“First, the physicists in the persons of Faraday and Maxwell, proposed the "electromagnetic field" in contradistinction to matter, as a reality of a different category. Then, during the last century, the mathematicians, … secretly undermined belief in the evidence of Euclidean Geometry. And now, in our time, there has been unloosed a cataclysm which has swept away space, time, and matter hitherto regarded as the firmest pillars of natural science, but only to make place for a view of things of wider scope and entailing a deeper vision. This revolution was promoted essentially by the thought of one man, Albert Einstein.”

— Hermann Weyl

Introduction
Space—Time—Matter (1952)

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Hermann Weyl 28

German mathematician 1885–1955

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“Euclidean geometry can be easily visualized; this is the argument adduced for the unique position of Euclidean geometry in mathematics. It has been argued that mathematics is not only a science of implications but that it has to establish preference for one particular axiomatic system. Whereas physics bases this choice on observation and experimentation, i. e., on applicability to reality, mathematics bases it on visualization, the analogue to perception in a theoretical science. Accordingly, mathematicians may work with the non-Euclidean geometries, but in contrast to Euclidean geometry, which is said to be "intuitively understood," these systems consist of nothing but "logical relations" or "artificial manifolds". They belong to the field of analytic geometry, the study of manifolds and equations between variables, but not to geometry in the real sense which has a visual significance.”

Hans Reichenbach (1891–1953) American philosopher

The Philosophy of Space and Time (1928, tr. 1957)

“In electromagnetism… the law of the inverse square had been supreme, but, as a consequence of the work of Faraday and Maxwell, it was superseded by the field. And the same change took place in the theory of gravitation. By and by the material particles, electrically charged bodies, and magnets which are the things that we actually observe come to be looked upon only as "singularities" in the field. So far this transformation from the force to the potential, from the action at a distance to the field, is only a purely mathematical operation.”

Willem de Sitter (1872–1934) Dutch cosmologist

Kosmos (1932), Above is Beginning Quote of the Last Chapter: Relativity and Modern Theories of the Universe -->

“Although the Special Theory of Relativity does not account for electromagnetic phenomena, it explains many of their properties. General Relativity, however, tells us nothing about electromagnetism. In Einstein's space-time continuum gravitational forces are absorbed in the geometry, but the electromagnetic forces are quite unaffected. Various attempts have been made to generate the geometry of space-time so as to produce a unified field theory incorporating both gravitational and electromagnetic forces.”

Gerald James Whitrow (1912–2000) British mathematician

p, 125
The Structure of the Universe: An Introduction to Cosmology (1949)

“Mathematicians have constructed a very large number of different systems of geometry, Euclidean or non-Euclidean, of one, two, three, or any number of dimensions. All these systems are of complete and equal validity. They embody the results of mathematicians' observations of their reality, a reality far more intense and far more rigid than the dubious and elusive reality of physics. The old-fashioned geometry of Euclid, the entertaining seven-point geometry of Veblen, the space-times of Minkowski and Einstein, are all absolutely and equally real. …There may be three dimensions in this room and five next door. As a professional mathematician, I have no idea; I can only ask some competent physicist to instruct me in the facts.
The function of a mathematician, then, is simply to observe the facts about his own intricate system of reality, that astonishingly beautiful complex of logical relations which forms the subject-matter of his science, as if he were an explorer looking at a distant range of mountains, and to record the results of his observations in a series of maps, each of which is a branch of pure mathematics. …Among them there perhaps none quite so fascinating, with quite the astonishing contrasts of sharp outline and shade, as that which constitutes the theory of numbers.”

G. H. Hardy (1877–1947) British mathematician

"The Theory of Numbers," Nature (Sep 16, 1922) Vol. 110 https://books.google.com/books?id=1bMzAQAAMAAJ p. 381

“Our understanding of the four basic concepts of Physics -- space, time, matter and force -- has undergone radical change in the course of work on unification, starting with Maxwell's unification of electricity with magnetism, all the way to present day string theory. What started as four independent concepts, with space and time postulated and the possible forms of matter and force arbitrarily chosen, now appear as different aspects of a rich and novel dynamically determined structure.”

Peter Freund (1936–2018) American physicist

Physics and Geometry, a paper written for the Symposium on Theoretical Physics at the University of Helsinki, Helsinki, Finland on August 28, 2003 and at the Freydoon Mansouri Memorial Session of the 3rd International Symposium on Quantum Theory and Symmetries at the University of Cincinnati, Cincinnati, OH, on September 13, 2003. Report #EFI03-47.

“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.

“It is not that science proves that the world is not moral. Science as such is not concerned with values, and proves nothing of the sort. Nevertheless, as a matter of history, it is science which has suggested this view to the modern mind by its insistence on mechanistic categories.”

Walter Terence Stace (1886–1967) British civil servant, educator and philosopher.

“The science of arithmetic may be called the science of exact limitation of matter and things in space, force, and time.”

Francis Wayland Parker (1837–1902) Union Army officer

Source: Talks on Pedagogics, (1894), p. 64. Reported in Robert Edouard Moritz. Memorabilia mathematica; or, The philomath's quotation-book https://archive.org/stream/memorabiliamathe00moriiala#page/81/mode/2up, (1914), p. 263

“My main professional interest during the 1970s has been in the dramatic change of concepts and ideas that has occurred in physics during the first three decades of the century, and that is still being elaborated in our current theories of matter. The new concepts in physics have brought about a profound change in our world view; from the mechanistic conception of Descartes and Newton to a holistic and ecological view, a view which I have found to be similar to the views of mystics of all ages and traditions.”

Fritjof Capra book The Turning Point

Preface.
The Turning Point (1982)

“Philosophy's position with regard to science, which at one time could be designated with the name "theory of knowledge," has been undermined by the movement of philosophical thought itself. Philosophy was dislodged from this position by philosophy.”

Jürgen Habermas book Knowledge and Human Interests

Source: Knowledge and Human Interests, 1971, p. 4

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