A mathematician, like a painter or a poet, is a maker of patterns. If his patterns are more permanent than theirs, it is because they are made with ideas.
A person's first duty, a young person's at any rate, is to be ambitious, and the noblest ambition is that of leaving behind something of permanent value.
A science is said to be useful if its development tends to accentuate the existing inequalities in the distribution of wealth, or more directly promotes the destruction of human life.
A science or an art may be said to be 'useful' if its development increases, even indirectly, the material well-being and comfort of men, it promotes happiness, using that word in a crude and commonplace way.
Asked if he believes in one G-d, a mathematician answered: 'Yes, up to isomorphism'.
Beauty is the first test: there is no permanent place in the world for ugly mathematics.
Chess problems are the hymn-tunes of mathematics.
Cricket is the only game where you are playing against eleven of the other side and ten of your own.
For any serious purpose, intelligence is a very minor gift.
Good work is not done by 'humble' men.
I believe that mathematical reality lies outside us, that our function is to discover or observe it, and that the theorems which we prove, and which we describe grandiloquently as our 'creations,' are simply the notes of our observations.
If a man has any genuine talent, he should be ready to make almost any sacrifice in order to cultivate it to the full.
If I could prove by logic that you would die in five minutes, I should be sorry you were going to die, but my sorrow would be very much mitigated by pleasure in the proof.
In these days of conflict between ancient and modern studies, there must surely be something to be said for a study (mathematics) which did not begin with Pythagoras, and will not end with Einstein, but is the oldest and the youngest of all.
It is hardly possible to maintain seriously that the evil done by science is not altogether outweighed by the good. For example, if ten million lives were lost in every war, the net effect of science would still have been to increase the average length of life.
It is not worth an intelligent man's time to be in the majority. By definition, there are already enough people to do that.
Many people, of course, use 'sentimentalism' as a term of abuse for other people's decent feelings, and 'realism' as a disguise for their own brutality.
Most people are so frightened of the name of mathematics that they are ready, quite unaffectedly, to exaggerate their own mathematical stupidity.
Most people can do nothing at all well.
No one should ever be bored. One can be horrified, or disgusted, but one can't be bored.
Sometimes one has to say difficult things, but one ought to say them as simply as one knows how.
The fact is there are few more popular subjects than mathematics. Most people have some appreciation of mathematics, just as most people can enjoy a pleasant tune.
Young men should prove theorems, old men should write books.
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