“It is the constant aim of the mathematician to reduce all his expressions to their lowest terms, to retrench every superfluous word and phrase, and to condense the Maximum of meaning into the Minimum of language.” (James J Sylvester, 1877)
"The union of the mathematician with the poet, fervor with measure, passion with correctness, this surely is the ideal." (William James, "Clifford's Lectures and Essays", 1879)
"[...] it is true that a mathematician who is not somewhat of a poet, will never be a perfect mathematician." (Karl Weierstrass, [Letter to Sofia Kovalevskaya] 1883)
"A scientist worthy of the name, above all a mathematician, experiences in his work the same impression as an artist; his pleasure is as great and of the same nature.... we work not only to obtain the positive results which, according to the profane, constitute our one and only affection, as to experience this esthetic emotion and to convey it to others who are capable of experiencing it." (Henri Poincaré, "Notice sur Halphen", Journal de l'École Polytechnique, 1890)
"There is probably no other science which presents such different appearances to one who cultivates it and one who does not, as mathematics. To [the noncultivator] it is ancient, venerable, and complete; a body of dry, irrefutable, unambiguous reasoning. To the mathematician, on the other hand, his science is yet in the purple bloom of vigorous youth, everywhere stretching out after the 'attainable but unattained', and full of the excitement of nascent thoughts; its logic is beset with ambiguities, and its analytic processes, like Bunyan's road, have a quagmire on one side and a deep ditch on the other, and branch off into innumerable by-paths that end in a wilderness." (Charles H Chapman, Bulletin of the New York Mathematical Society 2, 1892)
"The union of the mathematician with the poet, fervor with measure, passion with correctness, this surely is the ideal." (William James, "Clifford's Lectures and Essays", 1879)
"[...] it is true that a mathematician who is not somewhat of a poet, will never be a perfect mathematician." (Karl Weierstrass, [Letter to Sofia Kovalevskaya] 1883)
"A scientist worthy of the name, above all a mathematician, experiences in his work the same impression as an artist; his pleasure is as great and of the same nature.... we work not only to obtain the positive results which, according to the profane, constitute our one and only affection, as to experience this esthetic emotion and to convey it to others who are capable of experiencing it." (Henri Poincaré, "Notice sur Halphen", Journal de l'École Polytechnique, 1890)
"There is probably no other science which presents such different appearances to one who cultivates it and one who does not, as mathematics. To [the noncultivator] it is ancient, venerable, and complete; a body of dry, irrefutable, unambiguous reasoning. To the mathematician, on the other hand, his science is yet in the purple bloom of vigorous youth, everywhere stretching out after the 'attainable but unattained', and full of the excitement of nascent thoughts; its logic is beset with ambiguities, and its analytic processes, like Bunyan's road, have a quagmire on one side and a deep ditch on the other, and branch off into innumerable by-paths that end in a wilderness." (Charles H Chapman, Bulletin of the New York Mathematical Society 2, 1892)
“With our notion of the essence of intuition, an intuitive treatment of figurative representations will tend to yield a certain general guide on which mathematical laws apply and how their general proof may be structured. However, true proof will only be obtained if the given figures are replaced with figures generated by laws based on the axioms and these are then taken to carry through the general train of thought in an explicit case. Dealing with sensate objects gives the mathematician an impetus and an idea of the problems to be tackled, but it does not pre-empt the mathematical process itself. (Felix Klein, “Nicht-Euklidische Geometrie I: Vorlesung gehalten während des Wintersemesters 1889–90”, 1892)
"The contemplation of the various steps by which mankind has come into possession of the vast stock of mathematical knowledge can hardly fail to interest the mathematician. He takes pride in the fact that his science, more than any other, is an exact science, and that hardly anything ever done in mathematics has proved to be useless." (Florian Cajori, "A History of Mathematics", 1893)
"Mathematics is the most abstract of all the sciences. For it makes no external observations, nor asserts anything as a real fact. When the mathematician deals with facts, they become for him mere ‘hypotheses’; for with their truth he refuses to concern himself. The whole science of mathematics is a science of hypotheses; so that nothing could be more completely abstracted from concrete reality." (Charles S Peirce, "The Regenerated Logic", The Monist Vol. 7 (1), 1896)
"The union of the mathematician with the poet, fervor with measure, passion with correctness, this surely is the ideal." (William James, "Clifford's Lectures and Essays", 1897)
"Mathematicians will do well to observe that a reasonable acquaintance with theoretical physics at its present stage of development, to mention only such broad subjects as electricity, elastics, hydrodynamics, etc., is as much as most of us can keep permanently assimilated. It should also be remembered that the step from the formal elegance of theory to the brute arithmetic of the special case is always humiliating, and that this labor usually falls to the lot of the physicist." (Carl Barus, "The Mathematical Theory of the Top", 1898)
"Mathematicians will do well to observe that a reasonable acquaintance with theoretical physics at its present stage of development, to mention only such broad subjects as electricity, elastics, hydrodynamics, etc., is as much as most of us can keep permanently assimilated. It should also be remembered that the step from the formal elegance of theory to the brute arithmetic of the special case is always humiliating, and that this labor usually falls to the lot of the physicist." (Carl Barus, "The Mathematical Theory of the Top", 1898)
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