"Equiprobability in the physical world is purely a hypothesis. We may exercise the greatest care and the most accurate of scientific instruments to determine whether or not a penny is symmetrical. Even if we are satisfied that it is, and that our evidence on that point is conclusive, our knowledge, or rather our ignorance, about the vast number of other causes which affect the fall of the penny is so abysmal that the fact of the penny’s symmetry is a mere detail. Thus, the statement 'head and tail are equiprobable' is at best an assumption." (Edward Kasner & James R Newman, "Mathematics and the Imagination", 1940)
"In moderation, gambling possesses undeniable virtues. Yet it presents a curious spectacle replete with contradictions. While indulgence in its pleasures has always lain beyond the pale of fear of Hell’s fires, the great laboratories and respectable insurance palaces stand as monuments to a science originally born of the dice cup." (Edward Kasner & James R Newman, "Mathematics and the Imagination", 1940)
"Mathematics is an activity governed by the same rules imposed upon the symphonies of Beethoven, the paintings of DaVinci, and the poetry of Homer. Just as scales, as the laws of perspective, as the rules of metre seem to lack fire, the formal rules of mathematics may appear to be without lustre. Yet ultimately, mathematics reaches pinnacles as high as those attained by the imagination in its most daring reconnoiters. And this conceals, perhaps, the ultimate paradox of science. For in their prosaic plodding both logic and mathematics often outstrip their advance guard and show that the world of pure reason is stranger than the world of pure fancy." (Edward Kasner & James R Newman, "Mathematics and the Imagination", 1940)
"Mathematics is often erroneously referred to as the science of common sense. Actually, it may transcend common sense and go beyond either imagination or intuition. It has become a very strange and perhaps frightening subject from the ordinary point of view, but anyone who penetrates into it will find a veritable fairyland, a fairyland which is strange, but makes sense, if not common sense." (Edward Kasner & James R Newman, "Mathematics and the Imagination", 1940)
"Mathematics is the science which uses easy words for hard ideas." (Edward Kasner & James R Newman, "Mathematics and the Imagination", 1940)
"One of the difficulties arising out of the subjective view of probability results from the principle of insufficient reasons. This principle [...] holds that if we are wholly ignorant of the different ways an event can occur and therefore have no reasonable ground for preference, it is as likely to occur one way as another." (Edward Kasner & James R Newman, "Mathematics and the Imagination", 1940)
"Perhaps the greatest paradox of all is that there are paradoxes in mathematics [...] because mathematics builds on the old but does not discard it, because its theorems are deduced from postulates by the methods of logic, in spite of its having undergone revolutionary changes we do not suspect it of being a discipline capable of engendering paradoxes." (Edward Kasner & James R Newman, "Mathematics and the Imagination", 1940)
"Puzzles are made of the things that the mathematician, no less than the child, plays with, and dreams and wonders about, for they are made of things and circumstances of the world he [or she] live in." (Edward Kasner & James R Newman, "Mathematics and the Imagination", 1940)
"Statements about impossibility in mathematics are of a wholly different character. A problem in mathematics which may not be solved for centuries to come is not always impossible. 'Impossible' in mathematics means theoretically impossible, and has nothing to do with the present state of our knowledge." (Edward Kasner & James R Newman, "Mathematics and the Imagination", 1940)
"The curves treated by the calculus are normal and healthy; they possess no idiosyncrasies. But mathematicians would not be happy merely with simple, lusty configurations. Beyond these their curiosity extends to psychopathic patients, each of whom has an individual case history resembling no other; these are the pathological curves in mathematics." (Edward Kasner & James R Newman, "Mathematics and the Imagination", 1940)
"The infinite in mathematics is always unruly unless it is properly treated." (Edward Kasner & James R Newman, "Mathematics and the Imagination", 1940)
"The mathematician is still regarded as the hermit who knows little of the ways of life outside his cell, who spends his time compounding incredible and incomprehensible theorems in a strange, clipped, unintelligible jargon." (Edward Kasner & James R Newman, "Mathematics and the Imagination", 1940)
"There is a famous formula, perhaps the most compact and famous of all formulas developed by Euler from a discovery of de Moivre: It appeals equally to the mystic, the scientist, the philosopher, the mathematician." (Edward Kasner & James R Newman, "Mathematics and the Imagination", 1940)
"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)
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