On Geometry (1950-1974)

"Probability is a mathematical discipline with aims akin to those, for example, of geometry or analytical mechanics. In each field we must carefully distinguish three aspects of the theory: (a) the formal logical content, (b) the intuitive background, (c) the applications. The character, and the charm, of the whole structure cannot be appreciated without considering all three aspects in their proper relation." (William Feller, "An Introduction to Probability Theory and Its Applications", 1950)

"Whoever thinks algebra is a trick in obtaining unknowns has thought it in vain. No attention should be paid to the fact that algebra and geometry are different in appearance. Algebras are geometric facts which are proved." (Omar Khayyam [quoted by J.J. Winter and W. Arafat, "The Algebra of ‘Umar Khayyam’", Journal of the Royal Asiatic Society of Bengal, Volume 41, 1950)

"Geometrical truth is a product of reason; that makes it superior to empirical truth, which is found through generalization of a great number of instances." (Hans Reichenbach, "The Rise of Scientific Philosophy", 1954)

“[…] no branch of mathematics competes with projective geometry in originality of ideas, coordination of intuition in discovery and rigor in proof, purity of thought, logical finish, elegance of proofs and comprehensiveness of concepts. The science born of art proved to be an art.” (Morris Kline, “Projective Geometry”, Scientific America Vol. 192 (1), 1955)

 "Conventionalism as geometrical and mathematical truths are created by our choices, not dictated by or imposed on us by scientific theory. The idea that geometrical truth is truth we create by the understanding of certain conventions in the discovery of non-Euclidean geometries." (Clifford Singer, "Engineering a Visual Field", 1955)

"Geometry, whatever others may think, is the study of different shapes, many of them very beautiful, having harmony, grace and symmetry. […] Most of us, if we can play chess at all, are content to play it on a board with wooden chess pieces; but there are some who play the game blindfolded and without touching the board. It might be a fair analogy to say that abstract geometry is like blindfold chess – it is a game played without concrete objects." (Edward Kasner & James R Newman, "New Names for Old", 1956)

"Geometry exists in its own right, and by its own strength. It can now treat accurately and coherently a range of forms and spaces that far exceeds anything that terrestrial space can provide. Today it is geometry that contains the terrestrial forms, and not vice versa, for the terrestrial forms are merely special cases in an all-embracing geometry. [...] Geometry now acts as a framework on which all terrestrial forms can find their natural place, with the relations between the various forms readily appreciable." (W Ross Ashby, "An Introduction to Cybernetics", 1956)

"A logic machine is a device, electrical or mechanical, designed specifically for solving problems in formal logic. A logic diagram is a geometrical method for doing the same thing. […] A logic diagram is a two-dimensional geometric figure with spatial relations that are isomorphic with the structure of a logical statement. These spatial relations are usually of a topological character, which is not surprising in view of the fact that logic relations are the primitive relations underlying all deductive reasoning and topological properties are, in a sense, the most fundamental properties of spatial structures. Logic diagrams stand in the same relation to logical algebras as the graphs of curves stand in relation to their algebraic formulas; they are simply other ways of symbolizing the same basic structure." (Martin Gardner, "Logic Machines and Diagrams", 1958)

"[...] it is clear that differential geometry, analysis and physics prompted the early development of differential topology (it is this that explains our admitted bias toward differential topology, that it lies close to the main stream of  mathematics). On the other hand, the combinatorial approach to manifolds was started because it was believed that these means would afford a useful attack on the differentiable case." (Steven Smale, "A survey of some recent developments in differential topology", 1961)

"Nature does not seem full of circles and triangles to the ungeometrical; rather, mastery of the theory of triangles and circles, and later of conic sections, has taught the theorist, the experimenter, the carpenter, and even the artist to find them everywhere, from the heavenly motions to the pose of a Venus." (Clifford Truesdell, "Six Lectures on Modern Natural Philosophy", 1966)

"After all, Greek thought is expressed not only mythically, in fiction, but also directly, in theorems. The gate through which the Greek world may be discussed - and without the knowledge of which, in my opinion, one’s culture can not be deemed complete - is not necessarily Homer. Greek geometry is a wider gate, through which the eye might grasp an austere, yet essential landscape." (Dan Barbilian, 1967)

"An instance of such a 'non-Euclidean' geometry is projective geometry, concerned with those properties of figures which do not change under projective transformations. Projective geometry is not merely not Euclidean geometry; it is 'very much non-Euclidean'." (Isaak Yaglom, "Geometric Transformations", 1973)

"If indeed one tries to clarify the notion of equality, which is introduced right at the beginning of Geometry, one is led to say that two figures are equal when one can go from one to the other by a specific geometric operation, called a motion. This is only a change of words; but the axiom according to which two figures equal to a third are equal to one another, subjects those operations called motions to a certain law; that is, that an operation which is the result of two successive motions is itself a motion. It is this law that mathematicians express by saying that motions form a group. Elementary Geometry can then be defined by the study of properties of figures which do not change under the operations of the group of motions." (Élie Cartan, "Notice sur les travaux scientifiques", 1974)

"[…] it is the whole logical structure of elementary Geometry which is contained in the group of motions and even, in a more precise manner, in the law according to which operations of that group compose with each other, independently of the nature of the objects on which these operations act. This law constitutes what we call the group structure." (Élie Cartan, "Notice sur les travaux scientifiques", 1974)

"The effect of magnitude or absolute size as a determinant of form shows again how space shapes the things around us. In studying polyhedrons we are unconcerned with magnitude. We assume that a cube is a cube no matter what its size. We find, however, that the geometric relations that arise from a difference in size affect structural behavior, and that a large cube is relatively weaker than a small cube. We also find, as a corollary, that in order to maintain the same structural characteristics a difference in size must be accompanied by a difference in shape." (Peter B Stevens, "Patterns in Nature", 1974)