“The progress of mathematics has been most erratic, and… intuition has played a predominant rôle in it…. It was the function of intuition to create new forms; it was the acknowledged right of logic to accept or reject these new forms, in whose birth in had no part…. the children had to live, so while waiting for logic to sanctify their existence, they throve and multiplied.”

—  Tobias Dantzig

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

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Tobias Dantzig 25
American mathematician 1884–1956

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“The progress of mathematics has been most erratic, and… intuition has played a predominant rôle in it. …It was the function of intuition to create new forms; it was the acknowledged right of logic to accept or reject these new forms, in whose birth in had no part.”

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...the children had to live, so while waiting for logic to sanctify their existence, they throve and multiplied.
Number: The Language of Science (1930)

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“It is upon the foundation of this general principle, that I purpose to establish the Calculus of Logic, and that I claim for it a place among the acknowledged forms of Mathematical Analysis,”

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“It is by logic that we prove, but by intuition that we discover. To know how to criticize is good, to know how to create is better.”

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“There are many ways in which the intuition can be drawn into activity, and one of the most useful and potent is the study and interpretation of symbols. Symbols are the outer and visible forms of the inner spiritual realities, and when facility in discovering the reality behind any specific form has been gained, that very fact will indicate the awakening of the intuition.”

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“So intimate is the union between Mathematics and Physics that probably by far the larger part of the accessions to our mathematical knowledge have been obtained by the efforts of mathematicians to solve the problems set to them by experiment, and to create for each successive class phenomena a new calculus or a new geometry, as the case might be, which might prove not wholly inadequate to the subtlety of nature. Sometimes the mathematician has been before the physicist, and it has happened that when some great and new question has occurred to the experimentalist or the observer, he has found in the armory of the mathematician the weapons which he needed ready made to his hand. But much oftener, the questions proposed by the physicist have transcended the utmost powers of the mathematics of the time, and a fresh mathematical creation has been needed to supply the logical instrument requisite to interpret the new enigma.”

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As quoted in The Century: A Popular Quarterly (1874) ed. Richard Watson Gilder, Vol. 7, pp. 508-509, https://books.google.com/books?id=ceYGAQAAIAAJ&pg=PA508 "Relations of Mathematics to Physics". Earlier quote without citation in Nature, Volume 8 (1873), page 450.
Also quoted partially in Michael Grossman and Robert Katz, Calculus http://babel.hathitrust.org/cgi/mb?a=listis;c=216746186|Non-Newtonian (1972) p. iv. ISBN 0912938013.

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“I presume that few who have paid any attention to the history of the Mathematical Analysis, will doubt that it has been developed in a certain order, or that that order has been, to a great extent, necessary -- being determined, either by steps of logical deduction, or by the successive introduction of new ideas and conceptions, when the time for their evolution had arrived.”

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Source: 1850s, A treatise on differential equations (1859), p. v; cited in: Quotations by George Boole http://www-history.mcs.st-and.ac.uk/Quotations/Boole.html, MacTutor History of Mathematics, August 2010.

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