Monday, May 7, 2007

"random" and "determinate"

Do not be fooled by the dichotomy "random" vs. "law-governed." Randomness is not a property of events, it is not correlated with a particular type of cause or lack thereof. Randomness is a property of data. Consider three bodies moving in accordance with the laws of Newtonian gravity, with no additional forces acting upon them. Suppose two are rotating mutually in a plane while the third crosses this plane (between the other two) repeatedly in the course of its movement. Now, examine the system at suitably coarse, but regular intervals, marking a 1 in one's logbook if the third body is on this side of the other two and a 0 in one's logbook if it is on that side. The sequence of 0's and 1's you are left with is data about the system. As it turns out, this sequence of 0's and 1's will be indistinguishable from one which has been generated by tossing a "fair coin." In other words, there will be no discernible pattern in the data, in this sense it is random. Yet the three body system is completely determinate, every development is predictable with an arbitrary degree of precision given a precise enough knowledge of the initial conditions. Suppose one stumbles across such a system in nature, and observes it for many many years, generating a long string of data. No amount of knowledge of the laws which govern the system can allow one to precisely predict the next piece of data. This is because there exists an infinite number of solutions to the problem posed by the data one has collected, each corresponding to a different set of initial conditions and, mutatis mutandis, to a different subsequent development of the three-body system. We call a complex system with this characteristic chaotic. Most natural systems are chaotic in this sense.

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