Math, infinity and madness

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Mathematicians think differently about infinity than most people. For non-mathematicians, infinity is sort of a term of last resort.

Book Review "The Mystery of the Aleph: Mathematics, the Kabbalah, and the Human Mind" By Amir D. Aczel. Four Walls Eight Windows, $24.95

We don't necessarily think about the properties of infinity; we just use the term when we need to express a massively huge amount, larger than any other number we can conceive of. Mathematicians, however, realize that one should not use infinity so indiscriminately. First discovered by the Greeks between the 5th and 6th centuries B.C., the concept of infinity was "so overwhelming, so bizarre, so contrary to every human intuition, that it confounded the ancient philosophers and mathematicians who discovered it, causing pain, insanity and at least one murder," writes Amir D. Aczel in his new book, "The Mystery of the Aleph."

The ancients were not alone in finding pain with infinity. In examining the history of infinity, Aczel, a mathematician himself and the author of "Fermat's Last Theorem" and "God's Equation," focuses on George Cantor, a brilliant mathematician whose insights into infinity profoundly changed the world of math.

Born in Russia in 1845, Cantor became interested in numbers while still an adolescent. After receiving his degree, he obtained a position at the University of Halle, in Germany. Within several years he had his first nervous breakdown. These attacks continued and worsened as Cantor delved deeper into the power of infinity, eventually leading to lengthy stays at the local asylum, where he died in 1918.

For Cantor, "infinity was the realm of God," writes Aczel. He was not alone in relating the infinite and God; this concept was central to religious philosophers such as Thomas Aquinas, Rabbi Akiva, whose teachings were critical to Jewish mysticism, and St. Augustine, who wrote "individual numbers are finite but as a class are infinite. Does that mean that God does not know all the numbers, because of their infinity? No one could be insane enough to say that."

Although Aczel avoids heavy use of jargon and ably describes Cantor's theories and many other theories of math, most defy easy comprehension. What makes this book fascinating is not the math but the concepts drawn from the numbers and the stories of those who created them. It reveals a world that few enter but for those who do, who understand, and who survive what is going on, it must be rather compelling.