We all take our pleasures where we find them, and everyone is different, with different sources to draw upon. It will seem peculiar to many people that others could take pleasure in mathematics. Children usually learn to be bored or frightened by math, but there isn't any reason for this, other than incompetent teaching. As an attempt at remedy, husband and wife team Robert and Ellen Kaplan in 1994 began the Math Circle, Saturday morning sessions for kids who just wanted to find out more about mathematics. (The sessions were changed to Sunday morning when soccer practice interfered). Some kids (especially those who were pushed into the classes by their parents) dropped out, but some have come back, year after year, and the Kaplans have found that posing questions, inviting conjectures, asking for examples, and even suggesting ways towards proofs can be something children can enjoy. Mathematicians have been telling us for centuries about the beauty of the objects and systems that they have explored. The Math Circle seems to have taught math in a way to at least some kids who have caught the spirit of the quest for mathematical beauty. In _The Art of the Infinite: The Pleasures of Mathematics_ (Oxford University Press), the Kaplans have put some of those lessons into book form, concentrating on infinities of various kinds. This is a book for adults, or kids who hanker to think about math like adults ought to, but it is full of a sense of play.
As you might expect, things start simple and get very complicated, and this is true right off in the first chapter, considering more and more complicated numbers. The Natural Numbers are introduced with patterns, as if you had stones to position on a table. 1, 3, 6, and 10 stones make pleasing equilateral triangles, and 1, 4, 9, and 16 make pleasing squares. We move from these to zero and negative numbers: "Certainly zero and the negatives have all the marks of human artifice: deftness, ambiguity, understatement." Are these numbers invented or discovered? The profundity of this question is plumbed throughout the book. Rationals, irrationals, and finally the complex numbers are all included. As the numbers mount up, the irregularity and regularity of the primes is considered, one of the most fruitful arenas of number theory. Euclid had to make an assumption about the infinite, his famous fifth postulate; but it is only an assumption; assuming that parallel lines meet eventually produces also a worthy geometry that tells us much about how the Einsteinian universe works. But there is no need to look into these strange worlds to find wonders; before leaving Euclid's terra firma, we are reintroduced to the triangle, and are presented with some astonishing revelations of secret points within and around the simple three sides that will remind you that no matter how simple things look, or even how simple things are, everything is more complicated than you can imagine.
And if you want your infinities more complicated still, the final chapter has to do with Cantor's work. Common sense tells us there must be half as many even numbers as there are whole numbers, but Cantor showed that the infinity of both was equal. He showed that the infinite number of points in a line as long as your finger was equal to the infinite number in a line as long as from here to the Sun. In fact, the number of points on a line is equal to the number of points in a plane. And yet, some infinities are bigger than others. This is strange territory indeed, and requires some concentration to understand and enjoy, even with the Kaplan's literate, witty, and clear explanations. This is a fine introduction to different aspects of serious mathematics; true to its subtitle, it is a book full of pleasures.
