Just a Couple of Guys Sitting Around Crunching Numbers

Mind Tools: The Five Levels of Mathematical Reality by Rudy Rucker (Houghton Mifflin: $17.95; 328 pages)

Numberland: A Fable by George Weinberg (St. Martin’s: $9.99; 119 pages)

A reviewer approaches each new book with an open mind, but also with some expectations. Popular books on mathematics tend to strike my fancy, so I very much wanted to like Rudy Rucker’s “Mind Tools,” a romp through the simplicities and complexities of contemporary mathematics and its relationship to thought and the world.

If originality were the sole criterion for judging a book, Rucker would score very high. His premise, alluded to in the subtitle (“The Five Levels of Mathematical Reality”) is that mathematics has five ways of thinking: numbers, space, logic, infinity and information, each deeper than the one before.

He explores each of these levels, though it is not always clear where the discussion is leading. What’s more, the quality of his presentation varies widely, as does his assumption about the reader’s ability to follow along.

The level changes abruptly. Some of it is crystal clear, and some of it is opaque, and the two are inextricably interlaced. Rucker’s discussion of Church’s Theorem, which says that there is no easy way to predict in advance whether a sentence can be proved from a set of assumptions, is quite good. But his discussion of the so-called halting problem, which says that “there is no computer program that can always correctly predict in advance which programs will work properly,” left me befuddled.

The good parts of Rucker’s book are mixed with too much that is not so good and, worse, seemingly irrelevant.

Rucker is a computer scientist at San Jose State University (as well as being a popular science and science fiction writer), and his aim in this book is to show that information (in the computer sense) is the basis of everything. Fine.

But I didn’t understand what the beginning of the book--number and space--had to do with it. I kept wondering where Rucker was heading. He got going in the second half, though, leading inexorably into the bedeviling paradoxes that characterize all attempts to systematize thought.

There is, after all, the paradox of knowledge itself: What can we know? Using contemporary ideas in mathematics and computers, Rucker shows that there is no way that we cannot know everything. But we can know some things, and the trick is to gather as much as we can and pretend that there is a complete theory of the whole while acknowledging that there isn’t.

This, of course, is not too far from the basic, eternal and unanswerable problems of philosophy itself: What is life? What is knowledge? How do we know what we know? And that old chestnut, why is there something rather than nothing? The best answer is to shrug and go on to something else.

Rucker dances around these questions and others, and in the very last paragraph, after a lot of huffing and puffing, he tells you what he thinks:

“So what is reality, one more time? An incompressible computation by a fractal CA (cellular automata) of inconceivable dimensions. And where is this huge computation taking place? Everywhere; it’s what we’re made of.”

Folks, there is less here than meets the naked eye.

Which brings us to the other book under consideration today, “Numberland: A Fable,” a slim volume by George Weinberg, a psychotherapist who writes on psychology and has co-authored a textbook on statistics.

This book is about the world told from the point of view of the numbers: 1, 2, 3, 4 and so forth. Honest. Its protagonist is Six, and the plot, such as it is, involves Six’s efforts to tell his fellow numbers that they are not immortal and the effect this knowledge has on them.

In the climax, such as it is, Six declares, “My friends, I am here to tell you the most important fact of your life. It is a terrible fact. But it is a wonderful fact too. Oh my friends, you are going to die! . . . It doesn’t matter how big you are. We exist only because humans create us. When the last human mortal dies, we, too, must come to an end. Our society will be no more. We are their children.” So this book, such as it is, is about the meaning of death.