Euclid taught me that without assumptions there is no proof. Therefore, in any argument, examine the assumptions.
Wherever groups disclosed themselves, or could be introduced, simplicity crystallized out of comparative chaos.
It is the perennial youthfulness of mathematics itself which marks it off with a disconcerting immortality from the other sciences.
The Handmaiden of the Sciences.
[Book by that title.]
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.
"Obvious" is the most dangerous word in mathematics.
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.
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.
Any impatient student of mathematics or science or engineering who is irked by having algebraic symbolism thrust upon him should try to get along without it for a week.
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.
The longer mathematics lives the more abstract -- and therefore, possibly also the more practical -- it becomes.
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.
If "Number rules the universe" as Pythagoras asserted, Number is merely our delegate to the throne, for we rule Number.
I have always hated machinery, and the only machine I ever understood was a wheelbarrow, and that but imperfectly.
Archimedes, Newton, and Gauss, these three, are in a class by
themselves among the great mathematicians, and it is not for
ordinary mortals to attempt to range them in order of merit.
Had Poincaré been as strong in practical science as he was in theoretical he might have made a fourth with the incomparable
three, Archimedes, Newton, and Gauss.
[As a young teenager] Galois read [Legendre's] geometry from cover to cover as easily as other boys read a pirate yarn.
Even stranger things have happened; and perhaps the strangest of all is the marvel that mathematics should be possible to a race akin to the apes.
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.'