You can only be twice someone’s age once.

Man had been given a brain that could think in numbers, and it could not be coincidence that the world was unlocked by that very tool. To understand any aspect of the cosmos was to look on the face of God: not directly, but by a species of triangulation, because to think mathematically was to feel the action of God in oneself.

People who don't like math always accuse mathematicians of trying to make math complicated. (...) But anyone who does love math knows it's really the opposite: math rewards simplicity, and mathematicians value it above all else. So it's no surprise that Walter's favourite axiom was also the most simple in the realm of mathematics: the axiom of the empty set.

The axiom of the empty set is the axiom of zero. it states that there must be a concept of nothingness, that there must be the concept of zero: zero value, zero items. Math assumes there's a concept of nothingness, but is it proven? No. But it must exist.

And if we're being philosophical—which we today are—we can say that life itself is the axiom of the empty set. It begins in zero and ends in zero. We know that both states exist, but we will not be conscious of either experience: they are states that are necessary parts of life, even as they cannot be experienced as life. We assume the concept of nothingness, but we cannot prove it. But it must exist. So I prefer to think that Walter has not died but has instead proven for himself the axiom of the empty set, that he has proven the concept of zero. I know nothing else would have made him happier. An elegant mind wants elegant endings, and Walter had the most elegant mind. So I wish him goodbye; I wish him the answer to the axiom he so loved.

In a materialistic society, an employed boy is older than an unemployed man.

We envy people who are extremely old because we wish to live that long, not because we want to be that old.

Mathematics is one of the major modern mysteries. Perhaps it is the leading one, occupying a place in our society similar to the religious mysteries of another age. If we want to know something about what our age is all about, we should have some understanding of what mathematics is, and of how the mathematician operates and thinks.