# „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.“

## Citações relacionadas

### „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 German polymath, linguist and mathematician 1809 - 1877

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

### „Development of Western Science is based on two great achievements, the invention of the formal logical system (in Euclidean geometry) by the Greek philosophers, and the discovery of the possibility to find out causal relationships by systematic experiment (Renaissance). In my opinion one has not to be astonished that the Chinese sages have not made these steps. The astonishing thing is that these discoveries were made at all.“

— Albert Einstein German-born physicist and founder of the theory of relativity 1879 - 1955

Letter to J.S. Switzer (23 April 1953), quoted in The Scientific Revolution: a Hstoriographical Inquiry By H. Floris Cohen (1994), p. 234 http://books.google.com/books?id=wu8b2NAqnb0C&lpg=PP1&pg=PA234#v=onepage&q&f=false, and also partly quoted in The Ultimate Quotable Einstein edited by Alice Calaprice (2010), p. 405 http://books.google.com/books?id=G_iziBAPXtEC&lpg=PP1&pg=PA405#v=onepage&q&f=false

### „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 British mathematician 1877 - 1947

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

### „What is the true geometry of the plate?... Anyone examining the situation will prefer Poincaré's common-sense solution... to attribute it Euclidean geometry, and to consider the measured deviations... as due to the actions of a force (thermal stresses in the rule).... On employing a brass rule in place of one of steel we would find that the local curvature is trebled—and an ideal rule (c = 0) would... lead to Euclidean geometry.“

— Howard P. Robertson American mathematician and physicist 1903 - 1961

### „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) Scottish publisher and writer 1802 - 1871

Robert Chambers, Chambers's Information for the People (1875) Vol. 2 https://books.google.com/books?id=vNpTAAAAYAAJ

### „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 British mathematician 1903 - 1975

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.

### „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 American historian 1874 - 1948

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 French philosopher, mathematician, and scientist 1596 - 1650

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

### „Equations are just the boring part of mathematics. I attempt to see things in terms of geometry.“

— Stephen Hawking British theoretical physicist, cosmologist, and author 1942

As quoted in Stephen Hawking: A Biography (2005) by Kristine Larsen, p. 43

### „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 German mathematician 1885 - 1955

Introduction<!-- p. 2 -->

### „More than any of his predecessors Plato appreciated the scientific possibilities of geometry... By his teaching he laid the foundations of the science, insisting upon accurate definitions, clear assumptions, and logical proof. His opposition to the materialists, who saw in geometry only what was immediately useful to the artisan and the mechanic is... clear.... That Plato should hold the view... is not a cause for surprise. The world's thinkers have always held it. No man has ever created a mathematical theory for practical purposes alone. The applications of mathematics have generally been an afterthought.“

— David Eugene Smith American mathematician 1860 - 1944

p. 90