Book Review : Exploring the Mysterious Heart of Logic

Forever Undecided: A Puzzle Guide to Goedel by Raymond Smullyan (Knopf: $16.95; 240 pages)

The loss of certainty is the hallmark of the 20th Century. The theory of relativity belongs to physics, but we have learned in the last hundred years that its name could be aptly applied to science in general and to all thought.

We now know that truth is relative and always must be and that perfect knowledge is theoretically impossible. In physics, this takes the form of Heisenberg’s Uncertainty Principle, which says that basic facts about the world can never be known. In mathematics, which should be the most certain of all disciplines, Kurt Goedel achieved an equally startling result. He proved that no mathematical system can be both complete and consistent.

Put another way, Goedel showed that mathematics, presumably the most pristine and theoretically perfect of all human thought, must be either inconsistent or incomplete. Though this discovery was made more than 50 years ago, its consequences are still reverberating through the worlds of science, math and philosophy. Mathematicians, who know that Goedel’s Theorem is true, still act as if it weren’t.

Exploring Logic

Raymond Smullyan is a logician and professor of philosophy by trade who for the last 10 years has turned out an extraordinary succession of books that explore the heart of logic and its contradictions. He is one of the most amazing people around.

“Forever Undecided” is the latest of Smullyan’s works, and it takes its place on my shelf alongside his 10 other books, each one more intriguing than the rest. (Question: Is that logically possible? Can they all be more intriguing than the rest?)

Some aesthetic experiences are so wonderful that they are too wonderful, and they leave you cold. Standing at the rim of the Grand Canyon, for example, or hearing the Liebestod from “Tristan und Isolde.” Nothing has a right to be that good.

Smullyan’s books are in that category. But let me hasten to add that they are tough. “Forever Undecided” demands a lot of work and thought. This is not a book for light reading before you go to sleep. If you tackle Smullyan, you’d better have your wits about you and be prepared to think hard about what he says.

Devilishly Clever

For Smullyan employs his standard technique of using devilishly clever logic puzzles to lead the reader from one level of complexity to the next, till he builds a structure that leads him to a demonstration of Goedel’s Theorem. Along the way, he introduces the reader to the notation and use of modern symbolic logic, and he uses that to work up to Goedel as well.

But this is no textbook. Smullyan writes with a charming style and verve that propel the reader onward even as the details of the argument become murkier and more difficult to follow. (This is more a statement about my limitations than about Smullyan’s.)

He is a whimsical man, and occasionally his whimsy emerges, as when he writes, “I phrased the above problem in a very misleading way. (Occasionally I feel like being a bit sneaky!)” The logic problems themselves, which spring from Smullyan’s extraordinarily fertile mind, are full of kings and knaves and liars and truth-tellers and men and women with a bevy of intriguing qualities.

Generally, it has been observed, artists have one point of contact with the world that they keep coming back to time and again. Their point of contact is slightly askew from the mainstream, though it has to be close enough that the rest of us feel a kinship with it yet far enough away that we find it interesting and novel.

Unlimited Imagination

Smullyan’s point of contact is the rules of logic, and he may well be the greatest popularizer of the subject since Lewis Carroll in the last century. His imagination seemingly knows no limit. His books are all different, yet they repeatedly come back to mine the same area, using his intriguing puzzles to explore logic’s inherent limitations and contradictions.

For those who are interested in this subject, there is probably no better, more accessible explanation of Goedel’s work and its intellectual power. Smullyan has probably made it as easy as it can be made, which, to repeat, does not mean this is an easy book. It isn’t.

But, alas, many important things aren’t easy, and one must simply decide which if any to tackle and which to let go by the wayside. After all, nobody can know everything, and it’s perfectly respectable to pick and choose from the smorgasbord of life. If your tastes run to logic and its problems, Smullyan is your man.

Or, as the French say, chacun a son mishugas. Everyone to his own insanity.