20 quotes
The so-called obvious was repeatedly scrutinized from every angle and was frequently found to be not obvious but false. "Obvious" is the most dangerous word in mathematics.
Mathematics: Queen and Servant of Science (1938)
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.
The Development of Mathematics (1940)
Nearly always it is the recondite and complicated which is elaborated first; and it is only when some relatively unsophisticated mind attacks a problem that its deep simplicity is revealed.
The Development of Mathematics (1940)
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.
Mathematics: Queen and Servant of Science (1938)
Science makes no pretension to eternal truth or absolute truth; some of its rivals do. That science is in some respects inhuman may be the secret of its success in alleviating human misery and mitigating human stupidity.
Mathematics: Queen and Servant of Science (1938)
Wherever groups disclosed themselves, or could be introduced, simplicity crystallized out of comparative chaos.
Mathematics: Queen and Servant of Science (1938)
All computation was greatly simplified early in the seventeenth century by the invention of logarithms.
The Development of Mathematics (1940)
There are no absolutes... in mathematics or in its history.
The Development of Mathematics (1940)
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.
The Search for Truth (1934)
The mistakes and unresolved difficulties of the past in mathematics have always been the opportunities of its future; and should analysis ever appear to be without or blemish, its perfection might only be that of death.
The Development of Mathematics (1940)
In the entire history of Greek mathematics, all but the incomparable Archimedes and a few of the more heterodox sophists appear to have hated or feared the mathematical infinite. Analysis was thwarted when it might have prospered.
The Development of Mathematics (1940)
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.
Euclid taught me that without assumptions there is no proof. Therefore, in any argument, examine the assumptions. Then, in the alleged proof, be alert for inexplicit assumptions. Euclid's notorious oversights drove this lesson home.
Something more than impeccable logic is required in mathematics. An expert logician will not necessarily be a passable mathematician for all his skill in logic, any more than a scholarly prosodist will be a respectable poet for all his mastery of meter.
Whatever its source, mathematics has come down to the present by the two main streams of number and form. The first carried along arithmetic and algebra, the second, geometry. In the seventeenth century these two united, forming the ever-broadening river of mathematical analysis.
The Development of Mathematics (1940)
Each of five men—Lobachewsky, Bolyai, Plücker, Riemann, Lie—invented as part of his lifework as much (or more) new geometry as was created by all the Greek mathematicians in the two or three centuries of their greatest activity.
The Development of Mathematics (1940)
Some of his deepest discoveries were reasoned out verbally with very few if any symbols, and those for the most part mere abbreviations of words. Any impatient student of mathematics or science or engineering who is irked by having algebraic symbolism thrust on him should try to get on without it for a week.
Mathematics: Queen and Servant of Science (1938)
Fashion as king is sometimes a very stupid ruler. As was observed a little way back, the kernel of Plücker's theory of geometric dimensionality is that the dimensionality of a given space is not an absolute constant, but depends upon the elements, accepted as irreducible, in terms of which the space is described.
Mathematics: Queen and Servant of Science (1938)
With the intrusion of irrational numbers to disrupt the integral harmonies of the Pythagorean cosmos, a controversy that has raged off and on for well over two thousand years began: is the mathematical infinite a safe concept in mathematical reasoning, safe in the sense that contradictions will not result from the use of this infinite, subject to certain prescribed conditions?
The Development of Mathematics (1940)
The pursuit of pretty formulas and neat theorems can no doubt quickly degenerate into a silly vice, but so also can the quest for austere generalities which are so very general indeed that they are incapable of application to any particular.
Men of Mathematics (1937)