“Phys. I know that it is often a help to represent pressure and volume as height and width on paper; and so geometry may have applications to the theory of gases. But is it not going rather far to say that geometry can deal directly with these things and is not necessarily concerned with lengths in space?
Math. No. Geometry is nowadays largely analytical, so that in form as well as in effect, it deals with variables of an unknown nature. …It is literally true that I do not want to know the significance of the variables x, y, z, t that I am discussing. …
Phys. Yours is a strange subject. You told us at the beginning that you are not concerned as to whether your propositions are true, and now you tell us you do not even care to know what you are talking about.
Math.”

— Arthur Stanley Eddington

That is an excellent description of Pure Mathematics, which has already been given by an eminent mathematician <nowiki>[</nowiki>Bertrand Russell<nowiki>]</nowiki>.
Space, Time and Gravitation (1920)

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Arthur Stanley Eddington105

British astrophysicist 1882–1944

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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)

“Geometry can in no way be viewed… as a branch of mathematics; instead, geometry relates to something already given in nature, namely, space. I… realized that there must be a branch of mathematics which yields in a purely abstract way laws similar to geometry.”

Hermann Grassmann (1809–1877) German polymath, linguist and mathematician

Forward, as quoted by Mario Livio, Is God a Mathematician? (2009)
Ausdehnungslehre (1844)

“M. Desargues puts me under obligations on account of the pains that it has pleased him to have in me, in that he shows that he is sorry that I do not wish to study more in geometry, but I have resolved to quit only abstract geometry, that is to say, the consideration of questions which serve only to exercise the mind, and this, in order to study another kind of geometry, which has for its object the explanation of the phenomena of nature… You know that all my physics is nothing else than geometry.”

René Descartes (1596–1650) French philosopher, mathematician, and scientist

Letter to Marin Mersenne (July 27, 1638) as quoted by Florian Cajori, A History of Mathematics (1893) letter dated in The Philosophical Writings of Descartes Vol. 3, The Correspondence (1991) ed. John Cottingham, Robert Stoothoff, Dugald Murdoch

“My research work was based in pictures. I needed to visualise things and projective geometry was often most useful e. g. in figuring out how a particular quantity transforms under Lorentz transf[ormation]. When I came to publish the results I suppressed the projective geometry as the results could be expressed more concisely in analytic form.”

Paul Dirac (1902–1984) theoretical physicist

"Recollections of an Exciting Era," three lectures given at Varenna, 5 August 1972, quoted in Peter Galison, "The Suppressed Drawing: Paul Dirac's Hidden Geometry", Representations, No. 72 (Autumn, 2000)

“Riemann has shewn that as there are different kinds of lines and surfaces, so there are different kinds of space of three dimensions; and that we can only find out by experience to which of these kinds the space in which we live belongs. In particular, the axioms of plane geometry are true within the limits of experiment on the surface of a sheet of paper, and yet we know that the sheet is really covered with a number of small ridges and furrows, upon which (the total curvature not being zero) these axioms are not true. Similarly, he says although the axioms of solid geometry are true within the limits of experiment for finite portions of our space, yet we have no reason to conclude that they are true for very small portions; and if any help can be got thereby for the explanation of physical phenomena, we may have reason to conclude that they are not true for very small portions of space.”

William Kingdon Clifford On the Space-Theory of Matter

Abstract
On the Space-Theory of Matter (read Feb 21, 1870)

“Euclidean geometry is only one of several congruence geometries… Each of these geometries is characterized by a real number K, which for Euclidean geometry is 0, for the hyperbolic negative, and for the spherical and elliptic geometries, positive. In the case of 2-dimensional congruence spaces… K may be interpreted as the curvature of the surface into the third dimension—whence it derives its name…”

Howard P. Robertson (1903–1961) American mathematician and physicist

Geometry as a Branch of Physics (1949)

“Newton… (after having remarked that geometry only requires two of the mechanical actions which it postulates, namely, to describe a straight line and a circle) says: geometry is proud of being able to achieve so much while taking so little from extraneous sources. One might say of metaphysics, on the other hand: it stands astonished, that with so much offered it by pure mathematics it can effect so little. In the meantime, this little is something which mathematics indispensably requires in its application to natural science, which, inasmuch as it must here necessarily borrow from metaphysics, need not be ashamed to allow itself to be seen in company with the latter.”

Immanuel Kant (1724–1804) German philosopher

Preface, Tr. Bax (1883) citing Isaac Newton's Principia
(1786)

“Vladimir Voevodsky and his collaborators have provided us with a very interesting candidate-category of motives: a category (of sheaves relative to an extraordinarily fine Grothendieck-style topology on the category of schemes) which in some intuitive sense “softens algebraic geometry” so as to allow for a good notion of homotopy in an algebro-geometric setup and is sufficiently directly connected to concrete algebraic geometry to have already yielded some extraordinary applications. The quest for a full theory of motives is a potent driving force in complex analysis, algebraic geometry, automorphic representation theory, the study of L functions, and arithmetic. It will continue to be so throughout the current century.”

Barry Mazur (1937) American mathematician

"WHAT IS...a Motive?" 2004

“The last thirty years [1925 to 1955] have seen an enormous improvement in the position of geometry as a branch of mathematics, or, rather, have seen the re-integration of geometry into the main fabric of mathematics. Indeed, one can go further and say that with the restoration of geometry to its rightful place in the mathematical scheme the process of fragmentation which had been doing so much harm to mathematics has been reversed, and we may look forward to the day in which there are no longer analysts, algebraists, geometers and so on, but simply mathematicians. Mathematical research has two aspects, motivation and technique, and when the latter gains control the result is apt to be excessive specialisation. The revolution of geometrical thought, and the reinstatement of geometry as one of the major mathematical disciplines, have helped to bring about a unification of mathematics which we may justly regard as one of the major contributions of the last quarter century to the subject.”

W. V. D. Hodge (1903–1975) British mathematician

W. V. D. Hodge, Changing Views of Geometry. Presidential Address to the Mathematical Association, 14th April, 1955, The Mathematical Gazette 39 (329) (1955), 177-183.

“Nevertheless, it remains conceivable that the measure relations of space in the infinitely small are not in accordance with the assumptions of our geometry [Euclidean geometry], and, in fact, we should have to assume that they are not if, by doing so, we should ever be enabled to explain phenomena in a more simple way.”

Bernhard Riemann (1826–1866) German mathematician

Memoir (1854) Tr. William Kingdon Clifford, as quoted by A. D'Abro, The Evolution of Scientific Thought from Newton to Einstein https://archive.org/details/TheEvolutionOfScientificThought (1927) p. 55.

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Arthur Stanley Eddington105

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