We often hear that mathematics consists mainly of "proving theorems." Is a writer's job mainly that of "writing sentences?"

God created infinity, and man, unable to understand infinity, had to invent finite sets.

Stan Ulam was lazy, ... He talked too much ... He was self-centered ... . He had an overpowering personality.

How did he do it? He must be a genius!

Mathematics is the study of analogies between analogies. All science is. Scientists want to show that things that don't look alike are really the same. That is one of their innermost Freudian motivations. In fact, that is what we mean by understanding.

Theorems are not to mathematics what successful courses are to a meal.

Combinatorics is an honest subject. No adèles, no sigma-algebras. You count balls in a box, and you either have the right number or you haven't. You get the feeling that the result you have discovered is forever, because it's concrete. Other branches of mathematics are not so clear-cut. Functional analysis of infinite-dimensional spaces is never fully convincing; you don't get a feeling of having done an honest day's work. Don't get the wrong idea - combinatorics is not just putting balls into boxes. Counting finite sets can be a highbrow undertaking, with sophisticated techniques.