Writing a first-rate math book that will appeal to non-mathematicians must be one of the toughest jobs around. Such books do not appear often. So it is a pleasure to call your attention to two of them.
Eli Maor is a mathematician with one hand firmly in mathematics, the other reaching out to the rest of us. His book, "To Infinity and Beyond," is the broader, more expansive, more scholarly of the two, but it is still very accessible to non-specialists.
His approach to his subject--infinity as a mathematical and cultural phenomenon--enables Maor to bridge the gap between mathematics and the world.
Maor explores the idea of infinity in mathematics and in art and argues that this is the point of contact between the two, best exemplified in our time by the work of the Dutch artist M. C. Escher, six of whose works are shown here in beautiful color plates.
But Escher, who died in 1972, was not the first to make the connection between geometry and art. His work, Escher said, was inspired by designs on early Islamic art that he saw on a visit to the Alhambra in Spain. Maor gives a good discussion of the abstract art of the Moslems and concludes here that "the interplay between geometry and art reaches its supreme level."
The first part of Maor's book contains more mathematics, and the second part is more cultural. But the two can be read independently if that makes one more comfortable. Maor writes with mathematical precision and logic, which comes through in both parts and gives a pleasing insight into the thought processes that mathematicians use.
His exploration of the idea of infinity cuts a wide swath through intellectual history and the history of science. He starts with the Greeks and comes up to the present day, and he links his discussion to the philosophy of mathematics and to more general questions of philosophy to boot.
Our telescopes tell us that we are on a planet that is circling a star that is part of a galaxy that is part of a cluster of galaxies that is part of a supercluster of clusters and that all of the superclusters together make up the universe.
But how do we know that there aren't other universes somewhere and that these other universes aren't part of a cluster of universes, and so on? We don't know, Maor writes, but "in science, we must confine ourselves to the observable world, and in this sense the universe must be regarded as the last rung in the ladder."
Interwoven with these heady ideas are discussions and explanations that provide a rigorous framework to the general subject of recreational mathematics. This is the subject of Herbert Kohl's "Mathematical Puzzlements," which strikes a different tone altogether from Maor's book.
Kohl has done an admirable job making clear to beginners what recreational mathematics is all about. He gives a basic primer on a smorgasbord of mathematical games and recreations from tiling to numbers to maps to logic. If you have ever wondered how to solve problems of this sort but have never been able to find out, this is the book for you.
Kohl acknowledges his debts to Martin Gardner, who inspired his interest in mathematical play, and to John Conway, Elwyn Berlekamp and Richard Guy, whose "Winning Ways for Your Mathematical Plays" (Academic Press, 1982), remains the alpha and omega of mathematical games.
Like his masters, Kohl's scope is enormous. But he writes on a more elementary level than they do, and he explains everything. The clear drawings and diagrams uniformly help in understanding the text.
But Kohl's book is not just a how-to book. By focusing on the principles behind the games, it displays mathematical thinking, and Kohl argues that the process of solving the games is more important than getting the right answer.
"Many of the puzzlements that occur to us have not yet been solved by any mathematician, and some may have no solutions," Kohl writes in the introduction. "It is more that questioning and inventing keep the mind alive."
This, of course, may be the driving force behind mathematics, science and thought. Both of these books will get the juices flowing.