The infinite has been a perennial source of mathematical and philosophical wonder, in part because of its enormity—anything that large is grand, and provokes awe and contemplation—and in part because of the paradoxes like Galileo’s. Infinity seems impossible to tame intellectually, and to bring within the confines of human understanding. I will argue, however, that Cantor has tamed it. The good news is that Cantor’s mathematics makes infinity clear and consistent but does nothing to reduce the awe-inspiring grandeur of it.

It is important to learn to be surprised by simple things—for example, by the fact that bodies fall down, not up, and that they fall at a certain rate; that if pushed, they move on a flat surface in a straight line, not a circle; and so on. The beginning of science is the recognition that the simplest phenomena of ordinary life raise quite serious problems: Why are they as they are, instead of some different way?