The Information - James Gleick [155]
12 | THE SENSE OF RANDOMNESS
(In a State of Sin)
“I wonder,” she said. “It’s getting harder to see the patterns, don’t you think?”
—Michael Cunningham (2005)♦
IN 1958, GREGORY CHAITIN, a precocious eleven-year-old New Yorker, the son of Argentine émigrés, found a magical little book in the library and carried it around with him for a while trying to explain it to other children—and then, he had to admit, trying to understand it himself.♦ It was Gödel’s Proof, by Ernest Nagel and James R. Newman. Expanded from an article in Scientific American, it reviewed the renaissance in logic that began with George Boole; the process of “mapping,” encoding statements about mathematics in the form of symbols and even integers; and the idea of metamathematics, systematized language about mathematics and therefore beyond mathematics. This was heady stuff for the boy, who followed the authors through their simplified but rigorous exposition of Gödel’s “astounding and melancholy” demonstration that formal mathematics can never be free of self-contradiction.♦
The vast bulk of mathematics as practiced at this time cared not at all for Gödel’s proof. Startling though incompleteness surely was, it seemed incidental somehow—contributing nothing to the useful work of mathematicians, who went on making discoveries and proving theorems. But philosophically minded souls remained deeply disturbed by it, and these were the sorts of people Chaitin liked to read. One was John von Neumann—who had been there at the start, in Königsberg, 1930, and then in the United States took the central role in the development of computation and computing theory. For von Neumann, Gödel’s proof was a point of no return:
It was a very serious conceptual crisis, dealing with rigor and the proper way to carry out a correct mathematical proof. In view of the earlier notions of the absolute rigor of mathematics, it is surprising that such a thing could have happened, and even more surprising that it could have happened in these latter days when miracles are not supposed to take place. Yet it did happen.♦
Why? Chaitin asked. He wondered if at some level Gödel’s incompleteness could be connected to that new principle of quantum physics, uncertainty, which smelled similar somehow.♦ Later, the adult Chaitin had a chance to put this question to the oracular John Archibald Wheeler. Was Gödel incompleteness related to Heisenberg uncertainty? Wheeler answered by saying he had once posed that very question to Gödel himself, in his office at the Institute for Advanced Study—Gödel with his legs wrapped in a blanket, an electric heater glowing warm against the wintry drafts. Gödel refused to answer. In this way, Wheeler refused to answer Chaitin.
When Chaitin came upon Turing’s proof of uncomputability, he thought this must be the key. He also found Shannon and Weaver’s book, The Mathematical Theory of Communication, and was struck by its upside-down seeming reformulation of entropy: an entropy of bits, measuring information on the one hand and disorder on the other. The common element was randomness, Chaitin suddenly thought. Shannon linked randomness, perversely, to information. Physicists had found randomness inside the atom—the kind of randomness that Einstein deplored by complaining about God and dice. All these heroes of science were talking about or around randomness.
It is a simple word, random, and everyone knows what it means. Everyone, that is, and no one. Philosophers and mathematicians struggled endlessly. Wheeler said this much, at least: “Probability, like time, is a concept invented by humans, and humans have to bear the responsibility for the obscurities that attend it.”♦ The toss of a fair coin is random, though every detail of the coin’s trajectory may be determined à la Newton. Whether the population of France is an