Deterministic Chaos and the Simulated Universe

After several months of absence, Chris Wenham has returned with a new essay entitled 2 + 2. In it, he explores a common idea:

Many have speculated that you could simulate a working universe inside a computer. Maybe it wouldn’t be exactly the same as ours, and maybe it wouldn’t even be as complex, either, but it would have matter and energy and time would elapse so things could happen to them. In fact, tiny little universes are simulated on computers all the time, for both scientific work and for playing games in. Each one obeys simplified laws of physics the programmers have spelled out for them, with some less simplified than others.

As always, the essay is well done and thought provoking, exploring the idea from several mathematical angles. But it makes the assumption that the universe is both deterministic and infinitely quantifiable. I am certainly no expert on chaos theory, but it seems to me that it bears an importance on this subject.

A system is said to be deterministic if its future states are strictly dependant on current conditions. Historically, it was thought that all processes occurring in the universe were deterministic, and that if we knew enough about the rules governing the behavior of the universe and had accurate measurements about its current state we could predict what would happen in the future. Naturally, this theory has proven very useful in modeling real world events such as flying objects or the wax and wane of the tides, but there have always been systems which were more difficult to predict. Weather, for instance, is notoriously tricky to predict. It was always thought that these difficulties stemmed from an incomplete knowledge of how the system works or inaccurate measurement techniques.

In his essay, Wenham discusses how a meteorologist named Edward Lorenz stumbled upon the essence of what is referred to as chaos (or nonlinear dynamics, as it is often called):

Lorenz’s simulation worked by processing some numbers to get a result, and then processing the result to get the next result, thus predicting the weather two moments of time into the future. Let’s call them result₁, which was fed back into the simulation to get result₂. result₃ could then be figured out by plugging result₂ into the simulation and running it again. The computer was storing result_n to six decimal places internally, but only printing them out to three. When it was time to calculate result₃ the following day, he re-entered result₂, but only to three decimal places, and it was this that led to the discovery of something profound.

Given just an eentsy teensty tiny little change in the input conditions, the result was wild and unpredictable.

This phenomenon is called “sensitive dependence on initial conditions.” For the systems in which we could successfully make good predictions (such as the path of a flying object), only a reasonable approximation of the initial state is necessary to make a reasonably accurate prediction. Sensitive dependence of a reasonable approximation of the initial state, however, yields unreasonable predictions. In a system exhibiting sensitive dependence, reasonable approximations of the initial state do not provide reasonable approximations of the future state.

So here comes the important part: For a chaotic system such as weather, in order to make useful long term predictions, you need measurements of initial conditions with infinite accuracy. What this means is that even a deterministic system, which in theory can be modeled by mathematical equations, can generate behavior which seems random and unpredictable. This manifests itself in nature all the time. Weather is the typical example, but there is also evidence that the human brain is also governed by deterministic chaos. Indeed, our brain’s ability to generate seemingly unpredictable behavior is an important component of both survival and creativity.

So my question is, if it is not possible to quantify the initial conditions of a chaotic system with infinite accuracy, is that system really deterministic? In a sense, yes, even though it is impossible to calculate it:

Michaelangelo claimed the statue was already in the block of stone, and he just had to chip away the unnecessary parts. And in a literal sense, an infinite number of universes of all types and states should exist in thin air, indifferent to whether or not we discover the rules that exactly reveal their outcome. Our own universe could even be the numerical result of a mathematical equation that nobody has bothered to sit down and solve yet.

But we’d be here, waiting for them to discover us, and everything we’ll ever do.

The answer might be there, whether we can calculate it or not, but even if it is, can we really do anything useful with it? In the movie Pi, a mathematician stumbles upon an enigmatic 216 digit number which is supposedly the representation of the infinite, the true name of God, and thus holds the key to deterministic chaos. But it’s just a number, and no one really knows what to do with it, not even the mathematician who discovered it (though he could make accurate predictions on for the stock market, though he could not understand why and it came at a price). In the end, it drove him mad. I don’t pretend to have any answers here, but I think the makers of Pi got it right.