The primes are the raw material out of which we have to build arithmetic, and Euclid's theorem assures us that we have plenty of material for the task.

Asked if he believes in one G-d, a mathematician answered: "Yes, up to isomorphism".

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.

[I was advised] to read Jordan's 'Cours d'analyse'; and I shall never forget the astonishment with which I read that remarkable work, the first inspiration for so many mathematicians of my generation, and learnt for the first time as I read it what mathematics really meant.

A chess problem is simply an exercise in pure mathematics.

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

In [great mathematics] there is a very high degree of unexpectedness, combined with inevitability and economy.