The last level of metaphor in the Alice books is this: that life, viewed rationally and without illusion, appears to be a nonsense tale told by an idiot mathematician.

In many cases a dull proof can be supplemented by a geometric analogue so simple and beautiful that the truth of a theorem is almost seen at a glance.

Mathematics is not only real, but it is the only reality. That is that entire universe is made of matter, obviously. And matter is made of particles. It's made of electrons and neutrons and protons. So the entire universe is made out of particles. Now what are the particles made out of They're not made out of anything. The only thing you can say about the reality of an electron is to cite its mathematical properties. So there's a sense in which matter has completely dissolved and what is left is just a mathematical structure.

If God creates a world of particles and waves, dancing in obedience to mathematical and physical laws, who are we to say that he cannot make use of those laws to cover the surface of a small planet with living creatures?

In no other branch of mathematics is it so easy for experts to blunder as in probability theory.

One would be hard put to find a set of whole numbers with a more fascinating history and more elegant properties surrounded by greater depths of mystery--and more totally useless--than the perfect numbers.

The sudden hunch, the creative leap of mind that "sees" in a flash how to solve a problem in a simple way, is something quite different from general intelligence.