“At present, when the prevailing forms of society have become hindrances to the free expression of human powers, it is precisely the abstract branches of science, mathematics and theoretical physics, which … offer a less distorted form of knowledge than other branches of science which are interwoven with the pattern of daily life, and the practicality of which seemingly testifies to their realistic character.”

— Max Horkheimer

Source: "The Latest Attack on Metaphysics" (1937), p. 133.

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Max Horkheimer 61

German philosopher and sociologist 1895–1973

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“The branches of mathematics are as various as the sciences to which they belong, and each subject of physical enquiry has its appropriate mathematics.”

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Context: The branches of mathematics are as various as the sciences to which they belong, and each subject of physical enquiry has its appropriate mathematics. In every form of material manifestation, there is a corresponding form of human thought, so that the human mind is as wide in its range of thought as the physical universe in which it thinks.

“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

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“A great department of thought must have its own inner life, however transcendent may be the importance of its relations to the outside. No department of science, least of all one requiring so high a degree of mental concentration as Mathematics, can be developed entirely, or even mainly, with a view to applications outside its own range. The increased complexity and specialisation of all branches of knowledge makes it true in the present, however it may have been in former times, that important advances in such a department as Mathematics can be expected only from men who are interested in the subject for its own sake, and who, whilst keeping an open mind for suggestions from outside, allow their thought to range freely in those lines of advance which are indicated by the present state of their subject, untrammelled by any preoccupation as to applications to other departments of science. Even with a view to applications, if Mathematics is to be adequately equipped for the purpose of coping with the intricate problems which will be presented to it in the future by Physics, Chemistry and other branches of physical science, many of these problems probably of a character which we cannot at present forecast, it is essential that Mathematics should be allowed to develop freely on its own lines.”

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

Source: Presidential Address British Association for the Advancement of Science, Section A (1910), p. 286; Cited in: Moritz (1914, 106): Modern mathematics.

“De Saussur… develops the concept of semiology as the science which studies the functioning of signs in society, and treats linguistics as a branch of such a general science of signs.”

Adam Schaff (1913–2006) Polish Marxist philosopher and theorist

Source: Introduction to semantics, 1962, p. 4

“Indubitably, Magick is one of the subtlest and most difficult of the sciences and arts. There is more opportunity for errors of comprehension, judgement and practice than in any other branch of physics.”

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“Geometry is that of mathematical science which is devoted to consideration of form and size, and may be said to be the best and surest guide to study of all sciences in which ideas of dimension or space are involved. Almost all the knowledge required by navigators, architects, surveyors, engineers, and opticians, in their respective occupations, is deduced from geometry and branches of mathematics. All works of art constructed according to the rules which geometry involves; and we find the same laws observed in the works of nature. The study of mathematics, generally, is also of great importance in cultivating habits of exact reasoning; and in this respect it forms a useful auxiliary to logic.”

Robert Chambers (publisher, born 1802) (1802–1871) Scottish publisher and writer

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“There is no branch of mathematics, however abstract, which may not some day be applied to phenomena of the real world.”

Nikolai Ivanovich Lobachevsky (1792–1856) Russian mathematician of Ukrainian origin

As quoted in George Edward Martin, The Foundations of Geometry and the Non-Euclidean Plane, Springer (1998 [1975]), p. 225; also in Stanley Gudder, A Mathematical Journey, McGraw-Hill (1976), p. 36.

“have enumerated, in a compendious and systematical form, the principal facts which have been discovered with respect to galvanic electricity; and I have fortunately been able to profit by Mr. Davy's most important experiments, which have lately been communicated to the Royal Society, and which have already given to this branch of science a much greater perfection, and a far greater extent, than it before possessed.”

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

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“It is absolutely in the interest of India to have a vigorous school of research in fundamental physics, for such a school forms the spearhead of research not only in less advanced branches of physics but also in problems of immediate practical application in industry.”

Homi J. Bhabha (1909–1966) 1909-1966, Indian nuclear physicist

In his proposal to the Sir Dorab Tata Trust for establishing a "vigorous school of research in fundamental physics [in India]", which would later be the Tata Institute of Fundamental Research; as quoted in the "Homi Jehangir Bhabha" profile at the Vigyan Prasar Science Portal.
Context: There is at the moment in India no big school of research in the fundamental problems of physics, both theoretical and experimental. There are, however, scattered all over India competent workers who are not doing as good work as they would do if brought together in one place under proper direction. It is absolutely in the interest of India to have a vigorous school of research in fundamental physics, for such a school forms the spearhead of research not only in less advanced branches of physics but also in problems of immediate practical application in industry. If much of the applied research done in India today is disappointing or of very inferior quality it is entirely due to the absence of sufficient number of outstanding pure research workers who would set the standard of good research and act on the directing boards in an advisory capacity … Moreover, when nuclear energy has been successfully applied for power production in say a couple of decades from now, India will not have to look abroad for its experts but will find them ready at hand. I do not think that anyone acquainted with scientific development in other countries would deny the need in India for such a school as I propose.
The subjects on which research and advanced teaching would be done would be theoretical physics, especially on fundamental problems and with special reference to cosmic rays and nuclear physics, and experimental research on cosmic rays. It is neither possible nor desirable to separate nuclear physics from cosmic rays since the two are closely connected theoretically.

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Leonardo Da Vinci (1452–1519) Italian Renaissance polymath

The Notebooks of Leonardo da Vinci (1883), I Prolegomena and General Introduction to the Book on Painting

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