If I am given a formula and I am ignorant of its meaning, it cannot teach me anything, but if I already know it what does the formula teach me?
A good puzzle should demand the exercise of our best wit and ingenuity, and although a knowledge of mathematics and of logic are often of great service in the solution of these things, yet it sometimes happens that a kind of natural cunning and sagacity is of considerable value.
Imagination is more important than knowledge.
The science of mathematics presents the most brilliant example of how pure reason may successfully enlarge its domain without the aid of experience.
When you can measure what you are talking about and express it in numbers, you know something about it.
The scientist does not study nature because it is useful; he studies it because he delights in it, and he delights in it because it is beautiful. If nature were not beautiful, it would not be worth knowing, and if nature were not worth knowing, life would not be worth living.
Besides language and music, [mathematics] is one of the primary manifestations of the free creative powers of the human mind, and it is the universal organ for world-understanding through theoretical construction. Mathematics must therefore remain an essential element of the knowledge and abilities which we have to teach, of the culture we have to transmit, to the next generation.