In brief, the whole world is the totality of mathematically expressible motions of objects in space and time, and the entire universe is a great, harmonious, and mathematically designed machine.

Mathematicians create by acts of insights and intuition. Logic then sanctions the conquests of intuition.

A elegantly executed proof is a poem in all but the form in which it is written.

Psychologically, the teaching of abstractions first is wrong. Indeed, a thorough understanding of the concrete must precede the abstract.

On all levels primary, and secondary and undergraduate - mathematics is taught as an isolated subject with few, if any, ties to the real world. To students, mathematics appears to deal almost entirely with things whlch are of no concern at all to man.

A proof tells us where to concentrate our doubts.

Mathematics is a body of knowledge, but it contains no truths.