The mathematician is in much more direct contact with reality. ... [Whereas] the physicist's reality, whatever it may be, has few or none of the attributes which common sense ascribes instinctively to reality. A chair may be a collection of whirling electrons.

Beauty is the first test: there is no permanent place in the world for ugly mathematics.

Real mathematics must be justified as art if it can be justified at all.

All analysts spend half their time hunting through the literature for inequalities which they want to use and cannot prove.

The mathematician is in much more direct contact with reality. ... [Whereas] the physicist's reality, whatever it may be, has few or none of the attributes which common sense ascribes instinctively to reality. A chair may be a collection of whirling electrons.

A mathematician, like a painter or poet, is a maker of patterns. If his patterns are more permanent than theirs, it is because they are made with ideas.

The mathematician's patterns, like the painter's or the poet's, must be beautiful.