“We see that experience plays an indispensable role in the genesis of geometry; but it would be an error thence to conclude that geometry is, even in part, an experimental science. If it were experimental it would be only approximative and provisional. And what rough approximation!
…The object of geometry is the study of a particular 'group'; but the general group concept pre-exists… in our minds. It is imposed on us, not as form of our sense, but as form of our understanding. Only, from among all the possible groups, that must be chosen… will be… the standard to which we shall refer natural phenomena.
Experience guides us in this choice without forcing it upon us; it tells us not which is the truest geometry, but which is the most convenient.
Notice that I have been able to describe the fantastic worlds… imagined without ceasing to employ the language of ordinary geometry.”

— Henri Poincaré , book Science and Hypothesis

Source: Science and Hypothesis (1901), Ch. IV: Space and Geometry, Conclusions (1905) Tr. https://books.google.com/books?id=5nQSAAAAYAAJ George Bruce Halstead

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Henri Poincaré 49

French mathematician, physicist, engineer, and philosopher … 1854–1912

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“The objects of abstract Geometry possess in absolute precision properties which are only approximately realized in the corresponding objects of physical Geometry.”

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

Source: Squaring the Circle (1913), p. 5

“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)

“From a very early age, we form concepts. Each concept is a particular idea or understanding we have about our world. These concepts allow us to make sense of and reason about the things in our world. These things to which our concepts apply are called objects.”

James Martin (author) (1933–2013) British information technology consultant and writer

James Martin (1993, p. 17) as cited in: " CIS330 Object Oriented Approach Ch2 http://webcadnet.blogspot.nl/2011/04/cis330-object-oriented-approach-text_3598.html" webcadnet.blogspot.nl. 2011/04/16

“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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“In string theory one studies strings moving in a fixed classical spacetime. …what we call a background-dependent approach. …One of the fundamental discoveries of Einstein is that there is no fixed background. The very geometry of space and time is a dynamical system that evolves in time. The experimental observations that energy leaks from binary pulsars in the form of gravitational waves—at the rate predicted by general relativity to the… accuracy of eleven decimal places—tell us that there is no more a fixed background of spacetime geometry than there are fixed crystal spheres holding the planets up.”

Lee Smolin (1955) American cosmologist

"Loop Quantum Gravity," The New Humanists: Science at the Edge (2003)

“The word science of administration has been used. There are many who object to the term. Now if by science is meant a conceptual scheme of things in which every particularity coveted may be assigned a mathematical value, then administration is not a science. In this sense only astro-physics may be called a science and it is well to remember that mechanical laws of the heavens tell us nothing about the color and composition of the stars and as yet cannot account for some of the disturbances and explosions which seem accidental. If, on the other hand, we may rightly use the term science in connection with a body of exact knowledge derived from experience and observation, and a body of rules or axioms which experience has demonstrated to be applicable in concrete practice, and to work out in practice approximately as forecast, then we may, if we please, appropriately and for convenience, speak of a science of administration. Once, when the great French mathematician, Poincaré, was asked whether Euclidean geometry is true, he replied that the question had no sense but that Euclidean geometry is and still remains the most convenient. The Oxford English Dictionary tells us that a science is, among other things, a particular branch of knowledge or study; a recognized department of learning.”

Charles A. Beard (1874–1948) American historian

Source: Philosophy, Science and Art of Public Administration (1939), p. 660-1

“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

“Nature is our kindest friend and best critic in experimental science if we only allow her intimations to fall unbiased on our minds.”

Michael Faraday (1791–1867) English scientist

Letter to John Tyndall (19 April 1851); letter 2411, edited by
Context: I have far more confidence in the one man who works mentally and bodily at a matter than in the six who merely talk about it — and I therefore hope and am fully persuaded that you are working. Nature is our kindest friend and best critic in experimental science if we only allow her intimations to fall unbiased on our minds. Nothing is so good as an experiment which, whilst it sets an error right, gives us (as a reward for our humility in being reproved) an absolute advancement in knowledge.

“But if some mind very different from ours were to look upon some property of some curved line as we do on the evenness of a straight line, he would not recognize as such the evenness of a straight line; nor would he arrange the elements of his geometry according to that very different system, and would investigate quite other relationships as I have suggested in my notes.
We fashion our geometry on the properties of a straight line because that seems to us to be the simplest of all. But really all lines that are continuous and of a uniform nature are just as simple as one another. Another kind of mind which might form an equally clear mental perception of some property of any one of these curves, as we do of the congruence of a straight line, might believe these curves to be the simplest of all, and from that property of these curves build up the elements of a very different geometry, referring all other curves to that one, just as we compare them to a straight line. Indeed, these minds, if they noticed and formed an extremely clear perception of some property of, say, the parabola, would not seek, as our geometers do, to rectify the parabola, they would endeavor, if one may coin the expression, to parabolify the straight line.”

Roger Joseph Boscovich (1711–1787) Croat-Italian physicist

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"Non-Newtonian Calculus" (1972) by Michael Grossman and Robert Katz.

“Visual forms are not perceived differently from colors or brightness. They are sense qualities, and the visual character of geometry consists in these sense qualities.”

Hans Reichenbach (1891–1953) American philosopher

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

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