Determinism, Chaos and the Lorenz Attractor

Can something be deterministic, yet unpredictable? This is the question at the heart of chaos theory. Previously we’ve played the Chaos Game. If I were to give you a set of initial coordinates and a list of functions that are sequentially applied to these coordinates you would probably be able to eventually figure out the final coordinates. But if the functions were chosen at random you would not. Even though the random function is actually not random, but has an underlying deterministic function. In theory, you could, in practice you won’t.

Another thing that would make it impossible to figure out the final coordinates is when you start at an unknown coordinate. To illustrate this: imagine that someone is listing the digits of Pi. When this person starts out and you witness this start you, will be able to follow along and predict the next digit. You could use a deterministic spigot algorithm to do this. The only thing you need to keep track of is the number of digits the person has already listed. Alternatively, you could use a lookup table. If you were really fanatic, however, you would eventually run out of digits and you will have the use the spigot algorithm to get to more digits.

Pi is infinite and transcendental. Every subsequence can occur multiple times. If you were to tune in somewhere in the middle of the person listing its digits you would not be able to predict the next.

What is the next digit in this sequence? This subsequence of six nines occurs for the first time at the 762nd decimal point, also known as the Feynman point. But it also occurs at the 193034th decimal point. Without any context, there’s no way of knowing after which of the infinite subsequences you are supposed to continue.

Given that you’re far enough into the sequence, the digits of Pi can be seen as a pseudo-random number generator. Many things that appear random have underlying deterministic mechanisms. But due to tuning in at an unknown moment, you might not ever figure out what the underlying mechanism is.

The ancient Greeks also thought about this concept. Democritus illustrated the idea with a story of two servants that are sent to get water from a well at the exact same time. The servants would view their meeting as random. Unbeknownst to them, their masters concocted this evil plan. This story would have been a lot better if the masters had the servants walk out to the middle of nowhere from separate routes at the same time. This would have felt a lot more random to them. Of course, this example isn’t totally the same. The servants can easily deduce the underlying reason. The story would be more representative if the servants were mute and did not know sign language or how to write.