Shaking Us Up
Among the many surprises that the 20th Century has produced is the discovery that there is no certainty--not even in mathematics. From the time of the ancient Greeks until 1931 it had been assumed that mathematics, a purely logical system based on self-evident axioms, was as solid as anything could be. Mathematics was taken as the epitome of thought and as a model for what thinking should be. At the turn of this century the great number theorist David Hilbert declared, “In mathematics there is no ignorabimus . . . . We must know, we will know.”
Boy, was he wrong. Thirty years later Kurt Goedel made one of the most amazing and unsettling discoveries in history. He showed that any formal system, like mathematics, contains true statements that cannot be proved and other statements whose truth cannot be decided. Goedel’s incompleteness theorem showed that it is impossible in principle to know everything.
Now comes Gregory J. Chaitin of the IBM Thomas J. Watson Research Center in Yorktown Heights, N.Y., who has taken Goedel’s work one step further. In an article in the July issue of Scientific American titled “Randomness in Arithmetic,” Chaitin writes, “I have shown that there is randomness in the branch of pure mathematics known as number theory. My work indicates that--to borrow Einstein’s metaphor--God sometimes plays dice with whole numbers!”
In other words, anyone who thought that Goedel’s theorem applied only to the fringes and unimportant areas of mathematics was wrong. The numbers themselves--the most fundamental and self-evident area of mathematics--are not safe. “Incompleteness and randomness are natural and pervasive,” Chaitin says. He speculates that perhaps there are new axioms about the whole numbers that have yet to be discovered. And he wonders whether a complete understanding of life itself is related to his proof of the irreducible randomness of arithmetic.
Chaitin’s work is part of a highly abstract field that he calls algorithmic information theory, and he has just published a book by that name that spells it all out (Cambridge University Press). We leave the details to those who can get through the book. But to understand the stakes involved it is worth repeating what Chaitin writes in Scientific American: “At the end of his life John von Neumann challenged mathematicians to find an abstract mathematical theory for the origin and evolution of life. This fundamental problem, like most fundamental problems, is magnificently difficult. Perhaps algorithmic information theory can help to suggest a way to proceed.”
Of course, 2 plus 2 still equals 4, the sun will rise tomorrow in the East, and the Lakers may or may not be dethroned by the Pistons as the champions of the NBA. But reading Chaitin’s article makes the world shake just a little. It is a reminder that things are never quite as they seem.
