The progress of mathematics can be viewed as progress from the infinite to the finite.

If we have no idea why a statement is true, we can still prove it by induction.

The pendulum of mathematics swings back and forth towards abstraction and away from it with a timing that remains to be estimated.

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.

Every lecture should state one main point and repeat it over and over, like a theme with variations. An audience is like a herd of cows, moving slowly in the direction they are being driven towards. If we make one point, we have a good chance that the audience will take the right direction; if we make several points, then the cows will scatter all over the field. The audience will lose interest and everyone will go back to the thoughts they interrupted in order to come to our lecture.

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

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