The Information - James Gleick [156]
“Chance is only the measure of our ignorance,”♦ Henri Poincaré famously said. “Fortuitous phenomena are by definition those whose laws we do not know.” Immediately he recanted: “Is this definition very satisfactory? When the first Chaldean shepherds watched the movements of the stars, they did not yet know the laws of astronomy, but would they have dreamed of saying that the stars move at random?” For Poincaré, who understood chaos long before it became a science, examples of randomness included such phenomena as the scattering of raindrops, their causes physically determined but so numerous and complex as to be unpredictable. In physics—or wherever natural processes seem unpredictable—apparent randomness may be noise or may arise from deeply complex dynamics.
Ignorance is subjective. It is a quality of the observer. Presumably randomness—if it exists at all—should be a quality of the thing itself. Leaving humans out of the picture, one would like to say that an event, a choice, a distribution, a game, or, most simply, a number is random.
The notion of a random number is full of difficulties. Can there be such thing as a particular random number; a certain random number? This number is arguably random:
10097325337652013586346735487680959091173929274945…♦
Then again, it is special. It begins a book published in 1955 with the title A Million Random Digits. The RAND Corporation generated the digits by means of what it described as an electronic roulette wheel: a pulse generator, emitting 100,000 pulses per second, gated through a five-place binary counter, then passed through a binary-to-decimal converter, fed into an IBM punch, and printed by an IBM model 856 Cardatype.♦ The process took years. When the first batch of digits was tested, statisticians discovered significant biases: digits, or groups of digits, or patterns of digits that appeared too frequently or not frequently enough. Finally, however, the tables were published. “Because of the very nature of the tables,” the editors said wryly, “it did not seem necessary to proofread every page of the final manuscript in order to catch random errors of the Cardatype.”
The book had a market because scientists had a working need for random numbers in bulk, to use in designing statistically fair experiments and building realistic models of complex systems. The new method of Monte Carlo simulation employed random sampling to model phenomena that could not be solved analytically; Monte Carlo simulation was invented and named by von Neumann’s team at the atomic-bomb project, desperately trying to generate random numbers to help them calculate neutron diffusion. Von Neumann realized that a mechanical computer, with its deterministic algorithms and finite storage capacity, could never generate truly random numbers. He would have to settle for pseudorandom numbers: deterministically generated numbers that behaved as if random. They were random enough for practical purposes. “Any one who considers arithmetical methods of producing random digits is, of course, in a state of sin,”♦ said von Neumann.
Randomness might be defined in terms of order—its absence, that is. This orderly little number sequence can hardly be called “random”:
00000
Yet it makes a cameo appearance in the middle of the famous million random digits. In terms of probability, that is to be expected: “00000” is as likely to occur as any of the other 99,999 possible five-digit strings. Elsewhere in the million random digits we find:
010101
This, too, appears patterned.
To pick out fragments of pattern in this jungle of digits requires work by an intelligent observer. Given a long enough random string, every possible short-enough