of the continuum, which, by its very nature, cannot at all be battered into a single set of elements. Not the relationship of an element to a set, but of a part to a whole ought to be taken as a basis for the analysis of a continuum." (Hermann Weyl, "Riemanns geometrische Ideen, ihre Auswirkungen und ihre Verknüpfung mit der Gruppentheorie", 1925)
"The analysis of variance is not a mathematical theorem, but rather a convenient method of arranging the arithmetic." (Sir Ronald A Fisher, Journal of the Royal Statistical Society Vol. 1, 1934)
"The mathematical machine works with unerring precision; but what we get out of it is nothing more than a rearrangement of what we put into it. In the last analysis observation - the actual contact with real events - is the only reliable way of securing the data of natural history." (William R Thompson, "Science and Common Sense", 1937)
"The mistakes and unresolved difficulties of the past in mathematics have always been the opportunities of its future; and should analysis ever appear to be without or blemish, its perfection might only be that of death." (Eric T Bell, "The Development of Mathematics", 1940)
"Mathematics as an expression of the human mind reflects the active will, the contemplative reason. and the desire for aesthetic perfection. Its basic elements are logic and intuition, analysis and construction, generality and individuality. Though different traditions may emphasize different aspects, it is only the interplay of these antithetic forces and the struggle for their synthesis that constitute the life, usefulness, and supreme value of mathematical science." (Richard Courant & Herbert Robbins,"What is Mathematics?", 1941)
"Analogies are useful for analysis in unexplored fields. By means of analogies an unfamiliar system may be compared with one that is better known. The relations and actions are more easily visualized, the mathematics more readily applied, and the analytical solutions more readily obtained in the familiar system." (Harry F Olson, "Dynamical Analogies", 1943)
One would describe this as pure madness; and yet the whole development of algebra and analysis would have been impossible without that fundament - which, of course, was, in the nineteenth century, established on solid and rigorous bases. It has been written that the shortest and best way between two truths of the real domain often passes through the imaginary one." (Jacque S Hadamard, "An Essay on the Psychology of Invention in the Mathematical Field", 1945)"
"Only by the analysis and interpretation of observations as they are made, and the examination of the larger implications of the results, is one in a satisfactory position to pose new experimental and theoretical questions of the greatest significance." (John A Wheeler, "Elementary Particle Physics", American Scientist, 1947)
"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,"Works of the Mind", 1947)
"Mathematicians are led to new problems not only by way of contact with the world of physical experience but also by introspective study of the methods which they have elected to use. The trend of classical analysis has been to break up the object of study into finer and finer elements without end." (Marston Morse, "Equilibria in Nature: Stable and Unstable", Proceedings of the American Philosophical Society Vol. 93" (3), 1949)
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