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The tantalizing and compelling pursuit of mathematical problems offers mental absorption, peace of mind amid endless challenges, repose in activity, battle without conflict, refuge from the goading urgency of contingent happenings, and the sort of beauty changeless mountains present to senses tried by the present day kaleidoscope of events.
One cannot escape the feeling that these mathematical formulas have an independent existence and an intelligence of their own, that they are wiser than we are, wiser even than their discoverers.
The infinite in mathematics is always unruly unless it is properly treated.
It is the duty of all teachers, and of teachers of mathematics in particular, to expose their students to problems much more than to facts.
Biographical history, as taught in our public schools, is still largely a history of boneheads; ridiculous kings and queens, paranoid political leaders, compulsive voyagers, ignorant general the flotsam and jetsam of historical currents. The men who radically altered history, the great scientists and mathematicians, are seldom mentioned, if at all.
Mathematics is no more computation than typing is literature.
I love mathematics...principally because it is beautiful; because man has breathed his spirit of play into it, and because it has given him his greatest game the encompassing of the infinite.
In most sciences one generation tears down what another has built and what one has established another undoes. In mathematics alone each generations adds a new story to the old structure.
...it is the greatest achievement of a teacher to enable his students to surpass him.
The only way to learn mathematics is to do mathematics.
Mathematics compares the most diverse phenomena and discovers the secret analogies that unite them.
If present trends continue, our country may soon find itself far behind many other nations in both science and technology nations where, if you inform strangers that you are a mathematician, they respond with admiration and not by telling you how much they hated math in school, and how they sure could use you to balance their checkbooks.
There is a distinction between what may be called a problem and what may be considered an exercise. The latter serves to drill a student in some technique or procedure, and requires little if any, original thought... No exercise, then, can always be done with reasonbable dispatch and with a miniumum of creative thinking. In contrast to an exercise, a problem, if it is a good one for its level, should require though on the part of the student.
It is impossible to overstate the imporance of problems in mathematics. It is by means of problems that mathematics develops and actually lifts itself by its own bootstraps... Every new discovery in mathematics, results from an attempt to solve some problem.
The value of a problem is not so much coming up with the answer as in the ideas and attempted ideas it forces on the would be solver.
The profound study of nature is the most fertile source of mathematical discovery.
For me it remains an open question whether [this work] pertains to the realm of mathematics or to that of art.
By keenly confronting the enigmas that surround us, and by considering and analyzing the observations that I had made, I ended up in the domain of mathematics. Although I am absolutely without training in the exact sciences, I often seem to have more in common with mathematicians than with my fellow artists.
Geometry is the foundation of all painting.
Although he may not always recognize his bondage, modern man lives under a tyranny of numbers.
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