Mathematicians - for what they do - are really poorly rewarded. And it's a very competitive field, almost as bad as being a concert pianist.

Our faith in Mathematics is not likely to wane if we openly acknowledge that the personalities of even the greatest mathematicians may be as flawed as those of anyone else.

Very little mathematics has direct applications - though fortunately most of it has plenty of indirect ones.

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.

Richard Feynman was fond of giving the following advice on how to be a genius. You have to keep a dozen of your favorite problems constantly present in your mind, although by and large they will lay in a dormant state. Every time you hear or read a new trick or a new result, test it against each of your twelve problems to see whether it helps. Every once in a while there will be a hit, and people will say, "How did he do it? He must be a genius!"

The advice we give others is the advice that we ourselves need.

Are mathematical ideas invented or discovered? This question has been repeatedly posed by philosophers through the ages and will probably be with us forever.