The Axiom of Choice is necessary to select a set from an infinite number of socks, but not an infinite number of shoes.

I cannot escape from the conclusion that the great ages of progress have depended upon a small number of individuals of transcendent ability.

Even if we could be certain that one of the world's religions were perfectly true, given the sheer number of conflicting faiths on offer, every believer should expect damnation purely as a matter of probability.

It seems to me now that mathematics is capable of an artistic excellence as great as that of any music, perhaps greater; not because the pleasure it gives (although very pure) is comparable, either in intensity or in the number of people who feel it, to that of music, but because it gives in absolute perfection that combination, characteristic of great art, of godlike freedom, with the sense of inevitable destiny; because, in fact, it constructs an ideal world where everything is perfect and yet true.

3 is prime, 5 is prime, 7 is prime, 9 is a paradox; as is a paradox why the number 1 is not prime if it has no other divisors besides himself.

... mathematical knowledge ... is, in fact, merely verbal knowledge. "3" means "2+1", and "4" means "3+1". Hence it follows (though the proof is long) that "4" means the same as "2+2". Thus mathematical knowledge ceases to be mysterious.