"The mathematician of to-day admits that he can neither square the circle, duplicate the cube or trisect the angle. May not our mechanicians, in like manner, be ultimately forced to admit that aerial flight is one of that great class of problems with which men can never cope. [...] I do not claim that this is a necessary conclusion from any past experience. But I do think that success must await progress of a different kind from that of invention." (Simon Newcomb, "Side-lights on Astronomy and Kindred Fields of Popular Science", 1906)
"Architecture is the masterly, correct and magnificent play of masses brought together in light. Our eyes are made to see forms in light; light and shade reveal these forms; cubes, cones, spheres, cylinders or pyramids are the great primary forms which light reveals to advantage; the image of these is distinct and tangible within us without ambiguity. It is for this reason that these are beautiful forms, the most beautiful forms. Everybody is agreed to that, the child, the savage and the metaphysician." (Charles-Edouard Jeanneret [Le Corbusier], "Towards a New Architecture", 1923)
"Geometry is the study of form and shape. Our first encounter with it usually involves such figures as triangles, squares, and circles, or solids such as the cube, the cylinder, and the sphere. These objects all have finite dimensions of length, area, and volume - as do most of the objects around us. At first thought, then, the notion of infinity seems quite removed from ordinary geometry. That this is not so can already be seen from the simplest of all geometric figures - the straight line. A line stretches to infinity in both directions, and we may think of it as a means to go 'far out' in a one-dimensional world."
"Bivalence trades accuracy for simplicity. Binary outcomes of yes and no, white and black, true and false simplify math and computer processing. You can work with strings of 0s and 1s more easily than you can work with fractions. But bivalence requires some force fitting and rounding off [...] Bivalence holds at cube corners. Multivalence holds everywhere else." (Bart Kosko, "Fuzzy Thinking: The new science of fuzzy logic", 1993)
"Fuzzy entropy measures the fuzziness of a fuzzy set. It answers the question 'How fuzzy is a fuzzy set?' And it is a matter of degree. Some fuzzy sets are fuzzier than others. Entropy means the uncertainty or disorder in a system. A set describes a system or collection of things. When the set is fuzzy, when elements belong to it to some degree, the set is uncertain or vague to some degree. Fuzzy entropy measures this degree. And it is simple enough that you can see it in a picture of a cube." (Bart Kosko, "Fuzzy Thinking: The new science of fuzzy logic", 1993)
"Topology is a geometry in which all lengths, angles, and areas can be distorted at will. Thus a triangle can be continuously transformed into a rectangle, the rectangle into a square, the square into a circle, and so on. Similarly, a cube can be transformed into a cylinder, the cylinder into a cone, the cone into a sphere. Because of these continuous transformations, topology is known popularly as 'rubber sheet geometry'. All figures that can be transformed into each other by continuous bending, stretching, and twisting are called 'topologically equivalent'." (Fritjof Capra, "The Systems View of Life: A Unifying Vision", 2014)
"Ontological mathematics is operating in such a way as to organize itself into a zero-entropy structure – mathematical perfection. The 'Big Bang' is equivalent to the total scrambling of a cosmic Rubik’s Cube. The task of ontological mathematics is then to unscramble the Cube and return it to its original, pristine configuration. Emotionally, this amounts to returning to perfect Love and Bliss. Intellectually, it means reaching a state of perfect logic and reason [...] thinking perfectly." (Thomas Stark, "God Is Mathematics: The Proofs of the Eternal Existence of Mathematics", 2018)
"This method of subjecting the infinite to algebraic manipulations is called differential and integral calculus. It is the art of numbering and measuring with precision things the existence of which we cannot even conceive. Indeed, would you not think that you are being laughed at, when told that there are lines infinitely great which form infinitely small angles? Or that a line which is straight so long as it is finite would, by changing its direction infinitely little, become an infinite curve? Or that there are infinite squares, infinite cubes, and infinities of infinities, one greater than another, and that, as compared with the ultimate infinitude, those which precede it are as nought. All these things at first appear as excess of frenzy; yet, they bespeak the great scope and subtlety of the human spirit, for they have led to the discovery of truths hitherto undreamt of." (Voltaire)
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