Great Throughts Treasury

"Euclid taught me that without assumptions there is no proof. Therefore, in any argument, examine the assumptions."

"Guided only by their feeling for symmetry, simplicity, and generality, and an indefinable sense of the fitness of things, creative mathematicians now, as in the past, are inspired by the art of mathematics rather than by any prospect of ultimate usefulness. "

"Fashion as King is sometimes a very stupid ruler. "

"Science makes no pretension to eternal truth or absolute truth. "

"Time makes fools of us all. Our only comfort is that greater shall come after us."

""Obvious" is the most dangerous word in mathematics."

"Abstractness, sometimes hurled as a reproach at mathematics, is its chief glory and its surest title to practical usefulness. It is also the source of such beauty as may spring from mathematics."

"I have always hated machinery, and the only machine I ever understood was a wheelbarrow, and that but imperfectly."

"[As a young teenager] Galois read [Legendre's] geometry from cover to cover as easily as other boys read a pirate yarn."

"If a lunatic scribbles a jumble of mathematical symbols it does not follow that the writing means anything merely because to the inexpert eye it is indistinguishable from higher mathematics."

"If "Number rules the universe" as Pythagoras asserted, Number is merely our delegate to the throne, for we rule Number."

"If indeed, as Hilbert asserted, mathematics is a meaningless game played with meaningless marks on paper, the only mathematical experience to which we can refer is the making of marks on paper."

"In his wretched life of less than twenty-seven years Abel accomplished so much of the highest order that one of the leading mathematicians of the Nineteenth Century (Hermite, 1822-1901) could say without exaggeration, 'Abel has left mathematicians enough to keep them busy for five hundred years.' Asked how he had done all this in the six or seven years of his working life, Abel replied, 'By studying the masters, not the pupils.'"

"It is the perennial youthfulness of mathematics itself which marks it off with a disconcerting immortality from the other sciences."

"'Obvious' is the most dangerous word in mathematics."

"Nevertheless, the consuming hunger of the uncritical mind for what it imagines to be certainty or finality impels it to feast upon shadows in the prevailing famine of substance"

"The cowboys have a way of trussing up a steer or a pugnacious bronco which fixes the brute so that it can neither move nor think. This is the hog-tie, and it is what Euclid did to geometry."

"Poincaré was a vigorous opponent of the theory that all mathematics can be rewritten in terms of the most elementary notions of classical logic; something more than logic, he believed, makes mathematics what it is."

"Out of fifty mathematical papers presented in brief at such a meeting, it is a rare mathematician indeed who really understands what more than half a dozen are about."

"The longer mathematics lives the more abstract - and therefore, possibly also the more practical - it becomes."

"The mistakes and unresolved difficulties of the past in mathematics have always been the opportunities of its future."

"The pursuit of pretty formulas and neat theorems can no doubt quickly degenerate into a silly vice, but so can the quest for austere generalities which are so very general indeed that they are incapable of application to any particular."

"The only royal road to elementary geometry is ingenuity."

"Wherever groups disclosed themselves, or could be introduced, simplicity crystallized out of comparative chaos."