“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.”

— Tobias Dantzig

...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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Tobias Dantzig 25

American mathematician 1884–1956

Tobias Dantzig (1884–1956) American mathematician

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

“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.”

Alice A. Bailey (1880–1949) esoteric, theosophist, writer

Source: Glamour: A World Problem (1950), Certain Preliminary Clarifications

“Intuition is a distinct form of experience.”

Sarvepalli Radhakrishnan (1888–1975) Indian philosopher and statesman who was the first Vice President and the second President of India

Internet Encyclopedia of Philosophy
Context: Intuition is a distinct form of experience. Intuition is of a self-certifying character (svatassiddha). It is sufficient and complete. It is self-established (svatasiddha), self-evidencing (svāsaṃvedya), and self-luminous (svayam-prakāsa). Intuition entails pure comprehension, entire significance, complete validity. It is both truth-filled and truth-bearing Intuition is its own cause and its own explanation. It is sovereign. Intuition is a positive feeling of calm and confidence, joy and strength. Intuition is profoundly satisfying. It is peace, power and joy.

“The square is not a subconscious form. It is the creation of intuitive reason. The face of the new art. The square is a living, regal infant. The first step of pure creation in art.”

Kazimir Malevich (1879–1935) Russian and Soviet artist of polish descent

Quote in 'From Cubism and Futurism to Suprematism: The New Realism in Painting', Kazimir Malevich, November 1916
1910 - 1920

“Many people think that the progress of the human race is based on experiences of an empirical, critical nature, but I say that true knowledge is to be had only through a philosophy of deduction. For it is intuition that improves the world, not just following a trodden path of thought. Intuition makes us look at unrelated facts and then think about them until they can all be brought under one law. To look for related facts means holding onto what one has instead of searching for new facts. Intuition is the father of new knowledge, while empiricism is nothing but an accumulation of old knowledge. Intuition, not intellect, is the 'open sesame' of yourself.”

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

Source: Attributed in posthumous publications, Einstein and the Poet (1983), p. 16

“With respect to the mathematical part of optics, the curvature of the images, formed by lenses and mirrors, has been correctly investigated, and the inaccuracy of some former estimations has been demonstrated.”

Thomas Young (scientist) (1773–1829) English polymath

Preface
A Course of Lectures on Natural Philosophy and the Mechanical Arts (1807)

“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.”

Henry John Stephen Smith (1826–1883) mathematician

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.

“The typographic logic created “the outsider,” the alienated mass, as the type of integral, that is, intuitive and irrational, man.”

Marshall McLuhan (1911–1980) Canadian educator, philosopher, and scholar-- a professor of English literature, a literary critic, and a …

Source: 1960s, The Gutenberg Galaxy (1962), p. 241

“A new idea comes suddenly and in a rather intuitive way. But intuition is nothing but the outcome of earlier intellectual experience.”

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

Letter to Dr. H. L. Gordon (May 3, 1949 - AEA 58-217) as quoted in Einstein: His Life and Universe (2007) by Walter Isaacson ISBN 9780743264730
1940s

“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,”

George Boole (1815–1864) English mathematician, philosopher and logician

Source: 1840s, The Mathematical Analysis of Logic, 1847, p. iii
Context: That to the existing forms of Analysis a quantitative interpretation is assigned, is the result of the circumstances by which those forms were determined, and is not to be construed into a universal condition of Analysis. 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, regardless that in its object and in its instruments it must at present stand alone.

“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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